localized fault-tolerant topology control in wireless ad hoc networks

Localized Fault-Tolerant Topology Controlin Wireless Ad Hoc Networks

Ning Li, Student Member, IEEE, and Jennifer C. Hou, Senior Member, IEEE

Abstract—Topology control algorithms have been proposed to maintain network connectivity while improving energy efficiency and

increasing network capacity. However, by reducing the number of links in the network, topology control algorithms actually decrease

the degree of routing redundancy. As a result, the derived topology is more susceptible to node failures or departures. In this paper, we

resolve this problem by enforcing k-vertex connectivity in the topology construction process. We propose a fully localized algorithm,

Fault-tolerant Local Spanning Subgraph (FLSS), that can preserve k-vertex connectivity and is min-max optimal among all strictly

localized algorithms (i.e., FLSS minimizes the maximum transmission power used in the network, among all strictly localized

algorithms that preserve k-vertex connectivity). It can also be proved that FLSS outperforms two other existing localized algorithms in

terms of reducing the transmission power. We also discuss how to relax several widely used assumptions in topology control to

increase the practical utility of FLSS. Simulation results indicate that, compared with existing distributed/localized fault-tolerant

topology control algorithms, FLSS not only has better power-efficiency, but also leads to higher network capacity. Moreover, FLSS is

robust with respect to position estimation errors.

Index Terms—Algorithm design and analysis, fault tolerance, localized algorithms, topology control, wireless ad hoc networks.

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1 INTRODUCTION

A wireless ad hoc network is a group of autonomouswireless devices that communicate with each other over

shared wireless channels. These wireless devices are eithermobile (e.g., in mobile ad-hoc networks, MANETs) or(semi)static (e.g., in wireless sensor networks). Communica-tion links are formed on the fly according to the distribution ofwireless nodes (and, in the case of MANETs, mobility of thesenodes), and their status is dependent on the status of otherlinks due to wireless interference and medium contention.The rapid development of wireless technology over the recentyears has posed many new challenges in the system designand analysis of wireless ad hoc networks, among whichenergy-efficiency [2] and network capacity [3] are the mostfundamental. Topology control algorithms [4], [5], [6], [7], [8],[9], [10], [11], [12] have been proposed to maintain networkconnectivity while reducing energy consumption and im-proving network capacity.

Instead of transmitting using the maximal power, nodesin a wireless ad hoc network collaboratively determine thetransmission power and define the network topology byforming proper neighbor relations. By using smallertransmission power that is connectivity-preserving, topol-ogy control algorithms actually decrease the number oflinks, which in turn reduces the number of possible routingpaths in the network. As a result, the network topology issusceptible to unpredictable events such as battery deple-tion and hardware failures. To design fault-tolerant topol-ogy control algorithms, we consider k-connectivity of the

network. A k-vertex connected network is k� 1 fault-tolerant, i.e., the failure of at most k� 1 nodes will notdisconnect the network.

We first propose a centralized algorithm, Fault-tolerantGlobal Spanning Subgraph (FGSSk), that preserves k-vertexconnectivity and is min-max optimal, i.e., it minimizes themaximum transmission power used by nodes in thenetwork. As will be discussed in Section 5, min-maxoptimality is a critical property to prolong the networklifetime. Based on the centralized algorithm, we thenpropose a fully localized algorithm, Fault-tolerant LocalSpanning Subgraph (FLSSk). We prove that FLSSk preservesk-vertex connectivity, and is min-max optimal among allstrictly localized algorithms. We also prove that FLSSkalways results in smaller transmission power for each node,compared with two other existing localized algorithms thatconsider k-connectivity, CBTC(2�

3k ) and YAOp;k. Moreover,we examine several widely used assumptions in topologycontrol (e.g., a common maximal transmission power for allthe nodes, obstacle-free communication channel, andavailability of position information) and discuss how torelax these assumptions.

Simulation results indicate that compared with thetopologies derived by other distributed or localized fault-tolerant algorithms, the topology derived by FLSSk has asmaller average node degree and smaller average transmis-sion power. FLSSk also improves network capacity andenergy efficiency. With modest adjustment, FLSSk canpreserve the connectivity in spite of position estimationerrors.

The rest of the paper is organized as follows: We firstdefine the network model in Section 2, and summarize relatedwork in Section 3. We then present FGSSk in Section 4, andFLSSk in Section 5. A discussion on how to relax severalassumptions is given in Section 6. Finally, we evaluate theperformance of FLSSk in Section 7, and conclude the paper inSection 8.

IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 17, NO. 4, APRIL 2006 307

. The authors are with the Department of Computer Science, University ofIllinois at Urbana-Champaign, Urbana, IL 61801-2302.E-mail: {nli, jhou}@cs.uiuc.edu.

Manuscript received 23 Feb. 2005; revised 16 Apr. 2005; accepted 9 June 2005;published online 24 Feb. 2006.Recommended for acceptance by I. Stojmenovic, S. Olariu, andD. Simplot-Ryl.For information on obtaining reprints of this article, please send e-mail to:[email protected], and reference IEEECS Log Number TPDSSI-0185-0205.

1045-9219/06/$20.00 � 2006 IEEE Published by the IEEE Computer Society

2 NETWORK MODEL

In this section, we define the network model. Consider ahomogeneous wireless ad hoc network where every node hasthe same maximal transmission power, which corresponds tothe common transmission range rmax. Later, we will relax thisassumption and consider heterogeneous wireless networksin which different nodes may have different maximaltransmission power. Let the network topology be representedby an undirected simple graph G ¼ ðV ðGÞ; EðGÞÞ in theplane, where V ðGÞ ¼ fv1; v2; . . . ; vng is the set of nodes (orequivalently, vertices) and EðGÞ is the set of links (edges) inthe network. Each node is assigned a unique identifier, id(such as an IP or MAC address).

Although G is usually assumed to be geometric in theliterature, here, we only assume that G is a general graph,i.e., EðGÞ ¼ fðu; vÞ : u and v can communicate with eachotherg. We also assume that the wireless channel issymmetric (i.e., both the sender and the receiver shouldobserve the same path loss) and obstacle-free, and eachnode is able to obtain its own location information via, forexample, several lightweight localization techniques forwireless networks [13], [14]. We will further discuss how torelax some of these assumptions in Section 6.

Definition 1 (Visible Neighborhood). The visible neighbor-hood NV

u is the set of nodes that node u can reach by using themaximum transmission power, i.e., NV

u ¼ fv 2 V ðGÞ : ðu; vÞ2 EðGÞg. For each node u 2 V ðGÞ, letGV

u ¼ ðV ðGVu Þ; EðGV

u ÞÞbe the induced subgraph of G such that V ðGV

u Þ ¼ NVu and

EðGVu Þ ¼ fðu; vÞ 2 EðGÞ : u; v 2 V ðGV

u Þg.

Each edge in EðGÞ is assigned a weight. Two points areworth mentioning here. First, to build a power efficientspanning subgraph, the weight of an edge is usually thepower consumption of a transmission between the two end-nodes. For the algorithms to be presented later in this paper,it suffices to use the Euclidean distance as the weightfunction. The resulting topology will be the same, since thepower consumption is, in general, of the form c0 � d� þ c1,� � 2, which is a strictly increasing function of theEuclidean distance. Second, to ensure that two edges withdifferent end-vertices have different weights, we use the idsof the end-nodes as tie-breakers.

