research article delay minimization topology control in...

Research ArticleDelay Minimization Topology Control in Planetary SurfaceNetwork An Autonomous Systems Approach

Wei Zhang Gengxin Zhang Liang Gou Bo Kong and Dongming Bian

College of Communication Engineering PLA University of Science and Technology Nanjing 210007 China

Correspondence should be addressed to Gengxin Zhang satlab126com

Received 3 April 2015 Accepted 5 August 2015

Academic Editor Kameswara Namuduri

Copyright copy 2015 Wei Zhang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper investigates the topology control problem in the planetary surface network (PSN) of Interplanetary Internet (IPN) usingan autonomous system (AS) approach We propose a delay minimization topology control (DMTC) algorithm to achieve low timedelay and strong connectivity in the planetary surface network Compared with the most existing approaches where either thepurely centralized or the purely distributed control method is adopted the proposed algorithm is a hybrid control method Inorder to reduce the cost of control the control message exchange is constrained among neighboring AS networks We prove thatthe proposed algorithm could achieve logical k-connectivity on the condition that the original physical topology is k-connectivitySimulation results validate the theoretic analysis and effectiveness of the DMTC algorithm

1 Introduction

The Interplanetary Internet (IPN) was proposed to satisfythe demands of deep space communications early in thiscentury As proposed in [1 2] the IPN includes a backbonenetwork external networks and planetary networks (PNs) APN is composed of an orbiter network (ON) and a planetarysurface network (PSN) The former is composed of orbiterscircling the planets and provides a relay service betweenthe surface network and the backbone network [3 4] Thelatter is composed of landers rovers astronauts and sensorsfrom different countries or space missions [5 6] Nodesin the planetary surface network autonomously connect toeach other in order to perform collaborative tasks Unlikewireless sensor network (WSN) on the earth the PSN hasdistinguishing characteristics [7] The scale of the PSN islarge Communication abilities of nodes in the PSN areusually stronger Nodes in the PSN have a wide and unevendistribution Some links among them are short and someare extremely long Excessive use of these long-distancelinks data not only brings additional delay but also reducesoperation efficiency What is more long links usually meanmore energy will be consumed Consequently how to makean efficient and reliable topology control is challenging

Existing works about network topology control mainlyfocus onmaintaining a specified topology and achieve a set ofnetwork-wide objectives such as reducing energy consump-tion guaranteeing the robustness increasing the networkcapacity and reducing end-to-end delay for example [8ndash16] But in most studies either the purely centralized orthe purely distributed control method is adopted Central-ized algorithms rely on a central entity which knows theconditions of all the nodes in order to calculate the optimaltopology [17ndash19] However these algorithms are not suitablefor large scale network such as PSN where excessive amountsof control messages need to be collected by one central entityControl information in the PSN is costly for long distance andtoomany hops On the other hand in distributed algorithmseach node collects the information from its neighboringnodes and autonomously computes which link should bepreserved [19ndash21] Consider that the information each nodeobtains is limited the final topology usually cannot achieveglobal optimization for large scale networks Thus it isimportant to develop a special topology control algorithm forthe PSN

Instead of using long links nodes in the PSN shouldcollaboratively determine which links will be used anddefine the network topology though forming proper neighbor

Hindawi Publishing CorporationInternational Journal of Distributed Sensor NetworksVolume 2015 Article ID 726274 13 pageshttpdxdoiorg1011552015726274

2 International Journal of Distributed Sensor Networks

relations That is topology control algorithms of the PSNactually remove unnecessary long links As a result thenetwork topology is susceptible to unpredictable events suchas hardware failures in such a harsh environment Thereforeto design robust topology control algorithms 119896-connectivityof the network is considered where a 119896-connected network is119896 minus 1 fault-tolerant that is the failure of less than 119896 minus 1 nodeswill not disconnect the whole network

In this paper we study the topology control problem inthe PSN using an autonomous system (AS) approach An ASnetwork is a collection of nodes with similar properties forexample nodes distributed in the same region The reasonsfor using the AS approach are twofold Firstly the complexPSN is decoupled into a series of small AS networks andcentralized method can be used in each AS to ensure strongconnectivity Secondly distributed method is used amongAS networks thus the topology control message exchangecan be constrained among neighboring AS networks Wepropose a delay minimization topology control (DMTC)algorithm using such a hybrid approach DMTC preserves119896-connectivity and is min-max delay optimal The min-maxcriterion tries to minimize the maximum end-to-end delaybetween any pair of nodes in the network [22] Brieflythe DMTC algorithm consists of three phases (i) nodesin the PSN autonomously form AS networks and elect AScores (ii) with the topology information gathered from themembers of its AS network each AS core minimizes themaximum link delay used by all the nodes and guaranteesstrong connectivity using a centralized method (iii) eachAS core selects a set of border nodes shares topologyinformation with neighboring AS cores and computes lowtime delay links between neighboring AS networks using adistributed methodThemain contributions of this paper aresummarized as follows

(1) An AS network model of PSN is proposed Thelarge scale and complex PSN is decoupled into smallAS networks with similar nodes to achieve strongconnectivity with low cost control messages

(2) A delay minimization topology control (DMTC)algorithm is proposed to achieve low time delay It isa hybrid algorithm within an AS network and amongneighboring AS networks

(3) The strong connectivity of DMTC algorithm provedthat the algorithm could achieve logical 119896-connec-tivity on the condition that the original physicaltopology is 119896-connectivity

The rest of this paper is organized as follows In Section 2we define the network model and provide some definitionsIn Section 3 we propose an AS based algorithm DMTC toachieve low time delay and strong connectivity Then thevalidity of DMTC is proved in Section 4 and the messagecomplexity of our algorithm is analyzed in Section 5 InSection 6 simulation results and discussion are presentedFinally we make conclusion in Section 7

2 Network Model

In this section the network model of AS network is definedAs presented above the PSN is a self-organizing systemconstituted by various nodes For example as demonstratedin Figure 1 the Mars PSN is a part of the IPN Nodes inthe PSN have a wide and uneven distribution They work indifferent areas with either mobile (eg rovers) or static (egsensors) statuses If we apply a unified strategy to managethe whole PSN it will induce low efficiency and even cannotmaintain the normal operation of the network with toomuchcontrol information So as shown in Figure 2 we divide thePSN into a series of AS networks according to the property ofthe nodes Each AS network can adopt independent topologycontrol strategy to achieve strong connectivity And thecontrol message exchange is constrained among neighboringAS networks to reduce the cost of control

Considering that the properties of nodes in the PSN aresimilar except few nodes we assume that all the nodes arehomogeneous They have the same maximal transmissionrange 119877max Let the PSN network topology be representedby undirected simple graph 119866 = (119881 119864) where 119881 =

1199061 1199062 119906

119899 is the set of nodes (or equivalently vertices)

and 119864 = (119906119894 119906119895) | (119906119894 119906119895isin 119881) and (119903(119906

119894 119906119895) le 119877max) is the set

of links (edges) 119903(119906119894 119906119895) is the distance between nodes 119906

119894and

119906119895 Each node is assigned a unique identifier (ID) according

to its property such as MAC addressWe assume that 119866 is a general graph that is if (119906 V) isin

119864 119906 and V can exchange information with each other Wealso assume that the link is symmetric and obstacle-free andeach node is able to obtain its location by some means (egcelestial navigation [23] initial navigation [24] and visionnavigation [25]) We then define several graphs related termsin the following which will be used in both algorithms andproofs For all definitions we refer to graph 119866 = (119881 119864) andsubgraphs 119866

119894= (119881119894 119864119894) and 119866

119895= (119881119895 119864119895)

Definition 1 (weight function) For edge 119890 = (119906 V) theweight function is 119908(119906 V) = (119889(119906 V)min(119868119863(119906) 119868119863(V))max(119868119863(119906) 119868119863(V))) where119889(119906 V) is the time delay between 119906

and Vwhen exchanging information Given (1199061 V1) (1199062 V2) isin

119864 the relationship between119908(1199061 V1) and119908(119906

2 V2) is given as

119908 (1199061 V1) gt 119908 (119906

2 V2) lArrrArr 119889 (119906

1 V1) gt 119889 (119906

2 V2) or

(119889 (1199061 V1) = 119889 (119906

2 V2)) and (min (119868119863 (119906

1) 119868119863 (V

1))

gtmin (119868119863 (1199062) 119868119863 (V

2))) or

(119889 (1199061 V1) = 119889 (119906

2 V2)) and (min (119868119863 (119906

1) 119868119863 (V

1))

=min (119868119863 (1199062) 119868119863 (V

2)))

and (max (119868119863 (1199061) 119868119863 (V

1))

gtmax (119868119863 (1199062) 119868119863 (V

2)))

(1)

It is obvious that edges with the same vertices have equivalentweights However edges with different end-vertices havedifferent weights

International Journal of Distributed Sensor Networks 3

Earth

Satellite gateway

GEO

Mars

Earth station

IPN backbone

Other planets

Backbone nodes

Satellite gateway

Rover

Lander

Lander

Rover

Sensors

Sensors

Figure 1 The PSN is a part of the IPN and is a self-organizing system constituted by various nodes

Rover

Lander

Lander

Rover

Sensors

Sensors

AS-1 AS-2

AS-3

AS-4

Figure 2 The whole PSN is divided into a series of AS networks according to the property of the nodes

Definition 2 (119896-connected) In graph (topology) 119866 node119906 is said to be connected to node V if there existspath 119901 = 119906119909

11199092sdot sdot sdot 119909119898minus1

119909119898V where 119909

119894isin 119881 and

(119906 1199091) (119909119894 119909119895) (119909119898 V) isin 119864 And for any 119906 V isin 119881 if there

exist at least 119896 disjoint paths between them Graph 119866 is 119896-connected and denoted by 119862119874119873(119866 119896) If 119866 is 119896-connectedit follows that there does not exist a set of 119896 minus 1 vertices

whose removal will partition 119866 into two or more connectedcomponents

Definition 3 (neighboring 119896-connected subgraphs) For twodisjoint subgraphs 119866

119894and 119866

119895of 119866 if exist119906 isin 119881

119894 V isin 119881

119895

and exist(119906 V) isin 119864 119866119894and 119866

119895are neighboring subgraphs

denoted by 119873119861119877119866(119866119894 119866119895) If 119862119874119873(119866

119894 119896) and 119862119874119873(119866

119895 119896) and


exist(1199061 V1) (119906

119896 V119896) isin 119864 where 119906

1 119906

119896isin 119881119894and

V1 V

119896isin 119881119895 119866119894and 119866

119895are neighboring 119896-connected

subgraphs denoted by119873119861119877119866(119866119894 119866119895 119896)

Definition 4 (multihop 119896-connected subgraphs) Let 1198661 1198662

119866119899be partitioning of 119866 If exist119866

119897subject to 119873119861119877

119866(119866119894

119866119897 119896)and119873119861119877

119866(119866119897 119866119895 119896)119866

119894and119866

119895aremultihop 119896-connected


3 Algorithms for Topology Control

Recall from Introduction that the design aims of the DMTCalgorithm are twofold (1) to provide min-max delay optimalthrough an AS approach and (2) to achieve strong connec-tivity in the resulting network The DMTC algorithm doesnot require the global topology of the PSN network to beknown by any entity On the contrary DMTC relies on ASnetworks where nodes autonomously form groups and selecta core for each AS network It is a hybrid of centralizedalgorithm and distributed algorithm A centralized topologycontrol algorithm is applied to each AS network to achievethe desired connectivity within the AS while the desiredconnectivity between adjacent AS networks is achieved vialocalized information sharing between adjacentAS coresThefollowing subsections detail the three phases of the DMTCalgorithm

31 Phase 1 AS Network Formation The main function ofPhase 1 is to select a minimal number of nodes as cores thatdominate the AS networks by using only 1-hop transmissionAnd these cores will take the main responsibility for thesubsequent two phases

Step 1 (broadcasting hello messages) When starting upeach node broadcasts hello messages periodically in orderto let them discover each other in the surrounding areaA hello message is of the form (119873119900119889119890119868119863 119871119900119888119886119905119894119900119899 119862119900119903119890119868119863

119863119890119892119903119890119890 119863119890119897119886119910) The explanation of each field is as follows(1) 119873119900119889119890119868119863 the unique ID of each node (2) 119871119900119888119886119905119894119900119899 thelocation of each node (3) 119862119900119903119890119868119863 the ID of the core withwhich the sending node is currently associated if the sendingnode does not associate with any core it is zero note that acore node uses its own ID for this field (4) 119863119890119892119903119890119890 the degreeof connectivity (the number of neighbors) (5) 119863119890119897119886119910 timedelay to each neighbor when exchanging information It maycontain processing transmission and propagation delay inpractice In order to facilitate the analysis we only considerpropagation delay in this paper

Step 2 (core selection process) The core selection process ofeach node begins after it has broadcasted hello messages for acertain waiting timeThewaiting time should be long enoughto allow this node to receive at least one hello message fromevery immediate neighbor In this process every node willdecide whether it is suitable as a core of an AS or become amember of an AS by checking for its local optimality Eachnode computes its own height from its current states Theheight metric should be chosen to suit the design goals ofthe PSN topology control algorithm As a result we use

(119863119890119897119886119910119863119890119892119903119890119890119873119900119889119890119868119863) as the height metric 119873119900119889119890119868119863 isincluded in the metric calculation to break ties The heightfunction is ℎ119890119894119892ℎ119905(119906) = (ℎ(119906) 119868119863(119906)) In order to balance thefactor of119863119890119897119886119910 and119863119890119892119903119890119890 we formulate ℎ(119906) as

ℎ (119906) = 119891 (119863119890119892119903119890119890 (119906) 119863119890119897119886119910 (119906 V119894) 120572) (2)

where 119891(sdot) denotes the balance function and 120572 is the balancefactor The relationship between ℎ119894119892ℎ119905(119906) and ℎ119894119892ℎ119905(V) isgiven by

ℎ119894119892ℎ119905 (119906) gt ℎ119894119892ℎ119905 (V) lArrrArr ℎ (119906) gt ℎ (V) or

(ℎ (119906) = ℎ (V)) and (119873119900119889119890119868119863 (119906) gt 119873119900119889119890119868119863 (V)) (3)

Then if a node has the highest height among its neighborsit is considered as a local optimal node and should serve as acore After this process the first batch of cores is selected andall consequent hello messages will be changed accordingly

Step 3 (supplement of cores) After Step 2 each node checksif there are cores in the range 119877max If cores exist it will regardthe core that has the least 119863119890119897119886119910 between them as its parentThat is this node will be the member of the AS dominated byits parent core Then nodes update the 119862119900119903119890119868119863 in their hellomessages with their parent coresrsquo ID Note that a core nodeuses its own ID for this field After that nodes whose119862119900119903119890119868119863

are zero without parent calculate their height functions Andthe node that has the highest height among its neighborswithout parent in the range 119877max should serve as a core

Step 4 (optimization andmaintenance process) Consideringnodesrsquo mobility and in order to keep the number of cores aslow as possible if a core detects there are other cores in therange 119877max (from the hello process) it will check whether ithas the highest height among these cores If not it will turninto a member of the highest height core and its membernodes will turn into nodes without parent If there existnodes without parent in the PSN process will turn to Step 3Finally there are only two kinds of nodes cores andmembersAnd this optimization and maintenance process will keepmonitoring the PSN For instance if a new node is added tothe PSN the process will take this node as a node withoutparent and turn to Step 3

32 Phase 2 Intra-AS Topology Control In this phasewe present a centralized algorithm for intra-AS networkEach core will calculate the links for all of the mem-bers of its AS such that the resulting topology of the ASmeets the given topology constraint (min-max delay and119896-connectivity) The intra-AS topology control algorithm isdescribed in Algorithm 1 where 119866 represents the PSN andlet 1198661 1198662 119866

119899(AS) be partitioning of 119866

For each AS Algorithm 1 ensures that 119866119896preserve the 119896-

connectivity of 119866119904 that is 119862119874119873(119866

119904 119896) rArr 119862119874119873(119866

119896 119896) And

the maximum end-to-end delay among all edges in the ASnetwork is minimized by Algorithm 1 that is let 119863max(119866119896)be the maximum delay of all edges in the AS minimizedby Algorithm 1 and let 119878

119896(119866119904) be the set of all kinds of 119896-

connected subgraphs of 119866119904with the same vertices 119881

119904 then


Input (at AS 119866119904= (119881119904 119864119904))

119896 (required connectivity)Output

119866119896= (119881119896 119864119896)

Begin119881119896larr 119881119904 119864119896larr 0

Sort all edges in 119864119904in ascending order of weight (as defined in Definition 1)

for all edge (119906119894 V119894) in the order do

if 119906119894is not k-connected to V

119894then

119864119896larr 119864119896cup (119906119894 V119894)

end ifend forfor all edge (119906

119895 V119895) of 119864

119896in the descending order do

if 119906119895is still k-connected to V

119895with the disconnection of edge (119906

119895 V119895) then

119864119896larr 119864119896minus (119906119895 V119895)

end ifend for

Return 119866119896

Algorithm 1 Intra-AS topology control

we have 119863max(119866119896) = min119863max(119866119894) | 119866119894

isin 119878119896(119866119904) The

correctness of Algorithm 1 is provided in Section 4

33 Phase 3 Inter-AS Topology Control In this phase con-nectivity between adjacent AS networks is considered Inorder to allow adjacent AS networks to discover each otherevery node continues broadcasting hello message (119873119900119889119890119868119863

