IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 9 ...dtipper/3350/April_Paper8.pdfFault-Tolerant Relay Node Placement in Heterogeneous Wireless Sensor Networks Xiaofeng Han, Xiang Cao,

Fault-Tolerant Relay Node Placement inHeterogeneous Wireless Sensor Networks

Xiaofeng Han, Xiang Cao, Errol L. Lloyd, and Chien-Chung Shen, Member, IEEE

Abstract—Existing work on placing additional relay nodes in wireless sensor networks to improve network connectivity typically

assumes homogeneous wireless sensor nodes with an identical transmission radius. In contrast, this paper addresses the problem of

deploying relay nodes to provide fault tolerance with higher network connectivity in heterogeneous wireless sensor networks, where

sensor nodes possess different transmission radii. Depending on the level of desired fault tolerance, such problems can be categorized

as: 1) full fault-tolerant relay node placement, which aims to deploy a minimum number of relay nodes to establish kðk � 1Þ vertex-

disjoint paths between every pair of sensor and/or relay nodes and 2) partial fault-tolerant relay node placement, which aims to deploy

a minimum number of relay nodes to establish kðk � 1Þ vertex-disjoint paths only between every pair of sensor nodes. Due to the

different transmission radii of sensor nodes, these problems are further complicated by the existence of two different kinds of

communication paths in heterogeneous wireless sensor networks, namely, two-way paths, along which wireless communications exist

in both directions; and one-way paths, along which wireless communications exist in only one direction. Assuming that sensor nodes

have different transmission radii, while relay nodes use the same transmission radius, this paper comprehensively analyzes the range

of problems introduced by the different levels of fault tolerance (full or partial) coupled with the different types of path (one-way or two-

way). Since each of these problems is NP-hard, we develop Oð�k2Þ-approximation algorithms for both one-way and two-way partial

fault-tolerant relay node placement, as well as Oð�k3Þ-approximation algorithms for both one-way and two-way full fault-tolerant relay

node placement (� is the best performance ratio of existing approximation algorithms for finding a minimum k-vertex connected

spanning graph). To facilitate the applications in higher dimensions, we also extend these algorithms and derive their performance

ratios in d-dimensional heterogeneous wireless sensor networks ðd � 3Þ. Finally, heuristic implementations of these algorithms are

evaluated via QualNet simulations.

Index Terms—Heterogeneous wireless sensor networks, relay node placement, approximation algorithms.

Ç

1 INTRODUCTION

HETEROGENEOUS wireless sensor networks (H-WSNs) arecomposed of a large number of wireless devices

equipped with different communication and computingcapabilities. In comparison with homogeneous wirelesssensor networks, where all the devices possess the samecommunication and computing capability, H-WSNs accom-modate a variety of operating environments, and hence, areuseful in many civil or military applications. One classicalexample of H-WSNs is the underwater sensor networks [4],which usually consist of multiple types of sensors deployedat different depths within the water, as well as diverseonshore, surface, and underwater data sinks. Beingequipped with different functionalities, these heteroge-neous wireless devices collaboratively form a scalablemultipurpose sensor network and facilitate many sensingtasks, such as oceanographic data collection, pollutionmonitoring, offshore exploration, and tactical surveillance.

However, in real applications, unpredictable events suchas battery depletion and environmental impairment maycause these wireless devices to fail, partitioning the net-work, and disrupting normal network functions. Therefore,

fault tolerance becomes a critical issue for the successfuldeployment of wireless sensor networks. One approach toachieving fault tolerance in wireless sensor networks is todeploy a small number of additional relay nodes to providekðk � 2Þ vertex-disjoint paths between every pair of func-tioning devices (including both sensors and data sinks,together termed as target nodes in this paper) so that thenetwork can survive the failure of fewer than k nodes. Thisproblem is well known as relay node placement in theliterature [1], [5], [13], [6], [7], [8], [9].

1.1 Previous Work on Homogeneous WirelessSensor Networks

Most of the existing work considers relay node placement inthe context of homogeneous wireless sensor networkswhere both target nodes and relay nodes use an identicaltransmission radius (see [3] for a comprehensive survey).Before reviewing these results, we first introduce a keydefinition of approximation algorithm that will be usedthroughout this paper.

Definition 1.1 (Approximation Algorithm [10]). A poly-nomial-time algorithm solving a minimization problem is a�-approximation algorithm (or has a performance ratio of �), ifthe result provided by the algorithm is no more than � timesthe result produced by the optimal solution.

Relay node placement has been well studied for the caseof k ¼ 1, i.e., using a minimum number of relay nodes tobridge a partitioned network. For instance, Lin and Xue [1]

IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 9, NO. 5, MAY 2010 643

. The authors are with the Department of Computer and InformationSciences, University of Delaware, Newark, DE 19716.E-mail: {han, cao, elloyd, cshen}@cis.udel.edu.

Manuscript received 6 Aug. 2008; revised 9 Apr. 2009; accepted 25 June 2009;published online 26 Aug. 2009.For information on obtaining reprints of this article, please send e-mail to:[email protected], and reference IEEECS Log Number TMC-2008-08-0309.Digital Object Identifier no. 10.1109/TMC.2009.161.

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proved this problem to be NP-hard and proposed aminimum spanning tree (MST)-based 5-approximationalgorithm. Chen et al. [5] showed that the performanceratio of the algorithm described in [1] is actually 4, and theyalso proposed a 3-approximation algorithm for this pro-blem. In [13], Cheng et al. proposed a faster 3-approxima-tion algorithm and a randomized algorithm1 with aperformance ratio of 2.5.

Recently, work on relay node placement has also beendone for the general case of k � 2. For example, Bredin et al.[7] studied the full fault-tolerant relay node placement (FFRP),which aims to deploy a minimum number of relay nodes tocreate a full k-vertex connected network such that theresulting network contains k vertex-disjoint paths betweenevery pair of target and/or relay nodes. Fig. 1a gives anexample of a full 2-vertex connected network. The authorsof [7] presented a �ð9k4 þ 36ðk3 þ k2ÞÞ-approximation algo-rithm for FFRP. Here, � is the best performance ratio ofexisting approximation algorithms for finding a minimumk-vertex connected spanning graph (see Section 2.4 for thedefinition). In [8], Kashyap et al. proposed approximationalgorithms for deploying relay nodes to create partialk-edge (vertex) connected networks for the cluster headsin wireless sensor networks. The problem defined in [8] istermed as partial fault-tolerant relay node placement (PFRP)where the k edge (vertex)-disjoint paths are only guaranteedbetween every pair of target nodes. Fig. 1b shows anexample of a partial 3-vertex connected network. Theauthors of [8] proved that the performance ratios of theiralgorithms are 10 when k ¼ 2. However, the performanceratios of their algorithms for the case of k > 2 remain open.

In addition to the above work assuming that both targetnodes and relay nodes use an identical transmissionradius, relay node placement has also been studied intwo-tiered homogeneous wireless sensor networks underthe assumption that all of the relay nodes use transmissionradius R and can communicate with either relay or targetnodes, while all of the target nodes use transmissionradius r and only communicate with relay nodes. Relaynode placement in two-tiered homogeneous wirelesssensor networks aims to create a backbone containingonly relay nodes to provide k vertex-disjoint pathsbetween every pair of target nodes. For this particular

problem, Tang et al. [9] proposed 4.5-approximationalgorithms for the cases of k ¼ 1 and k ¼ 2, provided thatR � 4r. For the more general situation of R � r, Lloyd andXue [6]2 gave a ð5þ "Þ-approximation algorithm for thecase of k ¼ 1. Recently, Zhang et al. [11] extended the workin [6] to a more complicated case where k ¼ 2.

1.2 The Problems We Study

To the best of our knowledge, this paper is the first effort toaddress relay node placement in the context of H-WSNs. InH-WSNs, target nodes may have different transmissionradii,3 while all of the relay nodes use an identicaltransmission radius. The different transmission radii ofthe target nodes introduce asymmetric communication linksbetween neighboring nodes, which raise two nontrivialissues. First, asymmetric links result in the existence of twokinds of paths in H-WSNs, namely, one-way paths and two-way paths. For a two-way path, wireless communicationsexist in both directions, while for a one-way path, wirelesscommunications exist only in one direction. Second, sinceno constraints are imposed on the relation between thetransmission radius of relay nodes and the transmissionradii of target nodes, as discussed further in Section 2.2,placing relay nodes in the network to connect two particulartarget nodes becomes much more complicated.

Given a set of target nodes V in the context of H-WSNs,and a desired connectivity level kðk � 1Þ, assuming that thecardinality of V is larger than k, this paper systematicallyaddresses the following problems:

. One-way partial fault-tolerant relay node placement (One-way PFRP). We seek to deploy a minimum number ofrelay nodes to form a one-way partial k-vertexconnected network for V such that the resultingnetwork contains k vertex-disjoint one-way pathsfrom any target node to any other target node.

. Two-way partial fault-tolerant relay node placement(Two-way PFRP). We seek to deploy a minimumnumber of relay nodes to form a two-way partialk-vertex connected network for V such that theresulting network contains k vertex-disjoint two-waypaths between every pair of target nodes.