Definition 2 (Weight Function). For an edge e ¼ ðu; vÞ, theweight function w : E 7!R3 maps to a 3-tuple, i.e., wðu; vÞ ¼ðdðu;vÞ;maxfidðuÞ;idðvÞg;minfidðuÞ; idðvÞgÞ. Given ðu1; v1Þ;ðu2; v2Þ 2 E,

wðu1; v1Þ > wðu2; v2Þ , dðu1; v1Þ > dðu2; v2Þor ðdðu1; v1Þ ¼ dðu2; v2Þ

&& maxfidðu1Þ; idðv1Þg >maxfidðu2Þ; idðv2ÞgÞ

or ðdðu1; v1Þ ¼ dðu2; v2Þ&& maxfidðu1Þ; idðv1Þg ¼

maxfidðu2Þ; idðv2Þg&& minfidðu1Þ; idðv1Þg >

minfidðu2Þ; idðv2ÞgÞ:

It is obvious that two edges with different end-verticeshave different weights, and two edges with the same end-vertices have the same weight, i.e., wðu; vÞ ¼ wðv; uÞ.Definition 3 (Neighbor Set). Node v is a neighbor of node u

under an algorithmALG (denoted u �!ALG v), if and only if thereexists a directed edge ðu; vÞ in the topology generated by thealgorithm. In particular, we use u! v to denote the neighborrelation in G. u !ALG v if and only if u �!ALG v and v �!ALG u. Theneighbor set of node u is NALGðuÞ ¼ fv 2 V ðGÞ : u �!ALG vg.

Definition 4 (Radius). Ru, the radius of node u is defined asthe Euclidean distance between u and its farthest neighbor, i.e,Ru ¼ maxv2NALGðuÞfdðu; vÞg.

Definition 5 (Degree). The degree of a node u under analgorithm ALG, denoted degALGðuÞ, is the number ofneighbors, i.e., degALGðuÞ ¼ jNALGðuÞj.

What we just defined is often referred as the logical nodedegree. It is often necessary to consider the physical nodedegree, i.e., the number of nodes within the transmissionradius.

Definition 6 (Topology). The topology generated by analgorithm ALG is a directed graph GALG ¼ ðV ðGALGÞ;EðGALGÞÞ, where V ðGALGÞ ¼ V ðGÞ, and EðGALGÞ ¼ fðu; vÞ2 EðGÞ : u �!ALG vg.

Definition 7 (Connectivity). For any topology generated by analgorithm ALG, node u is said to be connected to node v(denoted u) v) if there exists a path ðp0 ¼ u; p1; . . . ; pm�1;

pm ¼ vÞ such that pi �!ALG

piþ1; i ¼ 0; 1; . . . ;m� 1, where pk 2V ðGALGÞ; k ¼ 0; 1; . . . ;m. It follows that u) v if u) p andp) v for some p 2 V ðGALGÞ.

Definition 8 (Bidirectionality). A topology generated by analgorithm ALG is bidirectional, if for any two nodesu; v 2 V ðGALGÞ, u 2 NALGðvÞ if and only if v 2 NALGðuÞ.

Definition 9 (Bidirectional Connectivity). For any topologygenerated by an algorithm ALG, node u is said to bebidirectionally connected to node v (denoted u, v) if thereexists a path ðp0 ¼ u; p1; . . . ; pm�1; pm ¼ vÞ such that pi !

ALG

piþ1; i¼0; 1; . . . ;m�1, where pk2V ðGALGÞ; k ¼ 0; 1; . . . ;m.It follows thatu, v ifu, p andp, v for somep 2 V ðGALGÞ.

A bidirectionally connected network topology facilitateslink level acknowledgment, which is critical to packettransmissions over unreliable wireless channels. Bidirec-tionality is also very important to floor acquisition mechan-isms such as RTS/CTS in IEEE 802.11.

Definition 10 (Addition and Removal). The Additionoperation is to add an extra edge ðv; uÞ into GALG if ðu; vÞ 2EðGALGÞ and ðv; uÞ =2 EðGALGÞ. The Removal operation is todelete any edge ðu; vÞ 2 EðGALGÞ if ðv; uÞ =2 EðGALGÞ.

Both Addition and Removal operations attempt to create abidirectional topology by either removing unidirectionaledges or converting unidirectional edges into bidirectionalones. The resulting topology after Addition or Removal isalway bidirectional, if the maximal transmission power foreach node is the same. If the maximal transmission powerfor each node is not the same, the result of Removal is still

308 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 17, NO. 4, APRIL 2006

bidirectional, while the result of Addition may not bebidirectional (see [12] for more discussion).

Definition 11 (k-vertex connectivity). A graph G isk-vertex connected if for any two vertices v1; v2 2 V ðGÞ,there are k pairwise-vertex-disjoint paths from v1 to v2. Or,equivalently, a graph is k-vertex connected if the removal ofany k� 1 nodes (and all the related links) does not partitionthe network.

k-edge connectivity can also be defined accordingly.Regarding fault tolerance in wireless ad hoc networks, weare more concerned with k-vertex connectivity since thefailure of at most k� 1 nodes will not disconnect aoriginally k-vertex connected network. In this paper, wewill concentrate on k-vertex connectivity, and usek-connectivity to refer to k-vertex connectivity.

3 RELATED WORK

Since the problem of finding a minimum-cost k-connectedsubgraph is NP-hard [15], many approximation algorithmswere proposed (see [15], [16] for a summary). Althoughmost topology control algorithms (see [12] for a summary)do not take fault tolerance into consideration, there havebeen several research efforts recently on studying theproperties of k-connected topologies [17], [18], devisingalgorithms to construct such topologies [19], [16], [5], [20],or both [21].

3.1 Work that Studies the Properties ofFault-Tolerant Topologies

Penrose [17] studied k-connectivity in a geometric randomgraph of n nodes derived by adding an edge between eachpair of nodes at most r apart. He proved that the minimumvalue of r at which the graph is k-connected is equal to theminimum value of r at which the graph has the minimumdegree of k, with probability 1 as n goes to infinity. Thesignificance of this result is that it links k-connectivity, aglobal property of the graph, to node degree, a localparameter. It is, hence, possible to come up with localizedalgorithms that can preserve asymptotic k-connectivity.However, the minimum value of r is not given in thiswork. Bettstetter [18] also investigated the relation betweenthe minimum node degree and k-connectivity for geometricrandom graphs. The analytical expression of the requiredrange r0 for the almost surely k-connected network isderived, and then verified by simulations.

Li et al. [21] extended Penrose’s work and gave the lowerbound and the upper bound on the minimum value of r atwhich the graph is k-connected with high probability. Theanalysis shows that, for a unit-area square region, theprobability that the network of n nodes is k-connected is atleast e�e

��, if the common transmission radius rn satisfies

�r2n � lnnþ ð2k� 3Þ ln lnn� 2 lnðk� 1Þ!þ 2�, for k > 0 and

sufficiently large n, where � is any real number. Under thehomogeneous network assumption, they also proposed alocalized topology control algorithm that preservesk-connectivity. The proposed structure, YAOp;k, is basedon the Yao structure, and is constructed by having everynode u choose k closest neighbors in each of the p � 6 equal

cones around u. YAOp;k is proved to preserve k-connectivityand is a length spanner. It is not clear whether or not, andhow, the proposed algorithm can be extended to accom-modate heterogeneous networks in which different nodesmay have different maximal transmission power.

3.2 Work that Devises Algorithms to ConstructFault-Tolerant Topologies

Bahramgiri et al. [19] augmented the CBTC algorithm [6] toprovide fault tolerance. Specifically, let Dð�Þ, the directedsubgraph of G, be the output of CBTCð�Þ algorithm, and letGð�Þ be the result of applying Removal on Dð�Þ. It is provedthat Gð2�3kÞ preserves the k-connectivity of G. As the work isextended from the CBTC algorithm, it shares the sameassumption of a homogeneous network, which may notalways hold in practice [12].

Hajiaghayi et al. [16] presented three approximationalgorithms to find the minimum power k-connected sub-graph. Two global algorithms are based on existingapproaches. The first algorithm gives an Oðk�Þ-approximation, where � is the best approximation factorfor the k-UPVCS problem defined in the paper. The secondalgorithm improves the approximation factor to OðkÞ forgeneral graphs. The third algorithm, Distributed k-UPVCS,is a distributed algorithm that gives a kOðcÞ-approximation,where c is the exponent in the propagation model. For 2-connectivity, it first computes the minimum spanning tree(MST) of the input graph by using a distributed algorithm,and then adds an arbitrary path among the neighbors ofeach node in the returned tree. Since this distributedalgorithm is based on the distributed MST algorithm, it isnot fully localized, i.e., it relies on information that ismultiple hops away to construct the MST. This impliesmore maintenance overhead and delay when the topologyhas to be adjusted in response to node mobility or failure.Moreover, a closer investigation of the distributed algo-rithm reveals that the neighbors of a node on the minimumspanning tree may not be able to communicate with eachother due to the limited transmission power. As a result,the “arbitrary path connecting neighbors” in the algorithm(see [16]) may not exist in a network of low density. Acounter-example can be found in [1].