119871119900119888119886119905119894119900119899 119862119900119903119890119868119863119863119890119892119903119890119890 119863119890119897119886119910) as in Phase 1 periodicallyWhen node 119906 receives a hello message from node V thatbelongs to a different AS (eg they have different 119862119900119903119890119868119863)119906will place Vrsquos information in its border listThen this borderlist is reported to the nodersquos parent core With these borderlists we present a distributed algorithm for inter-AS Thisalgorithm is described inAlgorithm 2where119866 represents thePSN and let 119866

1 1198662 119866


In this algorithm the core of AS 119860 checks whether thereexist 119896 disjoint links from this AS to each adjacent AS 119861 Thatis accomplished by applying an algorithm (119872119886119909119872119886119905119888ℎ119894119899119892)[26] that computes a matching of maximum cardinality ina bipartite graph defined by the nodes in respective ASnetworks and the edges with one vertex in each AS If 119896

does not exceed the size of maximum cardinality matchingthe core of AS 119860 selects 119896 disjoint links that meet the min-max delay optimal When there do not exist 119896 disjoint linksbetween 119860 and 119861 (only 119896

119898disjoint links) the core preserves

the 119896119898-connectivity between these two AS networks and

minimizes the maximum delay between them Note that thisconnectivity preservation (119896

119898-connectivity) cannot guaran-

tee 119896-connectivity between AS 119860 and 119861 However global 119896-connectivity can be guaranteed after Phase 3 is completedwhen connectivity with other neighboring AS networks isalready established This will be proved in Section 4

Parameter119863119868119860(1198661 1198662) in Algorithm 2 is used to perform

an optimization which removes unnecessary links betweencertain adjacent AS networks while preserving the connec-tivity of the resulting topology 119863

119868119860(1198661 1198662) is the maximum

delay of the selected 119896 links However when the number 119896119898of

disjoint links between two adjacentASnetworks is less than 119896119863119868119860(1198661 1198662) isinfinThenAS119860will not connect to neighboring

AS 119861 directly if it observes that there exists another AS 119862where 119862 is also a neighbor of 119861 and both 119863

119868119860(119866119860 119866119862) and

119863119868119860(119866119861 119866119862) are less than119863

119868119860(119866119860 119866119861)

After Phase 3 is completed each node is assigned a linklist and nodes connect to each other according to theselists This topology will be maintained by every node withhello message periodically and always preserve the objectiveconnectivity of the network

4 Proof of Strong Connectivity

In this section we prove the strong connectivity of Algo-rithms 1 and 2 [27] The results are given as the followingtheorems

41 Strong Connectivity of Algorithm 1

Theorem 5 Algorithm 1 can preserve 119896-connectivity of AS 119866119904

that is 119862119874119873(119866119904 119896) rArr 119862119874119873(119866

119896 119896) And the maximum delay

among all nodes in the network is minimized by Algorithm 1

Before proving the correctness ofTheorem 5 two lemmasare first provided Let 119901 = 119906119909


119909119898V be the path

fromnode 119906 to V (as defined inDefinition 2) Let themaximalset of disjoint paths from node 119906 to V in graph 119866

119904be


Input (at AS 119866119896= (119881119896 119864119896))

119896 (required connectivity)OutputLinks for all nodes in 119866

119896rsquos border list

Begin119866119896119894

= (119881119896119894 119864119896119894) 119881119896119894

larr 119881119896 119864119896119894

larr 0

for all 119866119894subject to 119873119861119877

119866(119866119896 119866119894) do

1198811015840larr V | (V isin 119866

119894) and (V is adjacent to 119866

119896)

119881119896119894

larr 119881119896119894

cup 1198811015840

119864119896119894

larr (119906 V) | (119906 isin 119881119896) and (V isin 119881

1015840) and (119903(119906 V) le 119877max)

119872 larr 0

119864119886larr sort all edges in 119864

119896119894in ascending order of weight (as defined in Definition 1)

119896119898

larr |119872119886119909119872119886119905119888ℎ119894119899119892(119866119896119894)|

|119872119886119909119872119886119905119888ℎ119894119899119892(119866119896119894)| is the number of edges in 119872119886119909119872119886119905119888ℎ119894119899119892(119866

119896119894)

if 119896119898

ge 119896 thenfor all edges 119890

119905= (119906119905 V119905) isin 119864119886in the order do

Find the smallest 119905 subject to |119872| ge 119896 where 119872 larr 119872119886119909119872119886119905119888ℎ119894119899119892(119866119905= (119881119896119894 119864119886(119905))) and |119872| is the number of

edges in 119872 119864119886(119905) = 119890

1 119890

119905

end for119863119868119860(119866119896 119866119894) larr |119890

119905| where |119890

119905| is the weight of 119890

119905

119871(119866119896 119866119894) larr 119872

elsefor all edges 119890


Find the smallest 119905 subject to |119872| ge 1198962 and 119872 larr 119872119886119909119872119886119905119888ℎ119894119899119892(119866

119905= (119881119896119894 119864119886(119905)))

end for119863119868119860(119866119896 119866119894) larr infin

119871(119866119896 119866119894) larr 119872

end ifSend 119863

119868119860(119866119896 119866119894) to neighbor AS

end forCollect 119863

119868119860from neighboring AS

119871119868119878119879 larr 0


119866(119866119896 119866119901) do

if there does not exist 119866119902subject to

119873119861119877119866(119866119896 119866119902) and 119873119861119877

119866(119866119896 119866119901)and

(119863119868119860(119866119896 119866119902) lt 119863

119868119860(119866119896 119866119901))and then

(119863119868119860(119866119901 119866119902) lt 119863

119868119860(119866119896 119866119901))

119871119868119878119879 larr 119871119868119878119879 cup 119871(119866119896 119866119894)

end ifend for

Return 119871119868119878119879

Algorithm 2 Inter-AS topology control

represented by 119875119906V(119866119904) that is forall119901119898 119901119899 isin 119875

119906V(119866119904) 119901119898 cap119901119899=

119906 V If edge 1198900= (119906 V) let 119866

119904minus 1198900be the resulting graph by

removing the edge 1198900from 119866

119904

Lemma6 Let 119906 and V be two vertices in the 119896-connected graph119866119904 if 119906 and V are still 119896-connected after the removal of edge

1198900= (119906 V) then 119862119874119873(119866

119904minus 1198900 119896)

Proof of Lemma 6 In order to prove 119862119874119873(119866119904minus 1198900 119896) we

prove that 1198661015840119904= 119866119904minus 1198900is connected with the removal of any

119896 minus 1 vertices from 1198661015840

119904 We already know that 119906 and V are 119896-

connected in 1198661015840

119904 Thus considering any two vertices 119906

1 V1

we assume that 1199061 V1 cap 119906 V = 0 We only need to prove

that 1199061is still connected to V

1after the removal of set 119896 minus 1

vertices 119883 = 1199091 119909

119896minus1 where 119909

119894isin (119881(119866

1015840

119904) minus 119906

1 V1) If

(1199061 V1) is an edge in119866

1015840

119904 that is obviously true Hence we only

consider the case that there is no direct edge from 1199061to V1

Since 119862119874119873(119866119904 119896) we have |119875

1199061V1

(119866119904)| ge 119896 where

|1198751199061V1

(119866119904)| is the number of paths in the set 119875

1199061V1

(119866119904) Let

1199031be the number of paths in 119875

1199061V1

(1198661015840

119904) that are broken after

the removal of vertices in the set of 119883 that is 1199031

= 119901 isin

1198751199061V1

(1198661015840

119904) | (119909

119894isin 119883) and (119909

119894isin 119901) We know that paths

in 1198751199061V1

(1198661015840

119904) are disjoint so the removal of any one vertex

in 119883 can only break at most one path in 1198751199061V1

(1198661015840

119904) Given

|119883| = 119896 minus 1 we have 1199031le 119896 minus 1

Let 11986610158401015840

119904be the resulting graph by removing 119883 from 119866

1015840

119904

If |1198751199061V1

(1198661015840

119904)| ge 119896 we have |119875

1199061V1

(11986610158401015840

119904)| ge (|119875

1199061V1

(1198661015840

119904)| minus

1199031) ge 1 that is 119906

1is still connected to V

1in 11986610158401015840

119904 Otherwise

|1198751199061V1

(1198661015840

119904)| lt 119896 it occurs only if the removal of edge


1198900

= (119906 V) breaks one path 119901119895

isin 1198751199061V1

(119866119904) Without loss

of generality let the order of vertices in the path 119901119895be

1199061 119906 V V

1 Since the paths in 119875

1199061V1

(119866119904) are disjoint

the removal of edge 1198900breaks at most one path that is

|1198751199061V1

(119866119904) minus 119901

119895| ge 119896 minus 1 So we have |119875

1199061V1

(1198661015840

119904)| = 119896 minus 1

If 1199031lt 119896minus1 it is obvious that (|119875

1199061V1

(1198661015840

119904)|minus1199031) ge 1 Hence

|1198751199061V1

(11986610158401015840

119904)| ge 1 That is 119906


1in 11986610158401015840

119904

Otherwise if 1199031= 119896 minus 1 every vertex in the set 119883 belongs to

the paths in1198751199061V1

(1198661015840

119904)We know that119901

119895isin 1198751199061V1

(119866119904) is disjoint

with the paths in 1198751199061V1

(1198661015840

119904) so we have 119901

119895cap119883 = 0 Hence no

vertex in 1199061 119906 V V

1is removed with the removal of

119883 So with the removal of 1198900 1199061is still connected to 119906 and V

is still connected to V1in11986610158401015840

119904 With the assumption that 119906 and

V are still 119896-connected after the removal of edge 1198900= (119906 V) in

Lemma 6 it is obvious that 119906 is still connected to V in 11986610158401015840

119904 So


1in 11986610158401015840

119904

We have proved that for any two vertices 1199061 V1 isin 1198661015840

119904 1199061

is connected to V1with the removal of any 119896minus 1 vertices from

119881(1198661015840

119904) minus 119906

1 V1 Hence 119862119874119873(119866

1015840

119904 119896)

Lemma 7 Let 119866119904and 119866

119904be two graphs where 119862119874119873(119866

119904 119896)

and 119881(119866119904) = 119881(119866

119904) If every edge subject to (119906 V) isin (119864(119866

119904) minus

119864(119866119904)) satisfies that 119906 is still 119896-connected to V in graph 119866

119904minus

(1199061015840 V1015840) isin 119864(119866

119904) | 119908(119906

1015840 V1015840) ge 119908(119906 V) then 119862119874119873(119866

119904 119896)

Proof of Lemma 7 Without loss of generality let 1198901 1198902

119890119898 = 119864(119866

119904)minus119864(119866

119904) = (119906

1 V1) (1199062 V2) (119906

119898 V119898) be a set

of edges subject to 119908(1198901) gt 119908(119890

2) gt sdot sdot sdot gt 119908(119890

119898) We define a

series of subgraphs of 119866119904 1198660119904= 119866119904 and 119866

119894

119904= 119866119894minus1

119904minus 119890119894 where

119894 = 1 2 119898Then119864(119866119898

119904) sube 119864(119866

119904) Herewe prove Lemma 7

by induction

Base Obviously we have 1198660

119904= 119866119904and 119862119874119873(119866

0

119904 119896)

Induction If119862119874119873(119866119894minus1

119904 119896) we prove that119862119874119873(119866

119894

119904 119896) where

119894 = 1 2 119898 Since 119866119904minus (1199061015840 V1015840) isin 119864(119866

119904) | 119908(119906

1015840 V1015840) ge

119908(119906119894 V119894) sube 119866

119894minus1

119904minus (119906119894 V119894) and from the assumption of

Lemma 7 (119906119894is 119896-connected to V

119894in graph 119866

119904minus (1199061015840 V1015840) isin

119864(119866119904) | 119908(119906

1015840 V1015840) ge 119908(119906

119894 V119894)) we obtain that 119906

119894is 119896-

connected to V119894in graph 119866

119894minus1

119904minus (119906119894 V119894) Applying Lemma 6

to 119866119894minus1

119904 it is obvious that 119862119874119873(119866

119894minus1

119904minus (119906119894 V119894) 119896) That is

119862119874119873(119866119894

119904 119896)

By induction we have 119862119874119873(119866119898

119904 119896) Since 119864(119866

119898

119904) sube

119864(119866119904) hence 119862119874119873(119866

119904 119896)

Finally we prove the correctness ofTheorem 5 as follows

Proof ofTheorem 5 In Algorithm 1 we place all edges into119866119896

in the ascending order Whether (119906 V) should be placed into119866119896depends on the connection of 119906 and V and edges of smaller

weights That is every edge (119906 V) isin 119864(119866119904) minus 119864(119866

119896) should

satisfy that 119906 is 119896-connected to V in 119866119904minus (1199061015840 V1015840) isin 119864(119866

119904) |

119908(1199061015840 V1015840) ge 119908(119906 V) Applying Lemma 7 here then we can

prove that 119862119874119873(119866119904 119896) rArr 119862119874119873(119866

119896 119896)

Recall that 119863max(119866119896) is the maximum delay of all edgesin the AS minimized by Algorithm 1 and 119878

119896(119866119904) is the set

of all kinds of 119896-connected subgraphs of 119866119904with the same

vertices 119881119904 The maximum delay among all edges in the

network isminimized byAlgorithm 1which can be describedas 119863max(119866119896) = min119863max(119866119894) | 119866

119894isin 119878119896(119866119904)

Let (119906119898 V119898) be the last edge that is placed into 119866

119896 It

is obvious that (119906119898 V119898) cannot be removed from 119864(119866

119896)

in the process of Algorithm 1 that is 119908(119906119898 V119898) =

max(119906V)isin119864(119866

119896)119908(119906 V) Let1198661015840

119896= 119866119896minus(119906119898 V119898) thenwe obtain

that |119875119906119898V119898

(1198661015840

119896)| lt 119896 Now we assume that there is graph

119867119904

= (119881(119867119904) 119864(119867

119904)) where 119881(119867

119904) = 119881(119866

119904) and 119864(119867

119904) =

(119906 V) isin 119864(119866119904) | 119908(119906 V) lt 119908(119906

119898 V119898) If we can prove

that 119862119874119873(119867119904 119896) is not true we will obtain that any 119866

119894isin

119878119896(119866119904) should have at least one edge equal to or heavier than

(119906119898 V119898) That is 119863max(119866119896) = min119863max(119866119894) | 119866

119894isin 119878119896(119866119904)

We prove that 119862119874119873(119867119904 119896) is not true by contradiction in the

followingAssume that 119862119874119873(119867

119904 119896) hence |119875

119906119898V119898

(119867119904)| ge 119896 We

have 119867119904minus 1198661015840

119896= 0 Since all edges are placed into 119866

1015840

119896in the

ascending order forall(119906 V) isin 119867119904minus 1198661015840

119896should satisfy that 119906 is 119896-

connected to V in119867119904minus(1199061015840 V1015840) isin 119864(119867

119904) | 119908(119906

1015840 V1015840) ge 119908(119906 V)

Applying Lemma 7 here we obtain that 119862119874119873(1198661015840

119896 119896) That is

|119875119906119898V119898

(1198661015840

119896)| ge 119896 which is a contradiction


Theorem 8 Let 119866 = (119881 119864) be the initial topology of the PSNLet1198661015840 = (119881 119864

1015840) be the topology after Algorithm 2 is completed

Then we have 119862119874119873(119866 119896) hArr 119862119874119873(1198661015840 119896)

Before proving the correctness of Theorem 8 severallemmas used in that proof are first provided

Lemma 9 Let 119866119894= (119881119894 119864119894) and 119866

119895= (119881119895 119864119895) be two sub-

graphs of graph 119866 If119873119861119877119866(119866119894 119866119895 119896) then 119862119874119873(119866

119894cup119866119866119895 119896)

Proof of Lemma 9 In order to prove 119862119874119873(119866119894cup119866119866119895 119896) we

prove 119866119894cup119866119866119895is connected with the removal of any 119896 minus 1

vertices from it Since 119873119861119877119866(119866119894 119866119895 119896) we have 119862119874119873(119866

119894 119896)

and 119862119874119873(119866119895 119896) that is consider any 119906 V isin 119866

119894or 119906 V isin 119866

119895

119906 is 119896-connected to V Then we only need to consider the case(119906 isin 119866

119894) and (V isin 119866

119895)

Since119873119861119877119866(119866119894 119866119895 119896) exist119906

0isin 119866119894 V0isin 119866119895 1199060is connected

to V0with the removal of any 119896 minus 1 vertices from 119881

119894cup 119881119895minus

1199060 V0 With 119862119874119873(119866

119894 119896) and 119862119874119873(119866

119895 119896) we know that

119906 is connected to 1199060 and V is connected to V

0 Hence 119906 is

connected to VThat is119866119894cup119866119866119895is connectedwith the removal

of any 119896 minus 1 vertices from it

Corollary 10 Let subgraphs 1198661 1198662 119866

119899be partitioning

of 119866 Let 119878119898be the maximal set of subgraphs subject to the

following forall119866119894 119866119895isin 119878119898 exist119872119862119874119873

119866(119866119894 119866119895 119896) Then cup

119866119866119894|

119866119894isin 119878119898 is 119896-connected

Lemma 11 Let 119866119904be a subgraph of 119866 and let 119866

1015840

119904be edges

reduction of 119866119904 Let 119866

10158401015840= (119881 119864

1015840) = (119866 minus 119866

119904)cup1198661198661015840

119904 If

119862119874119873(119866119904 119896) and 119862119874119873(119866

1015840

119904 119896) and 119862119874119873(119866 119896) then 119862119874119873(119866

10158401015840 119896)


Proof of Lemma 11 In order to prove 119862119874119873(11986610158401015840 119896) we prove

that forall119906 V isin 11986610158401015840 is connected with the removal of any 119896 minus 1

vertices from 11986610158401015840 Without loss of generality three cases are

considered in the following

(1) 119906 V isin 119881119904 it is obviously true because of 119862119874119873(119866