. One-way full fault-tolerant relay node placement (One-way FFRP). We seek to deploy a minimum numberof relay nodes to form a one-way full k-vertexconnected network for V such that the resultingnetwork contains k vertex-disjoint one-way pathsfrom any node to any other node.

. Two-way full fault-tolerant relay node placement (Two-way FFRP). We seek to deploy a minimum numberof relay nodes to form a two-way full k-vertexconnected network for V such that the resultingnetwork contains k vertex-disjoint two-way pathsbetween every pair of nodes.

644 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 9, NO. 5, MAY 2010

Fig. 1. Hollow circles stand for target nodes and solid circles stand forrelay nodes. (a) A full 2-vertex connected network which contains twovertex-disjoint paths between every pair of target and/or relay nodes.(b) A partial 3-vertex connected network. In this network, there existthree vertex-disjoint paths between every pair of target nodes, whilefor some pairs of relay nodes, like node u and node v, there exist onlytwo vertex-disjoint paths between them.

1. A randomized algorithm with a performance ratio � for aminimization problem is an algorithm that provides a solution no morethan � times the optimal solutions with a positive probability. Although thealgorithm does not necessarily guarantee providing such a solution, byrepeated applications, the probability can be made arbitrarily close to 1.

2. The authors of [6] also presented an MST-based 7-approximationalgorithm, provided that target nodes are also capable of communicatingwith each other.

3. This assumption reflects the diverse transmission radii in H-WSNs. Infact, some target nodes in H-WSNs may have the same transmission radius.However, this fact would not reduce the difficulty of relay node placementin H-WSNs. Moreover, even though some target nodes are initiallydeployed with the same transmission radius, noise, interference, residualenergy, and other environmental impairment present in realistic operatingenvironments may still affect their effective transmission radius over time.


In a nutshell, each of our solutions to the problemsmentioned above consists of two phases. Phase one solvesthe Minimum k-vertex Connected Spanning Graph problem(MKCSG, see Section 2.4 for detailed information aboutMKCSG) for the initial sensor networks. Phase two placesadditional relay nodes, if necessary, along the edges in theMKCSG computed in phase one to create valid communica-tion paths. However, there are two facts worth mentioning.First, although any solution to FFRP (either one-way or two-way) is obviously a solution (not necessarily optimal) toPFRP, this does not mean that FFRP is easier than PFRP. Dueto the diverse fault-tolerant levels between different pairs ofnodes, FFRP is believed by some researchers [7], [12] to besignificantly harder than PFRP. Second, although anysolution to Two-way FFRP (PFRP) is obviously a solutionto One-way FFRP (PFRP), it is not productive to solve One-way PFRP (PFRP) using approximation algorithms for Two-way PFRP (PFRP). This is so because certain networks mayalready be a one-way k-vertex connected network and donot need any additional relay nodes, while Two-way PFRP(FFRP) algorithms may nonetheless add relay nodes, andhence, cause the performance ratios to become infinite.

The contributions of this paper are summarized as follows:

1. We give an Oð�k2Þ-approximation algorithm forOne-way PFRP. Here, � is the best performanceratio of existing approximation algorithms for find-ing a minimum k-vertex connected spanning graph.

2. We give an Oð�k3Þ-approximation algorithm forOne-way FFRP.

3. We give an Oð�k2Þ-approximation algorithm forTwo-way PFRP.

4. We give an Oð�k3Þ-approximation algorithm forTwo-way FFRP.

5. We extend each of these algorithms to networks ind-dimensional ðd � 3Þ metric space, and generalizethe approximation ratios of the extended algorithms.

6. We evaluate heuristic implementations of the pro-posed algorithms with realistic sensor networkscenarios, and show that their performance is muchbetter than the proven performance ratios suggest.

The remainder of this paper is organized as follows:Section 2 presents the network model and preliminaries.Section 3 first describes the algorithm for One-way PFRP,then proves its approximation ratio. Section 4 gives thealgorithm for Two-way PFRP, as well as the proof of itsperformance ratio. Section 5 provides approximation algo-rithms for both One-way and Two-way FFRP, and analyzestheir approximation ratios. Section 6 extends the approx-imation algorithms to higher dimensional networks andderives the corresponding performance ratios. Section 7discusses heuristics for the practical implementations of thepresented approximation algorithms. These heuristic im-plementations are simulated in QualNet [14] and the resultsare evaluated in Section 8. Finally, Section 9 concludes thepaper with future research directions.

2 NETWORK MODEL AND PRELIMINARIES

This section describes the model of heterogeneous wirelesssensor networks, as well as some basic operations andpreliminary knowledge that will be used in this paper.

2.1 Model of Heterogeneous Wireless SensorNetworks

We consider stationary heterogeneous wireless sensor

networks with omnidirectional transceivers. Each target

node x possesses a (possibly different) transmission radius

T ðxÞ, and all of the relay nodes use the same transmission

radius T ðrelayÞ. T ðminÞ and T ðmaxÞ represent the mini-

mum and the maximum transmission radius among all of

the relay nodes and target nodes. Correspondingly, we

define constants � ¼ dT ðmaxÞT ðminÞe, � ¼ dT ðmaxÞT ðrelayÞe, and � ¼ dT ðrelayÞT ðminÞ e.

Furthermore, each wireless node x has a transmission range

which is a circle in a 2D plane (or a sphere in a 3D space)

centered at x with radius T ðxÞ. Given these terms and

notations, we model an H-WSN as a directed graph G ¼(V [R, E), where V , R, and E are the set of target nodes, the

set of additionally deployed relay nodes, and the set of

directed edges, respectively. For any two nodes u and v in

V [R, there is a directed edge uv�! from u to v in E if and

only if v is in u’s transmission range.Relative to an arbitrary graph G ¼ ðN;EÞ, either directed

or undirected, where N is the set of nodes and E is the set ofedges, and two nodes u and v in N , Table 1 lists the terms,notations, and their semantics used in this paper. Noticethat the term neighbor is defined and used for undirectedgraphs, and the terms in-neighbor and out-neighbor aredefined and used for directed graphs.

2.2 Steinerization of Edges

For two target nodes u and v, one common scenario in thispaper is that we want to create a one-way path from u to v, ora two-way path between u and v, using as few relay nodes aspossible, while ignoring all the other target nodes and anypreviously deployed relay nodes. To facilitate creating suchpaths, we define the following two operations:

1. One-way Steinerization. We create an edge uv�! andOne-way Steinerize uv�! as follows: Compute theweight of uv�! using (1):

weightð uv�!Þ ¼ 0; if T ðuÞ � juvj;juvj�T ðuÞT ðrelayÞ

l m; if T ðuÞ < juvj:

(ð1Þ

If weightð uv�!Þ � 1, then place one relay node x onthe straight line between u and v such thatjuxj ¼ T ðuÞ, and then, evenly place weightð uv�!Þ � 1relay nodes along the straight line between x and v.

HAN ET AL.: FAULT-TOLERANT RELAY NODE PLACEMENT IN HETEROGENEOUS WIRELESS SENSOR NETWORKS 645

TABLE 1Terms, Symbols, and Their Semantics


In this way, we create a one-way path from u to v.Fig. 2a depicts this operation.

2. Two-way Steinerization: We define � ¼ minfT ðuÞ;T ðvÞg, � ¼ minfT ðuÞ; T ðrelayÞg, and ! ¼ minfT ðvÞ;T ðrelayÞg. We create an edgecuvand Two-way Steinerizecuv as follows: Compute the weight of cuv using (2):

weightðcuvÞ ¼ 0; if juvj � �;juvj��!T ðrelayÞ

l mþ 1; if juvj > �:

(ð2Þ

If weightðcuvÞ ¼ 1, then place one relay node x on thestraight line between u and v such that juxj ¼ �. IfweightðcuvÞ � 2, then place two relay nodes x and y onthe straight line betweenuandv such that juxj ¼ �andjvyj ¼ !, and then, evenly distribute weightðcuvÞ � 2

relay nodes along the straight line between x and y. Inthis way, we create a two-way path between u and v.Fig. 2b depicts this operation.

2.3 Segmentation of Neighborhood

Another operation frequently used in this paper is to dividea certain neighborhood area of a particular node into smallregions such that the nodes within the same region areconnected by directed or undirected one-hop communica-tion links. Specifically, for a node u, we segment itsneighborhood with a pair of positive values (r1, r2) in a 2Dplane as follows: We first create a circle centered at u withradius r1, along with a square just large enough toencompass the circle. Then, we evenly segment this square(from top to bottom and from left to right) into small squarecells where the length of the diagonal of each cell equals r2.4

As demonstrated in Figs. 3a and 3b, the operation ofsegmentation guarantees that the euclidean distance be-tween every pair of nodes in the same cell is no more than r2.

2.4 Minimum k-Vertex Connected Spanning Graph

An important problem related to the relay node placementproblem is to find an MKCSG. This concept depends on thefollowing definition:

Definition 2.1 (k-Vertex connected graph). Let G ¼ ðN;EÞbe a graph, where N and E stand for the node set and the edge

set, respectively. Then, G is a k-vertex connected graph (or G

is k-vertex connected for short), if for any two nodes u and v in

N , there exist k vertex-disjoint paths between u and v in G (or

there exist k vertex-disjoint directed paths from u to v in G,

provided that G is a directed graph).

