Crescco CRESCCOIST-2001-33135Critical Resource Sharing for Cooperation inComplex Systems

Workpackage 1: Efficient Resource Assignment and Communication Protocols in Wire-less Networks

Deliverable D1.4Algorithmic Solutions and Technical Recommendations for Wireless Networks III

Responsible Partner: Computer Technology Institute (GR)

Report Version: 1.0Report Preparation Date: 15/03/05Classification: PUBContract Start Date: 01/01/02 Duration: 41 monthsProject Co-ordinator: University of Patras

Partners:Computer Technology Institute (GR)University of Geneva (CH)Centre National de la Recherche Scientific (F)Universite de Nice-Sophia Antipolis (F)Christian-Albrechts-Universitaet zu Kiel (D)Universita degli studi di Salerno (IT)Universita degli studi di Roma “Tor Vergata” (IT)

Project funded by the European Commu-nity under the “Information Society Tech-nologies” Programme (1998-2002)

Contents

1 Introduction 31.1 Frequency assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Energy consumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Sensor networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Satellite networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 Internet access to radio networks . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.6 Dynamic networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Frequency Assignment in Wireless Networks 62.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Algorithmic Solutions and Technical Recommendations . . . . . . . . . . . . . . 7

2.2.1 Radiocoloring Succinct Graphs, [7] . . . . . . . . . . . . . . . . . . . . . . 72.2.2 Radiocolorings in Periodic Planar Graphs, [29] . . . . . . . . . . . . . . . 82.2.3 Generating and Radiocoloring Families of Perfect Graphs, [8] . . . . . . . 82.2.4 Efficient use of frequency spectrum in ad hoc wireless networks, [14], [15] 82.2.5 List coloring, [36] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.6 Fault tolerance in wireless networks, [43] . . . . . . . . . . . . . . . . . . . 10

3 Energy Consumption 103.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Algorithmic Solutions and Technical Recommendations . . . . . . . . . . . . . . 11

3.2.1 On the Power Assignment Problem in Radio Networks, [21] . . . . . . . . 113.2.2 The Range Assignment Problem in Non-Homogeneous Static Ad-hoc Net-

works [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2.3 Efficient Algorithms for Low-Energy Bounded-Hop Broadcast in Ad-hoc

Wireless Networks, [3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2.4 Experimental Analysis of Practically Efficient Algorithms for Bounded-

Hop Accumulation in Ad-hoc Wireless Networks, [16] . . . . . . . . . . . 123.2.5 On the Approximability of the Range Assignment Problem on Radio

Networks in Presence of Selfish Agents, [4] . . . . . . . . . . . . . . . . . . 133.2.6 Round Robin is Optimal for Fault-Tolerant Broadcasting on Wireless Net-

works, [19] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2.7 An analytical model for energy minimization, [58] . . . . . . . . . . . . . 133.2.8 Energy efficiency or Minimum energy broadcast routing problem, [27, 54, 47] 14

4 Efficient Communication in wireless sensor networks 144.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.2 Algorithmic Solutions and Technical Recommendations . . . . . . . . . . . . . . 15

4.2.1 A Forward Planning Protocol for Scalable, Energy Efficient and Fault-tolerant Data Propagation in Wireless Sensor Networks, [20] . . . . . . . 15

4.2.2 Efficient Data Propagation Protocols in Wireless Sensor Networks, [10] . . 164.2.3 An integer programming heuristic for the dual power management problem

in wireless sensor networks, [59] . . . . . . . . . . . . . . . . . . . . . . . . 164.2.4 An Adaptive Blind Algorithm for Energy Balanced Data Propagation in

Wireless Sensors Networks, [51] . . . . . . . . . . . . . . . . . . . . . . . . 16

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4.2.5 Towards a dynamical model for wireless sensor networks, [52] . . . . . . . 174.2.6 Energy conservation in sensor networks, [26, 50, 44] . . . . . . . . . . . . 17

5 Satellite Networks 185.1 Algorithmic Solutions and Technical Recommendations . . . . . . . . . . . . . . 18

5.1.1 Satellite boarded fault tolerant networks, [12, 57] . . . . . . . . . . . . . . 185.1.2 Resource allocation for a geostationary satellite, [1, 49, 33] . . . . . . . . 18

6 Internet Access to Radio Networks 196.1 Algorithmic Solutions and technical Recommendations . . . . . . . . . . . . . . . 19

6.1.1 Radio networks: Internet in villages, [11, 46, 56] . . . . . . . . . . . . . . 19

7 Dynamic networks 207.1 Algorithmic Solutions and Technical Recommendations . . . . . . . . . . . . . . 20

7.1.1 Connectivity in evolving graphs, [24, 39, 25] . . . . . . . . . . . . . . . . . 207.1.2 Internet/Web algorithms, [18, 17, 37] . . . . . . . . . . . . . . . . . . . . . 20

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1 Introduction

In the third year of the CRESCCO project in Workpackage 1: “ Efficient Resource Assignmentand Communication Protocols in Wireless Networks”, we worked on the following issues:

1.1 Frequency assignment

The efficient bandwidth utilization consists a critical task for the overall performance of wirelessnetworks. The need of efficient frequency assignment becomes even more significant due to theincreased communication demands of wireless users in contrast to the strictly limited bandwidthavailable. The Problem of Frequency Assignment (FAP) consists of assigning frequencies to thetransmitters of a wireless network exploiting frequency reusability in order to save bandwidth,while keeping the interference caused when nearby stations transmit in the same or close fre-quency, in acceptable levels. The call admission control problem is to compute a set of networknodes of maximum size so that no signal interference is caused by assigning the same frequencyto all the nodes in the set.

Assuming that all transmitters have circular range, the graph reflecting possible interferencebetween pairs of transmitters is a disk graph. We study the problems of frequency assignmentand call admission call in the disk graph modelling of the network.

Moreover, we consider more complicate abstractions of FAP, capturing realistic scenariosof wireless networks. The Radiocoloring problem (RCP), describes interference constraintscaused by the high transmission power of stations and the fact that they do not transmit in theexact predefined frequencies. We also study RCP in very large networks defined via succinctspecification. Alternatively, they represent large wireless networks, highly met in large cities,consisting of sets of highly connected sub-networks representing town centers or networks thataccept periodic in time, requests for frequency assignment. For all these practical instances, weestimate the computational complexity of the problem providing hardness results and providepolynomial time approximations for each case. We also address the problem experimentally. Westudy the RCP on some important network topologies that are typically met in practice. Wealgorithmically and experimentally investigate the existence of hard or common instances of thenetwork families considered, that need ‘high’ or ’moderate’ number of colors, respectively, inorder to be radiocolored. Also, we experimentally evaluate known RCP algorithms, as well asthe new heuristics introduced, on various interesting network topologies.

1.2 Energy consumption

The last decade, wireless networks faced tremendous development in the networking applica-tions. Moreover, new networks designs are introduced enabling new networking possibilities;Ad-hoc wireless networks require no infrastructure. The network is simply a collection of trans-mitters/reveivers and a message communication between two transmitters is accomplished viaa multi-hop transmission through intermediate hosts. In some cases hosts are portable deviceswhich benefits only limited power resources. One of the main benefits of ad-hoc power controllednetworks is the ability of the hosts to vary the power used in the transmission (and therefore thetransmission range) in order to avoid interference problems and reduce the power consumption.We consider a fundamental problem in such networks: the Range Assignment problem. Thistask consists of assigning suitable power transmission to the transmitters of the network, mini-mizing the energy consumption so that a communication patterns is satisfied. Some importantsuch properties include connectivity, broadcast and multicast.