1015840

119904 119896)

(2) 119906 isin 119881119904and V isin 119881 minus 119881

119904 since 119862119874119873(119866 119896) 119906 is

connected to V in path 119901with the removal of any 119896minus1

vertices in 119866 If 119901 sube 119864 minus 119864119904 119901 also exists in 119866

10158401015840 119906is connected to V by removing those 119896 minus 1 verticesOtherwise exist(119886 isin 119901) and (119886 isin 119881

119904) and 119886 is connected to

V in 119866minus119866119904 Since 119862119874119873(119866

1015840

119904 119896) 119906 is connected to 119886 by

removing those 119896 minus 1 vertices Then 119906 is connected toV with the removal of any 119896 minus 1 vertices in 119866

10158401015840(3) 119906 V isin 119881 minus 119881

119904 similarly since 119862119874119873(119866 119896) 119906 is


vertices in 119866 If 119901 sube 119864 minus 119864119904 119906 is 119896-connected to V

in 11986610158401015840 Otherwise exist(119886

1 1198862

isin 119901) and (1198861 1198862

isin 119881119904) 119906 is

connected to 1198861 and 119886

2is connected to V in 119866 minus 119866

119904

Since 119862119874119873(1198661015840

119904 119896) 119886

1is connected to 119886

2by removing

those 119896 minus 1 vertices Then 119906 is connected to V with theremoval of any 119896 minus 1 vertices in 119866

10158401015840

Corollary 12 Let 1198661 1198662 119866

119899be 119896-connected subgraphs of

119896-connected graph 119866 Let 11986610158401 1198661015840

2 119866

1015840

119899be edges reduction of

1198661 1198662 119866

119899 and 119866

1015840

1 1198661015840

2 119866

1015840

119899are 119896-connected Then

11986610158401015840

= (119866 minus

119899

⋃

119894=1

119866119866119894)cup119866(

119899

⋃

119894=1

1198661198661015840

119894) (4)

is 119896-connected

Lemma 13 Let 119866 = (119881 119864) be the initial topology of the PSNLet1198661015840 = (119881 119864


Let 119866119894= (119881119894 119864119894) be the AS networks resulting from Phase 1 in

the topology control where 119894 = 1 119899 and 119864119894= (119906 V) isin 119864 |

119906 V isin 119881119894 Let 1198661015840

119894= (119881119894 1198641015840

119894) where 119864

1015840

119894= 119864119894cap 1198641015840 Then forall119894 119895

subject to 1 le 119894 le 119895 le 119899 we have that 119872119862119874119873119866(119866119894 119866119895 119896) rArr

1198721198621198741198731198661015840(1198661015840

119894 1198661015840

119895 119896)

Proof of Lemma 13 Since nodes of any intra-AS are 119896-connected we take an AS as a node here Formally we rep-resent graph 119866 as 119866 = (119881

119878 119864119878) where 119881

119878= 1198661 1198662 119866

119899

and 119864119878= (119866119894 119866119895) | 119873119861119877

119866(119866119894 119866119895 119896) Actually edge (119866

119894 119866119895)

contains at least 119896 disjoint paths between 119866119894and 119866

119895 Let

1198661015840

= (119881119878 1198641015840

119878) be the AS level representation of 119866

1015840 where1198641015840

119878= (119866

1015840

119894 1198661015840

119895) | 119873119861119877

1198661015840(1198661015840

119894 1198661015840

119895 119896)We use119881

119878to represent the

set of AS networks in 1198661015840 because we do not need to consider

the topology of intra-AS (both 119866119894and 119866

1015840

119894are 119896-connected)

We take all of them as nodes so we consider (119866119894 119866119895) and

(1198661015840

119894 1198661015840

119895) as the same edge Recall that in Algorithm 2 each

edge (119866119894 119866119895) isin 119864119878has weight 119863

119868119860(119866119894 119866119895)

In order to prove Lemma 13 it suffices to show thatforall119866119894 119866119895isin 119866 119866


119895in 1198661015840 We order all edges

in 119866 in the ascending sequence of weights and then judge

whether an edge should be placed into 1198661015840 Without loss of

generality let the ordering be (1198901 1198902 119890

119898) where119898 = |119864

119878|

Then we prove Lemma 13 by induction

Base Obviously the pair of AS networks corresponding toedge 119890

1should always be placed into 119866

1015840 that is 1198901isin 1198641015840

119878

Induction forall119905 le 119898 if for all 119902 lt 119905 the pair of AS networkscorresponding to 119890

119902are connected in 119866

1015840 (either directly orindirectly) And suppose 119890

119905= (119866119894 119866119895) FromAlgorithm 2 the

only reason why 119890119905notin 1198641015840

119878(119866119894is not directly connected to119866

119895in

1198661015840) is that there exists another AS 119866

119897 where both119863

119868119860(119866119894 119866119897)

and 119863119868119860(119866119897 119866119895) are less than 119863

119868119860(119866119894 119866119895) However edges

(119866119894 119866119897) and (119866

119897 119866119895) come before (119866

119894 119866119895) in the ascending

order From path 119866119894119866119897119866119895 119866119894is connected to 119866

119895in 1198661015840

By induction we prove that 119866119894is connected to 119866

119895in 1198661015840

and then119872119862119874119873119866(119866119894 119866119895 119896) rArr 119872119862119874119873

1198661015840(1198661015840

119894 1198661015840

119895 119896)

Finally we prove the correctness of Theorem 8 In theproof 119866

119894and 119866

1015840

119894have the same definition in Lemma 13

Proof of Theorem 8 For every AS 119866119894 we know that

119862119874119873(119866119894 119896) is true after Algorithm 1Thenwe partition those

AS networks into sets1198601 119860

119904 where each set contains AS

networks which are multihop 119896-connected in 119866 that is forall119903 =

1 119904 then (119866119894isin 119860119903) and (119872119862119874119873

119866(119866119894 119866119895 119896)) rArr 119866

119895isin 119860119903

Then we define sets 1198601015840

1 119860

1015840

119904 where forall119894 119866

119894isin 119860119903rArr 1198661015840

119894isin

1198601015840

119903 Applying Lemma 13 here for every 119860

1015840

119903= 1198661015840

1199031

1198661015840

119903119898

forall1 le 119894 lt 119895 le 119898 we have 119872119862119874119873

1198661015840(1198661015840

119903119894

1198661015840

119903119895

119896) Take 1198601015840

119903as a

subgraph of 1198661015840 1198601015840119903= (1198811198601015840

119903

1198641198601015840

119903

) where 1198811198601015840

119903

= V | V isin 1198601015840

119903

and 1198641198601015840

119903

= (119906 V) | (119906 V isin 1198601015840

119903) and ((119906 V) isin 119864

1015840) Since

1198601015840

119903only contains multihop 119896-connected subgraphs applying

Corollary 10 here we have that 1198601015840

119903is 119896-connected Then

applying Corollary 12 here we have that

1198661015840= (119866 minus (

119904

⋃

119903=1

119866119860119903))cup119866(

119904

⋃

119903=1

1198661198601015840

119903) (5)

is 119896-connected Then 119862119874119873(119866 119896) hArr 119862119874119873(1198661015840 119896)

5 Control Message Complexity Analysis

We study the control message complexity here by computingthe total number of control messages exchanged during thethree phases of theDMTC algorithmThe following terms areused in the complexity analysis

Let 119873 be the total number of nodes in the PSN Let 119878 bethe number ofAS networks and let119873

119878be the average number

of nodes per AS that is 119873119878

= 119873119878 Let 119877119861be the average

probability of nodes that are border nodes in an AS where0 lt 119877

119861lt 1 Let 119878

119873be the average number of neighboring AS

networks for each AS that is 0 lt 119878119873

lt 119878Table 1 shows the average control messages utilized in

each phase to complete the topology algorithm for eachAS We partition each phase into major steps Hence from


Table 1 Average message complexity in each phase of an AS

Steps in each phaseNumber ofcontrolmessages

Phase 1Each node announces its existence 119873

119904

Core of the AS is selected with 120582 cycles 120582119873119904

Each node announces its current role 119873119904

Phase 2Core node computes the intra-AS topology 0Phase 3All border nodes report their border lists to theparent core 119873

119904sdot 119877119861

Core node distributes 119863119868119860

vector to its bordernodes 1

Border nodes send 119863119868119860

vector to border nodes ofother AS networks 119878

119873

Border nodes of other AS networks report 119863119868119860

vector to their parent core 119878119873

Core node sends the link list to the AS members 1

Table 1 the total number of control messages required in thePSN is 119878((2 + 120582 + 119877

119861)119873119878+ 2119878119873

+ 2) It can be simplified as(2+120582+119877

119861)119873+2119878

119873119878+2119878 which is 119900(119873)+119900(119878

119873119878) in the worst

case

6 Simulation Results and Discussions

In this section we present several sets of simulation results toevaluate the effectiveness of the proposed DMTC algorithmRecall that the proposed algorithm is a hybrid of centralizedalgorithm and distributed algorithm We compare it withtypical centralized algorithm FGSS

119896[19] and distributed

algorithm FLSS119896[19]We chose these two algorithms because

they are also min-max optimal as our algorithm Thesesimulations were carried out using the NS2 simulator

In this simulation study the wireless channel is symmet-ric (ie both the sender and the receiver should observe thesame channel fading) and obstacle-free and each node has anequal maximal transmission range119877max = 450 km Nodes arerandomly distributed in a 2500 times 2500 km2 region In orderto study the effect of AS size on the resulting topologies wevary the number of nodes in the region among 125 150 175200 225 and 250

For each network we consider

(1) 119896-connectivity 119896 = 1 and 119896 = 2(2) algorithms the proposed hybrid algorithm DMTC

centralized algorithm FGSS119896 and distributed algo-

rithm FLSS119896

(3) 1000 Monte Carlo simulations

Relative to DMTC recall that in Phase 1 of AS networkformation we configure that each node is at most one hopaway from its parent core In our simulations algorithm inPhase 1 generates AS networks where the average number of

nodes per AS is 639 748 851 969 and 1069 (results of 1000simulations) respectively Note that by varying the numberof nodes in the network while maintaining other parameterssuch as the region size and maximal transmission range ofnodes we implicitly adjust the node degree of these topologycontrol algorithms

Before providing the experimental results regarding timedelay we first observe the actual topologies for one simulatednetwork using DMTC algorithm Four figures are given here

(1) Figure 3(a) shows the original physical topologywith-out topology control All nodes communicate withthe maximal transmission range 119877max

(2) Figure 3(b) shows the topology after applying algo-rithm of Phase 1 Nodes of the PSN are divided into17 AS networks where the average number of nodesper AS is 735

(3) Figure 3(c) is the topology resulting from the intra-AStopology control algorithm of Phase 2 when 119896 = 2

(4) Figure 3(d) shows the topology after applying inter-AS topology control algorithmofPhase 3 when 119896 = 2The inter-AS links are represented by black color

In Figure 4 we show average and maximum delaybetween two nodes which are obtained from three topologycontrol algorithms (the proposed hybrid algorithm DMTCcentralized algorithm FGSS

119896[19] and distributed algorithm

FLSS119896[19]) Note that we only consider link propagation

delay in this simulation It is evident from those results thatDMTC is very effective in reducing the delay between nodesRecall that the maximal transmission range 119877max of onenode is 450 km The corresponding delay is 1501ms When119896 = 1 (Figure 4(a)) DMTC reduces the maximum delay to1106ms when there are 125 nodes in the PSN and as low as0703ms when there are 225 nodes The maximum delay isapproximately 136 to 201 lower than FLSS

1distributed

algorithm and 61 to 186 higher than FGSS1centralized

algorithm For the average delay DMTC reduces the delay to0656ms when there are 125 nodes in the PSN and as low as0451ms when there are 225 nodes which is approximately52 to 103 lower than FLSS

1distributed algorithm and

85 to 109 higher than FGSS1centralized algorithm

When 119896 = 2 (Figure 4(b)) both the maximum andaverage delay resulting from DMTC FGSS

2 and FLSS

2

are all higher than those when 119896 = 1 That is expectedbecause 2-connected connectivity is a stronger property than1-connected connectivityWhat ismore the difference amongthe three algorithms when 119896 = 2 is in a greater range thanwhen 119896 = 1 This is the consequence of having to maintainanother higher delay link between adjacent AS networksand one more additional disjoint path from each node toother nodes within all AS networks The maximum delay isapproximately 185 to 209 lower than FLSS

2distributed


algorithmThe average delay is approximately 125 to 186lower than FLSS

2distributed algorithm and 82 to 156

higher than FGSS2centralized algorithm

The delay performance of the proposed algorithmDMTCfalls in between FGSS

119896and FLSS

119896 This is expected because


00

500 1000 1500 2000 2500

500

1000

1500

2000

2500

x (km)

y (k

m)

(a)

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

x (km)

y (k

m)

(b)

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

x (km)

y (k

m)

(c)

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

x (km)

y (k

m)

(d)

Figure 3 Network topologies of 125 nodes with different topology control settings (a)Without topology control (b) After applying algorithmof Phase 1 (c) 119896 = 2 after applying algorithm of Phase 2 (d) 119896 = 2 after applying algorithm of Phase 3

DMTC is a hybrid of centralized algorithm and distributedalgorithm Even though centralized algorithm has betterdelay performance (less than 20) they are not suitable forlarge scale networks Because excessive amounts of controlmessages need to be collected by one central entity and longdelay makes the control messages exchanged with remotenodes costly However the control message exchange inDMTC is constrained among neighboring AS networks andthe delay performance is better than distributed algorithm inthe simulation resultThus the proposedDMTC algorithm isbetter than centralized algorithm and distributed algorithmfor PSN

Figure 4(c) shows the average node degrees producedby DMTC versus a network without topology control It isobvious that the node degree of a network with DMTC doesnot depend on the size or density of the network

Figure 5 illustrates the number of messages exchangesrequired per node to complete DMTC in our simulationenvironment Recall that the message complexity of theDMTCalgorithm is 119900(119873)+119900(119878

119873119878) For each node the average

number of messages required is (119900(119873) + 119900(119878119873119878))119873 = 119900(1)

The result validates the analysis When the number of nodesin the PSN increases from 125 to 225 the average number ofmessages required per node in DMTCdoes not increaseThisshows that the DMTC algorithm has little extra overhead

7 Conclusion

We studied the topology control problem in the PSN using anAS approachThemotivation was that the AS network modeldecouples the complex PSN into simple AS networks Thenwe proposed the DMTC algorithm to minimize time delay


120 140 160 180 200 22002

04

06

08

1

12

14

16

Number of nodes in random topology

Tim

e del

ay (m

s)

DMTC maxFGSS1 maxFLSS1 max

DMTC avgFGSS1 avgFLSS1 avg

(a)

120 140 160 180 200 22002

04

06

08

1

12

14

16

18

2


Tim

e del

ay (m

s)

DMTC max DMTC avgFGSS2 maxFLSS2 max

FGSS2 avgFLSS2 avg

(b)

120 140 160 180 200 2200

2

4

6

8

10

12

14

16

18

20


Aver

age n

ode d

egre

e

DMTC k = 1

DMTC k = 2

Without control

(c)

Figure 4 Results from three topology control algorithms (DMTC FGSS119896 and FLSS

119896showing average and maximum link delay when (a)

119896 = 1 and (b) 119896 = 2 and (c) average node degree)

in the PSN Compared with most existing approaches whereeither the purely centralized or the purely distributed controlmethod is adopted DMTC utilizes a hybrid method In thisway not only is the control message exchange constrainedamong local neighboring AS networks but also the strongconnectivity of the network is preserved Our simulationresults validated the theoretic analysis and effectiveness of theDMTC algorithm

Although the assumptions stated in Sections 2 and 6 arewidely used in existing topology algorithms some of themmay not be practical Our future work will focus on howto relax these constraints (eg nodes in the PSN are homo-geneous obstacle-free channel and equal 119877max) for DMTCalgorithm so as to improve its practicality in real applicationsIn addition we find that the proposed ldquohybrid approachrdquo isa general method It can be extended to solve the control


120 140 160 180 200 2200

2

4

6

8

10

12


Aver

age n

umbe

r of m

essa

ges p

er n

ode

DMTC phase 1DMTC phase 3DMTC

Figure 5 Number of messages exchanges per node in DMTCwhenthe number of nodes in the PSN increases

problem of many other large scale networks for examplemachine-to-machine (M2M) network and space informationnetwork (SIN) Different topology control algorithms canbe applied within AS network and between adjacent ASnetworks depending on the optimization objective And eachAS network can be further separated into sub-AS networksWe will study these issues in the near future

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by NSF of China under Grants nos91338201 and 91438109

References

[1] I F Akyildiz O B Akan C Chen J Fang andW Su ldquoThe stateof the art in interplanetary internetrdquo IEEE CommunicationsMagazine vol 42 no 7 pp 108ndash118 2004

[2] J Mukherjee and B Ramamurthy ldquoCommunication technolo-gies and architectures for space network and interplanetaryinternetrdquo IEEE Communications Surveys and Tutorials vol 15no 2 pp 881ndash897 2013

[3] G Araniti I Bisio and M De Sanctis ldquoInterplanetary net-works architectural analysis technical challenges and solutionsoverviewrdquo in Proceedings of the IEEE International Conferenceon Communications pp 1ndash5 2010

[4] L Gou G-X Zhang D-M Bian F Xue and J Hu ldquoEfficientbroadcast retransmission based on network coding for inter-planetary internetrdquoChinaCommunication vol 10 no 8 pp 111ndash124 2013