Then, the problem of finding a minimum k-vertex

connected spanning graph is: Given a weighted complete

graph G ¼ ðV ;EÞ where each edge carries a nonnegative

weight, and a positive integer k, find a k-vertex connected

spanning graph G0 of G with minimum total weight. For

undirected graphs, when k ¼ 1, the problem is exactly that

of finding the minimum spanning tree; when k � 2, the

problem is NP-hard and following results are known. Ravi

and Williamson [15] claimed the first constant approxima-

tion algorithm with performance ratio ð2Pk

i¼11iÞ. However,

their proof was later found [16] to contain an error. In [17],

Kortsarz and Nutov presented a k-approximation algorithm

and a ðkþ12 Þ-approximation algorithm for the case of k � 7.

Most recently, Cheriyan et al. [18] developed an OðlgkÞ-approximation algorithm, provided that the complete graph

contains at least 6k2 nodes. For directed graphs, the problem

is NP-hard when k � 1. Kortsarz and Nutov [17] presented

a (1þ k)-approximation algorithm, along with a (2þ kkvk )-

approximation algorithm if the edge weights satisfy the

triangle inequality.In addition to the above algorithms with provable

performance guarantees, Li and Hou [19] proposed aneasily implemented greedy heuristic, which works bysorting the edges of graph G according to their weight,and then, iteratively adding edges to G0 in increasing orderof their weight until G0 is k-vertex connected. Bredin et al.[7] improved the algorithm of [19] by imposing anoptimization step, which removes any of the added edgeswhose removal does not destroy the k-vertex connectivity(i.e., each added edge is tested in decreasing order of weightto see if it can be safely removed). The simulation results in[7] show that the solutions produced by the heuristicalgorithms are typically close to the optimal.

3 ONE-WAY PARTIAL FAULT-TOLERANT RELAY

NODE PLACEMENT

This section presents an approximation algorithm for One-way PFRP in H-WSNs and analyzes the quality of the resultproduced by the algorithm with respect to the optimalsolution. Recall that in One-way PFRP, the objective is todeploy a minimum number of relay nodes to form a one-way partial k-vertex connected network such that in theresulting network, there exist k vertex-disjoint one-waypaths from any target node to any other target node.


Fig. 2. (a) One-way Steinerize the directed edge ~uv and create a one-way path from u to v. (b) Two-way Steinerize the undirected edge cuv andcreate a two-way path between u and v.

Fig. 3. Segmentation of the neighborhood of node u with (r1, r2). (a) Thesquare encompassing the circle can be evenly segmented by cellswhere the length of the diagonal equals r2. (b) The square encompass-ing the circle cannot be evenly segmented by cells where the length ofthe diagonal equals r2.

4. In the case where the encompassing square cannot be evenlysegmented by the square cells, the diagonal of some square cells can be lessthan r2.


3.1 Algorithm for One-Way PFRP

In brief, given a set of target nodes V , the algorithm firstcomputes a directed MKCSG S of the complete graph overV , and then, One-way Steinerizes each edge in S. We leavethe choice of an approximation algorithm for computingMKCSG as an open option, and assume that the approx-imation ratio of the selected algorithm is � in our analysis.The complete algorithm is described in Algorithm 1, andTheorem 3.1 states its performance ratio.

Algorithm 1. Algorithm for One-way PFRP

1: INPUT: Integer k and a set of target nodes V .

2: OUTPUT: A set of relay nodes R.

3: R � (empty set);4: W f uv�! j u; vðu 6¼ vÞ 2 V g;5: Define the weight of each edge uv�! 2W according

to Equation 1;

6: C ðV ;WÞ;7: Compute an approximate directed minimum k-vertex

connected spanning graph S of C using a

�-approximation algorithm;

8: One-way Steinerize each edge uv�! 2 S and place therelay nodes into R;

9: Output R;

Theorem 3.1. Let V be a set of target nodes. Algorithm 1 is anOð�k2Þ-approximation algorithm in terms of the number ofrelay nodes required to form a one-way partial k-vertexconnected network for V .

3.2 Algorithm Analysis

Although Algorithm 1 is relatively straightforward, theanalysis of its performance ratio is complicated. Theanalysis begins with two definitions.

Definition 3.2 (One-way partial k-vertex connectedgraph). Let V be a set of target nodes and G ¼ ðV [R;EÞ bea directed graph, whereR is the set of additionally deployed relaynodes and E is the set of directed edges induced by the relay andtarget nodes. Then, G is a one-way partial k-vertex connectedgraph for V , if for every pair of target nodes u and v in V , thereexist k vertex-disjoint one-way paths from u to v inG (i.e., thereexists at least one one-way path inG fromu to v after the removalof no more than k� 1 arbitrary nodes other than u and v).

Definition 3.3 (Superpath [6]). Let V be a set of target nodesand G ¼ ðV [R;EÞ be a one-way partial k-vertex connectedgraph for V . Then, a one-way path PGðu; v�!Þ in G is asuperpath if u and v are target nodes and every interior node (ifany) of PGðu; v�!Þ is a relay node.

From the description of Algorithm 1, we have animportant observation that: every relay node placed byAlgorithm 1 is on exactly one superpath. We use this fact tohelp establish the quantitative relationship between theresult produced by Algorithm 1 and the optimal solution.The entire analysis consists of three steps.

3.2.1 Step One

In this step, we prove that if there exists a one-way partialk-vertex connected graph G ¼ ðV [R;EÞ for V where eachrelay node in R is on exactly one superpath, the number ofrelay nodes computed by Algorithm 1 on V is at most �kRk.

The analysis proceeds by first establishing Lemmas 3.4 and3.5, which show that duplicate superpaths from the samestarting node to the same ending node are unnecessary,provided that each relay node is on exactly one superpath.Finally, Lemma 3.6 proves the result of step one.

Lemma 3.4. Let G ¼ ðN;EÞ be a directed k-vertex connectedgraph. For any two nodes u and v in N , if there exist multipledirected edges from u to v in G, then if we keep one of theseedges and remove the others, the resulting graph G0 ¼ ðN;E0Þis still a directed k-vertex connected graph. (A detailed proof ofthis lemma is presented in the Appendix).

Lemma 3.5. Let V be a set of target nodes, and G ¼ ðV [R;EÞbe a one-way partial k-vertex connected graph for V whereeach relay node in R is on exactly one superpath. For any pairof target nodes u and v in R, if in G, there exist multiplesuperpaths from u to v, then if we keep one of these superpathsand remove the others, and denote the resulting graph asG0 ¼ ðV [R0; E0Þ, then G0 remains a one-way partial k-vertexconnected graph for V .

Proof. Since each relay node in R is on exactly onesuperpath, by treating each superpath as an edge, all ofthe target nodes and superpaths in G form a directedk-vertex connected graph. From Lemma 3.4, it followsthat after removing the redundant superpaths, the targetnodes and the remaining superpaths in G0 still form adirected k-vertex connected graph. This means that, inG0, there exist k vertex-disjoint one-way paths from anytarget node to any other target node. Therefore, G0 is aone-way partial k-vertex connected graph for V . tu

Lemma 3.6. LetV be a set of target nodes, andG ¼ ðV [R;EÞ be aone-way partial k-vertex connected graph for V where each relaynode inR is on exactly one superpath inG. Then, the number ofrelay nodes computed by Algorithm 1 on V is at most �kRk.

Proof. For any pair of target nodes u and v in R, if thereexist in G multiple superpaths from u to v, we keep oneof these superpaths and remove others, and denote theresulting graph as G0 ¼ ðV [R0; E0Þ. By Lemma 3.5, G0

remains a one-way partial k-vertex connected graph forV , and we have kR0k � kRk.

Now, we consider the result produced by Algorithm 1.For two arbitrary target nodes u and v in V , (1) defines theminimum number of relay nodes required to create aone-way path from u to v. Furthermore, Algorithm 1 usesa �-approximation algorithm to compute an approximatedirected MKCSG S. Therefore, the number of relay nodescomputed by Algorithm 1 is at most �kR0k � �kRk. tu

3.2.2 Step Two

Let V be a set of target nodes, and let Go ¼ ðV [Ro;EoÞ bean optimal one-way partial k-vertex connected graph forV . Motivated by the analysis in [7], in this step, weperform a sequence of transformations on Go and create anew one-way partial k-vertex connected graph Gfinal ¼ðV [Rfinal; EfinalÞ for V where each relay node in Rfinal ison exactly one superpath. Then, we prove that kRfinalk �ðð32��2 þ 1

2Þk2 þ 3kþ 4ÞkRok (Note that � and � aredefined in Section 2.1).

Our analysis uses two transformation rules which wedefine in the paragraphs below.



Transformation Rule 1. Given a relay node r, wesegment its neighborhood with (T ðmaxÞ, T ðminÞ) and get8�2 square cells with the length of the diagonal being equalto T ðminÞ. In each square cell, node r randomly selectsk target in-neighbors and k target out-neighbors (or selectsall of the target in-neighbors and target out-neighbors ifthere are fewer than k of them) as its new target in-neighbors and out-neighbors, as well as all of the edgesassociated with the selected target nodes.