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We investigate trade-off issues of the range assignment problem between the power consump-tion and the number of hops h, needed in order to ensure communication between the networkstations. We provide lower and constructive upper bounds on the minimum power consumptionof stations on the plane for number of hops h. Moreover, we study the computational complexityof the problem. We prove that restricted cases the problem admit a polynomial time approxima-tion algorithm, while when h is unbounded, we prove that the problem becomes APX-complete.We also introduce and study the weighted version of the range assignment problem in which thecost a station s pays to transmit to another station depends on the distance between the stationsand on the energy cost of station s. For such a case, we present a polynomial-time algorithm forfinding an optimal range assignment to perform a 2-hop broadcast from a given source station.

Finally, we consider the range assignment problem in ad-hoc wireless networks in the contextof selfish agents. We investigate the existence of payment schemes which induce the stationsto follow the decisions of a network manager in computing a range assignment, that is, truthfulmechanisms for the range assignment problem.

1.3 Sensor networks

Wireless sensor networks are comprised of a vast number of ultra-small fully autonomous com-puting, communication and sensing devices, with very restricted energy and computing capa-bilities, that co-operate to accomplish sensing and information services. Such networks can bevery useful in practice i.e. in the local detection of remote crucial events and the propagationof data reporting their realization to a control center. Sensor networks have important appli-cations, including military (like forces and equipment monitoring), environmental (such as firedetection) and health applications (like telemonitoring of human physiological data).

A critical aspect in the design and efficient implementation of wireless sensor networks is tosave energy and keep the network functional for as long as possible. In this context, we address afundamental communication problem of data propagation in such networks. We propose energyefficient scalable and fault tolerance data propagating protocols. We also study the problem ofenergy balanced data propagation in wireless sensor networks and generalize previous works byallowing realistic energy assignment.

Moreover, we study a particular instance of the optimal power assignment in wireless sensornetworks, called dual power management problem (DPMP). Also, we introduce and investigatea dynamical model for wireless sensor networks, which consists one of the first works describingdynamic aspects of broadcast in wireless sensor networks.

Our research addresses also issues of distributed motion coordination in systems consistingof cooperative mobile robotics. We study the problem of the robots positioning themselves toform a circle. The significance of positioning the robots may be useful for various tasks, suchas in bridge building, in forming adjustable buttresses to support collapsing buildings, satelliterecovery, or tumor excision.

1.4 Satellite networks

Modern telecommunication satellites are very complex to design. Components are often proneto failure, and so providing robustness at the lowest possible cost is an important issue for themanufacturers. A key component of telecommunication satellites is an interconnection networkwhich allows to redirect signals received by the satellite to a set of amplifiers where the signalswill be retransmitted. We consider on-board networks in satellites interconnecting enteringsignals to amplifiers.

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Inside a telecommunication satellite, audio and video signals are routed through a switchingnetwork to amplifiers. Since it is impossible to repair a satellite, we choose to multiply thecomponents that may be faulty, that is amplifiers and switches, showing how to design a lowcost network where the network cost is proportional to the total number of switches used .We also study the problem of resource allocation in satellite networks. We present algorithmsfor time/frequency planning aiming at the minimization of the number of frequencies used andthe guarantee that the different types of demands that use different bandwidth will be satisfied.Furthermore, we consider the access to a constellation of satellites by using ideas of geographicalreservations. Our results guarantee the connection for a long time and give better precisionsthan preceding technics.

1.5 Internet access to radio networks

In the context of radio networks, we study the problem of designing efficient strategies to provideInternet access using wireless devices. Typically, in one village several houses wish to access agateway (a satellite antenna) and to use multi-hop wireless relay routing to do so. Based onthe demand of each user to transmit and the existing interference constraints, we study differ-ent network topologies where the communication requests may be of different size, bandwidthdemands and duration.

We also present work on the improvement of the norm 802.11b. In the case when all the sta-tions see each other, we have studied a memory-based process where the stations automaticallyadapt to their environment to avoid too frequent emissions (which generates collisions) or toorare ones (which results in a loss of bandwidth).

1.6 Dynamic networks

New technologies and the deployment of mobile and nomadic services naturally engender newroute-discovery problems under changing conditions over dynamic networks. Unfortunately, thetemporal variations in the topology of dynamic networks are hard to be effectively captured ina classical graph model. We used evolving graphs, which helps capture the dynamic character-istics of such networks, in order to show that computing different types of strongly connectedcomponents in dynamic networks is NP-complete, and investigated the concepts of journeys inEvolving Graphs, which captures both space and time constraints in routing problems.

We further investigated the connected components problem in dynamic networks with specialtopologies and prove the problem is still NP-complete when the topology is composed of unitdisc graphs and the nodes are placed on a grid and present a polynomial-time algorithm for treetopologies.

We address the the design of algorithms and protocols that are energy aware, using the theevolving graph combinatorial model and present an algorithm that minimizes the maximumenergy used by any one node of a mobile network, thus maximizing the life-time of a wirelesscommunication in it. We study algorithms for the Web Graph, applying a particular analysistechnique which can be used to characterize the asymptotic behaviour of a number of dynamicprocesses related to the web. Furthermore, we study the the trade-offs between the push andpull mechanisms used in information distribution and retrieval in the Internet. We present someinitial complexity results and analyze several heuristics in the context of more realistic scenarioscharacterized by high uncertainty and changing information request and update patterns.

We also present work on modelling of Peer-to-Peer networks for the design of dynamicnetworks with connectivity, low diameter and constant vertex degree. The network we obtain is

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robust when faults happen (for example, adversarial deletion of vertices and edges) and activelyreconnects itself.

2 Frequency Assignment in Wireless Networks

2.1 Introduction

We study the efficient use of the frequency spectrum in wireless and ad hoc networks utilizingthe Frequency Division Multiplexing (FDM) technology. Such networks are composed of a setof transmitters scattered in a geographical region. Each transmitter may select one specificfrequency from a spectrum of available frequencies and has a circular range around it where itcan transmit messages using the selected frequency. Two nodes may successfully (i.e., withoutsignal interference) transmit messages simultaneously either if their ranges do not overlap or ifthey use different frequencies. Since the spectrum of available frequencies is a scarce resource,the engineering problems of frequency assignment and call admission control are extremely im-portant. The Frequency Assignment Problem (FAP) is to assign frequencies to the nodes of thenetwork so that signal interference is avoided and the total number of frequencies used in thenetwork is minimized. The call admission control problem is to compute a set of network nodesof maximum size so that no signal interference is caused by assigning the same frequency to allthe nodes in the set.

The FAP can be modelled via graph-theory terms via an interference graph G(V, E). Thevertices of the graph (set V ) represent the transmitters of the wireless network and its edges(set E) represent possible interference between nearby transmitters when using same or nearbyfrequencies. The set of the available frequencies are represented by colors or integers. Using suchabstractions, the FAP consists of assigning colors (frequencies) to the vertices (transmitters) ofa graph (network), resulting to no interference between vertices using a minimum number of dis-tinct colors (frequencies). A usual model used is as a vertex coloring problem in a correspondinggraph G.

Assuming that all transmitters have circular range, the graph reflecting possible interferencebetween pairs of transmitters is a disk graph. We study the problems of frequency assignmentand call admission call in the disk graph modelling of the network.