Ramanathan and Rosales-Hain [5] presented two centra-lized algorithms, CONNECT and BICONN-AUGMENT, tominimize the maximal power used per node while maintain-ing the (bi)connectivity of the network. Both are simplegreedy algorithms that iteratively merge different compo-nents until only one remains. Calinescu and Wan [20] alsostudied the Min-Power Symmetric Biconnectivity problem.They first proved that the Min-Power Symmetric 2-Node (2-Edge) Connectivity problem is NP-hard. Then they proposeda new algorithm, MST-Augmentation, for 2-connectivity,which has a constant approximation ratio.

Although FGSSk and FLSSk bear some similarity toCONNECT and BICONN-AUGMENT in the way thetopology is derived (i.e., different components are mergediteratively), they differ from the latter in that

1. FGSSk is more general, i.e., FGSSk preservesk-connectivity, while BICONN-AUGMENT onlypreserves 2-connectivity;

LI AND HOU: LOCALIZED FAULT-TOLERANT TOPOLOGY CONTROL IN WIRELESS AD HOC NETWORKS 309

2. the correctness of BICONN-AUGMENT is onlymentioned but not formally proved in [5], while aformal treatment of the correctness of FGSSk is givenin this paper;

3. CONNECT and BICONN-AUGMENT are bothcentralized algorithms that require collection anddistribution of global information, while FLSSk isfully decentralized and localized; and

4. CONNECT and BICONN-AUGMENT operate un-der the assumption of homogeneous networks,while FGSSk and FLSSk can be (as will be formallyproved in Section 6) applied to heterogeneousnetworks where the maximal transmission powerof each node may be different.

4 FGSSk: FAULT-TOLERANT GLOBAL SPANNING

SUBGRAPH

In this section, we present a centralized greedy algorithm,Fault-tolerant Global Spanning Subgraph (FGSSk), that buildsk-connected spanning subgraphs. FGSSk is a generalizedversion of Kruskal’s algorithm for k � 2, where Kruskal’salgorithm [22] is a well-known algorithm for constructingthe minimum spanning tree (1-connected spanning sub-graph) of a given graph. The FGSSk algorithm is describedin Fig. 1.

By using network flow techniques [23], a query onwhether two vertices are k-connected can be answered inOðm

ffiffiffi

npÞ time, where n is the number of vertices and m is

the number of edges in the graph. Therefore, the timecomplexity of FGSSk is Oðm2

ffiffiffi

npÞ.

Although FGSSk is a generalized version of Kruskal’salgorithm, the techniques for the proof of correctness arecompletely different. Before proving the correctness ofFGSSk, we first provide two lemmas, which are also crucialto the proofs in this section and Section 6.1.

Let the path from node u to node v in G be representedby a set p of vertices on the path, i.e., p ¼ fu;w1; w2;

. . . ; wl; vg. Let SuvðF Þ be a maximal set of pairwise-vertex-disjoint paths from u to v in a graph F . Thus, for

8p1; p2 2 SuvðF Þ, we have p1 \ p2 ¼ fu; vg. Let F � ðu1; u2Þbe the resulting graph by removing an edge ðu1; u2Þ from F .

Lemma 1. Let u1 and u2 be two vertices in a k-connectedundirected graph F . If u1 and u2 are k-connected after theremoval of edge ðu1; u2Þ, then F � ðu1; u2Þ is still k-connected.

Proof. Equivalently, we prove that F 0 ¼ F � ðu1; u2Þ isconnected after the removal of any other k� 1 verticesfromF 0. Consider any two vertices v1 and v2 inF 0. Withoutloss of generality, we assume fu1; u2g \ fv1; v2g ¼ ; (othercases can be proved using a similar approach). We nowprove that v1 is still connected to v2 after removing a set ofany k� 1 vertices W ¼ fw1; w2; . . . ; wk�1g, where wi 2V ðF 0Þ � fv1; v2g. This is obvious true if ðv1; v2Þ is an edge inF . Therefore, we only consider the case where there is noedge from v1 to v2 in F . Since F is k-connected,jSv1v2

ðF Þj � k.Let F 00 be the resulting graph after ðu1; u2Þ and W

(and related edges) are removed from F , and let s1 bethe number of paths in Sv1v2

ðF 0Þ that are broken due tothe removal of vertices in W , i.e., s1 ¼ jfp 2 Sv1v2

ðF 0Þ : 9w 2W;w 2 pgj. Since the paths in Sv1v2ðF 0Þ are

pairwise-vertex-disjoint, the removal of any one vertexin W breaks at most one path in the set. GivenjW j ¼ k� 1, we have s1 � k� 1.

If jSv1v2ðF 0Þj � k, then jSv1v2

ðF 00Þj � jSv1v2ðF 0Þj � s1 � 1,

i.e., v1 is still connected to v2 in F 00. Now, we consider thecase where jSv1v2

ðF 0Þj < k. This occurs only when theremoval of ðu1; u2Þ breaks one path p0 2 Sv1v2

ðF Þ. With-out loss of generality, let the order of vertices on the pathbe v1, u1, u2, v2. Since the removal of ðu1; u2Þ reduces thenumber of pairwise-vertex-disjoint paths between v1 andv2 by at most one, jSv1v2

ðF Þ � fp0gj � k� 1. Hence,jSv1v2

ðF 0Þj ¼ k� 1. Now, we consider two cases:

1. s1 < k� 1: jSv1v2ðF 00Þj � jSv1v2

ðF 0Þj � s1 � 1, i.e., v1

is still connected to v2 in F 00.2. s1 ¼ k� 1: hence, every vertex in W belongs to

some path inSv1v2ðF 0Þ. Since p0 is internally-disjoint

with all paths in Sv1v2ðF 0Þ, we have p0 \W ¼ ;.

Thus, v1 is connected to u1 and u2 is connected to v2

inF 00. Let s2 be the number of paths inSu1u2ðF 0Þ that

are broken due to the removal of vertices inW , i.e.,s2 ¼ jfp 2 Su1u2

ðF 0Þ : 9w 2W;w 2 pgj. Since jSu1u2

ðF 0Þj � k and s2 � k� 1, jSu1u2ðF 00Þj � 1, i.e., u1 is

still connected to u2 in F 00. Therefore, v1 is stillconnected to v2 in F 00.

We have proved that for any two vertices v1; v2 2 F 0,v1 is connected to v2 after the removal of any k� 1vertices from F 0 � fv1; v2g. Therefore, F 0 is k-connected.tu

Lemma 2. Let G and G0 be two undirected simple graphs suchthat V ðGÞ ¼ V ðG0Þ. If G is k-connected, and every edgeðu; vÞ 2 EðGÞ �EðG0Þ satisfies that u is k-connected to v inG� fðu0; v0Þ 2 EðGÞ : wðu0; v0Þ � wðu; vÞg, then G0 is alsok-connected.

Proof. Let E ¼ EðGÞ � EðG0Þ ¼ fðu1; v1Þ; ðu2; v2Þ; . . . ; ðum;vmÞg be a set of edges satisfying wðu1; v1Þ � wðu2; v2Þ �� wðum; vmÞ. We define a series of graphs that aresubgraphs ofG:G0 ¼ G, andGi ¼ Gi�1 � ðui; viÞ, i ¼ 1; 2;. . . ;m. Now, we prove by induction.


Fig. 1. FGSSk algorithm.

