Power Optimization for Power Optimization for Connectivity ProblemsConnectivity Problems

MohammadTaghi Hajiaghayi, Guy MohammadTaghi Hajiaghayi, Guy

Kortsarz, Vahab S. Mirrokni, Zeev Nutov Kortsarz, Vahab S. Mirrokni, Zeev Nutov

IPCO 2005IPCO 2005

  Power Optimization in Power Optimization in Fault-Tolerant Topology Fault-Tolerant Topology

ControlControl Wireless multihop networksWireless multihop networks

Simple low-power devicesSimple low-power devices Radio transmittersRadio transmitters

Power is the Power is the main limitationmain limitation Power assignmentPower assignment

A power setting for each deviceA power setting for each device Defines possible communication linksDefines possible communication links

Power versus distance:Power versus distance: It takes power It takes power rrcc to to transmit a message to distance transmit a message to distance rr for some for some power attenuation exponentpower attenuation exponent c c between between 22 and and 44..

GoalGoal:: Minimize power usage while maintaining Minimize power usage while maintaining key network propertieskey network properties

Connectivity:Connectivity: There is a communication path There is a communication path between any pair of nodes between any pair of nodes

k-Fault tolerancek-Fault tolerance: Connectivity is maintained in : Connectivity is maintained in light of at mostlight of at most k-1 k-1 failures failures Device failures Device failures (our focus)(our focus) Communication link failuresCommunication link failures

By By k-Fault tolerancek-Fault tolerance,, we also have k-disjoint we also have k-disjoint paths and thus higher network capacitypaths and thus higher network capacity

Power Optimization in Power Optimization in Fault-Tolerant Topology Fault-Tolerant Topology

ControlControl

ModelModel

A A wireless networkwireless network is modeled as a graph is modeled as a graph G(V,E)G(V,E) with cost functions with cost functions d d and and pp on on EE V V is the set of is the set of mobile devicesmobile devices EE is the set of pairs of devices which can is the set of pairs of devices which can

communicatecommunicate bi-directionally bi-directionally dduvuv is the is the distancedistance between device between device uu and and vv ppuvuv is the is the powerpower needed to transmit between needed to transmit between

device device uu and and v v (usually it is distance to the (usually it is distance to the power attenuation exponent)power attenuation exponent)

ModelModel

Conversely, a subgraph Conversely, a subgraph H=(V,E’) H=(V,E’) of the of the network graph network graph GG defines an defines an assignment of power settingsassignment of power settings: device : device uu transmits at transmits at

p(u) = max p(u) = max {(u,v) in E’}{(u,v) in E’} p puvuv

The The power power used by a wireless network used by a wireless network with power settings defined by with power settings defined by HH is is

P(H) = P(H) = ΣΣ u in Vu in V p(u) p(u)

Problem FormulationProblem Formulation

GivenGiven A wireless networkA wireless network

FindFind An assignment of power settings that An assignment of power settings that

guarantees guarantees kk-fault tolerance while -fault tolerance while minimizing power usageminimizing power usage

Recall Recall kk-fault tolerance-fault tolerance means the network means the network remains connected even when up to remains connected even when up to k-1k-1

devices (or communication links) faildevices (or communication links) fail

Related Results for Power Related Results for Power MinimizationMinimization

ConnectivityConnectivity Cone-based local Cone-based local heuristicsheuristics

[Rodoplu, Meng ’99; Wattenhofer, Li, Bahl, Halpern, Wang ’02][Rodoplu, Meng ’99; Wattenhofer, Li, Bahl, Halpern, Wang ’02]

A A 22-approximation based on -approximation based on minimum weight minimum weight spanning treespanning tree

[Kerousis, Kranakis, Krizanc, Pelc ’00][Kerousis, Kranakis, Krizanc, Pelc ’00]

A A 1.691.69-approximation based on -approximation based on minimum minimum weight Steiner treeweight Steiner tree and a more and a more practicalpractical 1.8751.875-approximation-approximation [Calinescu, Mandoiu, Zelikovsky [Calinescu, Mandoiu, Zelikovsky ’02]’02]

Related Results for Related Results for Power MinimizationPower Minimization

22-Fault tolerance -Fault tolerance Heuristic Heuristic to minimize maximum transmit to minimize maximum transmit

power power

[Ramanathan, Rosales-Hain ’00][Ramanathan, Rosales-Hain ’00] (the only (the only previous result)previous result)

Fault tolerance for general Fault tolerance for general kk Pioneered in Pioneered in [Bahramgiri, Hajiaghayi, [Bahramgiri, Hajiaghayi,

Mirrokni, WINET’02] Mirrokni, WINET’02] andand [Hajiaghayi, [Hajiaghayi, Immorlica, Mirrokni, MOBICOM’03]Immorlica, Mirrokni, MOBICOM’03]

Cone-Based HeuristicCone-Based Heuristic Algorithm:

Input: A set of nodes on the plane, with max. power P Each node increases its power until the angle between any

two consecutive neighbors is less than some threshold or it reaches its maximum power P.