Moreover, we consider more complicated abstractions of FAP. In many practical cases of realnetworks, the stations transmission power is quite high, thus they interfere when transmitting inthe same frequency, not only with adjacent transmitters but also with transmitters located nextto their neighbours. Moreover, in real networks, the stations transmission is not always accurate;they do not operate in the exact frequencies but may use, beside the predefined frequency,adjacent to that frequencies. The vertex coloring model fails to describe such realistic scenariosof practical wireless networks, because it considers only one simple frequency constraint, ofneighbor vertices not to get the same frequencies. Henceforth, a number of generalizations ofthe vertex coloring problem have been introduced and investigated in the past, towards thisdirection [35, 60].

The Radiocoloring problem, captures these realistic constraints of FAP and is studied here.Given a graph G(V,E), the Radiocoloring Problem consists of assigning integers to vertices ofthe graph so that (i) any two vertices of distance one get integers that differ by at least twoand (ii) any two vertices of distance two get different integers. The usual objectives of such anassignment are either to minimize the range of integers (frequencies) used (min span RCP) orthe number of integers (distinct frequencies) used (min order RCP).

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We study min order RCP on networks that can be represented by three families of perfectgraphs: permutation, chordal and split graphs. Permutation graphs model well networks wheretwo groups of independent transmitters want to have communication with transmitters from theopposite group. Split graphs model networks where a set of transmitters, which are disconnectedto each other, want to communicate with transmitters forming a strongly connected subnetwork.

We also study FAP in huge but structured wireless networks. A usual case of wirelessnetworks, highly met in large cities, are networks consisting of sets (of highly connected) sub-networks (representing town centers). The sub-networks are connected to each other such thatall stations of a sub-network have the same neighbours. The VMG graphs model well suchnetworks, when RCP is of concern and are addressed in this study. These graphs describe alsowell the task of FAP in wireless networks with increased communication requirements on eachtransmitter, due to either multiple communication tasks of the transmitter or the need to providecommunication to a number of (mobile) users under its coverage.

We also deal with an interesting variation of the radiocoloring problem: that of satisfyingfrequency assignment requests which exhibit some periodic behavior. In this case, the inter-ference graph (modelling interference between transmitters) is some (infinite) periodic graph.We study the optimization version of RCP where the objective is to minimize the order of theassignment.

2.2 Algorithmic Solutions and Technical Recommendations

2.2.1 Radiocoloring Succinct Graphs, [7]

We study an interesting variation of the Frequency Assignment Problem, the RadiocoloringProblem on VMG graphs ([48]) and the Maximum Independent Set (MIS) problem on squares ofordinary graphs and VMG-represented graphs. The square of a graph G, G2, is the same graphas G and there is an edge between two vertices of G2 if and only if their distance in G is at mosttwo. Note that the graph L(G) described by such a model can be much larger than the (basis)graph G used in the succinct representation. We relate here the above problems (stated forVMG graphs) to similar problems stated on the basis graph. We then give tight lower boundsfor the minimum number of colors needed to radiocolor a VMG graph (called, radiochromaticnumber).

Also, constant ratio approximations for radiocoloring VMG graphs when the basis graph G

is either a k-tree or a planar graph are provided by this work. For this task, we employ knownalgorithms for radiocoloring the graph G.

For the weighted MIS problem on squares of k-trees (or induced subgraphs of k-trees), weprovide a new, polynomial time exact algorithm when k ≤ logn. For the same problem (weightedMIS) we give a PTAS for squares of planar graphs. Using the above results, we give a polynomialtime algorithm for MIS in squares of VMG graphs when the basis is either an induced subgraphof a k-tree or a planar graph. For the same graph classes, we show how to compute efficientlystronger lower bounds on the radiochromatic number via fractional coloring.

A technical recommendation extracted through this study is that one could address theRCP problem on VMG-represented graphs via fractional coloring combined with a roundingmethod on a corresponding ordinary graph. Moreover, using our results, the latter problem canalso become tractable using previous knowledge, (i.e. [34] and [41]), reducing it to weightedindependent set problems.

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2.2.2 Radiocolorings in Periodic Planar Graphs, [29]

We deal with an interesting variation of min order RCP: that of satisfying frequency assignmentrequests which exhibit some periodic behavior. In this case, the interference graph (modellinginterference between transmitters) is some (infinite) periodic graph. A periodic graph G is definedby an infinite two-way sequence of repetitions of the same finite graph Gi(Vi, Ei). The edge setof G is derived by connecting the vertices of each iteration Gi to some of the vertices of the nextiteration Gi+1, the same for all Gi. We focus on planar periodic graphs, because in many casesreal networks are planar and also because of their independent mathematical interest.

In a previous work of ours [28] we proved that min span RCP is PSPACE-complete forperiodic planar graphs and provided efficient approximations for the problem on such graphs.This work proves that the methodology of [28] can be adapted when studying the same problemin such graphs, when the objective is to minimize the span rather than the order of the assign-ment. We prove that min order RCP is PSPACE-complete for periodic planar graphs. Also,we provide an O(n(∆(Gi) + σ)) time algorithm, (where |Vi| = n, ∆(Gi) is the maximum degreeof the graph Gi and σ is the number of edges connecting each Gi to Gi+1), which obtains aradiocoloring of a periodic planar graph G that approximates the minimum order within a ratiowhich tends to 5

3 as ∆(Gi) + σ tends to infinity.

2.2.3 Generating and Radiocoloring Families of Perfect Graphs, [8]

In this work we experimentally study the min order Radiocoloring problem on Chordal, Split andPermutation graphs, which are three basic families of perfect graphs. RCP is an NP-Completeproblem on chordal and split graphs [13]. There are known upper bounds for the number ofcolors needed by an optimal radiocoloring assignment and approximation algorithms [13, 55].

In our study we design and implement radiocoloring heuristics for graphs of above families,which are based on the greedy heuristic. Also, for each one of above families, we investigatewhether there exists graph instances requiring a number of colors in order to be radiocolored,close to the best known upper bound for the family. Towards this goal, we present a numbergenerators that produce graphs of the above families that require either (i) a large number ofcolors (compared to the best upper bound), in order to be radiocolored, called “extremal” graphsor (ii) a small number of colors, called “non-extremal”instances. The experimental evaluationshowed that random generated graph instances are, in the most of the cases, “non-extremal”graphs. Also, that greedy like heuristics performs very well in the most of the cases, especiallyfor “non-extremal” graphs.

2.2.4 Efficient use of frequency spectrum in ad hoc wireless networks, [14], [15]

We study the problems of frequency assignment and call admission call in ad-hoc wireless net-works. We model the network as a disk graph, i.e., a graph representing overlaps of disks in theplane. Formally, given disks in the Euclidean plane, their intersection graph is the graph havinga node for each disk and an edge if and only if the two disks corresponding to the endpoints ofthe edge overlap. A graph is called disk graph if it is the intersection graph of some set of disksin the plane (called the disk representation of the disk graph). A disk graph is called unit diskgraph if it has a disk representation with disks of the same radius while it is called σ-boundeddisk graph if the ratio of the radius of the largest disk over the radius of the smallest disk in itsdisk representation is at most σ.