1. Base: G0 ¼ G is k-connected.2. Induction: If Gi�1 is k-connected, we prove that Gi

is k-connected, where i ¼ 1; 2; . . . ;m. Since

G� fðu0; v0Þ 2 EðGÞ : wðu0; v0Þ � wðui; viÞg� Gi�1 � ðui; viÞ; ui

is k-connected to vi in Gi�1 � ðui; viÞ. ApplyingLemma 1 to Gi�1, we can prove that Gi ¼Gi�1 � ðui; viÞ is still k-connected.

Now we have proved that Gm is k-connected. SinceEðGmÞ � EðG0Þ, G0 is also k-connected. tu

Theorem 1. FGSSk can preserve k-connectivity of G, i.e., Gk isk-connected if G is k-connected.

Proof. Since edges are inserted intoGk in an ascending order,whether u is k-connected to v at the moment before ðu; vÞ isinserted depends only on the edges of smaller weights.Therefore, every edge ðu; vÞ 2 E0 ¼ EðGÞ �EðGkÞ satis-fies that u is k-connected to v in G� fðu; vÞ 2 EðGÞ :wðu; vÞ � wðu0; v0Þg. We can prove that Gk preservesk-connectivity of G by applying Lemma 2 to Gk. tu

Let �ðF Þ be the largest radius of all nodes in F , i.e.,�ðF Þ ¼ maxu2V ðF ÞfRug. Now, we prove that FGSSk achievesthe min-max optimality, i.e., let SSkðGÞ be the set of allk-connected spanning subgraphs of G, then �ðGkÞ ¼ minf�ðF Þ : F 2 SSkðGÞg. This optimality is proved in [5] fork ¼ 2. Here, we extend the result to arbitrary k.

Theorem 2. The maximum transmission radius (or, equivalently,power) among all nodes in the network is minimized byFGSSk, i.e., �ðGkÞ ¼ minf�ðF Þ : F 2 SSkðGÞg.

Proof. Suppose G is k-connected. By Theorem 1, Gk is alsok-connected. Let ðu; vÞ be the last edge that is insertedinto Gk. We have wðu; vÞ ¼ maxðu0;v0Þ2EðGkÞ fwðu0; v0Þgand Ru ¼ �ðGkÞ. Le t G0k ¼ Gk � ðu; vÞ, we havejSuvðG0kÞj < k; otherwise, according to the algorithm inFig. 1, ðu; vÞ should not be included in Gk. Now,consider a graph H ¼ ðV ðHÞ; EðHÞÞ, where V ðHÞ ¼V ðGÞ and EðHÞ ¼ fðu0; v0Þ 2 EðGÞ : wðu0; v0Þ < wðu; vÞg.If we can prove that H is not k-connected, we will beable to conclude that any F 2 SSkðGÞ must have atleast one edge equal to or longer than ðu; vÞ, whichmeans �ðGkÞ ¼ minf�ðF Þ : F 2 SSkðGÞg.

Now, we prove by contradiction that H is notk-connected. Assume H is k-connected and, hence,jSuvðHÞj � k. We have EðHÞ � EðG0kÞ; o therwise ,jSuvðG0kÞj � jSuvðHÞj � k. Therefore, E0 ¼ EðHÞ � EðG0kÞ6¼ ;. Since edges are inserted intoG0k in an ascending order,8ðu1; v1Þ 2 E0 satisfies that u1 is k-connected to v1 inH�fðu0; v0Þ 2 EðHÞ : wðu0; v0Þ � wðu1; v1Þg. By Lemma 2,we can prove that G0k is k-connected. This meansjSuvðG0kÞj � k, which is a contradiction. tu

The min-max optimality of FGSSk is an importantfeature. The network lifetime is usually defined as the timeit takes for the first node to deplete its energy. Assume astatic network where each node has the same initial energy.If the traffic pattern is random and each node forwards

approximately equal amount of traffic, then the energyconsumption of each node is roughly proportional to itstransmission power. Therefore, the network lifetime is verylikely to be determined by the node that uses the maximumtranmission power among all nodes. By minimizing themaximum transmission power, FGSSk achieves the max-imum network lifetime.

FGSSk is a centralized algorithm that requires the knowl-edge of global information. Since there is, in general, nocentral authority in a wireless multihop network, it is verydifficult to collect and distribute global information, and bydoing so, the power-saving capability of topology controlmay be impaired. It is more desirable to devise distributedalgorithms where each node makes its own decision. To beless susceptible to topology changes (as a result of energydepletion, environmental interference, and limited mobility),it is also desirable that the algorithm depends only on theinformation collected locally. We will present a localizedalgorithm based on FGSSk in the next section.

5 FLSSk: FAULT-TOLERANT LOCAL SPANNING

SUBGRAPH

In this section, we present a localized, fault toleranttopology control algorithm, Fault-tolerant Local SpanningSubgraph (FLSSk). The topology is derived by having eachnode build its neighbor set and adjust its transmissionpower based on locally collected information.

FLSSk consists of three phases:

1. Information Collection: Each node u collects localinformation of neighbors such as positions and ids,and identifies the Visible Neighborhood NV

u by ex-changing Hello messages.

2. Topology Construction: Each node u defines, based onthe information in NV

u , the proper list of neighborsfor the final topology.

3. Construction of Topology with Only Bi-Directional Links(Optional): Each node u adjusts its list of neighborsto make sure that all the edges are bidirectional.

We will only explain the process of topology constructionhere, since the other two steps are similar to those in [11], [12].

Given the visible neighborhood NVu , each node u builds

its local spanning subgraph Su ¼ ðV ðSuÞ; EðSuÞÞ over NVu

using the algorithm described in Fig. 2. Note that Su will notbe k-connected if GV

u is not k-connected.

Definition 12 (Neighbor Relation in FLSSk). In Fault-

tolerant Local Spanning Subgraph (FLSSk), node v is a

neighbor of node u, denoted u �!FLSS v, if and only if

ðu; vÞ 2 EðSuÞ. That is, v is a neighbor of u if and only if v

is an immediate neighbor on u’s local spanning subgraph Su.

Definition 13 (Topology GFLSS). The topology,GFLSS , derived

under FLSSk is a directed graph GFLSS ¼ ðVGFLSS; EGFLSS

Þ,where VGFLSS

¼ V ðGÞ, EGFLSS¼ fðu; vÞ 2 EðGÞ : u �!FLSS vg.

The network topology under FLSSk is all the nodes in

V ðGÞ and their individually determined neighbor relations.

Note that the topology is not a simple superposition of all

local spanning subgraphs. In addition, the neighbor relation


defined above is not symmetric, i.e., u �!FLSS v does not

necessarily imply v �!FLSS u (an example for k ¼ 1 can

actually be found in [11, Fig. 1]). We can apply Addition or

Removal to enforce every edge to be bidirectional. The new

topologies GþFLSS and G�FLSS can be defined, respectively.

Definition 14 (Topology GþFLSS). The topology GþFLSS is an

undirected graph GþFLSS ¼ ðV ðGþFLSSÞ; EðGþFLSSÞÞ , where

V ðGþFLSSÞ ¼ V ðGFLSSÞ, a n d EþFLSS ¼ fðu; vÞ : ðu; vÞ 2EðGFLSSÞ or ðv; uÞ 2 EðGFLSSÞg.

Definition 15 (Topology G�FLSS). The topology G�FLSS is an

undirected graph G�FLSS ¼ ðV ðG�FLSSÞ; EðG�FLSSÞÞ , where

V ðG�FLSSÞ ¼ V ðGFLSSÞ, a n d E�FLSS ¼ fðu; vÞ : ðu; vÞ 2EðGFLSSÞ and ðv; uÞ 2 EðGFLSSÞg.

Theorem 3 (Connectivity of FLSSk). If G is k-connected, then

GFLSS , GþFLSS and G�FLSS are all k-connected.