Denote graph G0o ¼ ðV [R;E0oÞ as the resulting graph byperforming Transformation Rule 1 on Go. We have thefollowing lemma:

Lemma 3.7. The graph G0o ¼ ðV [R;E0oÞ is a one-way partialk-vertex connected graph for V , and each relay node inG0o has atmost 8�2k target in-neighbors and 8�2k target out-neighbors.

Proof. Let G ¼ ðN;EÞ be a graph where N and E are thesets of nodes and edges, respectively, and let X be a setof nodes (or edges). We define G nX as the resultinggraph by removing the nodes in N \X plus theirincident edges from G (or removing the edges in E \Xfrom G, if X is an edge set).

For two target nodes u and v in V , and k� 1 arbitrarynodes A ¼ fn1; n2; . . . ; nk�1g other than u and v in G0o, weprove that there is a path from u to v in G0o nA.

Obviously, there is a one-way path PGoðu; v�!Þ from u to v

in Go nA. For an arbitrary hop on PGoðu; v�!Þ, say xy�!, if

both x and y are target nodes or relay nodes, then xy�! stillexists in G0o nA. Now, consider the situation where x is arelay node and y is a target node: 1) If x chooses y as itsnew out-neighbor inG0o, then xy�! exists inG0o nA. 2) If y isnot chosen, then in the cell of x where y resides, x mustselect k other target out-neighbors, and at least one ofthese target out-neighbors is not in A. We denote thisparticular out-neighbor as z. Since jzyj � T ðminÞ, zy! existsin G0o nA. Therefore, we replace xy�! with xz�! and zy! inG0o nA. We can perform a similar replacement for thesituation where x is a target node and y is a relay node.

Because each hop on PGoðu; v�!Þ is valid in G0o nA, there

is a one-way path from u to v in G0o nA. Therefore, G0o is aone-way partial k-vertex connected graph for V . More-over, since there are at most k target in-neighbors andk target out-neighbors in each segmenting cell, each relaynode in G0o has at most 8�2k target in-neighbors and8�2k target out-neighbors. tu

Before stating the second transformation rule, we firstprovide three additional definitions.

Definition 3.8 (Relay component). Let V be a set of targetnodes, and G ¼ ðV [R;EÞ be a one-way partial k-vertexconnected graph for V . For a relay node r 2 R, the relaycomponent of r can be derived as follows: We start at r, travelalong each edge incident to r in G (when we travel, we omit thedirection of each edge, and traverse the edge in eitherdirection). If we meet a relay node, we repeat the process; ifwe meet a target node, we stop. Finally, all of the nodes (targetor relay) and edges visited during this recursive process formthe relay component of r. Furthermore, each relay componentis a subgraph of G, and intuitively, all of the boundary nodesin the relay component are target nodes and all the interiornodes are relay nodes.

Definition 3.9 (Undirected spanning tree of relay compo-

nent). Let Ci ¼ ðVi [Ri; EiÞ be a relay component, where Vi,Ri, and Ei are the sets of target nodes, relay nodes, anddirected edges in Ci, respectively. We first create a newundirected graph C0i from Ci by omitting the direction of eachedge in Ei. Then, we find a spanning tree of C0i rooted at anarbitrary relay node and having all of the target nodes in Vi asleaves. Such a spanning tree, denoted as TreeðCiÞ, is anundirected spanning tree of Ci.

Definition 3.10 (Harary graph). Let V be a set of nodes. Wecan construct a Harary graph of V as follows: We place all ofthe nodes in V into a circular doubly linked list L. For eachnode x in L, we add undirected edges between node x andk nearest nodes of x in L. A Harary graph is a k-vertexconnected graph. If we replace each undirected edge in aHarary graph with a pair of opposite directed edges, theresulting graph is a k-vertex connected directed graph.

Assume that G0o ¼ ðV [R;E0oÞ has m relay components,denoted as Ci ¼ ðVi [Ri; EiÞð1 � i � mÞ. Now, we definethe second transformation rule, which is performed on eachrelay component.

Transformation Rule 2. For each relay componentCi ¼ ðVi [Ri; EiÞ: 1) if kVik > k, we first make a clockwiseEulerian tour of TreeðCjÞ and place the target nodes in acircular doubly linked list L in the order in which they arevisited in this Eulerian tour. Then, we remove all the relaynodes in Ci and create a Harary Graph in Ci by connectingeach target node with k nearest target nodes in L. 2) ifkVik � k, we first remove all of the relay nodes in Cj andcreate a complete undirected graph in Ci by connectingeach target node with all of the other target nodes. Then, weadd k� kVik þ 1 duplicate edges for each edge in thiscomplete graph. After creating a Harary graph or acomplete graph with duplicate edges, we replace each edgewith two opposite directed edges, and One-way Steinerizeall of the directed edges.

Let C0i ¼ ðVi [R0i; E0iÞð1 � i � mÞ be the graph that resultsby performing Transformation Rule 2 on Ci, and let graphG00o ¼ ðV þR00o ; E00o Þ be the graph that results by performingTransformation Rule 2 on each relay component in G0o. Wehave the following lemma:

Lemma 3.11. The graph G00o is a one-way partial k-vertexconnected graph for V , and each relay node in G00o is on exactlyone superpath.

Proof. For two target nodes u and v in V , and k� 1 arbitrary

nodes A ¼ fn1; n2; . . . ; nk�1g other than u and v in G00o , we

prove that there is a path in G00o nA from u to v. By

Lemma 3.7, there is a one-way path PG0oðu; v�!Þ from u to v

in G0o nA. We partition PG0oðu; v�!Þ into multiple subpaths,

where each subpath is a superpath. For an arbitrary

subpath PG0oðx; y�!Þ: 1) If PG0oðx; y

�!Þ does not contain any

relay nodes, then PG0oðx; y�!Þ exists in G00o nA. 2) If PG0oðx; y

�!Þcontains some relay nodes, x and y must be in the same

relay component Cj in G0o, and hence, x and y are in C0j.

Transformation Rule 2 guarantees that there is at least

one one-way path from x to y in C0j nA, which can be

used to replace the subpath PG0oðx; y�!Þ in G00o . Since each



subpath of PG0oðu; v�!Þ is still valid in G00o , there is a one-way

path from u to v in G00o nA, which means that G00o is a one-

way partial k-vertex connected graph for V . Moreover,

each relay node in G00o is on exactly one superpath. tuFinally, we generate the graph Gfinal ¼ ðV [Rfinal; EfinalÞ

as follows: For two arbitrary target nodes u and v in

G00o ¼ ðV [R00o ; E00o Þ, if there are multiple superpaths from u to

v in G00o , we keep one of those superpaths and remove the

others. By Lemma 3.5, Gfinal remains a one-way partial

k-vertex connected graph for V . Furthermore, each relay

node in Gfinal is on exactly one superpath. Lemma 3.12

presents the result of Step Two.

Lemma 3.12. kRfinalk � ðð32��2 þ 12Þk2 þ 3kþ 4ÞkRok.

Proof. For an arbitrary relay component Ci ¼ ðVi [Ri; EiÞin G0o, we have kVik � 16�2kkRik from Lemma 3.7.

After applying Transformation Rule 2 on Ci, we get

C0i ¼ ðVi [R0i; E0iÞ.When kVik > K;Gfinal contains C0i. We use TreeðCiÞ to

count the number of relay nodes in R0i. For an arbitrary

superpath from u to v added by Transformation Rule 2 in

C0i, we spread the weight of uv�! on the path PTreeðCiÞðdu; vÞbetweenuandv inTreeðCiÞbychargingone to each interior

hop. For each edge cxy in TreeðCiÞ: 1) If cxy is incident to a

target node, cxy is totally charged 2k�. 2) Otherwise, we

denote the target descendants of x in TreeðCiÞ from left to

right as ft1; t2; . . . ; tng. For the nodes from t1 to tdk2e, cxy is

charged 2dk2e, 2ðdk2e � 1Þ; . . . ; 1, respectively. And for the

nodes from tn to tn�dk2eþ1, cxy is also charged 2dk2e;2ðdk2e � 1Þ; . . . ; 1, respectively. Therefore, cxy is charged at

most ð12 k2 þ 3kþ 4Þ. As a result, all of the edges inTreeðCiÞare charged a total of at most 2k�kVik þ 1

2 k2 þ 3kþ 4kRik,

which is no more than ðð32��2 þ 12Þk2 þ 3kþ 4ÞkRik.

When kVik � k;Gfinal keeps a one-way path from any

target node to any other target node. In this case, all of

the edges in TreeðCiÞ are charged a total of no more thanðð32��2 þ 1

2Þk2 þ 3kþ 4ÞkRik.Summing up all of the relay components, we have

kRfinalk � ðð32��2 þ 12Þk2 þ 3kþ 4ÞkRok. tu

3.2.3 Step Three

Proof of Theorem 3.1. By combining the results of Lemma

3.6 and Lemma 3.12, the number of relay nodes added in

Algorithm 1 is at most �ðð32��2 þ 12Þk2 þ 3kþ 4Þ times

the optimal solution. Therefore, Algorithm 1 is an

Oð�k2Þ-approximation algorithm. tu

4 TWO-WAY PARTIAL FAULT-TOLERANT RELAY

NODE PLACEMENT

This section provides an approximation algorithm for Two-way PFRP and derives the performance ratio of thealgorithm by following the framework used for analyzingAlgorithm 1.