It is clear that by considering the frequencies as colors, the problems of frequency assignment

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and call admission call are equivalent to minimum coloring and maximum independent setproblems in the disk graph modeling the network. The information about the coordinates ofthe disk centers and their radii may or may not be given together with the disk graph. Solvingproblems on disk graphs when the disk representation is not given is inherently more difficultsince the decision problem whether a graph is a disk graph is an NP-complete problem. In [14]and [15], we study the online version of the maximum independent set and the minimum coloringproblems where the nodes of the disk graph are revealed in steps and must be immediately coloredor included in an independent set, respectively. We use competitive analysis and express theperformance of online algorithms in terms of their competitive ratio.

For the online independent set problem, we investigate for the first time whether the useof randomization can help in achieving online algorithms with improved competitive ratios andeven beating the corresponding lower bounds for deterministic ones. We show that this is notthe case for online algorithms that do not use the disk representation by showing that no onlineindependent set algorithm that does not use the disk representation can achieve competitiveratio better than Ω(minn, σ2) in σ-bounded disk graphs with n nodes. This means that theintuitive deterministic algorithm First-Fit is optimal within constant factors. When the diskrepresentation is given, algorithms that achieve competitive ratios logarithmic in σ do exist. Inunit disk graphs, we have presented an online independent set algorithm with competitive ratio4.41. This is the first online independent set algorithm in unit disk graphs with competitiveratio better than 5. These results (together with new lower bounds for online independent setalgorithms in unit disk graphs) have appeared in [14].

For online coloring algorithms we have provided algorithms that do not use the disk represen-tation and achieve competitive ratio O(minlog n, log σ), an upper bound which was previouslyachievable only by online coloring algorithms that use the disk representation. The first suchalgorithm was presented in [14] but its running time is huge and makes it impractical. In [15],we show that the naive algorithm First-Fit achieves the above competitive ratio. First-Fit is alsoproved to be optimal within the class of deterministic online algorithms that do not use the diskrepresentation by proving a matching Ω(minlog n, log σ) lower bound. For online coloring inunit disk graphs, we show that no online algorithm that does not use the disk representationcan achieve competitive ratio better than 2.5 improving a previously known lower bound of 2.

Technical recommendations Concerning the applications of the two graph problems to adhoc wireless networks, our online algorithms are suitable for frequency assignment and calladmission control in such networks in realistic scenarios where communication requests appearin a dynamic manner. The information about the disk representation corresponds to accurateinformation on the location and range of the transmitters in the ad hoc network.

Applying our results for the independent set problem in disk graphs to call admission controlin ad hoc wireless networks indicates that in the case where the disk representation cannot beused, intuitive deterministic solutions, such as First-Fit, have optimal performance. In the casewhere the information about disk representation is provided (or can be computed), the useof randomization leads to significantly better solutions. For frequency assignment, our resultsestablish that First-Fit is optimal among all deterministic algorithms. It remains to investigatewhether randomized online coloring/frequency assignment algorithms may have provably betterperformance.

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2.2.5 List coloring, [36]

Satellites send information to receivers on earth, each of which is listening on a frequency.Technically it is impossible to focus the signal sent by the satellite exactly on receiver. So partof the signal is spread in an area around it creating noise for the other receivers displayed in thisarea and listening on the same frequency. A receiver is able to distinguish the signal directed toit from the extraneous noises it picks up if the sum of the noises does not become too big, i.e.does not exceed a certain threshold T . The problem is to assign frequencies to the receivers insuch a way that each receiver gets its dedicated signal properly. We investigate this problem inthe fundamental case where the noise area at a receiver does not depend on the frequency andwhere the “noise relation” is symmetric that is if a receiver u is in the noise area of a receiverv then v is in the noise area of u. Moreover the intensity I of the noise created by a signal isindependent of the frequency and the receiver. Hence to distinguish its signal from noises, areceiver must be in the noise area of at most k =

⌊TI

⌋receivers listening to signals on the same

frequency.Moreover, due to some practical reasons (as, for instance, the specific environment of a receiver),the frequency at each receiver must be chosen among a list of allowed ones for that receiver.

In [36], we model this problem in an improper list-colouring problem on some “noise graph”.We give some results for graphs with bounded density which generalizes planar graphs and alsocontains noise graphs of practical instances.

2.2.6 Fault tolerance in wireless networks, [43]

In [43], we study hardness results and approximation algorithms of k-tuple domination in graphs.The k-tuple domination problem is a generalization of the dominating set problem in graphsin which each vertex of the graph has to be dominated at least k times. A main applicationto network purposes of k-tuple domination is for fault tolerance or mobility in the followingsituations. Each vertex of the graph models a node of the network and edges are links. Nodeu can use a service (any read-only data base for example) only if it is replicated on u or on aneighbor of u. To ensure a certain degree of fault tolerance or to tolerate mobility of nodes, onecan imagine that any node u has in its (closed) neighborhood at least k copies of this serviceavailable. As each copy can cost a lot, the number of duplicated copies has to be minimized.This is the problem we study. In [43], we describe tight approximability and non-approximabilityresults for general graphs, graphs of constant degree and p-claw free graphs.

3 Energy Consumption

3.1 Introduction

Ad-hoc wireless networks have been extensively utilized in the recent years in order to describeimportant scenarios in which fixed wired infrastructure is not available. Such scenarios, includethe Internet or instance networks formed by satellites, ships, airplanes or networks connectingrescue teams in the case of earthquake or flood.

In such situations a collection of hosts with wireless network interfaces may form a temporarynetwork without the aid of any established infrastructure or centralized administration. Messagecommunication in these networks takes place by performing multi-hop transmissions. This typeof wireless networks are known as ad-hoc radio networks [31]. The network is simply a collectionof transmitters/reveivers located on a geographical area. A transmission range is assigned to

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each station any any other station within this range can directly (i.e. one hop) receive messagesfrom the other station. Communication between two stations that are not within their respectiveranges can be achieved via a multi-hop transmission through intermediate hosts. In some caseshosts are portable devices which benefits only limited power resources. One of the main benefitsof ad-hoc power controlled networks is the ability of the hosts to vary the power used in thetransmission (and therefore the transmission range) in order to avoid interference problems andreduce the power consumption.

Thus, deciding the transmission power of the single hosts in order to guarantee a “good”communication between hosts and minimize the overall power consumption of the network raisethe necessity of developing new algorithmic solutions. In particular, these two aspects yielda class of fundamental optimization problems, denoted as range assignment problems. Rangeassignment problem consists of assigning suitable power transmission to the transmitters of thenetwork, minimizing the energy consumption so that a communication patterns is satisfied. Someimportant such properties, studied here, include broadcast, h-broadcast, all-to-one operations,h-strong connectivity and strong connectivity. We consider the range assignment problem in2-dimensional and d (d > 2) Euclidean space. Moreover, we consider variations of the problemwhere the operation should be performed within unbounded or bounded number of hops (h).When both these parameters are given the problem is denoted as Min dDim h-range assignmentproblem.

Our research is also extended in the challenging task of fast and reliable communication instatic ad-hoc wireless networks. A useful paradigm of wireless communication is the structuringof communication into synchronous time-slots. A fundamental communication task in suchnetworks is the broadcast operation. It consists of transmitting a message from one source nodeto all the nodes. Most of the proposed broadcast protocols in wireless networks concern the casein which the network is fault-free. However, wireless networks are typically adopted in scenarioswhere unpredictable node and link faults happen very frequently. We study the completiontime of broadcast operations on Static ad-hoc Wireless Networks in presence of unpredictableand dynamical faults.