Proof. We only need to prove that G�FLSS preserves

k-connectivity of G, for EðG�FLSSÞ � EðGFLSSÞ � E

ðGþFLSSÞ. Since G�FLSS is bidirectional, we can treat it as

an undirected graph. Let E ¼ EðGÞ � EðG�FLSSÞ. For any

edge e ¼ ðu; vÞ 2 E, at least one of ðu; vÞ and ðv; uÞwas not

in GFLSS , since e =2 EðG�FLSSÞ. Without loss of generality,

assume ðu; vÞwas not inGFLSS . Thus, in the process of local

topology construction of nodeu,uwas alreadyk-connected

to v before ðu; vÞwas inspected. Since edges are inserted in

the ascending order, whether u is k-connected to v at the

moment before ðu; vÞ is inspected depends only on the

edges of smaller weights. Therefore,u is k-connected to v in

G�fðu0; v0Þ 2 EðGÞ : wðu0; v0Þ > wðu; vÞg. Let G0 ¼G�FLSS ,

we can conclude that G�FLSS is k-connected by Lemma 2.tuDefinition 16 (Strictly Localized Algorithms). An algorithm

is strictly localized if its operation on any node u is based only

on the information that is originated from the nodes in NVu .

Let LSSkðGÞ be the set of all k-connected spanning

subgraphs of G that are constructed by strictly localized

algorithms. Now, we prove that FLSS achieves the min-max

optimality among all strictly localized algorithms, i.e.,�ðGFLSSÞ ¼ minf�ðF Þ : F 2 LSSkðGÞg.Theorem 4. Among all strictly localized algorithms, FLSSk

minimizes the maximum transmission radius (or power) of nodes

in the network, i.e., �ðGFLSSÞ ¼ minf�ðF Þ : F 2 LSSkðGÞg.Proof. Suppose G is k-connected. Let ðu; vÞ be the last edge

added in GFLSS . We have wðu; vÞ ¼ maxðu0;v0Þ2EðGFLSSÞfwðu0; v0Þg and Ru ¼ �ðGFLSSÞ. Let G0 be the inducedsubgraph of GFLSS where V ðG0Þ ¼ NV

u , and let G00 ¼G0 � fðu; vÞg. We have jSuvðG00Þj < k; otherwise, ðu; vÞshould not be included in G0. Also, define H0 ¼ ðV ðH0Þ;EðH0ÞÞ, where V ðH0Þ ¼ V ðGV

u Þ and EðH0Þ ¼ fðu0; v0Þ 2EðGV

u Þ : wðu0; v0Þ < wðu; vÞg.To prove that H0 is not k-connected, we replace G, Gk,

G0k, and H with GVu , G0, G00, and H0, respectively, and

follow the corresponding proof in Theorem 2. Afterproving H0 is not k-connected, we consider the followingcases:

1. u is k-connected to v in GVu : since H0 is not

k-connected, any F 2 LSSkðGÞ should have had atleast one edge equal to or longer than ðu; vÞ;

2. u is not k-connected to v in GVu : to preserve the

connectedness as much as possible, any F 2LSSkðGÞ should have included ðu; vÞ.

In both cases, �ðF Þ � �ðGVu Þ ¼ �ðGFLSSÞ, i.e., �ðGFLSSÞ

¼ minf�ðF Þ : F 2 LSSkðGÞg. tu

Let RFLSSu , RCBTC

u , and RYAOu be the radius of any node u

under FLSSk, CBTCð2�3kÞ [19], and YAOp;k [21], respectively.We derive the relationship of RFLSS

u , RCBTCu , and RYAO

u .

Lemma 3. For three nodes u; v1; v2 2 G, if ffv1uv2 � �=3 andwðu; v2Þ < wðu; v1Þ, then wðv1; v2Þ < wðu; v1Þ.

Proof. If wðu; v2Þ < wðu; v1Þ, then dðu; v2Þ � dðu; v1Þ. Wehave dðv1; v2Þ � dðu; v1Þ, since ffv1uv2 � �=3. Considerthe following two cases (Fig. 3):

1. dðv1; v2Þ < dðu; v1Þ. It is obvious that wðv1; v2Þ <wðu; v1Þ.

2. dðv1; v2Þ ¼ dðu; v1Þ. This only occurs when dðv1; v2Þ¼ dðu; v1Þ ¼ dðu; v2Þ. Sincewðu; v2Þ< wðu; v1Þ, u, v1,and v2 satisfy one of the three scenarios (out of sixpossible scenarios): idðuÞ < idðv2Þ < idðv1Þ, idðv2Þ


Fig. 3. Illustrations for Theorems 5 and 6. (a) Lemma 3 and(b) Theorems 5 and 6.

Fig. 2. FLSSk algorithm.

< idðuÞ < idðv1Þ, and idðv2Þ < idðv1Þ < idðuÞ. Wecan check for all three scenarios, wðu; v2Þ <wðu; v1Þ. tu

Theorem 5. RFLSSu � RCBTC

u for any node u 2 G.

Proof. We prove by contradiction. Let vFLSS be the farthestneighbor of u in GFLSS . Suppose RFLSS

u ¼ dðu; vFLSSÞ >RCBTCu . Recall that there are two stages in CBTC. In the first

stage, u increases its power until the maximum anglebetween two consecutive neighbors is at most 2�

3k . If thecone coverage cannot be satisfied when u is alreadytransmitting with the maximal power, u will attempt toreduce its power subject to maintaining the cone coveragein the second stage.

Consider the number of neighbors of u in the shadedarea S in Fig. 3b (which is an open, minor sector with aradius of RCBTC

u and an angle of 2�3 at the center). If the

CBTC algorithm stops in the first stage, u has at least2�3 =

2�3k ¼ k nodes in S in order to fulfill the coverage

requirement. If the algorithm proceeds to the secondstage, u has at least k neighbors in S in order to shrinkback and maintain k-connectivity at the same time [19]. Inboth cases, u has at least k neighbors in S. Let thosenodes be vi; i ¼ 1; 2; . . . ; L, where L � k.

Now, consider the local topology construction processof u in FLSS. Since dðu; viÞ < dðu; vFLSSÞ � rmax anddðvi; vFLSSÞ<dðu; vFLSSÞ�rmax (by Lemma 3), i ¼ 1; 2;. . . ; L, there are at least k pairwise-vertex-disjoint pathsfrom u to vFLSS inG (i.e., ðu; vi; vFLSS; i ¼ 1; 2; . . . ; L � k)).As we have proved that FLSS does not affect thek-connectivity between u and vFLSS , u and vFLSS arealready k-connected when edge ðu; vFLSSÞ is examined inFLSS. Therefore, vFLSS cannot be a neighbor of u inGFLSS ,which contradicts with our assumption. tu

Theorem 6. RFLSSu � RYAO

u for any node u 2 G.

Proof. We also prove by contradiction. Let vFLSS be thefarthest neighbor of u in GFLSS . Suppose RFLSS

u ¼ dðu; vFLSSÞ > RYAO

u . Recall that in YAOp;k, each node uincrease it power until it has k closest neighbors in each ofp � 6 equal cones around u. Since RYAO

u < dðu; vFLSSÞ, uhas at least kneighbors that are closer to itself than vFLSS inthe cone where vFLSS resides. Given that p � 6, there are atleast k neighbors of u in S (Fig. 3b). Using an argumentsimilar to that in Theorem 5, we can conclude that vFLSS

cannot be a neighbor of u inGFLSS , which contradicts withour assumption. tu

Theorems 5 and 6 show that FLSSk outperforms bothCBTCð2�3kÞ and YAOp;k in terms of the transmission radius,which is also corroborated by the simulation study inSection 7. While both CBTCð2�3kÞ and YAOp;k impose certaincoverage constraints in each individual cone, FLSSk onlyimposes requirements on the entire visible neighborhood.Take a node u in CBTCð2�3kÞ, for example, if we assume arelative high density of nodes, u has to have at least2�=ð2�3kÞ ¼ 3k neighbors, w.h.p., to ensure that the maximumangle between any two consecutive neighbors is at most 2�

3k .This bound is actually not very tight since most likely it willtake more neighbors to fulfill the requirement of cone

coverage. Similarly, the degree of a node for YAOp;k is atleast pk (k nodes in each of the p cones), w.h.p.