[5] R Alena B Gilbaugh B Glass and S P Braham ldquoCommu-nication system architecture for planetary explorationrdquo IEEEAerospace and Electronic Systems Magazine vol 16 no 11 pp4ndash11 2001

[6] X-J Zhai H-Y Jing and T Vladimirova ldquoMulti-sensor datafusion in Wireless Sensor Networks for Planetary Explorationrdquoin Proceedings of the NASAESA Conference on Adaptive Hard-ware and Systems (AHS rsquo14) pp 188ndash195 July 2014

[7] P Rodrigues A Oliveira F Alvarez et al ldquoSpace wirelesssensor networks for planetary exploration node and networkarchitecturesrdquo in Proceedings of the NASAESA Conference onAdaptive Hardware and Systems (AHS rsquo14) pp 180ndash187 July2014

[8] B-Y Guo Q-S Guan F R Yu S-M Jiang and V C MLeung ldquoEnergy-efficient topology control with selective diver-sity in cooperative wireless ad hoc networks a game-theoreticapproachrdquo IEEE Transactions onWireless Communications vol13 no 11 pp 6484ndash6495 2014

[9] X Ao F R Yu S Jiang Q-S Guan and V C M LeungldquoDistributed cooperative topology control for WANETs withopportunistic interference cancelationrdquo IEEE Transactions onVehicular Technology vol 63 no 2 pp 789ndash801 2014

[10] L Liu Y Liu and N Zhang ldquoA complex network approach totopology control problem in underwater acoustic sensor net-worksrdquo IEEE Transactions on Parallel and Distributed Systemsvol 25 no 12 pp 3046ndash3055 2014

[11] D Shang B Zhang Z Yao and C Li ldquoAn energy efficientlocalized topology control algorithm for wireless multihopnetworksrdquo Journal of Communications andNetworks vol 16 no4 pp 371ndash377 2014

[12] M Huang S Chen Y Zhu and YWang ldquoTopology control fortime-evolving and predictable delay-tolerant networksrdquo IEEETransactions on Computers vol 62 no 11 pp 2308ndash2321 2013

[13] M Li Z Li and A V Vasilakos ldquoA survey on topology controlin wireless sensor networks taxonomy comparative study andopen issuesrdquo Proceedings of the IEEE vol 101 no 12 pp 2538ndash2557 2013

[14] S Sardellitti S Barbarossa and A Swami ldquoOptimal topologycontrol and power allocation for minimum energy consump-tion in consensus networksrdquo IEEE Transactions on SignalProcessing vol 60 no 1 pp 383ndash399 2012

[15] OAwwadAAl-Fuqaha BKhan andG B Brahim ldquoTopologycontrol schema for better QoS in hybrid RFFSO mesh net-worksrdquo IEEE Transactions on Communications vol 60 no 5pp 1398ndash1406 2012

[16] AAAziz Y A Sekercioglu P Fitzpatrick andM Ivanovich ldquoAsurvey ondistributed topology control techniques for extendingthe lifetime of battery powered wireless sensor networksrdquo IEEECommunications Surveys andTutorials vol 15 no 1 pp 121ndash1442013

[17] R Ramanathan and R Rosales-Hain ldquoTopology control ofmultihop wireless networks using transmit power adjustmentrdquoin Proceedings of the 19th Annual Joint Conference of the IEEEComputer and Communications Societies (INFOCOM rsquo00) vol2 pp 404ndash413 IEEE Tel Aviv Israel 2000

[18] J Yu H Roh W Lee S Pack and D-Z Du ldquoTopologycontrol in cooperative wireless ad-hoc networksrdquo IEEE Journal


on Selected Areas in Communications vol 30 no 9 pp 1771ndash1779 2012

[19] N Li and J C Hou ldquoLocalized fault-tolerant topology controlin wireless ad hoc networksrdquo IEEE Transactions on Parallel andDistributed Systems vol 17 no 4 pp 307ndash320 2006

[20] R Wattenhofer L Li P Bahl and Y-M Wang ldquoDistributedtopology control for power efficient operation in multihopwireless ad hoc networksrdquo in Proceedings of the 20th AnnualJoint Conference of the IEEE Computer and CommunicationsSocieties pp 1388ndash1397 April 2001

[21] T M Chiwewe and G P Hancke ldquoA distributed topologycontrol technique for low interference and energy efficiencyin wireless sensor networksrdquo IEEE Transactions on IndustrialInformatics vol 8 no 1 pp 11ndash19 2012

[22] P Djukic and S Valaee ldquoDelay aware link scheduling for multi-hop TDMAwireless networksrdquo IEEE Transactions on IndustrialInformatics vol 8 no 1 pp 11ndash19 2012

[23] M-L Cao ldquoAlgorithms research of autonomous navigationand control of planetary exploration roverrdquo in Proceedings ofthe Control and Decision Conference pp 4359ndash4364 XuzhouChina May 2010

[24] X-N Ning and L-L Liu ldquoA two-mode INSCNS navigationmethod for lunar roversrdquo IEEE Transactions on Instrumentationand Measurement vol 63 no 9 pp 2170ndash2179 2014

[25] S B Goldberg MWMaimone and L Matthies ldquoStereo visionand rover navigation software for planetary explorationrdquo inProceedings of the IEEE Aerospace Conference pp 2025ndash2036IEEE 2002

[26] A Azad M Halappanavar S Rajamanickam E G BomanA Khan and A Pothen ldquoMultithreaded algorithms for max-imum matching in bipartite graphsrdquo in Proceedings of the 26thIEEE International Parallel amp Distributed Processing Symposium(IPDPS rsquo12) pp 860ndash872 IEEE Shanghai China May 2012

[27] J A Bondy and U S R Murty GraphTheory Springer 2008

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



relations That is topology control algorithms of the PSNactually remove unnecessary long links As a result thenetwork topology is susceptible to unpredictable events suchas hardware failures in such a harsh environment Thereforeto design robust topology control algorithms 119896-connectivityof the network is considered where a 119896-connected network is119896 minus 1 fault-tolerant that is the failure of less than 119896 minus 1 nodeswill not disconnect the whole network

In this paper we study the topology control problem inthe PSN using an autonomous system (AS) approach An ASnetwork is a collection of nodes with similar properties forexample nodes distributed in the same region The reasonsfor using the AS approach are twofold Firstly the complexPSN is decoupled into a series of small AS networks andcentralized method can be used in each AS to ensure strongconnectivity Secondly distributed method is used amongAS networks thus the topology control message exchangecan be constrained among neighboring AS networks Wepropose a delay minimization topology control (DMTC)algorithm using such a hybrid approach DMTC preserves119896-connectivity and is min-max delay optimal The min-maxcriterion tries to minimize the maximum end-to-end delaybetween any pair of nodes in the network [22] Brieflythe DMTC algorithm consists of three phases (i) nodesin the PSN autonomously form AS networks and elect AScores (ii) with the topology information gathered from themembers of its AS network each AS core minimizes themaximum link delay used by all the nodes and guaranteesstrong connectivity using a centralized method (iii) eachAS core selects a set of border nodes shares topologyinformation with neighboring AS cores and computes lowtime delay links between neighboring AS networks using adistributed methodThemain contributions of this paper aresummarized as follows

(1) An AS network model of PSN is proposed Thelarge scale and complex PSN is decoupled into smallAS networks with similar nodes to achieve strongconnectivity with low cost control messages

(2) A delay minimization topology control (DMTC)algorithm is proposed to achieve low time delay It isa hybrid algorithm within an AS network and amongneighboring AS networks

(3) The strong connectivity of DMTC algorithm provedthat the algorithm could achieve logical 119896-connec-tivity on the condition that the original physicaltopology is 119896-connectivity

The rest of this paper is organized as follows In Section 2we define the network model and provide some definitionsIn Section 3 we propose an AS based algorithm DMTC toachieve low time delay and strong connectivity Then thevalidity of DMTC is proved in Section 4 and the messagecomplexity of our algorithm is analyzed in Section 5 InSection 6 simulation results and discussion are presentedFinally we make conclusion in Section 7

2 Network Model

In this section the network model of AS network is definedAs presented above the PSN is a self-organizing systemconstituted by various nodes For example as demonstratedin Figure 1 the Mars PSN is a part of the IPN Nodes inthe PSN have a wide and uneven distribution They work indifferent areas with either mobile (eg rovers) or static (egsensors) statuses If we apply a unified strategy to managethe whole PSN it will induce low efficiency and even cannotmaintain the normal operation of the network with toomuchcontrol information So as shown in Figure 2 we divide thePSN into a series of AS networks according to the property ofthe nodes Each AS network can adopt independent topologycontrol strategy to achieve strong connectivity And thecontrol message exchange is constrained among neighboringAS networks to reduce the cost of control

Considering that the properties of nodes in the PSN aresimilar except few nodes we assume that all the nodes arehomogeneous They have the same maximal transmissionrange 119877max Let the PSN network topology be representedby undirected simple graph 119866 = (119881 119864) where 119881 =

1199061 1199062 119906

119899 is the set of nodes (or equivalently vertices)

and 119864 = (119906119894 119906119895) | (119906119894 119906119895isin 119881) and (119903(119906

119894 119906119895) le 119877max) is the set

of links (edges) 119903(119906119894 119906119895) is the distance between nodes 119906

119894and

119906119895 Each node is assigned a unique identifier (ID) according

to its property such as MAC addressWe assume that 119866 is a general graph that is if (119906 V) isin

119864 119906 and V can exchange information with each other Wealso assume that the link is symmetric and obstacle-free andeach node is able to obtain its location by some means (egcelestial navigation [23] initial navigation [24] and visionnavigation [25]) We then define several graphs related termsin the following which will be used in both algorithms andproofs For all definitions we refer to graph 119866 = (119881 119864) andsubgraphs 119866

119894= (119881119894 119864119894) and 119866

119895= (119881119895 119864119895)

Definition 1 (weight function) For edge 119890 = (119906 V) theweight function is 119908(119906 V) = (119889(119906 V)min(119868119863(119906) 119868119863(V))max(119868119863(119906) 119868119863(V))) where119889(119906 V) is the time delay between 119906

and Vwhen exchanging information Given (1199061 V1) (1199062 V2) isin

119864 the relationship between119908(1199061 V1) and119908(119906

2 V2) is given as

119908 (1199061 V1) gt 119908 (119906

2 V2) lArrrArr 119889 (119906

1 V1) gt 119889 (119906

2 V2) or

(119889 (1199061 V1) = 119889 (119906

2 V2)) and (min (119868119863 (119906

1) 119868119863 (V

1))

gtmin (119868119863 (1199062) 119868119863 (V

2))) or

(119889 (1199061 V1) = 119889 (119906

2 V2)) and (min (119868119863 (119906

1) 119868119863 (V

1))

=min (119868119863 (1199062) 119868119863 (V

2)))

and (max (119868119863 (1199061) 119868119863 (V

1))

gtmax (119868119863 (1199062) 119868119863 (V

2)))

(1)

It is obvious that edges with the same vertices have equivalentweights However edges with different end-vertices havedifferent weights


Earth

Satellite gateway

GEO

Mars

Earth station

IPN backbone

Other planets

Backbone nodes

Satellite gateway

Rover

Lander

Lander

Rover

Sensors

Sensors


Rover

Lander

Lander

Rover

Sensors

Sensors

AS-1 AS-2

AS-3

AS-4




119909119898V where 119909

119894isin 119881 and





119894and 119866

119895of 119866 if exist119906 isin 119881

119894 V isin 119881

119895



denoted by 119873119861119877119866(119866119894 119866119895) If 119862119874119873(119866

119894 119896) and 119862119874119873(119866

119895 119896) and


exist(1199061 V1) (119906

119896 V119896) isin 119864 where 119906

1 119906

119896isin 119881119894and

V1 V

119896isin 119881119895 119866119894and 119866





119897subject to 119873119861119877

119866(119866119894

119866119897 119896)and119873119861119877

119866(119866119897 119866119895 119896)119866

119894and119866










ℎ (119906) = 119891 (119863119890119892119903119890119890 (119906) 119863119890119897119886119910 (119906 V119894) 120572) (2)



(ℎ (119906) = ℎ (V)) and (119873119900119889119890119868119863 (119906) gt 119873119900119889119890119868119863 (V)) (3)









119904 119896) rArr 119862119874119873(119866

119896 119896) And




119904 then


Input (at AS 119866119904= (119881119904 119864119904))


119866119896= (119881119896 119864119896)

Begin119881119896larr 119881119904 119864119896larr 0




119894then

119864119896larr 119864119896cup (119906119894 V119894)


119895 V119895) of 119864




119895 V119895) then

119864119896larr 119864119896minus (119906119895 V119895)

end ifend for

Return 119866119896



isin 119878119896(119866119904) The




1 1198662 119866











119868119860(1198661 1198662) is the maximum




119868119860(119866119860 119866119862) and

119863119868119860(119866119861 119866119862) are less than119863

119868119860(119866119860 119866119861)






that is 119862119874119873(119866119904 119896) rArr 119862119874119873(119866







119904be


Input (at AS 119866119896= (119881119896 119864119896))



Begin119866119896119894

= (119881119896119894 119864119896119894) 119881119896119894

larr 119881119896 119864119896119894

larr 0


119866(119866119896 119866119894) do

1198811015840larr V | (V isin 119866


119896)

119881119896119894

larr 119881119896119894

cup 1198811015840

119864119896119894


1015840) and (119903(119906 V) le 119877max)

119872 larr 0



119896119898

larr |119872119886119909119872119886119905119888ℎ119894119899119892(119866119896119894)|


119896119894)

if 119896119898




edges in 119872 119864119886(119905) = 119890

1 119890

119905

end for119863119868119860(119866119896 119866119894) larr |119890

119905| where |119890


119905

119871(119866119896 119866119894) larr 119872




119905= (119881119896119894 119864119886(119905)))


119871(119866119896 119866119894) larr 119872

end ifSend 119863

119868119860(119866119896 119866119894) to neighbor AS



119871119868119878119879 larr 0


119866(119866119896 119866119901) do


119873119861119877119866(119866119896 119866119902) and 119873119861119877

119866(119866119896 119866119901)and

(119863119868119860(119866119896 119866119902) lt 119863

119868119860(119866119896 119866119901))and then

(119863119868119860(119866119901 119866119902) lt 119863

119868119860(119866119896 119866119901))

119871119868119878119879 larr 119871119868119878119879 cup 119871(119866119896 119866119894)

end ifend for

Return 119871119868119878119879



119906V(119866119904) 119901119898 cap119901119899=

119906 V If edge 1198900= (119906 V) let 119866



119904


1198900= (119906 V) then 119862119874119873(119866

119904minus 1198900 119896)







1 V1




vertices 119883 = 1199091 119909


119894isin (119881(119866

1015840

119904) minus 119906

1 V1) If

(1199061 V1) is an edge in119866

1015840



Since 119862119874119873(119866119904 119896) we have |119875

1199061V1

(119866119904)| ge 119896 where

|1198751199061V1


1199061V1

(119866119904) Let


1199061V1

(1198661015840



= 119901 isin

1198751199061V1

(1198661015840

119904) | (119909

119894isin 119883) and (119909


in 1198751199061V1

(1198661015840



(1198661015840

119904) Given


Let 11986610158401015840


1015840

119904

If |1198751199061V1

(1198661015840

119904)| ge 119896 we have |119875

1199061V1

(11986610158401015840

119904)| ge (|119875

1199061V1

(1198661015840

119904)| minus

1199031) ge 1 that is 119906


1in 11986610158401015840

119904 Otherwise

|1198751199061V1

(1198661015840



1198900

= (119906 V) breaks one path 119901119895

isin 1198751199061V1



1199061 119906 V V


1199061V1



|1198751199061V1

(119866119904) minus 119901


1199061V1

(1198661015840

119904)| = 119896 minus 1


1199061V1

(1198661015840

119904)|minus1199031) ge 1 Hence

|1198751199061V1

(11986610158401015840

119904)| ge 1 That is 119906


1in 11986610158401015840

119904



(1198661015840


119895isin 1198751199061V1

(119866119904) is disjoint


(1198661015840

119904) so we have 119901

119895cap119883 = 0 Hence no

vertex in 1199061 119906 V V







119904 So


1in 11986610158401015840

119904


119904 1199061


119881(1198661015840

119904) minus 119906

1 V1 Hence 119862119874119873(119866

1015840

119904 119896)

Lemma 7 Let 119866119904and 119866


119904 119896)

and 119881(119866119904) = 119881(119866


119904) minus


119904minus

(1199061015840 V1015840) isin 119864(119866

119904) | 119908(119906

1015840 V1015840) ge 119908(119906 V) then 119862119874119873(119866

119904 119896)


119890119898 = 119864(119866

119904)minus119864(119866

119904) = (119906

1 V1) (1199062 V2) (119906

119898 V119898) be a set



119898) We define a


119894

119904= 119866119894minus1

119904minus 119890119894 where

119894 = 1 2 119898Then119864(119866119898

119904) sube 119864(119866


by induction


119904= 119866119904and 119862119874119873(119866

0

119904 119896)


119904 119896) we prove that119862119874119873(119866

119894

119904 119896) where


119904) | 119908(119906

1015840 V1015840) ge

119908(119906119894 V119894) sube 119866

119894minus1



119894in graph 119866

119904minus (1199061015840 V1015840) isin

119864(119866119904) | 119908(119906

1015840 V1015840) ge 119908(119906

119894 V119894)) we obtain that 119906

119894is 119896-


119894minus1


to 119866119894minus1


119894minus1

119904minus (119906119894 V119894) 119896) That is

119862119874119873(119866119894

119904 119896)