Moreover, motivated by the fact that, for data intensive applications, a significant amount ofenergy is dissipated in the memory, we also focus on the memory energy dissipation. We studythe relation between the energy complexity and the memory activities in wireless (mobile) ad-hocnetworks.

3.2 Algorithmic Solutions and Technical Recommendations

3.2.1 On the Power Assignment Problem in Radio Networks, [21]

We consider the Min dDim h-range assignment problem, given by a radio network S locatedon a d-dimensional Euclidean space. We wish to ensure communication between any pair ofstations in at most h hops, 1 ≤ h ≤ |S| − 1. Two main issues related to this problem areconsidered in this work: the trade-off between the power consumption and the number of hops;the computational complexity of the problem.

As for the first question, we provide a lower bound on the minimum power consumption ofstations on the plane for constant h. The lower bound is a function of |S|, h and the minimumdistance over all the pairs of stations in S. Then, we derive a constructive upper bound as afunction of |S|, h and the maximum distance over all pairs of stations in S (i.e. the diameter ofS). It turns out that when the minimum distance between any two stations is “not too small”(i.e. well spread instances) the upper bound matches the lower bound. Previous results for this

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problem were known only for very special 1-dimensional configurations (i.e., when points arearranged on a line at unitary distance) [42].

As for the second question, we observe that the tightness of our upper bound implies that theMin 2Dim h-range assignment problem restricted to well spread instances admits a polynomialtime approximation algorithm. Then, we also show that the same approximation result can beobtained for random instances. On the other hand, we prove that for h = |S| − 1 (i.e. theunbounded case) Min 2Dim h-range assignment problem is NP-hard and Min 3Dim h-rangeis APX-complete.

3.2.2 The Range Assignment Problem in Non-Homogeneous Static Ad-hoc Net-works [2]

Range assignment problems in ad-hoc wireless networks have been the subject of several recentstudies. All these studies deal with the homogeneous case, i.e., all stations share the same energycost function. However, this assumption does not well model realistic scenarios in which theenergy cost of a station varies dramatically depending on the particular enviroment conditionsof its location. We introduce the weighted version of the range assignment problem in whichthe cost a station s pays to transmit to another station depends on the distance between thestations and on the energy cost of station s. Most of the algorithm results for the unweightedrange assignment problem can not be applied to the weighted version. We thus provide a set ofalgorithmic results for this version and discuss some interesting related open questions.

3.2.3 Efficient Algorithms for Low-Energy Bounded-Hop Broadcast in Ad-hocWireless Networks, [3]

We study the problem of computing a minimal energy-cost range assignment in a ad-hoc wirelessnetwork which allows a station s to perform a broadcast operation in at most h hops. The generalversion of the problem (i.e., when transmission costs are arbitrary) is known to be NP -hardeven for h = 2. This work considers the well-studied real case in which n stations are located onthe plane and the cost to transmit from station i to station j is proportional to the α-th powerof the distance between station i and j, where α is any positive constant. A polynomial-timealgorithm is presented for finding an optimal range assignment to perform a 2-hop broadcastfrom a given source station. The algorithm relies on dynamic programming and operates in(worst-case) time O(n7). Then, a polynomial-time approximation scheme (PTAS) is providedfor the above problem for any fixed h ≥ 1. For fixed h ≥ 1 and ε > 0, the PTAS has timecomplexity O(nµ) where µ = O((α2αhα/ε)αh

).

3.2.4 Experimental Analysis of Practically Efficient Algorithms for Bounded-HopAccumulation in Ad-hoc Wireless Networks, [16]

We study the minimum energy-cost range assignment in an ad-hoc wireless network located in the2-dimensional Euclidean space. We consider the version of the problem when the communicationoperation to be performed is the accumulation (all-to-one) operations towards some root stationb in at most h hops, denoted as Min 2Dim h-Accumulation range Assignment problem).

We experimentally investigate the behavior of fast and easy-to-implement heuristics for theMin 2Dim h-Accumulation problem on instances obtained by choosing at random n points ina square of side length L. We compare the performance of an easy-to-implement, very fastheuristic with those of three simple heuristics based on classical greedy algorithms (Prim’s andKruskal’s ones) defined for the Minimum Spanning Tree problem. The comparison is carried out

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over thousands of random instances in several different situations depending on: the distributionof the stations in the plane, their density, the energy cost function.

3.2.5 On the Approximability of the Range Assignment Problem on Radio Net-works in Presence of Selfish Agents, [4]

We consider the range assignment problem in ad-hoc wireless networks in the context of selfishagents: A network manager aims to assigning transmission ranges to the stations in order toachieve strong connectivity of the network within a minimal overall power consumption. Stationis not directly controlled by the manager and may refuse to transmit with a certain transmissionrange because it might be costly in terms of power consumption.

We investigate the existence of payment schemes which induce the stations to follow thedecisions of a network manager in computing a range assignment, that is, truthful mechanismsfor the range assignment problem. We provide both positive and negative results on the existenceof truthful VCG-based mechanisms for this NP-hard problem. We prove that (i) in general,every polynomial-time truthful VCG-based mechanism computes a solution of cost far-off theoptimum, unless P=NP, and (ii) there exists a polynomial-time truthful VCG-based mechanismachieving constant approximation for practically relevant, still NP-hard versions, i.e., the metricand the well-spread case.

3.2.6 Round Robin is Optimal for Fault-Tolerant Broadcasting on Wireless Net-works, [19]

We study the completion time of broadcast operations on Static ad-hoc Wireless Networks inpresence of unpredictable and dynamical faults.

Concerning oblivious fault-tolerant distributed protocols, we provide an Ω(Dn) lower boundwhere n is the number of nodes of the network and D is the source eccentricity in the fault-freepart of the network. Rather surprisingly, this lower bound implies that the simple Round-Robinprotocol, working in O(Dn) time, is an optimal fault-tolerant oblivious protocol. Then, wedemonstrate that networks of o(n/ log n) maximum in-degree admit faster oblivious protocols.Indeed, we derive an oblivious protocol having O(D minn,∆log n) completion time on anynetwork of maximum in-degree ∆.

Finally, we address the question whether adaptive protocols can be faster than obliviousones. We show that the answer is negative at least in the general setting: we indeed prove anΩ(Dn) lower bound when D = Θ(

√n). This clearly implies that no (adaptive) protocol can

achieve, in general, o(Dn) completion time.

3.2.7 An analytical model for energy minimization, [58]

Energy has emerged as a critical constraint in mobile computing because the power availability inmost of these systems is limited by the battery power of the device. In this work, we focus on thememory energy dissipation. This is motivated by the fact that, for data intensive applications,a significant amount of energy is dissipated in the memory. Advanced memory architectures likethe Mobile SDRAM and the RDRAM support multiple power states of memory banks, whichcan be exploited to reduce energy dissipation in the system. Therefore, it is important to designefficient controller policies that transition among power states. Since the addressed memorychip must be in the active state in order to perform a read/write operation, the key point isthe tradeoff between the energy reduction due to the use of low power modes and the energy

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overheads of the resulting activations. The lack of rigorous models for energy analysis is the mainmotivation of this work. Assuming regular transitions, we derive a formal model that capturesthe relation between the energy complexity and the memory activities. Given a predeterminednumber of activations, we approximate the optimal repartition among available power modes.We evaluate our model on the RDRAM and analyze the behavior of each parameter togetherwith the energy that can be saved or lost.