4.1 Algorithm for Two-Way PFRP

The algorithm for Two-way PFRP is presented in Algorithm 2,and Theorem 4.1 describes the performance ratio.

Algorithm 2. Algorithm for Two-way PFRP1: INPUT: Integer k and a set of target nodes V .

2: OUTPUT: A set of relay nodes R.

3: R �;

4: W fcuv j u; vðu 6¼ vÞ 2 V g;5: Define the weight of each edge cuv 2W according to (2);

6: C ðV ;WÞ;7: Compute an approximate undirected minimum k-vertex

connected spanning graph S of C using a�-approximation algorithm;

8: Two-way Steinerize each edge cuv 2 S and place the relay

nodes in R;

9: Output R;

Theorem 4.1. Let V be a set of target nodes. Algorithm 2 is anOð�k2Þ-approximation algorithm in terms of the number ofrelay nodes required to form a two-way partial k-vertexconnected network for V .

4.2 Algorithm Analysis

Algorithm 2 is based on undirected graphs. We cananalogously define two-way partial k-vertex connected graphand superpath for undirected graphs, and Lemma 4.2,Lemma 4.3, and Lemma 4.4 follow directly.

Lemma 4.2. Let G ¼ ðN;EÞ be an undirected k-vertex connectedgraph. For two nodes u and v in N , if there are multipleundirected edges between u and v in G, then we can keep one ofthese edges and remove the others, and the resulting graphG0 ¼ ðN;E0Þ remains an undirected k-vertex connected graph.

Lemma 4.3. Let V be a set of target nodes, and let G ¼ðV [R;EÞ be a two-way partial k-vertex connected graph forV where each relay node in R is only on one superpath. Forany pair of target nodes u and v in R, if there exist multiplesuperpaths between u and v in G, then if we keep one of thesesuperpaths and remove the others, and denote the resultinggraph as G0 ¼ ðV [R0; E0Þ, then G0 is a two-way partialk-vertex connected graph for V .

Lemma 4.4. Let V be a set of target nodes, and G ¼ ðV [R;EÞbe a two-way partial k-vertex connected graph for V , whereeach relay node in R is exactly on one superpath in G. Then,the number of relay nodes computed by Algorithm 2 on V is atmost �kRk.

Further, considering the optimal two-way partialk-vertex connected graph Go ¼ ðV [Ro;EoÞ for V , wedefine the following transformation rules:

Transformation Rule 3. For relay node r, we segment itsneighborhood with (Trelay, Tmin), and get 8�2 square cellswith the length of the diagonal being equal to Tmin. In eachcell, node r selects k arbitrary target neighbors (or select allof the target neighbors if there are fewer than k of them) asits new target neighbors.

Let graph G0o ¼ ðV [R;E0oÞ be the graph resulted byperforming Transformation Rule 3 on Go. We have thefollowing lemma:

Lemma 4.5. The graph G0o ¼ ðV [R;E0oÞ is a two-way partialk-vertex connected graph for V , and each relay node in G0o hasat most 8�2 target neighbors.



Assume that G0o ¼ ðV [R;E0oÞ has totally m relaycomponents, denoted as Ci ¼ ðVi [Ri; EiÞð1 � i � mÞ.Now, we define Transformation Rule 4, which is performedon each relay component.

Transformation Rule 4. For each relay component Ci ¼ðVi [Ri; EiÞ: 1) If kVik > k, we make a clockwise Euleriantour on TreeðCjÞ and place the target nodes in a circulardoubly linked list L in the order they are visited in theEulerian tour. Then, we remove all of the relay nodes in Ciand create a Harary Graph in Ci by connecting each targetnode with k nearest target nodes in L. 2) If kVik � k, weremove all of the relay nodes in Cj and create a completeundirected graph in Ci by connecting each target node withall of the other target nodes. Further, we add k� kVik þ 1duplicate edges for each edge in the complete graph. Finally,after creating a Harary graph or a complete graph withduplicate edges, we two-way Steinerize all of the edges.

Denote C0i ¼ ðVi [R0i; E0iÞ (1 � i � m) as the graph re-sulted by performing Transformation Rule 4 on Ci, anddenote graph G00o ¼ ðV þR00o ; E00o Þ as the graph resulted byperforming Transformation Rule 4 on each relay componentin G0o. We have the following lemma:

Lemma 4.6. G00o is a two-way partial k-vertex connected graph forV , and each relay node in G00o is on exactly one superpath.

Finally, we are able to generate a two-way partialk-vertex connected graph Gfinal ¼ ðV [Rfinal; EfinalÞ for Vby removing duplicate superpaths according to Lemma 4.3,and we have the following lemma holds:

Lemma 4.7. kRfinalk � ðð8�2 þ 14Þk2 þ 3

2 kþ 2ÞkRok.Proof. For an arbitrary relay component Ci ¼ ðVi [Ri; EiÞ

in G0o, we have kVik � 8�2kkRik from Lemma 4.5.After applying Transformation Rule 4 on Ci, we getC0i ¼ ðVi [R0i; E0iÞ.

We still use TreeðCiÞ to count the number of relay

nodes in R0i. When kVik > K, Gfinal contains C0i. For an

arbitrary superpath between u and v added by Transfor-

mation Rule 4 in C0i, we spread the weight of cuv on the

path PTreeðCiÞðdu; vÞ by charging one on each interior hop.

For each edge cxy in TreeðCiÞ: 1) If cxy is incident to a target

node, cxy is totally charged k, and 2) otherwise, we denote

the target descendants of x in TreeðCiÞ from left to right as

ft1; t2; . . . ; tng. For the nodes from t1 to tðdk2e, cxy is charged

dk2e; ðdk2e � 1Þ; . . . ; 1, respectively. And for the nodes from

tn to tn�dk2eþ1, cxy is also charged dk2e; ðdk2e � 1Þ; . . . ; 1,

respectively. Therefore, cxy is charged at most ð11 k2 þ32 kþ 2Þ. As a result, all of the edges in TreeðCiÞ are totally

charged at most k�kVik þ ð12 k2 þ 32 kþ 2ÞkRik, which is no

more than ðð8�2 þ 14Þk2 þ 3

2 kþ 2ÞkRik.When kVik � k, Gfinal keeps one one-way path from

any target node to any other target node. In this case, allof the edges in TreeðCiÞ are totally charged no more thanðð8�2 þ 1

4Þk2 þ 32 kþ 2ÞkRik.

Summing up all of the relay components, we havekRfinalk � ðð8�2 þ 1

4Þk2 þ 32 kþ 2ÞkRok. tu

Proof of Theorem 4.1. By combining the results ofLemma 4.4 and Lemma 4.7, the number of relay nodesadded in Algorithm 2 is at most �ðð8�2 þ 1

4Þk2 þ 32 kþ 2Þ

times the optimal solution. Therefore, Algorithm 2 is anOð�k2Þ-approximation algorithm in terms of the numberof relay nodes required to form a two-way partialk-vertex connected network for V . tu

5 ONE-WAY AND TWO-WAY FULL

FAULT-TOLERANT RELAY NODE PLACEMENT

In this section, based on the work for One-way PFRP andTwo-way PFRP, we propose approximation algorithms forboth One-way FFRP and Two-way FFRP, and present theanalysis of their performance ratios. The algorithms aredescribed in Algorithm 3.

Algorithm 3. Algorithm for One-way (Two-way) FFRP1: INPUT: Integer k and a set of target nodes V

2: OUTPUT: A set of relay nodes F

3: F �;

4: Run the One-way (Two-way) PFRP algorithm on V and

get a set of relay nodes R, as well as the resulting

network G ¼ ðV [R;EÞ;5: For each superpath PGðu; v�!Þ (PGðdu; vÞ for Two-way

FFRP) in G, at the position of every relay node inPGðu; v�!Þ (PGðdu; vÞ), place k� 1 additional relay nodes,

and put the new relay nodes into F ;

6: For each target node u in V , if u is the starting or the

ending node of a superpath containing relay nodes in

G, segment u’s neighborhood with (T ðuÞ; T ðrelayÞ), and

place a cluster of k� 1 relay nodes at the position of u,

and a cluster of k relay nodes at the center of each

segmenting cell. Put each cluster of relay nodes into F ;7: For each cluster of relay nodes deployed in Step 6

� remove all of the relay nodes in the cluster from F ;

� if the graph of the resulting network is not a directed

(undirected for Two-way PFRP) k-vertex connected

graph, restore all of the relay nodes in the cluster;

End For

8: Output F ¼ F [R;

We first analyze the performance of Algorithm 3 for One-way FFRP. Algorithm 3 executes the One-way PFRPalgorithm on V and produces a resulting networkG ¼ ðV [R;EÞ. Then, for each superpath PGðu; v�!Þ in G,Algorithm 3 replicates each relay node on PGðu; v�!Þ withk� 1 additional relay nodes. Furthermore, in Step 6,Algorithm 3 connects the relay nodes on PGðu; v�!Þ with u

and v as well as their in-neighbors and out-neighbors. Weterm the operation in Step 6 full-connection. Although full-connection is costly, it guarantees that there are k vertex-disjoint one-way paths from any node to any other node inthe resulting network. Fig. 4 depicts the operations Step 5and Step 6 on one one-way superpath from node u to node v.Finally, in Step 7, Algorithm 3 tests each cluster of relaynodes deployed in Step 6 and removes the cluster if thegraph of the resulting network remains a directed k-vertexconnected graph. Results of our simulations show thatStep 7, on average, removes 83.6 percent of the relay nodesdeployed in Step 6. Theorem 5.1 proves the correctness ofAlgorithm 3 for One-way FFRP, and Theorem 5.2 presentsthe approximation ratio of Algorithm 3 for One-way FFRP.