119904 119896) Since 119864(119866

119898

119904) sube

119864(119866119904) hence 119862119874119873(119866

119904 119896)





119896) should


119904) |


prove that 119862119874119873(119866119904 119896) rArr 119862119874119873(119866

119896 119896)


119896(119866119904) is the set




119894isin 119878119896(119866119904)


119896 It


119896)


max(119906V)isin119864(119866

119896)119908(119906 V) Let1198661015840


that |119875119906119898V119898

(1198661015840


119867119904

= (119881(119867119904) 119864(119867

119904)) where 119881(119867

119904) = 119881(119866

119904) and 119864(119867

119904) =

(119906 V) isin 119864(119866119904) | 119908(119906 V) lt 119908(119906



119894isin



119894isin 119878119896(119866119904)



119904 119896) hence |119875

119906119898V119898

(119867119904)| ge 119896 We

have 119867119904minus 1198661015840


1015840

119896in the




119904) | 119908(119906

1015840 V1015840) ge 119908(119906 V)


119896 119896) That is

|119875119906119898V119898

(1198661015840





Then we have 119862119874119873(119866 119896) hArr 119862119874119873(1198661015840 119896)


Lemma 9 Let 119866119894= (119881119894 119864119894) and 119866

119895= (119881119895 119864119895) be two sub-


119894cup119866119866119895 119896)




119894 119896)


119894or 119906 V isin 119866

119895


119894) and (V isin 119866

119895)

Since119873119861119877119866(119866119894 119866119895 119896) exist119906



119894cup 119881119895minus

1199060 V0 With 119862119874119873(119866

119894 119896) and 119862119874119873(119866

119895 119896) we know that


0 Hence 119906 is







119866(119866119894 119866119895 119896) Then cup

119866119866119894|



1015840

119904be edges

reduction of 119866119904 Let 119866

10158401015840= (119881 119864

1015840) = (119866 minus 119866

119904)cup1198661198661015840

119904 If

119862119874119873(119866119904 119896) and 119862119874119873(119866

1015840

119904 119896) and 119862119874119873(119866 119896) then 119862119874119873(119866

10158401015840 119896)







1015840

119904 119896)


119904 since 119862119874119873(119866 119896) 119906 is





V in 119866minus119866119904 Since 119862119874119873(119866

1015840



10158401015840(3) 119906 V isin 119881 minus 119881





1 1198862

isin 119901) and (1198861 1198862

isin 119881119904) 119906 is



119904

Since 119862119874119873(1198661015840

119904 119896) 119886


2by removing


10158401015840

Corollary 12 Let 1198661 1198662 119866



2 119866

1015840


1198661 1198662 119866

119899 and 119866

1015840

1 1198661015840

2 119866

1015840


11986610158401015840

= (119866 minus

119899

⋃

119894=1

119866119866119894)cup119866(

119899

⋃

119894=1

1198661198661015840

119894) (4)

is 119896-connected





119906 V isin 119881119894 Let 1198661015840

119894= (119881119894 1198641015840

119894) where 119864

1015840

119894= 119864119894cap 1198641015840 Then forall119894 119895


1198721198621198741198731198661015840(1198661015840

119894 1198661015840

119895 119896)


119878 119864119878) where 119881

119878= 1198661 1198662 119866

119899

and 119864119878= (119866119894 119866119895) | 119873119861119877

119866(119866119894 119866119895 119896) Actually edge (119866

119894 119866119895)


119895 Let

1198661015840

= (119881119878 1198641015840


1015840 where1198641015840

119878= (119866

1015840

119894 1198661015840

119895) | 119873119861119877

1198661015840(1198661015840

119894 1198661015840

119895 119896)We use119881




1015840



(1198661015840

119894 1198661015840



119868119860(119866119894 119866119895)







119898) where119898 = |119864

119878|




1015840 that is 1198901isin 1198641015840

119878







119895in


119897 where both119863

119868119860(119866119894 119866119897)

and 119863119868119860(119866119897 119866119895) are less than 119863

119868119860(119866119894 119866119895) However edges

(119866119894 119866119897) and (119866

119897 119866119895) come before (119866

119894 119866119895) in the ascending


119895in 1198661015840


119895in 1198661015840

and then119872119862119874119873119866(119866119894 119866119895 119896) rArr 119872119862119874119873

1198661015840(1198661015840

119894 1198661015840

119895 119896)


119894and 119866

1015840







1 119904 then (119866119894isin 119860119903) and (119872119862119874119873

119866(119866119894 119866119895 119896)) rArr 119866

119895isin 119860119903


1 119860

1015840

119904 where forall119894 119866

119894isin 119860119903rArr 1198661015840

119894isin

1198601015840


1015840

119903= 1198661015840

1199031

1198661015840

119903119898


1198661015840(1198661015840

119903119894

1198661015840

119903119895

119896) Take 1198601015840

119903as a

subgraph of 1198661015840 1198601015840119903= (1198811198601015840

119903

1198641198601015840

119903

) where 1198811198601015840

119903

= V | V isin 1198601015840

119903

and 1198641198601015840

119903

= (119906 V) | (119906 V isin 1198601015840

119903) and ((119906 V) isin 119864

1015840) Since

1198601015840





1198661015840= (119866 minus (

119904

⋃

119903=1

119866119860119903))cup119866(

119904

⋃

119903=1

1198661198601015840

119903) (5)









119861lt 1 Let 119878









119904




119904sdot 119877119861





119873





119861)119873119878+ 2119878119873


119861)119873+2119878

119873119878+2119878 which is 119900(119873)+119900(119878

119873119878) in the worst

case










rithm FLSS119896













1distributed






2 and FLSS

2


2distributed






119896and FLSS



00

500 1000 1500 2000 2500

500

1000

1500

2000

2500

x (km)

y (k

m)

(a)

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

x (km)

y (k

m)

(b)

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

x (km)

y (k

m)

(c)

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

x (km)

y (k

m)

(d)








7 Conclusion



120 140 160 180 200 22002

04

06

08

1

12

14

16


Tim

e del

ay (m

s)



(a)

120 140 160 180 200 22002

04

06

08

1

12

14

16

18

2


Tim

e del

ay (m

s)


FGSS2 avgFLSS2 avg

(b)

120 140 160 180 200 2200

2

4

6

8

10

12

14

16

18

20


Aver

age n

ode d

egre

e

DMTC k = 1

DMTC k = 2

Without control

(c)







120 140 160 180 200 2200

2

4

6

8

10

12


Aver

age n

umbe

r of m

essa

ges p

er n

ode






Acknowledgment


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Earth

Satellite gateway

GEO

Mars

Earth station

IPN backbone

Other planets

Backbone nodes

Satellite gateway

Rover

Lander

Lander

Rover

Sensors

Sensors


Rover

Lander

Lander

Rover

Sensors

Sensors

AS-1 AS-2

AS-3

AS-4




119909119898V where 119909

119894isin 119881 and





119894and 119866

119895of 119866 if exist119906 isin 119881

119894 V isin 119881

119895



denoted by 119873119861119877119866(119866119894 119866119895) If 119862119874119873(119866

119894 119896) and 119862119874119873(119866

119895 119896) and


exist(1199061 V1) (119906

119896 V119896) isin 119864 where 119906

1 119906

119896isin 119881119894and

V1 V

119896isin 119881119895 119866119894and 119866





119897subject to 119873119861119877

119866(119866119894

119866119897 119896)and119873119861119877

119866(119866119897 119866119895 119896)119866

119894and119866










ℎ (119906) = 119891 (119863119890119892119903119890119890 (119906) 119863119890119897119886119910 (119906 V119894) 120572) (2)



(ℎ (119906) = ℎ (V)) and (119873119900119889119890119868119863 (119906) gt 119873119900119889119890119868119863 (V)) (3)









119904 119896) rArr 119862119874119873(119866

119896 119896) And




119904 then


Input (at AS 119866119904= (119881119904 119864119904))


119866119896= (119881119896 119864119896)

Begin119881119896larr 119881119904 119864119896larr 0




119894then

119864119896larr 119864119896cup (119906119894 V119894)


119895 V119895) of 119864




119895 V119895) then

119864119896larr 119864119896minus (119906119895 V119895)

end ifend for

Return 119866119896



isin 119878119896(119866119904) The




1 1198662 119866











119868119860(1198661 1198662) is the maximum




119868119860(119866119860 119866119862) and

119863119868119860(119866119861 119866119862) are less than119863

119868119860(119866119860 119866119861)






that is 119862119874119873(119866119904 119896) rArr 119862119874119873(119866







119904be


Input (at AS 119866119896= (119881119896 119864119896))



Begin119866119896119894

= (119881119896119894 119864119896119894) 119881119896119894

larr 119881119896 119864119896119894

larr 0


119866(119866119896 119866119894) do

1198811015840larr V | (V isin 119866


119896)

119881119896119894

larr 119881119896119894

cup 1198811015840

119864119896119894


1015840) and (119903(119906 V) le 119877max)

119872 larr 0



119896119898

larr |119872119886119909119872119886119905119888ℎ119894119899119892(119866119896119894)|


119896119894)

if 119896119898




edges in 119872 119864119886(119905) = 119890

1 119890

119905

end for119863119868119860(119866119896 119866119894) larr |119890

119905| where |119890


119905

119871(119866119896 119866119894) larr 119872




119905= (119881119896119894 119864119886(119905)))


119871(119866119896 119866119894) larr 119872

end ifSend 119863

119868119860(119866119896 119866119894) to neighbor AS



119871119868119878119879 larr 0


119866(119866119896 119866119901) do


119873119861119877119866(119866119896 119866119902) and 119873119861119877

119866(119866119896 119866119901)and

(119863119868119860(119866119896 119866119902) lt 119863

119868119860(119866119896 119866119901))and then

(119863119868119860(119866119901 119866119902) lt 119863

119868119860(119866119896 119866119901))

119871119868119878119879 larr 119871119868119878119879 cup 119871(119866119896 119866119894)

end ifend for

Return 119871119868119878119879



119906V(119866119904) 119901119898 cap119901119899=

119906 V If edge 1198900= (119906 V) let 119866



119904


1198900= (119906 V) then 119862119874119873(119866

119904minus 1198900 119896)







1 V1




vertices 119883 = 1199091 119909


119894isin (119881(119866

1015840

119904) minus 119906

1 V1) If

(1199061 V1) is an edge in119866

1015840



Since 119862119874119873(119866119904 119896) we have |119875

1199061V1

(119866119904)| ge 119896 where

|1198751199061V1


1199061V1

(119866119904) Let


1199061V1

(1198661015840



= 119901 isin

1198751199061V1

(1198661015840

119904) | (119909

119894isin 119883) and (119909


in 1198751199061V1

(1198661015840



(1198661015840

119904) Given


Let 11986610158401015840


1015840

119904

If |1198751199061V1

(1198661015840

119904)| ge 119896 we have |119875

1199061V1

(11986610158401015840

119904)| ge (|119875

1199061V1

(1198661015840

119904)| minus

1199031) ge 1 that is 119906


1in 11986610158401015840

119904 Otherwise

|1198751199061V1

(1198661015840



1198900

= (119906 V) breaks one path 119901119895

isin 1198751199061V1



1199061 119906 V V


1199061V1



|1198751199061V1

(119866119904) minus 119901


1199061V1

(1198661015840

119904)| = 119896 minus 1


1199061V1

(1198661015840

119904)|minus1199031) ge 1 Hence

|1198751199061V1

(11986610158401015840

119904)| ge 1 That is 119906


1in 11986610158401015840

119904



(1198661015840


119895isin 1198751199061V1

(119866119904) is disjoint


(1198661015840

119904) so we have 119901

119895cap119883 = 0 Hence no

vertex in 1199061 119906 V V







119904 So


1in 11986610158401015840

119904


119904 1199061


119881(1198661015840

119904) minus 119906

1 V1 Hence 119862119874119873(119866

1015840

119904 119896)

Lemma 7 Let 119866119904and 119866


119904 119896)

and 119881(119866119904) = 119881(119866


119904) minus


119904minus

(1199061015840 V1015840) isin 119864(119866

119904) | 119908(119906

1015840 V1015840) ge 119908(119906 V) then 119862119874119873(119866

119904 119896)


119890119898 = 119864(119866

119904)minus119864(119866

119904) = (119906

1 V1) (1199062 V2) (119906

119898 V119898) be a set



119898) We define a


119894

119904= 119866119894minus1

119904minus 119890119894 where

119894 = 1 2 119898Then119864(119866119898

119904) sube 119864(119866


by induction


119904= 119866119904and 119862119874119873(119866

0

119904 119896)


119904 119896) we prove that119862119874119873(119866

119894

119904 119896) where


119904) | 119908(119906

1015840 V1015840) ge

119908(119906119894 V119894) sube 119866

119894minus1



119894in graph 119866

119904minus (1199061015840 V1015840) isin

119864(119866119904) | 119908(119906

1015840 V1015840) ge 119908(119906

119894 V119894)) we obtain that 119906

119894is 119896-


119894minus1


to 119866119894minus1


119894minus1

119904minus (119906119894 V119894) 119896) That is

119862119874119873(119866119894

119904 119896)


119904 119896) Since 119864(119866

119898

119904) sube

119864(119866119904) hence 119862119874119873(119866

119904 119896)





119896) should


119904) |


prove that 119862119874119873(119866119904 119896) rArr 119862119874119873(119866

119896 119896)


119896(119866119904) is the set




119894isin 119878119896(119866119904)


119896 It


119896)


max(119906V)isin119864(119866

119896)119908(119906 V) Let1198661015840


that |119875119906119898V119898

(1198661015840


119867119904

= (119881(119867119904) 119864(119867

119904)) where 119881(119867

119904) = 119881(119866

119904) and 119864(119867

119904) =

(119906 V) isin 119864(119866119904) | 119908(119906 V) lt 119908(119906



119894isin



119894isin 119878119896(119866119904)



119904 119896) hence |119875

119906119898V119898

(119867119904)| ge 119896 We

have 119867119904minus 1198661015840


1015840

119896in the




119904) | 119908(119906

1015840 V1015840) ge 119908(119906 V)


119896 119896) That is

|119875119906119898V119898

(1198661015840





Then we have 119862119874119873(119866 119896) hArr 119862119874119873(1198661015840 119896)


Lemma 9 Let 119866119894= (119881119894 119864119894) and 119866

119895= (119881119895 119864119895) be two sub-


119894cup119866119866119895 119896)




119894 119896)


119894or 119906 V isin 119866

119895


119894) and (V isin 119866

119895)

Since119873119861119877119866(119866119894 119866119895 119896) exist119906



119894cup 119881119895minus

1199060 V0 With 119862119874119873(119866

119894 119896) and 119862119874119873(119866

119895 119896) we know that


0 Hence 119906 is







119866(119866119894 119866119895 119896) Then cup

119866119866119894|



1015840

119904be edges

reduction of 119866119904 Let 119866

10158401015840= (119881 119864

1015840) = (119866 minus 119866

119904)cup1198661198661015840

119904 If

119862119874119873(119866119904 119896) and 119862119874119873(119866

1015840

119904 119896) and 119862119874119873(119866 119896) then 119862119874119873(119866

10158401015840 119896)







1015840

119904 119896)


119904 since 119862119874119873(119866 119896) 119906 is





V in 119866minus119866119904 Since 119862119874119873(119866

1015840



10158401015840(3) 119906 V isin 119881 minus 119881





1 1198862

isin 119901) and (1198861 1198862

isin 119881119904) 119906 is



119904

Since 119862119874119873(1198661015840

119904 119896) 119886


2by removing


10158401015840

Corollary 12 Let 1198661 1198662 119866



2 119866

1015840


1198661 1198662 119866

119899 and 119866

1015840

1 1198661015840

2 119866

1015840


11986610158401015840

= (119866 minus

119899

⋃

119894=1

119866119866119894)cup119866(

119899

⋃

119894=1

1198661198661015840

119894) (4)

is 119896-connected





119906 V isin 119881119894 Let 1198661015840

119894= (119881119894 1198641015840

119894) where 119864

1015840

119894= 119864119894cap 1198641015840 Then forall119894 119895


1198721198621198741198731198661015840(1198661015840

119894 1198661015840

119895 119896)


119878 119864119878) where 119881

119878= 1198661 1198662 119866

119899

and 119864119878= (119866119894 119866119895) | 119873119861119877

119866(119866119894 119866119895 119896) Actually edge (119866

119894 119866119895)


119895 Let

1198661015840

= (119881119878 1198641015840


1015840 where1198641015840

119878= (119866

1015840

119894 1198661015840

119895) | 119873119861119877

1198661015840(1198661015840

119894 1198661015840

119895 119896)We use119881




1015840



(1198661015840

119894 1198661015840



119868119860(119866119894 119866119895)







119898) where119898 = |119864

119878|




1015840 that is 1198901isin 1198641015840

119878







119895in


119897 where both119863

119868119860(119866119894 119866119897)

and 119863119868119860(119866119897 119866119895) are less than 119863

119868119860(119866119894 119866119895) However edges

(119866119894 119866119897) and (119866

119897 119866119895) come before (119866

119894 119866119895) in the ascending


119895in 1198661015840


119895in 1198661015840

and then119872119862119874119873119866(119866119894 119866119895 119896) rArr 119872119862119874119873

1198661015840(1198661015840

119894 1198661015840

119895 119896)