6 PRACTICAL CONSIDERATIONS

Although the assumptions stated in Section 2 are widelyused in existing topology control algorithms, some of themare made to facilitate algorithm design and analysis, andmay not be practical. In this section, we discuss how to relaxthese assumptions for FGSSk and FLSSk so as to improvetheir practicality in real applications.

6.1 Relaxing the Homogeneous NetworkAssumption

As mentioned in [12], the assumption of homogeneousnodes does not always hold in practice, since even devicesof the same type may have slightly different maximaltransmission power, let alone the fact that devices ofdifferent types possess dramatically different capabilities.Fortunately, FGSSk/FLSSk can be readily applied inheterogeneous networks, with only one modification: Adirected graph G ¼ ðV ðGÞ; EðGÞÞ in the 2D plane should beused in Section 2 to represent the original topology of aheterogeneous network.

Now, we prove that FGSSk preserves k-connectivity andis min-max optimal even in heterogeneous networks. Thefollowing results correspond to Lemma 1, Lemma 2,Theorem 1, and Theorem 2, respectively. The proof isliterally the same as that in Section 4, except that now weconsider directed graphs consisting of directed edges. Thisresemblance is by no means a coincidence, since we actuallyconsidered more general cases when we proved all thetheorems and lemmas in Section 4.

Lemma 1*. Let u1 and u2 be two vertices in a k-connecteddirected graph F . If u1 is k-connected to u2 after the removalof edge ðu1; u2Þ, then F � ðu1; u2Þ is still k-connected.

Lemma 2*. Let G and G0 be two directed simple graphs such thatV ðGÞ ¼ V ðG0Þ. If G is k-connected, and every edge ðu; vÞ 2EðGÞ � EðG0Þ satisfies that u is k-connected to v inG� fðu0; v0Þ 2 EðGÞ : wðu0; v0Þ � wðu; vÞg, then G0 is alsok-connected.

Theorem 1*. FGSSk can preserve k-connectivity in hetero-geneous networks, i.e., Gk is k-connected if G is k-connected,where G is a directed graph.

Theorem 2*. The maximum transmission radius (or power)among all nodes is minimized by FGSSk, i.e., �ðGkÞ ¼minf�ðF Þ : F 2 SSkðGÞg, where G is a directed graph andSSkðGÞ is the set of all k-connected spanning subgraphs of G.

The following theorem proves that FLSSk preservesk-connectivity. Note that G�FLSS can no longer preservek-connectivity for heterogeneous networks.

Theorem 3* (Connectivity of FLSSk). If G is k-connected,then GFLSS and GþFLSS are both k-connected.

Proof. We only need to prove that GFLSS preservesk-connectivity of G, since EðGFLSSÞ � EðGþFLSSÞ. LetE ¼ EðGÞ �EðGFLSSÞ. For any edge e ¼ ðu; vÞ 2 E, it isnot in EðGFLSSÞ because in the process of local topologyconstruction of node u, u was already k-connected to v


before ðu; vÞ was inserted. Since edges are inserted in anascending order, whether u is k-connected to v at themoment before ðu; vÞ is inserted depends only on the edgesof smaller weights. Therefore, u is k-connected to v inG�fðu0; v0Þ2EðGÞ : wðu0; v0Þ > wðu; vÞg. Let G0 ¼ GFLSS ,we conclude that GFLSS is k-connected by Lemma 2*. tu

The min-max optimality of FLSSk can be proved in astraightforward manner:

Theorem 4*. Among all strictly localized algorithms, FLSSkminimizes the maximum transmission radius (or power) of

nodes in the network, i.e., �ðGFLSSÞ ¼ minf�ðF Þ : F 2LSSkðGÞg, where G is a directed graph.

6.2 Relaxing the Obstacle-Free CommunicationChannel

We assume in Section 2 an obstacle-free communicationchannel, which is not always applicable, especially forindoor communications. In this section, we argue that thisassumption can be readily dismissed.

We have previously assumed that the original networktopology, G, is a general directed or undirected graph. Theinformation needed by FGSSk and FLSSk is the edges thatexist inG. An edge between nodesu and v is not formed in thenetwork, either because u and v are not within the transmis-sion range of each other, or because there exist obstacles inbetween. No matter what is the reason, nonexistent edges arenot considered in the construction of topologies, in FGSSkand FLSSk. As long as the original topology (which has takeninto consideration of the effect of obstacles) is k-connected,FGSSk and FLSSk can be applied to provide a min-maxoptimal solution to preserve the k-connectivity. Therefore, theassumption of an obstacle-free wireless channel can berelaxed without any modification to FLSSk.

6.3 Relaxing the Requirement on PositionInformation

It is assumed in Section 2 that each node is equipped withthe capability to obtain its own location information. In thissection, we discuss how to relax this assumption.

As mentioned in Section 6.2, what is required by FGSSkand FLSSk is the information of all the existing edges in thenetwork. In order to obtain such information, each node uincludes its node id and position in the Hello message. Withall the messages that reach a node v, node v can then inferthe local topology in the visible neighborhood NV

v with theposition information.

Note that our algorithms can still operate if the positioninformation is not available, as only the knowledge of all theexisting edges,EðGV Þ, is required.EðGV Þ can be constructedlocally as follows: First, each node periodically broadcasts,using its maximal transmission power, a very shortHi message which includes only its node id and its maximaltransmission power. Upon receiving such a message from aneighbor node v, each node u estimates the length of the edgeðu; vÞ based on the attenuation incurred in the transmission.Let the set of edges incident at u be denoted asETu ¼ fðu; vÞ : v 2 NV

u gÞ. After nodeu collects the informationon ET

u , it can then broadcast this information in an Edgemessage. Each node will be able to infer EðGV

u Þ based on theEdge messages received from all of its neighbors.

Although this solution may incur more communicationand computation overhead, it eliminates the need for theposition information and thus is better suited for wirelesssensor networks where the cost of each sensor should bekept as low as possible.

7 PERFORMANCE EVALUATION

In this section, we evaluate the performance of FLSSkagainst CBTCð2�3kÞ, YAOp;k, and Hajiaghayi’s algorithms [16],with respect to several metrics via simulations. We set p ¼ 6in YAOp;k in order to minimize the average power [21]. Forthe sake of fair comparison, we have to use severalassumptions that are common to all algorithms, e.g., theUnit Disk Graph (UDG) model. The performance of thecentralized algorithm FGSSk is also shown as a baseline. Aswill be shown in the following discussions, the performanceof FLSSk is only slightly worse than that of FGSSk.

7.1 Comparison between CBTCð2�3kÞ, YAOp;k, andFLSSk

In the first set of simulations, we study the performancewith respect to node degree, maximum radius, and energysaving. Nodes are uniformly distributed in a 1; 000�1; 000m2 region. The transmission range of all nodes is261:195m, which corresponds to a transmission power of0.28183815 watt under the Free Space propagation model.We vary the number of nodes in the region from 70 to 300.Each data point is the average of 50 simulation runs.

Node Degree: We compare the average physical nodedegree of the topologies derived under different algorithms.Recall that the physical degree is defined as the number ofnodes within the transmission radius of a node. Nodedegree is a good indication of the level of possible MACinterference (and, hence, the extent of spatial reuse).

Fig. 4a shows the average node degree of the topologiesderived under CBTCð2�3kÞ, YAO6;k, FLSSk, and FGSSk fork ¼ 3. The average degree under NONE (with no topologycontrol) increases almost linearly with the number of nodes.The average degree under CBTCð2�3kÞ and YAO6;k alsoincreases as the number of nodes increases. In contrast,the average degree under FGSSk and FLSSk actuallyslightly decreases. The average degrees of both CBTCð2�3kÞand YAO6;k are much higher than that of FLSSk since inFLSSk, nodes always has a smaller transmission radius (asproved in Section 5).

Fig. 4b shows the average maximum node degree in thetopologies derived by CBTCð2�3kÞ, YAO6;k, FLSSk, and FGSSkfor k ¼ 3. The average maximum node degree under FGSSkand FLSSk is significantly smaller than that underNONE=CBTCð2�3kÞ=YAO6;k. All results show that FLSS2 canachieve better spatial reuse, and the performance improve-ment becomes even more prominent when the networkdensity becomes higher.