Theorem 5.1. Let V be a set of target nodes. Algorithm 3computes a set of relay nodes F , whose deployment creates aOne-way full k-vertex connected network for V .

Proof. For two arbitrary nodes u and v in the resultingnetwork G ¼ ðV [ F;EÞ, if both u and v are target nodes;thus, there are k-vertex disjoint paths from u to v in G, aswell as k-vertex disjoint paths from v to u in G, sinceAlgorithm 3 is based on the One-way PFRP algorithm.Now, we consider the case where u is a target node and vis a relay node.

Note that relay node v is deployed by Algorithm 3either on a superpath, or in a segmented cell of the endingtarget nodes of a superpath. Assume that this superpathis PGðx; y�!Þ, where x is the starting target node and y is theending target nodes. Step 5 of Algorithm 3 guaranteesthat there are k-vertex disjoint paths from y to u in G, andk-vertex disjoint paths from u to x in G. Moreover, since xhas at least k incoming neighbors, and y has at leastk outgoing neighbors, there is at least one one-way pathfrom u to v, and one one-way path from v to u afterremoving fewer than k arbitrary nodes inG. Likewise, wecan get the same results for the case where v is a targetnode and u is a relay node, and the case where both u andv are relay nodes. Therefore, deploying F in V creates aone-way k-vertex connected network for V . tu

Theorem 5.2. Let V be a set of target nodes. Algorithm 3 is anOð�k3Þ-approximation algorithm in term of the number ofrelay nodes required to form a one-way full k-vertex connectednetwork for V .

Proof. Denote R and F as the sets of relay nodes that resultfrom running the One-way PFRP algorithm and the One-way FFRP algorithm on V , and denote G ¼ ðV [R;EÞ asthe resulting network by deploying R in V . Consider theoptimal set Fo of relay nodes for One-way FFRP.

Because Fo is also a solution for One-way PFRP, we

have kRk < �ðð32��2 þ 12Þk2 þ 3kþ 4ÞkFok. For an arbi-

trary superpath PGðu; v�!Þ in G, Algorithm 3 adds at most

4�2kþ weightð uv�!Þkþ 4�2k relay nodes. Therefore, for

all of the superpaths in G, Algorithm 3 totally adds at

most 4�2kkRk þ kRkkþ 4�2kkRk relay nodes. As a

result, we have kFk � k�ð8�2 þ 1Þðð32��2 þ 12Þk2 þ 3k þ

4ÞkFok. Therefore, Algorithm 3 is an Oð�k3Þ-approxima-

tion algorithm. tuThe analysis of Algorithm 3 for Two-way FFRP can be

conducted in a similar manner, and Theorem 5.3 statesthe result.

Theorem 5.3. Let V be a set of target nodes. Algorithm 3 is anOð�k3Þ-approximation algorithm in terms of the number of

relay nodes required to form a two-way full k-vertex connectednetwork for V .

6 EXTENSION TO HIGHER DIMENSIONS

Although 2D terrain is widely used in wireless sensornetwork research, emerging applications, such as under-water sensor networks, demand operations in 3D environ-ments. Motivated by this fact, this section extends all of theapproximation algorithms described in previous sections totackle relay node placement in higher dimensionalH-WSNs, and derives the approximation ratio of each ofthese extended algorithms.

For a particular node x, we can extend the method forsegmenting x’s neighbor with ðr1; r2Þ to the case ofd-dimensional (d � 3) metric space as follows: We firstcreate a d-dimensional sphere centered at x with radius r1.Then, we create a d-dimensional cube just large enough toencompass this sphere. Finally, we evenly segment thisd-dimensional cube into small d-dimensional cube cellswhere the length of the diagonal equals r2.

With this extended segmentation method, we can directlyapply the One-way PFRP and Two-way PFRP algorithms tosolve partial fault-tolerant relay node placement in thesituations where heterogeneous target nodes are deployed ind-dimensional metric space. By the d-dimensional Pythagor-ean Theorem,5 we have following two theorems:

Theorem 6.1. Let V be a set of target nodes. Algorithm 1 is anOð�ð2

ffiffiffidp

�Þdk2Þ-approximation algorithm in terms of thenumber of relay nodes required to form a one-way partialk-vertex connected network forV in d-dimensional metric space.

Theorem 6.2. Let V be a set of target nodes. Algorithm 2 is anOð�ð2

ffiffiffidp

�Þdk2Þ-approximation algorithm in terms of thenumber of relay nodes required to form a two-way partialk-vertex connected network forV in d-dimensional metric space.

Further, we extend the full connection operation ofAlgorithm 3 to d-dimensional metric space as follows: Foreach target node u in V , if u is the starting or the endingnode of a superpath containing relay nodes in G, we firstsegment u’s neighborhood with (T ðuÞ,

ffiffiffidp

T ðrelayÞ), andthen, place a cluster of k� 1 relay nodes at the position of u,and a cluster of k relay nodes at the center of eachsegmented cell. Denote the algorithm that is identical toAlgorithm 3 excepting using extended full connection asAlgorithm 3E, then the following theorems hold directly:

Theorem 6.3. Let V be a set of target nodes. Algorithm 3E is anOð�ð4d��Þdk3Þ-approximation algorithm in terms of thenumber of relay nodes required to form a one-way full k-vertexconnected network for V in d-dimensional metric space.

Theorem 6.4. Let V be a set of target nodes. Algorithm 3E is anOð�ð4d��Þdk3Þ-approximation algorithm in terms of thenumber of relay nodes required to form a two-way full k-vertexconnected network for V in d-dimensional metric space.

7 HEURISTIC IMPLEMENTATIONS

All of the approximation algorithms discussed in theprevious sections are based on a subroutine (approximation


Fig. 4. Examples of the operation of full connection on target nodes uand v for the case of k ¼ 2, and the operation of connecting relay nodeson superpath with the ending target nodes.

5. For a d-dimensional cube with diagonal of length of L, and the length

of d edges E1; E2; . . . ; Ed, this theorem shows that L2 ¼Pd

i¼1 E2i .


algorithm) to compute MKCSG. However, most existingconstant approximation algorithms for the MKCSG problemrely on the Frank and Tardos algorithm [2] or its variations,and are quite complicated and difficult to implement incomputation- and communication-constrained sensor net-works [7]. Thus, motivated by the heuristic algorithmsdescribed in [19] and [7], we present a greedy heuristicalgorithm in this section as a practical alternative to solvethe MKCSG problem. For the ease of description, we willterm our algorithm connectivity-first algorithm (CFA) and thealgorithm in [7] weight-first algorithm (WFA) in the rest ofthis paper.

The basic idea of CFA is to repeatedly add the edges thatcan best help to improve the graph connectivity until thegraph becomes a k-vertex connected graph, then remove theredundant edges if their removal does not break thek-vertex connectivity. To quantitatively measure the im-provement of each edge on the graph’s connectivity, wedefine the concept of the contribution of an edge.

Definition 7.1 (Contribution). In a graph that is not k-vertexconnected, there must exist some node pairs which can bepartitioned by the removal of few than k nodes, which meansthat the connectivity between such node pairs is lower than k.We define these node pairs as unsaturated node pairs. Thecontribution of an edge uv�! (or cuv) is defined as the numberunsaturated node pairs whose connectivity can be improved bythe deployment of edge uv�! (or cuv) in this graph.

The complete algorithm is stated in Algorithm 4. One

key step in Algorithm 4 is to check the connectivity between

node pairs and the connectivity of the entire graph. This can

be achieved by using a maximum network-flow-based

checking algorithm in [15]. Specifically, for a directed

graph G, the checking algorithm first constructs a directed

graph G0 from G as follows: First, split each node u in G into

two nodes ui and uo. Then, add edge uiuo��! in G0; also, for

each edge ux�! in G, add edge uoxi��! into G0; for each edge yu�!,

add edge youi��! into G0. Finally, assign each edge in G0 a

capacity of one. For the node pair u and v in G, the

connectivity between u and v in G is equal to the maximum

network flow between uo and vi in G0. The graph

connectivity can be checked by examining the connectivity

between every pair of nodes. For an undirected graph G, we

can first extend G to a directed graph by replacing each

undirected edge in G with a pair of opposite directed edges,

and then, apply the checking algorithm discussed above.