119894and 119866

1015840







1 119904 then (119866119894isin 119860119903) and (119872119862119874119873

119866(119866119894 119866119895 119896)) rArr 119866

119895isin 119860119903


1 119860

1015840

119904 where forall119894 119866

119894isin 119860119903rArr 1198661015840

119894isin

1198601015840


1015840

119903= 1198661015840

1199031

1198661015840

119903119898


1198661015840(1198661015840

119903119894

1198661015840

119903119895

119896) Take 1198601015840

119903as a

subgraph of 1198661015840 1198601015840119903= (1198811198601015840

119903

1198641198601015840

119903

) where 1198811198601015840

119903

= V | V isin 1198601015840

119903

and 1198641198601015840

119903

= (119906 V) | (119906 V isin 1198601015840

119903) and ((119906 V) isin 119864

1015840) Since

1198601015840





1198661015840= (119866 minus (

119904

⋃

119903=1

119866119860119903))cup119866(

119904

⋃

119903=1

1198661198601015840

119903) (5)









119861lt 1 Let 119878









119904




119904sdot 119877119861





119873





119861)119873119878+ 2119878119873


119861)119873+2119878

119873119878+2119878 which is 119900(119873)+119900(119878

119873119878) in the worst

case










rithm FLSS119896













1distributed






2 and FLSS

2


2distributed






119896and FLSS



00

500 1000 1500 2000 2500

500

1000

1500

2000

2500

x (km)

y (k

m)

(a)

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

x (km)

y (k

m)

(b)

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

x (km)

y (k

m)

(c)

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

x (km)

y (k

m)

(d)








7 Conclusion



120 140 160 180 200 22002

04

06

08

1

12

14

16


Tim

e del

ay (m

s)



(a)

120 140 160 180 200 22002

04

06

08

1

12

14

16

18

2


Tim

e del

ay (m

s)


FGSS2 avgFLSS2 avg

(b)

120 140 160 180 200 2200

2

4

6

8

10

12

14

16

18

20


Aver

age n

ode d

egre

e

DMTC k = 1

DMTC k = 2

Without control

(c)







120 140 160 180 200 2200

2

4

6

8

10

12


Aver

age n

umbe

r of m

essa

ges p

er n

ode






Acknowledgment


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










exist(1199061 V1) (119906

119896 V119896) isin 119864 where 119906

1 119906

119896isin 119881119894and

V1 V

119896isin 119881119895 119866119894and 119866





119897subject to 119873119861119877

119866(119866119894

119866119897 119896)and119873119861119877

119866(119866119897 119866119895 119896)119866

119894and119866










ℎ (119906) = 119891 (119863119890119892119903119890119890 (119906) 119863119890119897119886119910 (119906 V119894) 120572) (2)



(ℎ (119906) = ℎ (V)) and (119873119900119889119890119868119863 (119906) gt 119873119900119889119890119868119863 (V)) (3)









119904 119896) rArr 119862119874119873(119866

119896 119896) And




119904 then


Input (at AS 119866119904= (119881119904 119864119904))


119866119896= (119881119896 119864119896)

Begin119881119896larr 119881119904 119864119896larr 0




119894then

119864119896larr 119864119896cup (119906119894 V119894)


119895 V119895) of 119864




119895 V119895) then

119864119896larr 119864119896minus (119906119895 V119895)

end ifend for

Return 119866119896



isin 119878119896(119866119904) The




1 1198662 119866











119868119860(1198661 1198662) is the maximum




119868119860(119866119860 119866119862) and

119863119868119860(119866119861 119866119862) are less than119863

119868119860(119866119860 119866119861)






that is 119862119874119873(119866119904 119896) rArr 119862119874119873(119866







119904be


Input (at AS 119866119896= (119881119896 119864119896))



Begin119866119896119894

= (119881119896119894 119864119896119894) 119881119896119894

larr 119881119896 119864119896119894

larr 0


119866(119866119896 119866119894) do

1198811015840larr V | (V isin 119866


119896)

119881119896119894

larr 119881119896119894

cup 1198811015840

119864119896119894


1015840) and (119903(119906 V) le 119877max)

119872 larr 0



119896119898

larr |119872119886119909119872119886119905119888ℎ119894119899119892(119866119896119894)|


119896119894)

if 119896119898




edges in 119872 119864119886(119905) = 119890

1 119890

119905

end for119863119868119860(119866119896 119866119894) larr |119890

119905| where |119890


119905

119871(119866119896 119866119894) larr 119872




119905= (119881119896119894 119864119886(119905)))


119871(119866119896 119866119894) larr 119872

end ifSend 119863

119868119860(119866119896 119866119894) to neighbor AS



119871119868119878119879 larr 0


119866(119866119896 119866119901) do


119873119861119877119866(119866119896 119866119902) and 119873119861119877

119866(119866119896 119866119901)and

(119863119868119860(119866119896 119866119902) lt 119863

119868119860(119866119896 119866119901))and then

(119863119868119860(119866119901 119866119902) lt 119863

119868119860(119866119896 119866119901))

119871119868119878119879 larr 119871119868119878119879 cup 119871(119866119896 119866119894)

end ifend for

Return 119871119868119878119879



119906V(119866119904) 119901119898 cap119901119899=

119906 V If edge 1198900= (119906 V) let 119866



119904


1198900= (119906 V) then 119862119874119873(119866

119904minus 1198900 119896)







1 V1




vertices 119883 = 1199091 119909


119894isin (119881(119866

1015840

119904) minus 119906

1 V1) If

(1199061 V1) is an edge in119866

1015840



Since 119862119874119873(119866119904 119896) we have |119875

1199061V1

(119866119904)| ge 119896 where

|1198751199061V1


1199061V1

(119866119904) Let


1199061V1

(1198661015840



= 119901 isin

1198751199061V1

(1198661015840

119904) | (119909

119894isin 119883) and (119909


in 1198751199061V1

(1198661015840



(1198661015840

119904) Given


Let 11986610158401015840


1015840

119904

If |1198751199061V1

(1198661015840

119904)| ge 119896 we have |119875

1199061V1

(11986610158401015840

119904)| ge (|119875

1199061V1

(1198661015840

119904)| minus

1199031) ge 1 that is 119906


1in 11986610158401015840

119904 Otherwise

|1198751199061V1

(1198661015840



1198900

= (119906 V) breaks one path 119901119895

isin 1198751199061V1



1199061 119906 V V


1199061V1



|1198751199061V1

(119866119904) minus 119901


1199061V1

(1198661015840

119904)| = 119896 minus 1


1199061V1

(1198661015840

119904)|minus1199031) ge 1 Hence

|1198751199061V1

(11986610158401015840

119904)| ge 1 That is 119906


1in 11986610158401015840

119904



(1198661015840


119895isin 1198751199061V1

(119866119904) is disjoint


(1198661015840

119904) so we have 119901

119895cap119883 = 0 Hence no

vertex in 1199061 119906 V V







119904 So


1in 11986610158401015840

119904


119904 1199061


119881(1198661015840

119904) minus 119906

1 V1 Hence 119862119874119873(119866

1015840

119904 119896)

Lemma 7 Let 119866119904and 119866


119904 119896)

and 119881(119866119904) = 119881(119866


119904) minus


119904minus

(1199061015840 V1015840) isin 119864(119866

119904) | 119908(119906

1015840 V1015840) ge 119908(119906 V) then 119862119874119873(119866

119904 119896)


119890119898 = 119864(119866

119904)minus119864(119866

119904) = (119906

1 V1) (1199062 V2) (119906

119898 V119898) be a set



119898) We define a


119894

119904= 119866119894minus1

119904minus 119890119894 where

119894 = 1 2 119898Then119864(119866119898

119904) sube 119864(119866


by induction


119904= 119866119904and 119862119874119873(119866

0

119904 119896)


119904 119896) we prove that119862119874119873(119866

119894

119904 119896) where


119904) | 119908(119906

1015840 V1015840) ge

119908(119906119894 V119894) sube 119866

119894minus1



119894in graph 119866

119904minus (1199061015840 V1015840) isin

119864(119866119904) | 119908(119906

1015840 V1015840) ge 119908(119906

119894 V119894)) we obtain that 119906

119894is 119896-


119894minus1


to 119866119894minus1


119894minus1

119904minus (119906119894 V119894) 119896) That is

119862119874119873(119866119894

119904 119896)


119904 119896) Since 119864(119866

119898

119904) sube

119864(119866119904) hence 119862119874119873(119866

119904 119896)





119896) should


119904) |


prove that 119862119874119873(119866119904 119896) rArr 119862119874119873(119866

119896 119896)


119896(119866119904) is the set




119894isin 119878119896(119866119904)


119896 It


119896)


max(119906V)isin119864(119866

119896)119908(119906 V) Let1198661015840


that |119875119906119898V119898

(1198661015840


119867119904

= (119881(119867119904) 119864(119867

119904)) where 119881(119867

119904) = 119881(119866

119904) and 119864(119867

119904) =

(119906 V) isin 119864(119866119904) | 119908(119906 V) lt 119908(119906



119894isin



119894isin 119878119896(119866119904)



119904 119896) hence |119875

119906119898V119898

(119867119904)| ge 119896 We

have 119867119904minus 1198661015840


1015840

119896in the




119904) | 119908(119906

1015840 V1015840) ge 119908(119906 V)


119896 119896) That is

|119875119906119898V119898

(1198661015840





Then we have 119862119874119873(119866 119896) hArr 119862119874119873(1198661015840 119896)


Lemma 9 Let 119866119894= (119881119894 119864119894) and 119866

119895= (119881119895 119864119895) be two sub-


119894cup119866119866119895 119896)




119894 119896)


119894or 119906 V isin 119866

119895


119894) and (V isin 119866

119895)

Since119873119861119877119866(119866119894 119866119895 119896) exist119906



119894cup 119881119895minus

1199060 V0 With 119862119874119873(119866

119894 119896) and 119862119874119873(119866

119895 119896) we know that


0 Hence 119906 is







119866(119866119894 119866119895 119896) Then cup

119866119866119894|



1015840

119904be edges

reduction of 119866119904 Let 119866

10158401015840= (119881 119864

1015840) = (119866 minus 119866

119904)cup1198661198661015840

119904 If

119862119874119873(119866119904 119896) and 119862119874119873(119866

1015840

119904 119896) and 119862119874119873(119866 119896) then 119862119874119873(119866

10158401015840 119896)







1015840

119904 119896)


119904 since 119862119874119873(119866 119896) 119906 is





V in 119866minus119866119904 Since 119862119874119873(119866

1015840



10158401015840(3) 119906 V isin 119881 minus 119881





1 1198862

isin 119901) and (1198861 1198862

isin 119881119904) 119906 is



119904

Since 119862119874119873(1198661015840

119904 119896) 119886


2by removing


10158401015840

Corollary 12 Let 1198661 1198662 119866



2 119866

1015840


1198661 1198662 119866

119899 and 119866

1015840

1 1198661015840

2 119866

1015840


11986610158401015840

= (119866 minus

119899

⋃

119894=1

119866119866119894)cup119866(

119899

⋃

119894=1

1198661198661015840

119894) (4)

is 119896-connected





119906 V isin 119881119894 Let 1198661015840

119894= (119881119894 1198641015840

119894) where 119864

1015840

119894= 119864119894cap 1198641015840 Then forall119894 119895


1198721198621198741198731198661015840(1198661015840

119894 1198661015840

119895 119896)


119878 119864119878) where 119881

119878= 1198661 1198662 119866

119899

and 119864119878= (119866119894 119866119895) | 119873119861119877

119866(119866119894 119866119895 119896) Actually edge (119866

119894 119866119895)


119895 Let

1198661015840

= (119881119878 1198641015840


1015840 where1198641015840

119878= (119866

1015840

119894 1198661015840

119895) | 119873119861119877

1198661015840(1198661015840

119894 1198661015840

119895 119896)We use119881




1015840



(1198661015840

119894 1198661015840



119868119860(119866119894 119866119895)







119898) where119898 = |119864

119878|




1015840 that is 1198901isin 1198641015840

119878







119895in


119897 where both119863

119868119860(119866119894 119866119897)

and 119863119868119860(119866119897 119866119895) are less than 119863

119868119860(119866119894 119866119895) However edges

(119866119894 119866119897) and (119866

119897 119866119895) come before (119866

119894 119866119895) in the ascending


119895in 1198661015840


119895in 1198661015840

and then119872119862119874119873119866(119866119894 119866119895 119896) rArr 119872119862119874119873

1198661015840(1198661015840

119894 1198661015840

119895 119896)


119894and 119866

1015840







1 119904 then (119866119894isin 119860119903) and (119872119862119874119873

119866(119866119894 119866119895 119896)) rArr 119866

119895isin 119860119903


1 119860

1015840

119904 where forall119894 119866

119894isin 119860119903rArr 1198661015840

119894isin

1198601015840


1015840

119903= 1198661015840

1199031

1198661015840

119903119898


1198661015840(1198661015840

119903119894

1198661015840

119903119895

119896) Take 1198601015840

119903as a

subgraph of 1198661015840 1198601015840119903= (1198811198601015840

119903

1198641198601015840

119903

) where 1198811198601015840

119903

= V | V isin 1198601015840

119903

and 1198641198601015840

119903

= (119906 V) | (119906 V isin 1198601015840

119903) and ((119906 V) isin 119864

1015840) Since

1198601015840





1198661015840= (119866 minus (

119904

⋃

119903=1

119866119860119903))cup119866(

119904

⋃

119903=1

1198661198601015840

119903) (5)









119861lt 1 Let 119878









119904




119904sdot 119877119861





119873





119861)119873119878+ 2119878119873


119861)119873+2119878

119873119878+2119878 which is 119900(119873)+119900(119878

119873119878) in the worst

case










rithm FLSS119896













1distributed






2 and FLSS

2


2distributed






119896and FLSS



00

500 1000 1500 2000 2500

500

1000

1500

2000

2500

x (km)

y (k

m)

(a)

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

x (km)

y (k

m)

(b)

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

x (km)

y (k

m)

(c)

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

x (km)

y (k

m)

(d)








7 Conclusion



120 140 160 180 200 22002

04

06

08

1

12

14

16


Tim

e del

ay (m

s)



(a)

120 140 160 180 200 22002

04

06

08

1

12

14

16

18

2


Tim

e del

ay (m

s)


FGSS2 avgFLSS2 avg

(b)

120 140 160 180 200 2200

2

4

6

8

10

12

14

16

18

20


Aver

age n

ode d

egre

e

DMTC k = 1

DMTC k = 2

Without control

(c)







120 140 160 180 200 2200

2

4

6

8

10

12


Aver

age n

umbe

r of m

essa

ges p

er n

ode






Acknowledgment


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Input (at AS 119866119904= (119881119904 119864119904))


119866119896= (119881119896 119864119896)

Begin119881119896larr 119881119904 119864119896larr 0




119894then

119864119896larr 119864119896cup (119906119894 V119894)


119895 V119895) of 119864




119895 V119895) then

119864119896larr 119864119896minus (119906119895 V119895)

end ifend for

Return 119866119896



isin 119878119896(119866119904) The




1 1198662 119866











119868119860(1198661 1198662) is the maximum




119868119860(119866119860 119866119862) and

119863119868119860(119866119861 119866119862) are less than119863

119868119860(119866119860 119866119861)






that is 119862119874119873(119866119904 119896) rArr 119862119874119873(119866







119904be


Input (at AS 119866119896= (119881119896 119864119896))



Begin119866119896119894

= (119881119896119894 119864119896119894) 119881119896119894

larr 119881119896 119864119896119894

larr 0


119866(119866119896 119866119894) do

1198811015840larr V | (V isin 119866


119896)

119881119896119894

larr 119881119896119894

cup 1198811015840

119864119896119894


1015840) and (119903(119906 V) le 119877max)

119872 larr 0



119896119898

larr |119872119886119909119872119886119905119888ℎ119894119899119892(119866119896119894)|


119896119894)

if 119896119898




edges in 119872 119864119886(119905) = 119890

1 119890

119905

end for119863119868119860(119866119896 119866119894) larr |119890

119905| where |119890


119905

119871(119866119896 119866119894) larr 119872




119905= (119881119896119894 119864119886(119905)))


119871(119866119896 119866119894) larr 119872

end ifSend 119863

119868119860(119866119896 119866119894) to neighbor AS



119871119868119878119879 larr 0


119866(119866119896 119866119901) do


119873119861119877119866(119866119896 119866119902) and 119873119861119877

119866(119866119896 119866119901)and

(119863119868119860(119866119896 119866119902) lt 119863

119868119860(119866119896 119866119901))and then

(119863119868119860(119866119901 119866119902) lt 119863

119868119860(119866119896 119866119901))

119871119868119878119879 larr 119871119868119878119879 cup 119871(119866119896 119866119894)

end ifend for

Return 119871119868119878119879



119906V(119866119904) 119901119898 cap119901119899=

119906 V If edge 1198900= (119906 V) let 119866



119904


1198900= (119906 V) then 119862119874119873(119866

119904minus 1198900 119896)







1 V1




vertices 119883 = 1199091 119909


119894isin (119881(119866

1015840

119904) minus 119906

1 V1) If

(1199061 V1) is an edge in119866

1015840



Since 119862119874119873(119866119904 119896) we have |119875

1199061V1

(119866119904)| ge 119896 where

|1198751199061V1


1199061V1

(119866119904) Let


1199061V1

(1198661015840



= 119901 isin

1198751199061V1

(1198661015840

119904) | (119909

119894isin 119883) and (119909


in 1198751199061V1

(1198661015840



(1198661015840

119904) Given


Let 11986610158401015840


1015840

119904

If |1198751199061V1

(1198661015840

119904)| ge 119896 we have |119875

1199061V1

(11986610158401015840

119904)| ge (|119875

1199061V1

(1198661015840

119904)| minus

1199031) ge 1 that is 119906


1in 11986610158401015840

119904 Otherwise

|1198751199061V1

(1198661015840



1198900

= (119906 V) breaks one path 119901119895

isin 1198751199061V1



1199061 119906 V V


1199061V1



|1198751199061V1

(119866119904) minus 119901


1199061V1

(1198661015840

119904)| = 119896 minus 1


1199061V1

(1198661015840

119904)|minus1199031) ge 1 Hence

|1198751199061V1

(11986610158401015840

119904)| ge 1 That is 119906


1in 11986610158401015840

119904



(1198661015840


119895isin 1198751199061V1

(119866119904) is disjoint


(1198661015840

119904) so we have 119901

119895cap119883 = 0 Hence no

vertex in 1199061 119906 V V







119904 So


1in 11986610158401015840

119904


119904 1199061


119881(1198661015840

119904) minus 119906

1 V1 Hence 119862119874119873(119866

1015840

119904 119896)