3.2.8 Energy efficiency or Minimum energy broadcast routing problem, [27, 54, 47]

In [47], we present a new heuristic called Adaptive Broadcast Consumption (ABC for short) forthe Minimum-Energy Broadcast Routing (MEBR) problem. We first investigate the problemtrying to understand which are the main properties not taken into account by the classic andwell–studied MST and BIP heuristics, then we propose a new algorithm proving that it com-putes the MEBR with an approximation ratio less than or equal to MST, for which we prove anapproximation ratio of at most 12.15 instead of the well–known 12. Finally, we present exper-imental results supporting our intuitive ideas, comparing ABC with other heuristics presentedin the literature and showing its good performance on random instances even compared to theoptimum.

In [27, 54], we present new upper bounds on the approximation ratio of the Minimum Span-ning Tree heuristic for the basic problem on Ad-Hoc Networks given by the Minimum EnergyBroadcast Routing (MEBR) problem. In [27], we introduce a new analysis allowing to establisha 7.6-approximation ratio for the 2-dimensional case, thus significantly decreasing the previ-ously known 12 upper bound (actually corrected to 12.15 in [47]). We also extend our analysisto any number of dimensions d ≥ 2, obtaining a general approximation ratio of 3d − 1. Theimprovements of the approximation ratios are specifically significant in comparison with thelower bounds given by the kissing numbers, as these grow at least exponentially with respect tod. In [54], we introduce a new analysis allowing to establish a 6.33-approximation ratio in the2-dimensional case, thus decreasing the 7.6 upper bound from [27].

4 Efficient Communication in wireless sensor networks

4.1 Introduction

Wireless Sensor Networks are very large collections of tiny sensor nodes that form ad hoc dis-tributed sensing and data propagation wireless networks that collect quite detailed informationabout the physical environment. In a usual scenario, these networks are largely deployed inareas of interest (such as inaccessible terrains or disaster places) for fine grained monitoring invarious classes of applications [9]. Several issues are considered when dealing with wireless sen-sor networks: localization [6], positioning [38], network topology, resource allocation, networkconnectivity [53], reliability and security, routing protocols, and energy consumption [5], amongothers.

A challenging algorithmic and technological task in such networks is the efficient and robustrealization of such large, highly-dynamic, complex, non-conventional networking environments.Features including the huge number of sensor devices involved, the severe power, computa-tional and memory limitations, their dense deployment and frequent failures, pose new designand implementation aspects which are essentially different not only with respect to distributedcomputing and systems approaches but also to ad-hoc networking techniques.

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An important issue in this context, is the problem of propagating data reporting the re-alization of a series of events from the sensor devices that detected them towards the controlcenter. We present a new protocol that manages to increase the number of events reported tothe control center by operating with high energy efficiency and fault-tolerance. Also, we pro-pose a family of power conservation schemes for data propagation, that can be used to improvethe energy-efficiency of sensor networks. We also consider the problem of energy balanced datapropagation in wireless sensor networks and we generalize previous works by allowing realisticenergy assignment. Moreover, we introduce a dynamical model for wireless sensor networks.This work is one of the first works modelling dynamic aspects in such networks.

Another relevant critical task considered is a particular instance of the optimal power as-signment in wireless sensor networks, called dual power management problem (DPMP). In thiscase, each node of the network can be assigned high or low power during the network initializingstage. The goal is then to minimize the number of nodes that will be assigned high power,without breaking the network connectivity.

Our research in wireless ad-hoc systems considers also systems consisting of cooperative mobilerobotics. Such systems have received a lot of attention from various research institutes andindustries. An important research issue here is that of distributed motion coordination, sinceit allows the robots to form certain patterns and move in formation towards cooperating forthe achievement of certain tasks. The significance of positioning the robots based on somegiven patterns may be useful for various tasks, such as in bridge building, in forming adjustablebuttresses to support collapsing buildings, satellite recovery, or tumor excision. Distributedmotion planning algorithms for robotic systems are potentially useful in environments that areinhospitable to humans or are hard to directly control and observe (e.g. space, undersea). Weconsider a system of multiple robots that move on the plane and we study the problem of therobots positioning themselves to form a circle.

4.2 Algorithmic Solutions and Technical Recommendations

4.2.1 A Forward Planning Protocol for Scalable, Energy Efficient and Fault-tolerant Data Propagation in Wireless Sensor Networks, [20]

In this work we propose the Forward Planning Protocol (FPP) for scalable, energy efficientand fault tolerant data propagation in wireless sensor networks. To deal with the increasedcomplexity of such large-scale sensor systems, FPP employs a series of plan & forward phasesthrough which devices self-organize into clusters that propagate data over discovered paths. FPP

performs a limited number of long range, high power data transmissions to collect informationregarding the neighboring devices. The acquired information, allows to plan a (parameterizablelong) sequence of short range, low power transmissions between nearby particles, based on certainoptimization criteria. All particles that responded to these long range transmissions enter theforwarding phase during which information to the sink is propagated via the acquired plan. Thisrole-based approach where a selective number of devices do the high cost planning and the restof the network operates in a low cost state leads to systems that have increased energy efficiencyand high fault-tolerance since these long range planning phases allow to bypass obstacles (whereno sensors are available) or faulty sensors (that have been disabled due to power failure or othernatural events).

We implement the protocol and perform an extensive experimental evaluation and compari-son to (i) a representative protocol (LTP) and (ii) a protocol that employs variable transmissionrange (VTRP) using several important performance measures with a focus on energy consump-

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tion. Our findings indeed demonstrate that our protocol achieves significant improvements inenergy efficiency and network lifetime.

4.2.2 Efficient Data Propagation Protocols in Wireless Sensor Networks, [10]

In this work, we study the problem of multiple event detection and propagation, i.e. the localsensing of a series of crucial events and the energy efficient propagation of data reporting therealization of these events to a (fixed or mobile) control center. The control center could in factbe some human authorities responsible of taking action upon the realization of the crucial event.We use the term “sink” for this control center.

We propose extended versions of two data propagation protocols in wireless sensor networks:the Sleep-Awake Probabilistic Forwarding Protocol (SW-PFR) and the Hierarchical Thresholdsensitive Energy Efficient Network protocol (H-TEEN). These non-trivial extensions aim at im-proving the performance of the original protocols, by introducing sleep-awake periods in thePFR case to save energy, and introducing a hierarchy of clustering in the TEEN case to bettercope with large networks areas. Moreover, We implemented the two protocols and performed anextensive experimental comparison of various important measures of their performance with afocus on energy consumption, c) We investigate in detail the relative advantages and disadvan-tages of each protocol and discuss and explain their behavior, d) We discuss a possible hybridcombination of the two protocols towards optimizing certain goals, e) We propose a protocol thatmay vary the transmission range, towards achieving obstacle avoidance and energy balance.

4.2.3 An integer programming heuristic for the dual power management problemin wireless sensor networks, [59]

We seek an integer programming based heuristic for solving the dual power management prob-lem in wireless sensor networks. For a given network with two possible transmission powers (lowand high), the problem is to find a minimum size subset of nodes such that if they are assignedhigh transmission power while the others are assigned low transmission power, the network willbe strongly connected. The main purpose behind this efficient setting is to minimize the totalcommunication power consumption while maintaining the network connectivity. In a theoreticalpoint of view, the problem is known to be difficult to solve exactly. An approach to approximatethe solution is to work with a spanning tree of clusters. Each cluster is a strongly connectedcomponent when consider low transmission power. We follow the same approach, and we for-mulate the node selection problem inside clusters as an integer programming problem which issolved exactly using specialized codes. Experimental results show that our algorithm is efficientboth in execution time and solution quality.