Maximum Radius: The average maximum radius for thetopologies under CBTCð2�3kÞ, YAO6;k, FLSSk and FGSSk areshown in Fig. 4c for k ¼ 3. The average maximum radius ofCBTCð2�3kÞ or YAO6;k comes very close to that of NONE,which implies that CBTCð2�3kÞ and YAO6;k cannot reallyprolong the network lifetime (given the lifetime defined inSection 4). In contrast, the average maximum radius of


FLSSk is significantly smaller. Moreover, its performance isvery close to that of the centralized algorithm FGSSk.

Network Capacity and Energy Efficiency: In the secondset of simulations, we compare CBTCð2�3kÞ, YAO6;k, FLSSk,and FGSSk with respect to network capacity and energyefficiency using the ns-2 simulator [24]. In this set ofsimulations, n nodes are randomly distributed in a 150�20m2 region, with half of them being sources and the otherhalf being destinations. We use a rectangular region for thesame reasons described in [11].

In the simulation, the propagation model is the Two-Ray ground model, the MAC protocol is IEEE 802.11, therouting protocol is AODV, and the traffic sources areCBR and TCP traffic with bulk FTP sources. The starttime of each connection is chosen randomly from ½0s; 10s.Each simulation run lasts for 100 seconds.

For CBR traffic, Fig. 5 shows the total amount of datadelivered (bytes), the total energy consumption (Joules),and the energy efficiency (bytes/Joule), for k ¼ 3. The

difference between the results obtained under CBTC/YAO

and those under NONE is not significant. Meanwhile,

FLSSk not only significantly improves the network capacity,

but also is the most power-efficient. The results under TCP/

FTP traffic also exhibit the similar trends.

7.2 Comparison between k-UPVCS andFGSSk=FLSSk

In the third set of simulations, we compare FGSSk and FLSSkwith both the distributed and the centralized versions of

k-UPVCS [16] with regard to the average expended energy

ratio (EER), where EER is defined in [16] as

EER ¼ Eave

Emax� 100;

Eave is the average transmission power over all the nodes in

the network, and Emax is the maximal transmission power

that can reach the transmission range of 261:195m. Here, we


Fig. 4. Comparison of CBTCð2�3kÞ, YAO6;k, FLSSk and FGSSk with regard to geometric metrics (k ¼ 3). (a) Average node degree, (b) averagemaximum degree, and (c) average maximum radius.

use the free-space propagation model to calculate thetransmission power. The simulation is conducted in asimilar setting to that in [16]. As we are unable to accuratelycontrol the density of the original graph (with the maximaltransmission power), we compare the algorithms under thetopology of roughly the same average degree. Fig. 6 givesthe comparison results for both k ¼ 2 and k ¼ 3. FLSSperforms better than the distributed version of k-UPVCS inalmost every setting, though FGSS performs worse than theglobal version of k-UPVCS. The latter is probably due to thefact that FGSS is simply a greedy algorithm.

7.3 Trade-Off between Topology Robustness andPerformance

In the fourth set of simulations, we compare FLSS1 (i.e.,LMST [11]), FLSS2 and FLSS3. As shown in Fig. 7a, Fig. 7b,and Fig. 7c, as k increases, FLSSk renders topologies thathave larger average degrees, longer average radii, andlonger average maximum radii, and consume more power.

However, the topologies are also more robust and areresilient to k� 1 failures. This shows the tradeoff betweenthe robustness of the topology and the other performancemetrics (e.g., power consumption, network lifetime, spatialreuse, and MAC level interference).

We also compare FLSS1, FLSS2, and FLSS3 with respectto network capacity and energy efficiency. The simulationsettings are the same as those in the second set ofsimulations. Here, we only show the results with the useof TCP/FTP traffic (the results with CBR traffic exhibit thesimilar trends). The total amount of data delivered (Fig. 7d),the total energy consumption (Fig. 7e), and the energyefficiency (Fig. 7f) are shown. It can be observed that withthe increase in the level of network connectivity (in theorder of FLSS1, FLSS2, FLSS3, NONE), the total throughputdecreases, the total energy consumption increases, and theenergy efficiency decreases. This result again demonstratesthe trade-off between the robustness (or routing redun-dancy) and the network capacity/energy efficiency.


Fig. 5. Comparison of CBTCð2�3kÞ, YAO6;k, and FLSSk with regard to. energy efficiency under CBR traffic (k ¼ 3). (a) Total data delivery, (b) totalenergy consumption, and (c) energy efficiency.

7.4 Robustness with Regard to Position EstimationErrors

In Section 2, we assumed that each node can estimate its

own position by using various localization techniques. In

Section 6.3, we relaxed this assumption by having each

node estimate the length of edges it is incident to. Both

methods will introduce errors on the position estimation.

As a result, FLSS may not be able to preserve k-connectivity.

In the next set of simulations, we investigate to what extent

FLSS is robust with respect to the position estimation error.Suppose the real position of a node u is ðux; uyÞ, and the

estimated position of u is modeled to be randomly

distributed inside the disk centered at ðux; uyÞ. Given the

error rate Er, the radius of the disk is Er � rmax (where rmax is


Fig. 6. Comparison of FGSS/FLSS, and the global/distributed versions of k-UPVCS with regard to. EER. (a) k ¼ 2 and (b) k ¼ 3.

Fig. 7. Comparison of FLSS1, FLSS2, and FLSS3. (a) Average radius, (b) average degree, (c) EER, (d) total data delivery (bytes), (e) total energyconsumption (Joules), and (f) energy efficiency (bytes/J).

the maximal transmission range). For a given targetedconnectivity, FLSS is used to build the network topologybased on the estimated positions. To deal with the errorsincurred in the estimation, we increase the transmissionradius of each node, in the hope that the networkconnectivity can be repaired. The increase is proportionalto the transmission radius (subject to the maximal tranmis-sion range rmax). Other schemes [25] have also beenproposed to deal with the uncertainty of the node positions.

For this set of simulation, nodes are randomly distributed

in a square region of 2rmax by 2rmax, where rmax ¼ 261:195m.

The number of nodesN varies from 30 to 60, and the targeted

connectivity K varies from 3 to 6. The error rate varies from

2 percent to 20 percent. Thus, the maximum error varies from

2 percent �rmax ( 5:2m) to 20 percent �rmax ( 52m). The

increase in the transmission radius varies from 5 percent to

25 percent. Each data point is the average of 200 simulation

runs.

Fig. 8 gives the average connectivity under different error

rates on positions and different increases in the transmission

radius. We can observe that the estimation error does

influence the preservation of connectivity in FLSS. An modest

increase in the transmission radius, however, will boost the

connectivity to the desired level. We further look into the

standard deviation of the average connectivity for N ¼ 30

and N ¼ 60 in Fig. 9. Generally, if each node increases its

transmission radius by 15 percent, the targeted connectivity

can be achieved despite the position estimation errors, which

shows that FLSS is quite robust.

8 CONCLUSIONS

We consider fault-tolerant topology control algorithms in

wireless ad hoc networks. We propose a centralized

algorithm, FGSSk and a fully localized algorithm, FLSSk,

and show that they can preserve k-connectivity and are min-

max optimal. We further examine several widely used

assumptions in topology control, and discuss how to relax

these assumptions for FGSSk and FLSSk. Simulation results

indicate that FLSSk outperforms most distributed/localized

fault-tolerance centric topology control algorithms with

respect to node degree, maximum radius, power consump-

tion, energy efficiency, and network capacity. FLSSk is also

shown to be robust with respect to position estimation

errors.

Although FLSSk outperforms other localized algorithms

in random networks in terms of power consumption, we

were unable to give any performance bound on the power

consumption. The dominating reason for the lack of a

performance bound is that FLSSk is greedy and highly

localized. Although it performs very well in most cases, we

conjecture that the information available within the trans-

mission range of a node is not sufficient to upper-bound the

performance under some rare, extreme cases. As part of our

future research, we will extend FLSSk to utilize more

information in the network so as to provide some

performance bound.