OutputOutput: two nodes are connected if both can hear each : two nodes are connected if both can hear each other with the new power assignmentother with the new power assignment

Theorem [BHM’02]: If the network of max. powers is k-connected and the angle between any pair of adjacent neighbors is at most 2π/3k, then the new network is k-connected (2π/3k is almost tight)

Main disadvantage:Main disadvantage: The algorithm is local and The algorithm is local and thus does not give any bound on the global goal of thus does not give any bound on the global goal of minimizing sum of the powers (or the average minimizing sum of the powers (or the average power)power)

Approximating ConnectivityApproximating Connectivity

Recall the Recall the powerpower P(H)P(H) of subgraph of subgraph HH is is

P(H) = P(H) = ΣΣ u in Vu in V p(u) p(u)

where where p(u) = max p(u) = max {(u,v) in H(E)}{(u,v) in H(E)} p puvuv

Define the Define the weightweight W(H)W(H) of subgraph H of subgraph H asas

W(H) = W(H) = ΣΣ (u,v) in H(E)(u,v) in H(E) p puvuv

Approximating ConnectivityApproximating Connectivity TheoremTheorem [KKKP ’00]:[KKKP ’00]: The minimum weight The minimum weight

spanning tree spanning tree MSTMST of of GG uses at most twice uses at most twice as much power as the minimum power as much power as the minimum power connected subgraph connected subgraph OPTOPT of of GG..

Lemma 1Lemma 1: For any graph : For any graph GG, , P(G) ≤ 2W(G)P(G) ≤ 2W(G).. Lemma 2Lemma 2: For any tree : For any tree TT, , W(T) ≤ P(T)W(T) ≤ P(T).. Lemma 3Lemma 3: : OPTOPT is a tree is a tree

Proof (of Thm)Proof (of Thm): From the above lemmas,: From the above lemmas,P(MST) ≤ 2W(MST) ≤ 2W(OPT) ≤ 2P(OPT)P(MST) ≤ 2W(MST) ≤ 2W(OPT) ≤ 2P(OPT)..

Approximating k-Approximating k-ConnectivityConnectivity

Minimum weight k-connected Minimum weight k-connected subgraphsubgraph: : an an LP-based algorithmLP-based algorithm gives a solution of weight at most gives a solution of weight at most O(log k)O(log k) times optimal weight ( times optimal weight (nn is is at least at least 6k6k22) ) [Cheriyan, Vempala, Vetta, [Cheriyan, Vempala, Vetta, STOC’02], STOC’02], [Kortsarz, Nutov, STOC’04[Kortsarz, Nutov, STOC’04]]

Minimum Power k-connected Minimum Power k-connected subgraph: subgraph: Using the above Using the above algorithms an algorithms an O(k)-approximation O(k)-approximation can be derived can be derived [Hajiaghayi, Immorlica, [Hajiaghayi, Immorlica, Mirrokni MOBICOM’03]Mirrokni MOBICOM’03]

Distributed (Local) Distributed (Local) ApproximationApproximation

AlgorithmAlgorithm: : [Hajiaghayi, Immorlica, Mirrokni, [Hajiaghayi, Immorlica, Mirrokni, MOBICOM’03]MOBICOM’03]

Construct minimum weight spanning tree Construct minimum weight spanning tree with with O(n log n + m)O(n log n + m) messages messages [Gallager, [Gallager, Humbler, Spira, ’83]Humbler, Spira, ’83]

Use Use local augmentationlocal augmentation to create a to create a

k-connected sub-graph with k-connected sub-graph with O(n)O(n) messages messages TheoremTheorem: If : If ppuvuv= (d= (duvuv))cc for all pairs of nodes, for all pairs of nodes,

then the algorithm is an then the algorithm is an O(1)O(1)-approximation -approximation when k is constant.when k is constant.

Min Power K-cover Min Power K-cover

k-edge coverk-edge cover is a subgraph in which is a subgraph in which the degree of each vertex is at least the degree of each vertex is at least kk..