4.2.4 An Adaptive Blind Algorithm for Energy Balanced Data Propagation inWireless Sensors Networks, [51]

We consider the problem of energy balanced data propagation in wireless sensor networks andwe generalize previous works by allowing realistic energy assignment. A new modelization ofthe process of energy consumption as a random walk along with a new analysis are proposed.Two new algorithms are presented and analyzed. The first one is easy to implement and fast toexecute. However, it needs a priori assumptions on the process generating data to be propagated.The second algorithm overcomes this need by inferring information from the observation of theprocess. Furthermore, this algorithm is based on stochastic estimation methods and is adaptive

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to environmental changes. This represents an important contribution for propagating energybalanced data in wireless sensor networks due to their highly dynamic nature.

4.2.5 Towards a dynamical model for wireless sensor networks, [52]

We propose a model for describing dynamic aspects of broadcast in wireless sensor networks.To our knowledge previous works on modelling wireless sensor networks focus on static aspects,for instance the topology given by the underlying graph of connections [23], or focus on specificalgorithms such as routing, localization, etc. We obtain a convergent martingale for the broad-cast process in such networks. To our knowledge, such martingales were unknown previously.We look at a formal model using the formalisms of martingales, dynamical systems and Markovchains, each formalism providing complementary and coherent information with each other. Themodel is validated in its scope of application with numerical simulation of wireless sensor net-works, we informally make explicit the situations where the model is realistic. We also providean alternative dynamical model based on the hypothesis that the distribution of the locationof the emitting sensors is uniform. This hypothesis is more fulfilled when the emission radius r

becomes larger and emission angle α smaller in order to keep the expected number of connectionsENC constant. In the situations where the hypothesis is close to be fulfilled good agreementsbetween numerical experiments and results of the model are observed. The alternative approachleads to similar quantitative and qualitative results as the first model.

4.2.6 Energy conservation in sensor networks, [26, 50, 44]

Wireless sensor networks have recently posed many new system building challenges. One of themain problems is energy conservation since most of the sensors are devices with limited batterylife and it is infeasible to replenish energy via replacing batteries. An effective approach forenergy conservation is scheduling sleep intervals for some sensors, while the remaining sensorsstay active providing continuous service. In [26] we consider the problem of selecting a setof active sensors of minimum cardinality so that sensing coverage and network connectivityare maintained. We show that the greedy algorithm that provides complete coverage has anapproximation factor of Ω(log n), where n is the number of sensor nodes. Then we presentalgorithms that provide approximate coverage while the number of nodes selected is a constantfactor far from the optimal solution.

The X-rank of a sensor s is the number of sensors whose X-coordinate is less than the X-coordinate of s. In [50] we provide a theoretical foundation for sensor ranking, in the case wheresome sensors know their locations and other sensors determine their ranking only by exchanginginformation. We show that in one dimension we can solve the ranking problem in linear time.On the other hand, the ranking problem is NP-hard in R2.

Some of the first routing algorithms for position-aware wireless networks used the Delaunaytriangulation of the point-locations of the network nodes as the underlying connectivity graph.Later on these solutions were considered impractical because the Delaunay triangulation may ingeneral contain arbitrarily long edges and because calculating the Delaunay triangulation mayrequire a global view of the network. Many other algorithms were then suggested for geometricrouting, often assuming random placement of network nodes for analysis or simulation. But aswe show in [45], when the nodes are uniformly placed in the unit disk the Delaunay triangulationdoes not contain long edges, it is easy to compute locally and it is in many ways optimal forgeometric routing and flooding.

Motivated by the study of sensor networks we consider the following problem in [44]:

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Throw n points independently and randomly onto the n vertices of G. Remap the pointson G such that the load of each vertex is exactly 1, minimizing the maximal distance that anypoint has to move (on G).

We call it the Points and Vertices problem. It may be viewed as an extension of the classicalBalls into Bins problem, where m balls are thrown (independently and uniformly at random)into n bins, by adding graph-structural properties to the bins so that the bins become verticesand there is an edge between two vertices if they are “close” enough.

The interest in the Points and Vertices problem arises from the fact that it captures ina natural way the “distance” between the randomness of throwing points (independently anduniformly at random) onto the vertices of G, and the order of the points being evenly balancedon G. The problem also has important applications in several fields such as token distribution,geometric matching, wireless communications and robotics.

5 Satellite Networks

5.1 Algorithmic Solutions and Technical Recommendations

5.1.1 Satellite boarded fault tolerant networks, [12, 57]

Inside a telecommunication satellite, audio and video signals are routed through a switchingnetwork to amplifiers. Since it is impossible to repair a satellite, we choose to multiply thecomponents that may be faulty, that is amplifiers and switches.

The first problem is to build a valid network which allows to route n input signals, to n

amplifiers (outputs), arbitrarily chosen among n + k, and thus supporting k broken amplifiers.Each switch has 4 links and the routes followed by the signals must be disjoint. Thus foreconomical constraints, the objective is to build valid networks having the minimum number ofswitches.

In [12], we studied a variation of this problem where p of the input signals called prioritiesmust be routed to the p amplifiers providing the best quality of service.

In [57], we also considered the case where the number of failures is of the same order of n,typically the case k = n.

5.1.2 Resource allocation for a geostationary satellite, [1, 49, 33]

In some further work [1] we present an allocation algorithm for resources in satellite networks. Itdeals with the planning of the time/frequency plan for a set of terminals with a known geomet-ric configuration and with interference constraints. The objective is to minimize the size of thefrequency plan with the guarantee that the different types of demands are satisfied, each typeusing different bandwidth. The proposed algorithm uses two main techniques: the first generatesadmissible configurations for the interference constraints, the second uses mixed linear/integerprogramming with column generation. The obtained solution estimates a possible allocationplan with optimality guarantees and highlights the frequency interferences which degrade theconstruction of good solutions, improving drastically the simulated annealing approach previ-ously proposed (see also [49] ). In [33] we consider the access to a constellation of satellites byusing ideas of geographical reservations. We study in particular the case where the satellite’sprint is non rectangular. Ours results guarantee the connection for a long time and give betterprecisions than preceding technics. The method is particularly adapted to constellations likeGlobalstar.

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6 Internet Access to Radio Networks

6.1 Algorithmic Solutions and technical Recommendations

6.1.1 Radio networks: Internet in villages, [11, 46, 56]

In cooperation with France Telecom, we have studied the problem of designing efficient strategiesto provide Internet access using wireless devices. Typically, in one village several houses wishto access a gateway (a satellite antenna) and to use multi-hop wireless relay routing to do so.