Many topology control algorithms assume a UDG

model, i.e., the antenna pattern of a wireless device is a

perfect disk. This is also the underlying assumption for

algorithms that use explicit channel propagation models.

This is because the same models are applied to all directions

and, hence, the antenna patterns have to be isotropic and

the transmission area a perfect disk. The antenna model

does not affect the centralized algorithm, FGSSk, but will

influence the manner in which the information on NVu can

be collected in FLSSk. Given an arbitrary antenna pattern,

the information dissemination technique in Section 6.3 can

still be applied, but the estimation of edge lengths becomes

quite difficult. The reason is that the antenna pattern is not

necessarily isotropic, and power attenuation may vary in

different directions. How to tackle this issue is also one of

our research agendas.

ACKNOWLEDGMENTS

The authors would like to thank the anonymous reviewers

and the editor for their insightful comments. A preliminary

version was presented in the Proceedings of ACM MOBICOM

2004, Philadelphia, Pennsylvania, September 2004 [1].


Fig. 8. Robustness of FLSSk with regard to. position estimation errors. (a) N ¼ 30, K ¼ 3 and (b) N ¼ 60, K ¼ 6.

REFERENCES

[1] N. Li and J.C. Hou, “FLSS: A Fault-Tolerant Topology ControlAlgorithm for Wireless Networks,” Proc. 10th ACM Int’l Conf.Mobile Computing and Networking (MOBICOM), pp. 275-286, Sept.2004.

[2] C.E. Jones, K.M. Sivalingam, P. Agrawal, and J.C. Chen, “ASurvey of Energy Efficient Network Protocols for WirelessNetworks,” Wireless Networks, vol. 7, no. 4, pp. 343-358, Aug. 2001.

[3] P. Gupta and P.R. Kumar, “The Capacity of Wireless Networks,”IEEE Trans. Information Theory, vol. 46, no. 2, pp. 388-404, Mar. 2000.

[4] V. Rodoplu and T.H. Meng, “Minimum Energy Mobile WirelessNetworks,” IEEE J. Selected Areas of Comm., vol. 17, no. 8, pp. 1333-1344, Aug. 1999.

[5] R. Ramanathan and R. Rosales-Hain, “Topology Control ofMultihop Wireless Networks Using Transmit Power Adjustment,”Proc. IEEE INFOCOM 2000, vol. 2, pp. 404-413, Mar. 2000.

[6] L. Li, J.Y. Halpern, P. Bahl, Y.-M. Wang, and R. Wattenhofer,“Analysis of a Cone-Based Distributed Topology Control Algo-rithm for Wireless Multi-Hop Networks,” Proc. ACM Symp.Principles of Distributed Computing (PODC), pp. 264-273, Aug. 2001.

[7] S. Narayanaswamy, V. Kawadia, R.S. Sreenivas, and P.R. Kumar,“Power Control in Ad-Hoc Networks: Theory, Architecture,Algorithm and Implementation of the COMPOW Protocol,” Proc.Conf. European Wireless 2002, Next Generation Wireless Networks:Technologies, Protocols, Services and Applications, pp. 156-162, Feb.2002.

[8] V. Kawadia and P. Kumar, “Power Control and Clustering in AdHoc Networks,” Proc. IEEE INFOCOM 2003, vol. 1, pp. 459-469,Apr. 2003.

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[10] X.-Y. Li, G. Calinescu, and P.-J. Wan, “Distributed Construction ofPlanar Spanner and Routing for Ad Hoc Networks,” Proc. IEEEINFOCOM 2002, pp. 1268-1277, June 2002.

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[12] N. Li and J.C. Hou, “Localized Topology Control Algorithms forHeterogeneous Wireless Networks,” IEEE/ACM Trans. Network-ing, 2005.

[13] C. Savarese, J.M. Rabaey, and J. Beutel, “Location in Distributed Ad-Hoc Wireless Sensor Networks,” Proc. IEEE Int’l Conf. Acoustics,Speech, and Signal Processing, vol. 4, pp. 2037-2040, May 2001.

[14] T. He, C. Huang, B.M. Blum, J.A. Stankovic, and T. Abdelzaher,“Range-Free Localization Schemes for Large Scale Sensor Net-works,” Proc. Ninth ACM Int’l Conf. Mobile Computing andNetworking (MOBICOM), pp. 81-95, Sept. 2003.

[15] S. Khuller, “Approximation Algorithms for Finding HighlyConnected Subgraphs,” Approximation Algorithms for NP-HardProblems, D.S. Hochbaum, ed., Boston: PWS Publishing Co., 1996.

[16] M. Hajiaghayi, N. Immorlica, and V.S. Mirrokni, “PowerOptimization in Fault-Tolerant Topology Control Algorithms forWireless Multi-Hop Networks,” Proc. Ninth ACM Int’l Conf. MobileComputing and Networking (MOBICOM), pp. 300-312, Sept. 2003.

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Fig. 9. Average connectivity and standard deviation. (a) N ¼ 30, 2 percent error, (b) N ¼ 30, 10 percent error, (c) N ¼ 30, 20 percent error,(d) N ¼ 60, 2 percent error, (e) N ¼ 60, 10 percent error, and (f) N ¼ 60, 20 percent error.

[21] X.-Y. Li, P.-J. Wan, Y. Wang, and C.-W. Yi, “Fault TolerantDeployment and Topology Control in Wireless Networks,” Proc.Fourth ACM Symp. Mobile Ad Hoc Networking and Computing(MOBIHOC), pp. 117-128, June 2003.

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Ning Li received the BEng and MEng degreesfrom the Department of Automation, TsinghuaUniversity, Beijing, PR China, in 1998 and 1999,respectively, the MS degree in computer en-gineering from The Ohio State University,Columbus, Ohio, in 2001, and the PhD degreein computer science from University of Illinois atUrbana-Champaign, Urbana, Illinois. His re-search interests include design and analysis ofwireless mobile ad hoc networks and sensor

networks, large-scale network simulation and emulation, and distributedand mobile computing. He is a student member of the IEEE.

Jennifer C. Hou received the PhD degree fromThe University of Michigan, Ann Arbor, Michi-gan, in 1993. She is currently a professor in theDepartment of Computer Science at the Uni-versity of Illinois at Urbana-Champaign, Urbana,Illinois. Dr. Hou has been supervising severalfederally and industry funded projects in theareas of network modeling and simualtion,network measurement and diagnostics, enablingcommunication software for assisted living, and

both the theoretical and protocol design aspects of wireless sensornetworks. She has published (with her former advisor, students, andcolleagues) more than 125 papers and book chapters in archivedjournals and peer-reviewed conferences, and released a truly exten-sible, reusable, component-based, compositional network simulationand emulation package, J-Sim. She has also served on the programcommittee of several major networking, real-time, and distributedsystems conferences/symposiums, such as IEEE INFOCOM, IEEEICNP, IEEE ICDCS, IEEE RTSS, IEEE ICC, IEEE Globecome, ACMMobicom, and ACM Sigmetrics. She is the technical program cochair of27th IEEE INFOCOM 2008, First International Wireless InternetConference 2005, ACM Third Information Processing in SensorNetworks (IPSN 2004), and IEEE Real-time Technology and ApplicationSymposium (RTAS 2000). She is severing on the editorial board of IEEETransactions on Wireless Communications, IEEE Transactions onParallel and Distributed Systems, IEEE Wireless CommunicationMagazine, ACM/Kluwer Wireless Networks, Kluwer Computer Net-works, and ACM Transactions on Sensor Networks. Dr. Hou was arecipient of an ACM Recognition of Service Award in 2004, a LumleyResearch Award from The Ohio State University in 2001, a US NationalScience Foundation CAREER award from 1996-2000, and a Women inScience Intiative Award from The University of Wisconsin-Madison from1993-1995. Dr. Hou is a senior member of the IEEE.

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