Lemma 7 Let 119866119904and 119866


119904 119896)

and 119881(119866119904) = 119881(119866


119904) minus


119904minus

(1199061015840 V1015840) isin 119864(119866

119904) | 119908(119906

1015840 V1015840) ge 119908(119906 V) then 119862119874119873(119866

119904 119896)


119890119898 = 119864(119866

119904)minus119864(119866

119904) = (119906

1 V1) (1199062 V2) (119906

119898 V119898) be a set



119898) We define a


119894

119904= 119866119894minus1

119904minus 119890119894 where

119894 = 1 2 119898Then119864(119866119898

119904) sube 119864(119866


by induction


119904= 119866119904and 119862119874119873(119866

0

119904 119896)


119904 119896) we prove that119862119874119873(119866

119894

119904 119896) where


119904) | 119908(119906

1015840 V1015840) ge

119908(119906119894 V119894) sube 119866

119894minus1



119894in graph 119866

119904minus (1199061015840 V1015840) isin

119864(119866119904) | 119908(119906

1015840 V1015840) ge 119908(119906

119894 V119894)) we obtain that 119906

119894is 119896-


119894minus1


to 119866119894minus1


119894minus1

119904minus (119906119894 V119894) 119896) That is

119862119874119873(119866119894

119904 119896)


119904 119896) Since 119864(119866

119898

119904) sube

119864(119866119904) hence 119862119874119873(119866

119904 119896)





119896) should


119904) |


prove that 119862119874119873(119866119904 119896) rArr 119862119874119873(119866

119896 119896)


119896(119866119904) is the set




119894isin 119878119896(119866119904)


119896 It


119896)


max(119906V)isin119864(119866

119896)119908(119906 V) Let1198661015840


that |119875119906119898V119898

(1198661015840


119867119904

= (119881(119867119904) 119864(119867

119904)) where 119881(119867

119904) = 119881(119866

119904) and 119864(119867

119904) =

(119906 V) isin 119864(119866119904) | 119908(119906 V) lt 119908(119906



119894isin



119894isin 119878119896(119866119904)



119904 119896) hence |119875

119906119898V119898

(119867119904)| ge 119896 We

have 119867119904minus 1198661015840


1015840

119896in the




119904) | 119908(119906

1015840 V1015840) ge 119908(119906 V)


119896 119896) That is

|119875119906119898V119898

(1198661015840





Then we have 119862119874119873(119866 119896) hArr 119862119874119873(1198661015840 119896)


Lemma 9 Let 119866119894= (119881119894 119864119894) and 119866

119895= (119881119895 119864119895) be two sub-


119894cup119866119866119895 119896)




119894 119896)


119894or 119906 V isin 119866

119895


119894) and (V isin 119866

119895)

Since119873119861119877119866(119866119894 119866119895 119896) exist119906



119894cup 119881119895minus

1199060 V0 With 119862119874119873(119866

119894 119896) and 119862119874119873(119866

119895 119896) we know that


0 Hence 119906 is







119866(119866119894 119866119895 119896) Then cup

119866119866119894|



1015840

119904be edges

reduction of 119866119904 Let 119866

10158401015840= (119881 119864

1015840) = (119866 minus 119866

119904)cup1198661198661015840

119904 If

119862119874119873(119866119904 119896) and 119862119874119873(119866

1015840

119904 119896) and 119862119874119873(119866 119896) then 119862119874119873(119866

10158401015840 119896)







1015840

119904 119896)


119904 since 119862119874119873(119866 119896) 119906 is





V in 119866minus119866119904 Since 119862119874119873(119866

1015840



10158401015840(3) 119906 V isin 119881 minus 119881





1 1198862

isin 119901) and (1198861 1198862

isin 119881119904) 119906 is



119904

Since 119862119874119873(1198661015840

119904 119896) 119886


2by removing


10158401015840

Corollary 12 Let 1198661 1198662 119866



2 119866

1015840


1198661 1198662 119866

119899 and 119866

1015840

1 1198661015840

2 119866

1015840


11986610158401015840

= (119866 minus

119899

⋃

119894=1

119866119866119894)cup119866(

119899

⋃

119894=1

1198661198661015840

119894) (4)

is 119896-connected





119906 V isin 119881119894 Let 1198661015840

119894= (119881119894 1198641015840

119894) where 119864

1015840

119894= 119864119894cap 1198641015840 Then forall119894 119895


1198721198621198741198731198661015840(1198661015840

119894 1198661015840

119895 119896)


119878 119864119878) where 119881

119878= 1198661 1198662 119866

119899

and 119864119878= (119866119894 119866119895) | 119873119861119877

119866(119866119894 119866119895 119896) Actually edge (119866

119894 119866119895)


119895 Let

1198661015840

= (119881119878 1198641015840


1015840 where1198641015840

119878= (119866

1015840

119894 1198661015840

119895) | 119873119861119877

1198661015840(1198661015840

119894 1198661015840

119895 119896)We use119881




1015840



(1198661015840

119894 1198661015840



119868119860(119866119894 119866119895)







119898) where119898 = |119864

119878|




1015840 that is 1198901isin 1198641015840

119878







119895in


119897 where both119863

119868119860(119866119894 119866119897)

and 119863119868119860(119866119897 119866119895) are less than 119863

119868119860(119866119894 119866119895) However edges

(119866119894 119866119897) and (119866

119897 119866119895) come before (119866

119894 119866119895) in the ascending


119895in 1198661015840


119895in 1198661015840

and then119872119862119874119873119866(119866119894 119866119895 119896) rArr 119872119862119874119873

1198661015840(1198661015840

119894 1198661015840

119895 119896)


119894and 119866

1015840







1 119904 then (119866119894isin 119860119903) and (119872119862119874119873

119866(119866119894 119866119895 119896)) rArr 119866

119895isin 119860119903


1 119860

1015840

119904 where forall119894 119866

119894isin 119860119903rArr 1198661015840

119894isin

1198601015840


1015840

119903= 1198661015840

1199031

1198661015840

119903119898


1198661015840(1198661015840

119903119894

1198661015840

119903119895

119896) Take 1198601015840

119903as a

subgraph of 1198661015840 1198601015840119903= (1198811198601015840

119903

1198641198601015840

119903

) where 1198811198601015840

119903

= V | V isin 1198601015840

119903

and 1198641198601015840

119903

= (119906 V) | (119906 V isin 1198601015840

119903) and ((119906 V) isin 119864

1015840) Since

1198601015840





1198661015840= (119866 minus (

119904

⋃

119903=1

119866119860119903))cup119866(

119904

⋃

119903=1

1198661198601015840

119903) (5)









119861lt 1 Let 119878









119904




119904sdot 119877119861





119873





119861)119873119878+ 2119878119873


119861)119873+2119878

119873119878+2119878 which is 119900(119873)+119900(119878

119873119878) in the worst

case










rithm FLSS119896













1distributed






2 and FLSS

2


2distributed






119896and FLSS



00

500 1000 1500 2000 2500

500

1000

1500

2000

2500

x (km)

y (k

m)

(a)

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

x (km)

y (k

m)

(b)

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

x (km)

y (k

m)

(c)

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

x (km)

y (k

m)

(d)








7 Conclusion



120 140 160 180 200 22002

04

06

08

1

12

14

16


Tim

e del

ay (m

s)



(a)

120 140 160 180 200 22002

04

06

08

1

12

14

16

18

2


Tim

e del

ay (m

s)


FGSS2 avgFLSS2 avg

(b)

120 140 160 180 200 2200

2

4

6

8

10

12

14

16

18

20


Aver

age n

ode d

egre

e

DMTC k = 1

DMTC k = 2

Without control

(c)







120 140 160 180 200 2200

2

4

6

8

10

12


Aver

age n

umbe

r of m

essa

ges p

er n

ode






Acknowledgment


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Input (at AS 119866119896= (119881119896 119864119896))



Begin119866119896119894

= (119881119896119894 119864119896119894) 119881119896119894

larr 119881119896 119864119896119894

larr 0


119866(119866119896 119866119894) do

1198811015840larr V | (V isin 119866


119896)

119881119896119894

larr 119881119896119894

cup 1198811015840

119864119896119894


1015840) and (119903(119906 V) le 119877max)

119872 larr 0



119896119898

larr |119872119886119909119872119886119905119888ℎ119894119899119892(119866119896119894)|


119896119894)

if 119896119898




edges in 119872 119864119886(119905) = 119890

1 119890

119905

end for119863119868119860(119866119896 119866119894) larr |119890

119905| where |119890


119905

119871(119866119896 119866119894) larr 119872




119905= (119881119896119894 119864119886(119905)))


119871(119866119896 119866119894) larr 119872

end ifSend 119863

119868119860(119866119896 119866119894) to neighbor AS



119871119868119878119879 larr 0


119866(119866119896 119866119901) do


119873119861119877119866(119866119896 119866119902) and 119873119861119877

119866(119866119896 119866119901)and

(119863119868119860(119866119896 119866119902) lt 119863

119868119860(119866119896 119866119901))and then

(119863119868119860(119866119901 119866119902) lt 119863

119868119860(119866119896 119866119901))

119871119868119878119879 larr 119871119868119878119879 cup 119871(119866119896 119866119894)

end ifend for

Return 119871119868119878119879



119906V(119866119904) 119901119898 cap119901119899=

119906 V If edge 1198900= (119906 V) let 119866



119904


1198900= (119906 V) then 119862119874119873(119866

119904minus 1198900 119896)







1 V1




vertices 119883 = 1199091 119909


119894isin (119881(119866

1015840

119904) minus 119906

1 V1) If

(1199061 V1) is an edge in119866

1015840



Since 119862119874119873(119866119904 119896) we have |119875

1199061V1

(119866119904)| ge 119896 where

|1198751199061V1


1199061V1

(119866119904) Let


1199061V1

(1198661015840



= 119901 isin

1198751199061V1

(1198661015840

119904) | (119909

119894isin 119883) and (119909


in 1198751199061V1

(1198661015840



(1198661015840

119904) Given


Let 11986610158401015840


1015840

119904

If |1198751199061V1

(1198661015840

119904)| ge 119896 we have |119875

1199061V1

(11986610158401015840

119904)| ge (|119875

1199061V1

(1198661015840

119904)| minus

1199031) ge 1 that is 119906


1in 11986610158401015840

119904 Otherwise

|1198751199061V1

(1198661015840



1198900

= (119906 V) breaks one path 119901119895

isin 1198751199061V1



1199061 119906 V V


1199061V1



|1198751199061V1

(119866119904) minus 119901


1199061V1

(1198661015840

119904)| = 119896 minus 1


1199061V1

(1198661015840

119904)|minus1199031) ge 1 Hence

|1198751199061V1

(11986610158401015840

119904)| ge 1 That is 119906


1in 11986610158401015840

119904



(1198661015840


119895isin 1198751199061V1

(119866119904) is disjoint


(1198661015840

119904) so we have 119901

119895cap119883 = 0 Hence no

vertex in 1199061 119906 V V







119904 So


1in 11986610158401015840

119904


119904 1199061


119881(1198661015840

119904) minus 119906

1 V1 Hence 119862119874119873(119866

1015840

119904 119896)

Lemma 7 Let 119866119904and 119866


119904 119896)

and 119881(119866119904) = 119881(119866


119904) minus


119904minus

(1199061015840 V1015840) isin 119864(119866

119904) | 119908(119906

1015840 V1015840) ge 119908(119906 V) then 119862119874119873(119866

119904 119896)


119890119898 = 119864(119866

119904)minus119864(119866

119904) = (119906

1 V1) (1199062 V2) (119906

119898 V119898) be a set



119898) We define a


119894

119904= 119866119894minus1

119904minus 119890119894 where

119894 = 1 2 119898Then119864(119866119898

119904) sube 119864(119866


by induction


119904= 119866119904and 119862119874119873(119866

0

119904 119896)


119904 119896) we prove that119862119874119873(119866

119894

119904 119896) where


119904) | 119908(119906

1015840 V1015840) ge

119908(119906119894 V119894) sube 119866

119894minus1



119894in graph 119866

119904minus (1199061015840 V1015840) isin

119864(119866119904) | 119908(119906

1015840 V1015840) ge 119908(119906

119894 V119894)) we obtain that 119906

119894is 119896-


119894minus1


to 119866119894minus1


119894minus1

119904minus (119906119894 V119894) 119896) That is

119862119874119873(119866119894

119904 119896)


119904 119896) Since 119864(119866

119898

119904) sube

119864(119866119904) hence 119862119874119873(119866

119904 119896)





119896) should


119904) |


prove that 119862119874119873(119866119904 119896) rArr 119862119874119873(119866

119896 119896)


119896(119866119904) is the set




119894isin 119878119896(119866119904)


119896 It


119896)


max(119906V)isin119864(119866

119896)119908(119906 V) Let1198661015840


that |119875119906119898V119898

(1198661015840


119867119904

= (119881(119867119904) 119864(119867

119904)) where 119881(119867

119904) = 119881(119866

119904) and 119864(119867

119904) =

(119906 V) isin 119864(119866119904) | 119908(119906 V) lt 119908(119906



119894isin



119894isin 119878119896(119866119904)



119904 119896) hence |119875

119906119898V119898

(119867119904)| ge 119896 We

have 119867119904minus 1198661015840


1015840

119896in the




119904) | 119908(119906

1015840 V1015840) ge 119908(119906 V)


119896 119896) That is

|119875119906119898V119898

(1198661015840





Then we have 119862119874119873(119866 119896) hArr 119862119874119873(1198661015840 119896)


Lemma 9 Let 119866119894= (119881119894 119864119894) and 119866

119895= (119881119895 119864119895) be two sub-


119894cup119866119866119895 119896)




119894 119896)


119894or 119906 V isin 119866

119895


119894) and (V isin 119866

119895)

Since119873119861119877119866(119866119894 119866119895 119896) exist119906



119894cup 119881119895minus

1199060 V0 With 119862119874119873(119866

119894 119896) and 119862119874119873(119866

119895 119896) we know that


0 Hence 119906 is







119866(119866119894 119866119895 119896) Then cup

119866119866119894|



1015840

119904be edges

reduction of 119866119904 Let 119866

10158401015840= (119881 119864

1015840) = (119866 minus 119866

119904)cup1198661198661015840

119904 If

119862119874119873(119866119904 119896) and 119862119874119873(119866

1015840

119904 119896) and 119862119874119873(119866 119896) then 119862119874119873(119866

10158401015840 119896)







1015840

119904 119896)


119904 since 119862119874119873(119866 119896) 119906 is





V in 119866minus119866119904 Since 119862119874119873(119866

1015840



10158401015840(3) 119906 V isin 119881 minus 119881





1 1198862

isin 119901) and (1198861 1198862

isin 119881119904) 119906 is



119904

Since 119862119874119873(1198661015840

119904 119896) 119886


2by removing


10158401015840

Corollary 12 Let 1198661 1198662 119866



2 119866

1015840


1198661 1198662 119866

119899 and 119866

1015840

1 1198661015840

2 119866

1015840


11986610158401015840

= (119866 minus

119899

⋃

119894=1

119866119866119894)cup119866(

119899

⋃

119894=1

1198661198661015840

119894) (4)

is 119896-connected





119906 V isin 119881119894 Let 1198661015840

119894= (119881119894 1198641015840

119894) where 119864

1015840

119894= 119864119894cap 1198641015840 Then forall119894 119895


1198721198621198741198731198661015840(1198661015840

119894 1198661015840

119895 119896)


119878 119864119878) where 119881

119878= 1198661 1198662 119866

119899

and 119864119878= (119866119894 119866119895) | 119873119861119877

119866(119866119894 119866119895 119896) Actually edge (119866

119894 119866119895)


119895 Let

1198661015840

= (119881119878 1198641015840


1015840 where1198641015840

119878= (119866

1015840

119894 1198661015840

119895) | 119873119861119877

1198661015840(1198661015840

119894 1198661015840

119895 119896)We use119881




1015840



(1198661015840

119894 1198661015840



119868119860(119866119894 119866119895)







119898) where119898 = |119864

119878|




1015840 that is 1198901isin 1198641015840

119878







119895in


119897 where both119863

119868119860(119866119894 119866119897)

and 119863119868119860(119866119897 119866119895) are less than 119863

119868119860(119866119894 119866119895) However edges

(119866119894 119866119897) and (119866

119897 119866119895) come before (119866

119894 119866119895) in the ascending


119895in 1198661015840


119895in 1198661015840

and then119872119862119874119873119866(119866119894 119866119895 119896) rArr 119872119862119874119873

1198661015840(1198661015840

119894 1198661015840

119895 119896)