We can use the Push-relabel algorithm with dynamic trees

with time complexity OðjV kEjlogðjV j2

jEj ÞÞ (V and E are the

vertex and the edge sets, respectively) for checking the

maximum network flow between each pair of nodes.

Therefore, the time complexity of Algorithm 4 is

OðjV j3jEjlogðjV j2

jEj ÞÞ.

Algorithm 4. Connectivity-first Algorithm for MKCSG

1: INPUT: Integer k, an undirected (or directed)

weighted complete graph G ¼ ðV ;EÞ.2: OUTPUT: An undirected (or directed) k-vertex

connected spanning graph S of G.

3: M �;

4: S ðV ;MÞ;5: place all the edges with weight zero into M;

6: While M is not k-vertex connected

� add the edge in GnS with highest contribution to S

(If multiple edges has the same highest contribution,

add an edge with the least weight);

End While

7: Test each edge cuv (or uv�!) 2M in decreasing order of

weight, and delete cuv (or uv�!) from S if Sncuv (or uv�!) is

k-vertex connected;

8: Output S

Figs. 5 and 6 depict the performance of both WFA and

CFA for one-way and two-way MKCSG on the same

graphs. We can notice that the WFA always chooses edgeswith lower weights but the resulting graph may contain

more edges; while the CFA may choose some edges withhigher weight but the resulting graph contains fewer edges.

Such difference would make these two greedy algorithms

suitable for different applications, which is further dis-cussed in the following section.

On the other hand, the performance ratios of both CFA

and WFA can be arbitrarily large in some particular cases.For example, in the graph shown in Fig. 7a, to create a

2-vertex connected graph, WFA will choose edges cad and bce(both carry weight 1), and thus, find the optimal solution;while CFA will choose edge bcd with weight nðn > 2Þ and

result in a performance ratio of n2 . Furthermore, in the graphshown in Fig. 7b, to create a 2-vertex connected graph, CFA

will directly choose edge bcd with weight 2, which is the


Fig. 5. The 4-vertex connected undirected graphs computed using different greedy algorithms. Solid edges stand for the existing edges in the initialgraph. Dotted edges stand for the resulting edges chosen by different algorithms. (a) Initial graph. (b) 4-vertex connected graph computed by CFA.(c) 4-vertex connected graph computed by WFA.


optimal solution, while WFA will choose 4n edges withweight 1 and result in a performance ratio of 2n.

8 SIMULATION RESULTS

This section evaluates the performance of our algorithmsusing heuristic implementations described in Section 7. Weuse QualNet [14] as the simulation platform. In each ofthese simulations, we randomly place target nodes in a1;000 m� 1;000 m 2D terrain. To model an H-WSN, we setT ðminÞ ¼ 200 m and T ðmaxÞ ¼ 500 m, and let every targetnode uses a random transmission radius between T ðminÞand T ðmaxÞ in each simulation. Each of the resultspresented in this section is the average of 50 runs.

In the first simulation, we use T ðrelayÞ ¼ 350 m andgradually increase the number of target nodes in the networkfrom 5 to 50. Figs. 8 and 9 depict the performance of thepresented algorithm using different greedy subroutine forpartial fault tolerance and full fault tolerance, respectively.We have the following observations. 1) For partial faulttolerance, WFA uses slightly fewer relay nodes than doesCFA. This is so because WFA always tries to use the edgeswith lower weight first. For full tolerance, CFA outperformsweight-first subroutine since CFA uses few edges than doesWFA, and therefore, requires much fewer relay nodes in thefull-connection operation. 2) The number of relay nodescomputed by all the algorithms first increases, then decreasesrapidly when the network size goes beyond a threshold. Thisis so because when nodal density is below this threshold, theinitial network connectivity is low; therefore, when the scale

of networks increases, we need to deploy more relay nodes to

achieve the desired connectivity. However, when the nodal

density is beyond this threshold, the initial network con-

nectivity will increase when the nodal density increases, and

the number of relay nodes required also drops accordingly.

3) When the number of target nodes is small, two-way fault

tolerance always requires more relay nodes than one-way

fault tolerance. However, such difference will decrease when

the number of target nodes increases. We further continue

the first simulation by increasing the target nodes in the

network from 50 to 100, and the number of relay nodes

computed by all of the algorithms converges to small values

fewer than 9.In the second simulation, we gradually increase T ðrelayÞ

from T ðminÞ to T ðmaxÞ and evaluate the algorithms using

different subroutines for the connectivity level of k ¼ 4 in

two networks containing 20 and 60 target nodes, respec-

tively. As depicted in Figs. 10 and 11, the increase of

T ðrelayÞ can effectively improve the performance of all of


Fig. 6. The 4-vertex connected directed graphs computed using different greedy algorithms. Solid edges stand for the existing directed edges in theinitial graph. Dotted edges stand for the resulting edges chosen by different algorithms. (a) Initial graph. (b) 4-vertex connected directed graphcomputed by CFA. (c) 4-vertex connected directed graph computed by WFA.

Fig. 7. To create 2-vertex connected graph. Solid lines stand for existingedges. Dotted lines stand for available lines for choosing. (a) A particularcase where WFA finds the optimal solution, while CFA finds a solutionwith arbitrary approximation ratio. (b) A particular case where there existtotally nðn > 0Þ nodes between a and c as well as b and d. In this case,CFA finds the optimal solution, while WFA will find a solution witharbitrary approximation ratio.

Fig. 8. Performance of the presented algorithms for one-way and two-way partial fault tolerance. (a) Results computed using CFA. (b) Resultscomputed using WFA.

Fig. 9. Performance of the presented algorithms for one-way and two-way full fault tolerance. (a) Results computed using CFA. (b) Resultscomputed using WFA.


the described algorithms. Specifically, we have the follow-

ing observations:

1. WFA subroutine uses fewer relay nodes than doesthe CFA subroutine in partial fault tolerance, whileCFA subroutine outperforms WFA subroutine in fullfault tolerance.

2. In a sparse network with 20 target nodes, whenT(relay) increases fromT ðminÞ toT ðmaxÞ, the averageedge weight drops steadily, and the number of relaynodes computed by the One-way and Two-way PFRPalgorithms drops 29 and 41 percent, respectively.

3. In a dense network with 60 target nodes, most edgesselected by the One-way and the Two-way PFRPalgorithms carry weight one, and as a result, whenT ðrelayÞ increases, the performances of both One-way and Two-way PFRP algorithms remain stable.

4. For full fault tolerance, the increase of T ðrelayÞexponentially reduces the cost of the full-connectionoperation in Step 6 of Algorithm 3. Therefore, in asparse network, the number of relay nodes com-puted by the One-way FFRP and the Two-way FFRPdrops 70 and 65 percent, respectively; in a densenetwork, the number of relay nodes computed bythe One-way and the Two-way FFRP algorithmsdrops 82 and 75 percent, respectively.

5. We also noticed that for a given network, thenumber of relay nodes required to achieve full faulttolerance is always higher than that to achievepartial fault tolerance. This is also because full faulttolerance contains an extra step of full-connectionoperation in Step 6 of Algorithm 3, which requires alot of additional nodes, especially when the desirednetwork connectivity is high.

Overall, the simulation results show that WFA is moreefficient for partial fault tolerance, while CFA is more efficientfor full fault tolerance. Moreover, the expected behaviors ofthe described algorithms are much better than the perfor-mance ratios derived previously, which indicates that thesealgorithms work well in real sensor networking applicationswhere the target nodes are usually densely deployed.

9 CONCLUSION AND FUTURE WORK

This paper systematically addresses the problem of deploy-

ing a minimum number of relay nodes to achieve diverse

levels of fault tolerance in the context of heterogeneous

wireless sensor networks, where target nodes have different

transmission radii. The different transmission radii of the

target nodes introduce asymmetric communication links

between neighboring nodes, resulting in one-way and

two-way paths. The problem is further complicated by the

need to facilitate the desired fault-tolerant levels between

every pair of (target and/or relay) nodes, or every pair of

target nodes. Specifically, we develop Oð�k2Þ-approxima-

tion algorithms for the one-way and the two-way partial

fault-tolerant relay node placement, and Oð�k3Þ-approxima-

tion algorithms for the one-way and the two-way full fault-

tolerant relay node placement. Moreover, we analyze the

performance ratios of these algorithms in d-dimensional

networks ðd � 3Þ. To support realistic applications, we also

provide heuristic implementations of these algorithms, and

evaluate their performance via QualNet simulation. The

results show that performance of the proposed algorithms

is much better than the performance ratios derived,

suggesting that these algorithms work well in realistic

sensor networking scenarios. In ongoing work, we are

pursuing tighter performance ratios of the approximation

algorithms, as well as better heuristic implementations.

APPENDIX

PROOF OF LEMMA 3.4

Lemma 3.4. Let G ¼ ðN;EÞ be a directed k-vertex connected

graph. For two nodes u and v in N , if there are multiple

directed edges from u to v in G, we can keep one of these edges

and remove the others, and the resulting graph G0 ¼ ðN;E0Þ is

still a directed k-vertex connected graph.