Our ResultsOur Results Main ResultMain Result: An : An min(O(logmin(O(log44n)+2a,k(1+o(1)))-n)+2a,k(1+o(1)))-

approximation for minimum power k-connected approximation for minimum power k-connected subgraph where subgraph where aa is the approximation factor of is the approximation factor of minimum weight k-connected subgraph.minimum weight k-connected subgraph.

An An min(min(O(logO(log44n),k+1)-n),k+1)-approximation for min. approximation for min. power k-edge cover subgraph.power k-edge cover subgraph.

An An O(√n)-O(√n)-approximation for min. power k-edge approximation for min. power k-edge connected subgraph.connected subgraph.

APX-HardnessAPX-Hardness of k-edge cover and k- of k-edge cover and k-connectivity. connectivity.

Strong hardness of min. power k-edge disjoint Strong hardness of min. power k-edge disjoint paths. paths.

Useful Facts/LemmasUseful Facts/Lemmas

Fact 1:Fact 1: For any forest For any forest F, p(F)>= c(F)F, p(F)>= c(F).. Fact 2: Fact 2: For any graph For any graph G, p(G)<= 2c(G)G, p(G)<= 2c(G).. ThusThus, approximating , approximating cost cost and and power power for for

forests are the same within a constant forests are the same within a constant factor.factor.

Theorem:Theorem: If If G G is a k-edge cover and is a k-edge cover and F F is is an inclusion minimal edge set such that an inclusion minimal edge set such that G+F G+F is k-connected, then is k-connected, then FF is a forest. is a forest.

k-edge cover to k-connectedk-edge cover to k-connected

Theorem:Theorem: If there are If there are cc-approximation for min. weight k--approximation for min. weight k-

connected subgraph.connected subgraph. dd-approximation for min. power k-edge -approximation for min. power k-edge

covercover

Then we have Then we have 2c+d2c+d-approximation for -approximation for min power k-connected subgraph.min power k-connected subgraph.

c is in c is in O(log n)O(log n) by previous results. by previous results. d is in d is in O(logO(log44n)n) by a new involved by a new involved

combinatorial algorithm.combinatorial algorithm.

Algorithm for k-connectivityAlgorithm for k-connectivity

Find the Find the minimum power k-edge coverminimum power k-edge cover.. Set the weight of the edges in the k-Set the weight of the edges in the k-

edge cover to edge cover to zerozero.. Augment Augment the k-edge cover to a k-the k-edge cover to a k-

connected graph by finding the connected graph by finding the min. min. cost k-connected cost k-connected with the new weight with the new weight function.function.

Note that the Note that the augmentationaugmentation is a forest. is a forest.

Approximating k-edge coverApproximating k-edge cover

A simple A simple k+1k+1-approximation: pick -approximation: pick kk small small edges adjacent to each vertex. edges adjacent to each vertex.

An involvedAn involved O(log^4 n)O(log^4 n)-approximation:-approximation: If all weights are the same, then the If all weights are the same, then the

problem is easy.problem is easy. Devide the edges into Devide the edges into log(n)log(n) weight weight

classes. classes. Devide the vertices into Devide the vertices into log(n)log(n) subsets subsets

based on their deficiency….based on their deficiency….

O(√n)-O(√n)-approximation for min. approximation for min. power k-edge connectivitypower k-edge connectivity

Augmenting k-edge cover to k-edge Augmenting k-edge cover to k-edge connected graph with at most connected graph with at most n-1n-1 edges. edges.

Using Using O(log^4 n)-O(log^4 n)-approximation for min. approximation for min. power power kk-edge cover and -edge cover and 22-approximation -approximation for min. weight for min. weight kk-edge connected -edge connected subgraph to agument to subgraph to agument to kk-edge -edge connected.connected.

This gives This gives O(√n)-O(√n)-approximation for min. approximation for min. power k-edge connected.power k-edge connected.

Hardness ResultsHardness Results

APX-hardness:APX-hardness: Reduction from 4- Reduction from 4-bounded set cover to the minimum bounded set cover to the minimum power k-edge cover, k-connected power k-edge cover, k-connected and k-edge connected subgraphs.and k-edge connected subgraphs.

Stronger inapproximability for Stronger inapproximability for minimum power k edge disjoint paths minimum power k edge disjoint paths in directed graphs.in directed graphs.

Open ProblemsOpen Problems

Closing the gap between the Closing the gap between the inapproximability and approximation inapproximability and approximation factor of the minimum power k-factor of the minimum power k-connected and k-edge connected connected and k-edge connected subgraph.subgraph.

Better approximation for metric Better approximation for metric graphs.graphs.

Thank you. Thank you.