119894and 119866

1015840







1 119904 then (119866119894isin 119860119903) and (119872119862119874119873

119866(119866119894 119866119895 119896)) rArr 119866

119895isin 119860119903


1 119860

1015840

119904 where forall119894 119866

119894isin 119860119903rArr 1198661015840

119894isin

1198601015840


1015840

119903= 1198661015840

1199031

1198661015840

119903119898


1198661015840(1198661015840

119903119894

1198661015840

119903119895

119896) Take 1198601015840

119903as a

subgraph of 1198661015840 1198601015840119903= (1198811198601015840

119903

1198641198601015840

119903

) where 1198811198601015840

119903

= V | V isin 1198601015840

119903

and 1198641198601015840

119903

= (119906 V) | (119906 V isin 1198601015840

119903) and ((119906 V) isin 119864

1015840) Since

1198601015840





1198661015840= (119866 minus (

119904

⋃

119903=1

119866119860119903))cup119866(

119904

⋃

119903=1

1198661198601015840

119903) (5)









119861lt 1 Let 119878









119904




119904sdot 119877119861





119873





119861)119873119878+ 2119878119873


119861)119873+2119878

119873119878+2119878 which is 119900(119873)+119900(119878

119873119878) in the worst

case










rithm FLSS119896













1distributed






2 and FLSS

2


2distributed






119896and FLSS



00

500 1000 1500 2000 2500

500

1000

1500

2000

2500

x (km)

y (k

m)

(a)

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

x (km)

y (k

m)

(b)

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

x (km)

y (k

m)

(c)

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

x (km)

y (k

m)

(d)








7 Conclusion



120 140 160 180 200 22002

04

06

08

1

12

14

16


Tim

e del

ay (m

s)



(a)

120 140 160 180 200 22002

04

06

08

1

12

14

16

18

2


Tim

e del

ay (m

s)


FGSS2 avgFLSS2 avg

(b)

120 140 160 180 200 2200

2

4

6

8

10

12

14

16

18

20


Aver

age n

ode d

egre

e

DMTC k = 1

DMTC k = 2

Without control

(c)







120 140 160 180 200 2200

2

4

6

8

10

12


Aver

age n

umbe

r of m

essa

ges p

er n

ode






Acknowledgment


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1198900

= (119906 V) breaks one path 119901119895

isin 1198751199061V1



1199061 119906 V V


1199061V1



|1198751199061V1

(119866119904) minus 119901


1199061V1

(1198661015840

119904)| = 119896 minus 1


1199061V1

(1198661015840

119904)|minus1199031) ge 1 Hence

|1198751199061V1

(11986610158401015840

119904)| ge 1 That is 119906


1in 11986610158401015840

119904



(1198661015840


119895isin 1198751199061V1

(119866119904) is disjoint


(1198661015840

119904) so we have 119901

119895cap119883 = 0 Hence no

vertex in 1199061 119906 V V







119904 So


1in 11986610158401015840

119904


119904 1199061


119881(1198661015840

119904) minus 119906

1 V1 Hence 119862119874119873(119866

1015840

119904 119896)

Lemma 7 Let 119866119904and 119866


119904 119896)

and 119881(119866119904) = 119881(119866


119904) minus


119904minus

(1199061015840 V1015840) isin 119864(119866

119904) | 119908(119906

1015840 V1015840) ge 119908(119906 V) then 119862119874119873(119866

119904 119896)


119890119898 = 119864(119866

119904)minus119864(119866

119904) = (119906

1 V1) (1199062 V2) (119906

119898 V119898) be a set



119898) We define a


119894

119904= 119866119894minus1

119904minus 119890119894 where

119894 = 1 2 119898Then119864(119866119898

119904) sube 119864(119866


by induction


119904= 119866119904and 119862119874119873(119866

0

119904 119896)


119904 119896) we prove that119862119874119873(119866

119894

119904 119896) where


119904) | 119908(119906

1015840 V1015840) ge

119908(119906119894 V119894) sube 119866

119894minus1



119894in graph 119866

119904minus (1199061015840 V1015840) isin

119864(119866119904) | 119908(119906

1015840 V1015840) ge 119908(119906

119894 V119894)) we obtain that 119906

119894is 119896-


119894minus1


to 119866119894minus1


119894minus1

119904minus (119906119894 V119894) 119896) That is

119862119874119873(119866119894

119904 119896)


119904 119896) Since 119864(119866

119898

119904) sube

119864(119866119904) hence 119862119874119873(119866

119904 119896)





119896) should


119904) |


prove that 119862119874119873(119866119904 119896) rArr 119862119874119873(119866

119896 119896)


119896(119866119904) is the set




119894isin 119878119896(119866119904)


119896 It


119896)


max(119906V)isin119864(119866

119896)119908(119906 V) Let1198661015840


that |119875119906119898V119898

(1198661015840


119867119904

= (119881(119867119904) 119864(119867

119904)) where 119881(119867

119904) = 119881(119866

119904) and 119864(119867

119904) =

(119906 V) isin 119864(119866119904) | 119908(119906 V) lt 119908(119906



119894isin



119894isin 119878119896(119866119904)



119904 119896) hence |119875

119906119898V119898

(119867119904)| ge 119896 We

have 119867119904minus 1198661015840


1015840

119896in the




119904) | 119908(119906

1015840 V1015840) ge 119908(119906 V)


119896 119896) That is

|119875119906119898V119898

(1198661015840





Then we have 119862119874119873(119866 119896) hArr 119862119874119873(1198661015840 119896)


Lemma 9 Let 119866119894= (119881119894 119864119894) and 119866

119895= (119881119895 119864119895) be two sub-


119894cup119866119866119895 119896)




119894 119896)


119894or 119906 V isin 119866

119895


119894) and (V isin 119866

119895)

Since119873119861119877119866(119866119894 119866119895 119896) exist119906



119894cup 119881119895minus

1199060 V0 With 119862119874119873(119866

119894 119896) and 119862119874119873(119866

119895 119896) we know that


0 Hence 119906 is







119866(119866119894 119866119895 119896) Then cup

119866119866119894|



1015840

119904be edges

reduction of 119866119904 Let 119866

10158401015840= (119881 119864

1015840) = (119866 minus 119866

119904)cup1198661198661015840

119904 If

119862119874119873(119866119904 119896) and 119862119874119873(119866

1015840

119904 119896) and 119862119874119873(119866 119896) then 119862119874119873(119866

10158401015840 119896)







1015840

119904 119896)


119904 since 119862119874119873(119866 119896) 119906 is





V in 119866minus119866119904 Since 119862119874119873(119866

1015840



10158401015840(3) 119906 V isin 119881 minus 119881





1 1198862

isin 119901) and (1198861 1198862

isin 119881119904) 119906 is



119904

Since 119862119874119873(1198661015840

119904 119896) 119886


2by removing


10158401015840

Corollary 12 Let 1198661 1198662 119866



2 119866

1015840


1198661 1198662 119866

119899 and 119866

1015840

1 1198661015840

2 119866

1015840


11986610158401015840

= (119866 minus

119899

⋃

119894=1

119866119866119894)cup119866(

119899

⋃

119894=1

1198661198661015840

119894) (4)

is 119896-connected





119906 V isin 119881119894 Let 1198661015840

119894= (119881119894 1198641015840

119894) where 119864

1015840

119894= 119864119894cap 1198641015840 Then forall119894 119895


1198721198621198741198731198661015840(1198661015840

119894 1198661015840

119895 119896)


119878 119864119878) where 119881

119878= 1198661 1198662 119866

119899

and 119864119878= (119866119894 119866119895) | 119873119861119877

119866(119866119894 119866119895 119896) Actually edge (119866

119894 119866119895)


119895 Let

1198661015840

= (119881119878 1198641015840


1015840 where1198641015840

119878= (119866

1015840

119894 1198661015840

119895) | 119873119861119877

1198661015840(1198661015840

119894 1198661015840

119895 119896)We use119881




1015840



(1198661015840

119894 1198661015840



119868119860(119866119894 119866119895)







119898) where119898 = |119864

119878|




1015840 that is 1198901isin 1198641015840

119878







119895in


119897 where both119863

119868119860(119866119894 119866119897)

and 119863119868119860(119866119897 119866119895) are less than 119863

119868119860(119866119894 119866119895) However edges

(119866119894 119866119897) and (119866

119897 119866119895) come before (119866

119894 119866119895) in the ascending


119895in 1198661015840


119895in 1198661015840

and then119872119862119874119873119866(119866119894 119866119895 119896) rArr 119872119862119874119873

1198661015840(1198661015840

119894 1198661015840

119895 119896)


119894and 119866

1015840







1 119904 then (119866119894isin 119860119903) and (119872119862119874119873

119866(119866119894 119866119895 119896)) rArr 119866

119895isin 119860119903


1 119860

1015840

119904 where forall119894 119866

119894isin 119860119903rArr 1198661015840

119894isin

1198601015840


1015840

119903= 1198661015840

1199031

1198661015840

119903119898


1198661015840(1198661015840

119903119894

1198661015840

119903119895

119896) Take 1198601015840

119903as a

subgraph of 1198661015840 1198601015840119903= (1198811198601015840

119903

1198641198601015840

119903

) where 1198811198601015840

119903

= V | V isin 1198601015840

119903

and 1198641198601015840

119903

= (119906 V) | (119906 V isin 1198601015840

119903) and ((119906 V) isin 119864

1015840) Since

1198601015840





1198661015840= (119866 minus (

119904

⋃

119903=1

119866119860119903))cup119866(

119904

⋃

119903=1

1198661198601015840

119903) (5)









119861lt 1 Let 119878









119904




119904sdot 119877119861





119873





119861)119873119878+ 2119878119873


119861)119873+2119878

119873119878+2119878 which is 119900(119873)+119900(119878

119873119878) in the worst

case










rithm FLSS119896













1distributed






2 and FLSS

2


2distributed






119896and FLSS



00

500 1000 1500 2000 2500

500

1000

1500

2000

2500

x (km)

y (k

m)

(a)

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

x (km)

y (k

m)

(b)

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

x (km)

y (k

m)

(c)

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

x (km)

y (k

m)

(d)








7 Conclusion



120 140 160 180 200 22002

04

06

08

1

12

14

16


Tim

e del

ay (m

s)



(a)

120 140 160 180 200 22002

04

06

08

1

12

14

16

18

2


Tim

e del

ay (m

s)


FGSS2 avgFLSS2 avg

(b)

120 140 160 180 200 2200

2

4

6

8

10

12

14

16

18

20


Aver

age n

ode d

egre

e

DMTC k = 1

DMTC k = 2

Without control

(c)







120 140 160 180 200 2200

2

4

6

8

10

12


Aver

age n

umbe

r of m

essa

ges p

er n

ode






Acknowledgment


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















1015840

119904 119896)


119904 since 119862119874119873(119866 119896) 119906 is





V in 119866minus119866119904 Since 119862119874119873(119866

1015840



10158401015840(3) 119906 V isin 119881 minus 119881





1 1198862

isin 119901) and (1198861 1198862

isin 119881119904) 119906 is



119904

Since 119862119874119873(1198661015840

119904 119896) 119886


2by removing


10158401015840

Corollary 12 Let 1198661 1198662 119866



2 119866

1015840


1198661 1198662 119866

119899 and 119866

1015840

1 1198661015840

2 119866

1015840


11986610158401015840

= (119866 minus

119899

⋃

119894=1

119866119866119894)cup119866(

119899

⋃

119894=1

1198661198661015840

119894) (4)

is 119896-connected





119906 V isin 119881119894 Let 1198661015840

119894= (119881119894 1198641015840

119894) where 119864

1015840

119894= 119864119894cap 1198641015840 Then forall119894 119895


1198721198621198741198731198661015840(1198661015840

119894 1198661015840

119895 119896)


119878 119864119878) where 119881

119878= 1198661 1198662 119866

119899

and 119864119878= (119866119894 119866119895) | 119873119861119877

119866(119866119894 119866119895 119896) Actually edge (119866

119894 119866119895)


119895 Let

1198661015840

= (119881119878 1198641015840


1015840 where1198641015840

119878= (119866

1015840

119894 1198661015840

119895) | 119873119861119877

1198661015840(1198661015840

119894 1198661015840

119895 119896)We use119881




1015840



(1198661015840

119894 1198661015840



119868119860(119866119894 119866119895)







119898) where119898 = |119864

119878|




1015840 that is 1198901isin 1198641015840

119878







119895in


119897 where both119863

119868119860(119866119894 119866119897)

and 119863119868119860(119866119897 119866119895) are less than 119863

119868119860(119866119894 119866119895) However edges

(119866119894 119866119897) and (119866

119897 119866119895) come before (119866

119894 119866119895) in the ascending


119895in 1198661015840


119895in 1198661015840

and then119872119862119874119873119866(119866119894 119866119895 119896) rArr 119872119862119874119873

1198661015840(1198661015840

119894 1198661015840

119895 119896)


119894and 119866

1015840







1 119904 then (119866119894isin 119860119903) and (119872119862119874119873

119866(119866119894 119866119895 119896)) rArr 119866

119895isin 119860119903


1 119860

1015840

119904 where forall119894 119866

119894isin 119860119903rArr 1198661015840

119894isin

1198601015840


1015840

119903= 1198661015840

1199031

1198661015840

119903119898


1198661015840(1198661015840

119903119894

1198661015840

119903119895

119896) Take 1198601015840

119903as a

subgraph of 1198661015840 1198601015840119903= (1198811198601015840

119903

1198641198601015840

119903

) where 1198811198601015840

119903

= V | V isin 1198601015840

119903

and 1198641198601015840

119903

= (119906 V) | (119906 V isin 1198601015840

119903) and ((119906 V) isin 119864

1015840) Since

1198601015840





1198661015840= (119866 minus (

119904

⋃

119903=1

119866119860119903))cup119866(

119904

⋃

119903=1

1198661198601015840

119903) (5)









119861lt 1 Let 119878









119904




119904sdot 119877119861





119873





119861)119873119878+ 2119878119873


119861)119873+2119878

119873119878+2119878 which is 119900(119873)+119900(119878

119873119878) in the worst

case










rithm FLSS119896













1distributed






2 and FLSS

2


2distributed






119896and FLSS



00

500 1000 1500 2000 2500

500

1000

1500

2000

2500

x (km)

y (k

m)

(a)

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

x (km)

y (k

m)

(b)

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

x (km)

y (k

m)

(c)

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

x (km)

y (k

m)

(d)








7 Conclusion



120 140 160 180 200 22002

04

06

08

1

12

14

16


Tim

e del

ay (m

s)



(a)

120 140 160 180 200 22002

04

06

08

1

12

14

16

18

2


Tim

e del

ay (m

s)


FGSS2 avgFLSS2 avg

(b)

120 140 160 180 200 2200

2

4

6

8

10

12

14

16

18

20


Aver

age n

ode d

egre

e

DMTC k = 1

DMTC k = 2

Without control

(c)







120 140 160 180 200 2200

2

4

6

8

10

12


Aver

age n

umbe

r of m

essa

ges p

er n

ode






Acknowledgment


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













119904




119904sdot 119877119861





119873





119861)119873119878+ 2119878119873


119861)119873+2119878

119873119878+2119878 which is 119900(119873)+119900(119878

119873119878) in the worst

case










rithm FLSS119896













1distributed






2 and FLSS

2


2distributed






119896and FLSS



00

500 1000 1500 2000 2500

500

1000

1500

2000

2500

x (km)

y (k

m)

(a)

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

x (km)

y (k

m)

(b)

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

x (km)

y (k

m)

(c)

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

x (km)

y (k

m)

(d)








7 Conclusion



120 140 160 180 200 22002

04

06

08

1

12

14

16


Tim

e del

ay (m

s)



(a)

120 140 160 180 200 22002

04

06

08

1

12

14

16

18

2


Tim

e del

ay (m

s)


FGSS2 avgFLSS2 avg

(b)

120 140 160 180 200 2200

2

4

6

8

10

12

14

16

18

20


Aver

age n

ode d

egre

e

DMTC k = 1

DMTC k = 2

Without control

(c)







120 140 160 180 200 2200

2

4

6

8

10

12


Aver

age n

umbe

r of m

essa

ges p

er n

ode






Acknowledgment


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










00

500 1000 1500 2000 2500

500

1000

1500

2000

2500

x (km)

y (k

m)

(a)

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

x (km)

y (k

m)

(b)

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

x (km)

y (k

m)

(c)

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

x (km)

y (k

m)

(d)








7 Conclusion



120 140 160 180 200 22002

04

06

08

1

12

14

16


Tim

e del

ay (m

s)



(a)

120 140 160 180 200 22002

04

06

08

1

12

14

16

18

2


Tim

e del

ay (m

s)


FGSS2 avgFLSS2 avg

(b)

120 140 160 180 200 2200

2

4

6

8

10

12

14

16

18

20


Aver

age n

ode d

egre

e

DMTC k = 1

DMTC k = 2

Without control

(c)







120 140 160 180 200 2200

2

4

6

8

10

12


Aver

age n

umbe

r of m

essa

ges p

er n

ode






Acknowledgment


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










120 140 160 180 200 22002

04

06

08

1

12

14

16


Tim

e del

ay (m

s)



(a)

120 140 160 180 200 22002

04

06

08

1

12

14

16

18

2


Tim

e del

ay (m

s)


FGSS2 avgFLSS2 avg

(b)

120 140 160 180 200 2200

2

4

6

8

10

12

14

16

18

20


Aver

age n

ode d

egre

e

DMTC k = 1

DMTC k = 2

Without control

(c)







120 140 160 180 200 2200

2

4

6

8

10

12


Aver

age n

umbe

r of m

essa

ges p

er n

ode






Acknowledgment


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










120 140 160 180 200 2200

2

4

6

8

10

12


Aver

age n

umbe

r of m

essa

ges p

er n

ode






Acknowledgment


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article delay minimization topology control in...

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