On the one hand we have modeled the problem as follows. Each node (representing a house)is able to communicate to nodes not too far away (at distance at most d). On the other hand,there is interference between nodes (at distance at most d′). The distances can be measuredeither in terms of euclidean distances or number of hops. In our first study we have consideredthe special case where each node has one information (message) he wants to transmit to (oranalogously receive from) the gateway (gathering problem). We have in particular obtained theresults for specific topologies like paths or grids. For grids the problem is solved when d = 1 andthe gateway is at the center of the grid ([11]). In [46], we have considered the case where thereis permanent demand (systolic algorithms). That leads to the definition of a call schedulingproblem. In such networks the physical space is a common resource that nodes have to share,since concurrent transmissions cannot be interfering. We study how one can satisfy steadybandwidth demands according to this constraint. We show that it can be relaxed into a simplerproblem: The call weighting problem, which is almost a usual multi-commodity flow problem,but the capacity constraints are replaced by the much more complex notion of non interference.Not surprisingly, this notion involves independent sets, and we prove that the complexity ofthe call weighting problem is strongly related to the one of the independent set problem andits variants (max-weight, coloring, fractional coloring). The hardness of approximation followswhen the interferences are described by an arbitrary graph. We refine our study by consideringsome particular cases for which efficient polynomial algorithms can be provided: the Gatheringin which all the demand are directed toward the same sink, and specific interference relations:namely those induced by the dimension 1 and 2 Euclidean space, those cases are likely to be thepractical ones.

On the other hand, we have worked on the improvement of the norm 802.11b. In the casewhen all the stations see each other, we have studied a memory-based process where the stationsautomatically adapt to their environment to avoid too frequent emissions (which generatescollisions) or too rare ones (which results in a loss of bandwidth). When the number of stationsincreases, the 801.11b norms makes the global capacity of the system tend to zero. In [32], wedescribe and study several alternatives, and we propose a solution where the total capacity ofthe system does not tend to zero but stays relatively high. Simulations show that this systemsimproves by 40% the capacity of the channel for 100 stations, and by 8% when the RTS/CTSmechanism is used. This study has opened the way to several developments in the CORSOcontract.

In particular, in [56], we analyze the case where hidden stations are present through a Markovchain analysis. We obtain a precise analysis of the cases where communications can fail in thecontext of a chain of stations (i.e. each station sees at most two immediate neighbors).

In [40], we study distributed code allocation algorithms for ad-hoc cdma networks. Severalknown and new algorithms are simulated and compared with respect to the knowledge of theneighborhood that each node has.

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7 Dynamic networks

7.1 Algorithmic Solutions and Technical Recommendations

7.1.1 Connectivity in evolving graphs, [24, 39, 25]

New technologies and the deployment of mobile and nomadic services naturally engender newroute-discovery problems under changing conditions over dynamic networks. Unfortunately, thetemporal variations in the topology of dynamic networks are hard to be effectively captured ina classical graph model. We used evolving graphs, which helps capture the dynamic character-istics of such networks, in order to show that computing different types of strongly connectedcomponents in dynamic networks is NP-complete, and investigated the concepts of journeys inEvolving Graphs, which captures both space and time constraints in routing problems [24].

We further investigated the connected components problem in dynamic networks with spe-cial topologies. In a dynamic setting, the topology of a network derives from the set of all thepossible links, past and future. We proved that the strongly connected components problem isstill NP-complete when the topology is composed of unit disc graphs and the nodes are placedon a grid [39]. On the other hand, we also gave a polynomial-time algorithm, by dynamic pro-gramming, in order to compute a maximum strongly connected components when the topologyis a tree [39].

One of the new challenges facing research in wireless networks is the design of algorithmsand protocols that are energy aware. The minimum-energy broadcast routing problem, whichattracted a great deal of attention these past years, is NP-hard, even for a planar static network.The best approximation ratio for it is a solution proved to be within a factor 12 of the optimal.One popular way of achieving this ratio is based on finding a Minimum Spanning Tree of thestatic planar network. We used the evolving graph combinatorial model to prove that computinga Minimum Spanning Tree of a planar network in the presence of mobility is NP-Complete [25].We also gave a polynomial-time algorithm to build a rooted spanning tree of a mobile network,that minimizes the maximum energy used by any one node, thus maximizing the life-time of awireless communication network [25].

7.1.2 Internet/Web algorithms, [18, 17, 37]

In [18], we study the size of generalized dominating sets in two graph processes which are widelyused to model aspects of the world-wide web. On the one hand, we show that graphs generatedthis way have fairly large dominating sets (i.e. linear in the size of the graph). On the otherhand, we present efficient strategies to construct small dominating sets. The algorithmic resultsrepresent an application of a particular analysis technique which can be used to characterize theasymptotic behaviour of a number of dynamic processes related to the web.

In [37], we introduce the Push Tree problem which contains elements from both the SteinerTree and the Shortest Path problem. The Push Tree problem deals with the trade-offs betweenthe push and pull mechanisms used in information distribution and retrieval in the Internet.We present some initial complexity results and analyze several heuristics. Moreover, we discusswhat lessons can be learnt from the static and deterministic Push Tree problem for more real-istic scenarios characterized by high uncertainty and changing information request and updatepatterns.

In [17], we consider a randomized algorithm for assigning neighbours to vertices joining adynamic distributed network. The algorithm acts to maintain connectivity, low diameter and

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constant vertex degree. This is effected as follows: On joining each vertex donates a fixed numberof tokens to the network. The tokens contain the address of the donor vertex. Tokens makeindependent random walks in the network. A token can be used by any vertex it is visiting toestablish a connection to the donor vertex. This allows joining vertices to be allocated a randomset of neighbors although the overall membership of the network is unknown. The networkwe obtain in this way is robust under adversarial deletion of vertices and edges and activelyreconnects itself.

References

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[2] C. Ambuehl, A.E.F. Clementi, M. Di Ianni, A. Monti, G. Rossi, R. Silvestri. The Range As-signment Problem in Non-Homogeneous Static Ad-Hoc Networks. Proceedings of IPDPS-WMAN 2004.

[3] C. Ambuehl, A.E.F. Clementi, M. Di Ianni, N. Lev-Tov, A. Monti, D. Peleg, G. Rossi, R.Silvestri. Efficient Algorithms for Low-Energy Bounded-Hop Broadcast in Ad-Hoc WirelessNetworks. Proceedings of STACS 2004: 418-427

[4] C. Ambuehl, A.E.F. Clementi, P. Penna, G. Rossi, R. Silvestri. On the Approximability ofthe Range Assignment Problem on Radio Networks in Presence of Selfish Agents. Theor.Comput. Sci. to appear.

[5] L. Anderegg and S. Eidenbenz, Ad hoc-VCG: a truthful and cost-e.cient routing protocolfor mobile ad hoc networks with sel.sh agents, Proceedings of the 9th annual internationalconference on Mobile computing and networking, San Diego, CA, USA, 2003.

[6] J. Apnes, D. Goldenberg, and Y. Richard, On the computational complexity of sensornetwork localization, Algorithmic Aspects of Wireless Sensor Networks: First InternationalWorkshop, ALGOSENSORS 2004, Turku, Finland, July 16, 2004.

[7] M.I. Andreou M. Karpinski, V.G. Papadopoulou and P.G. Spirakis, ”RadiocoloringSuccinct Graphs and its relation to MIS on squares of ordinary Graphs”, (RA.CTI.TR2004/12/01), CRESCCO Technical Report 2004.

[8] M.I. Andreou V.G. Papadopoulou P.G. Spirakis, B. Theodorides and A. Xeros, ”Generat-ing and Radiocoloring Families of Perfect Graphs”, (RA.CTI. TR2004/12/01), CRESCCOTechnical Report 2004, submitted to WEA 2005.

[9] I.F. Akyildiz, W. Su, Y. Sankarasubramaniam, and E. Cayirci. Wireless sensor networks:a survey. Journal of Computer Networks 38, pp. 393422, 2002.

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