Proof. Since the vertex-disjoint paths between other pairs of

nodes, rather than u and v, only use uv�! once, the

removal of the duplicate edges from u to v only degrades

the connectivity from u to v. We prove that in addition to

the edge uv�!, we can find another k� 1 vertex-disjoint

paths from u to v in G.Besides node v, node u must have at least k� 1 other

out-neighbors, denoted as fv1; v2; . . . ; vk�1g. For v1, thereare k vertex-disjoint paths from v1 to v in G, and at leastone of them, say P 1

Gðv1; v��!Þ, does not contain any nodes in

fu; v2; . . . ; vk�1g (there are in total k� 1 nodes infu; v2; . . . ; vk�1g). Therefore, we find the first pathP 1Gðu; v�!Þ ¼ uv1

�!þ P 1Gðv1; v��!Þ from u to v, as depicted in

Fig. 12a.


Fig. 10. Performance of the presented algorithms for one-way and two-way partial fault tolerance. The connectivity level k equals 4. N standsfor the network size. (a) Results computed using CFA. (b) Resultscomputed using WFA.

Fig. 11. Performance of the presented algorithms for one-way and two-way full fault tolerance. The connectivity level k equals 4. N stands forthe network size. (a) Results computed using CFA. (b) Resultscomputed using WFA.


Similarly, for node v2, there exist two paths P 1Gðv2; v��!Þ

and P 2Gðv2; v��!Þ which do not contain any nodes in

fu; v3; v4; . . . ; vk�1g. If one of P 1Gðv2; v��!Þ and P 2

Gðv2; v��!Þ is

vertex-disjoint with P 1Gðv1; v��!Þ, say P 1

Gðv2; v��!Þ, then we find

the second path P 2Gðu; v�!Þ ¼ uv2

�!þ P 1Gðv2; v��!Þ from u to v.

Otherwise, P 1Gðv1; v��!Þ intersects with both P 1

Gðv2; v��!Þ and

P 2Gðv2; v��!Þ, and we assume that P 1

Gðv1; v��!Þ first intersects

with P 1Gðv2; v��!Þ at node x before intersecting with

P 2Gðv2; v��!Þ. In this case, we change P 1

Gðu; v�!Þ to be the

combination of uv1�!, the portion from v1 to x on P 1

Gðv1; v��!Þ,

and the portion from x to v on P 1Gðv2; v��!Þ. As a result, we

can still find the second path P 2Gðu; v�!Þ ¼ uv2

�!þ P 2Gðv2; v��!Þ

from u to v, which is vertex-disjoint with the new

P 1Gðu; v�!Þ, as depicted in Fig. 12b.

Inductively, assume that we find m� 1 vertex-disjointpaths PO ¼ fP 1

Gðu; v�!Þ; . . . ; Pm�1G ðu; v�!Þg from u to v, where

PiGðu; v�!Þð1 � i � m� 1Þ correspondingly goes through vi.

Now, we show that we can find an mth path PmG ðu; v�!Þ,

which goes through vm.

Node vm has m vertex-disjoint paths to v, denoted asPM ¼ fP 1

Gðvm; v��!Þ; P 2Gðvm; v��!Þ; . . . ; Pm

G ðvm; v��!Þg, which do notinvolve any nodes in fu; vmþ1; vmþ2; . . . ; vk�1g. If one pathin PM is vertex-disjoint with all of the paths in PO, thenwe trivially finish the proof. Otherwise, we create anempty set and consider them� 1 paths in PO one by one.

First, we traverse along P 1Gðu; v�!Þ until one of the

following two situations occurs: 1) We meet the node v,

and in this case, we let node v1 use the path P 1Gðu; v�!Þ, and

2) we intersect with a path PjGðvm; v��!Þ in PM and denote

the intersecting node as I1st1;j , which stands for the first

intersecting node of P 1Gðu; v�!Þ and Pj

Gðvm; v��!Þ. Then, we

move PjGðvm; v��!Þ from PM to VPSET, and mark I1st

1;j as the

candidate node of path PjGðvm; v��!Þ.

Then, assuming that we have traversed along all of

the preceding paths before PlGðu; v�!Þ, we start traversing

along PlGðu; v�!Þ until one of following three situations

happens: 1) We meet node v, and in this case, we let vluse Pl

Gðu; v�!Þ, 2) we meet a path PpGðvm; v��!Þ in the PM and

then mark intersecting node I1stl;p as the candidate node of

PpGðvm; v��!Þ and move Pp

Gðvm; v��!Þ from PM to VPSET, and

3) we intersect with a path PqGðvm; v��!Þ in VPSET at

node I1stl;q . In this case, we compare the candidate node,

say Iythh;q , of path PqGðvm; v��!Þ with I1st

l;q , to check which node

is closer to v along path PqGðvm; v��!Þ. If Iythh;q is closer, we

continue to traverse along PlGðu; v�!Þ, as depicted in

Fig. 12c; otherwise, we change the candidate node of

PqGðvm; v��!Þ to be I1st

l;q and continue the traverse along

PhGðu; v�!Þ, as depicted in Fig. 12d. No matter whether we

continue to traverse along PlGðu; v�!Þ or along a previously

considered path PhGðu; v�!Þ, we will finish this round of

traversal once either one of the first two situations (i.e.,

(1) and (2)) happens. Furthermore, we move at most l

paths from PM to VPSET at the end of this round.

After traversing along all of the m� 1 paths in PO in

order, we have at least one path left in PM, say

PgGðvm; v��!Þ. We let node vm use the path uvm

��!þ PgGðvm; v��!Þ.

For an arbitrary path PiGðu; v�!Þ in PO, we stop traversing

along it when we meet node v or stop at the candidate

node Izthi;e of a path in VPSET, say PeGðvm; v��!Þ. For the first

case, we let node vi use the path PiGðu; v�!Þ; for the second

case, we let node vi use a new path which consists of uvi�!,

the portion from vi to Izthi;e on PiGðu; v�!Þ, and the portion

from Izthi;e to v on PeGðvm; v��!Þ. As a result, we find m new

paths from u to v, denoted as PN ¼ fPN1G ðu; v�!Þ;

PN2G ðu; v�!Þ; . . . ; PNm

G ðu; v�!Þg, and each path PNiG ðu; v�!Þð1 �

i � mÞ correspondingly goes through vi.

Finally, we prove that the m new paths in PN are

vertex-disjoint. Obviously, the path PNmG ðu; v�!Þ used by

vm is vertex-disjoint with all other paths. For any two

newly found paths PNaG ðu; v�!Þ and PNb

G ðu; v�!Þ, we first

consider the situation where both PNaG ðu; v�!Þ and

PNbG ðu; v�!Þ are the combination of one path in PO and

one path in PM. Assuming that PNaG ðu; v�!Þ consists of

uva�!, a portion on a path in PO (part A1), and a portion

on a path in PM (part A2); and assuming that PNbG ðu; v�!Þ

consists of uvb�!, a portion on a path in PO (part B1), and

a portion on a path in PM (part B2). First, A1 and B1

are vertex-disjoint. Second, since there are no other

intersecting nodes in A2 and B2, and A1 and B2 are

vertex-disjoint, so are A2 and B1. Lastly, A2 and B2 are

obviously vertex-disjoint. Therefore, PNaG ðu; v�!Þ and

PNbG ðu; v�!Þ are vertex-disjoint. If one or both of

PNaG ðu; v�!Þ and PNb

G ðu; v�!Þ are selected from PO, it is easy

to get the same conclusion using a similar analysis.Thus, based on the induction, besides the edge uv�!,

we can find another k� 1 vertex-disjoint paths from uto v, where each of them goes through one out-neighbor of u from v1 to vk�1. This completes the proofof Lemma 3.4. tu

ACKNOWLEDGMENTS

The authors would like to thank the anonymous reviewersand editors for their valuable comments. The work ofX. Han, X. Cao, and C.-C. Shen was supported in part bythe US National Science Foundation under grant CNS-0347460. The work of E.L. Lloyd was prepared throughcollaborative participation in the Communications and


Fig. 12. Explanation of Lemma 3.4.


Networks Consortium sponsored by the US Army Re-search Laboratory under the Collaborative TechnologyAlliance Program, Cooperative Agreement DAAD19-01-2-0011. The US Government is authorized to reproduce anddistribute reprints for Government purposes not with-standing any copyright notation thereon. The views andconclusions contained in this document are those of theauthors and should not be interpreted as representing theofficial policies, either expressed or implied, of the USArmy Research Laboratory or the US Government.Preliminary work of this paper has appeared as a regularpaper with the same title in IEEE INFOCOM 2007.

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Xiaofeng Han received the BS degree fromthe Hefei University of Technology, the MSdegree from Shanghai Jiaotong University,and the PhD degree from the University ofDelaware, all in computer science. His re-search interests include diverse optimizationproblems in mobile ad hoc wireless networksand wireless sensor networks.

Xiang Cao received the bachelor’s and master’sdegrees in computer engineering from TsinghuaUniversity, Beijing, China, and the master’sdegrees in computer science from the Universityof Delaware. He is a software engineer atMicrosoft Corporation, Washington. Prior toMicrosoft, he served as a software engineer atMotorola, New Jersey. His research interest is inthe ad hoc wireless network and wireless sensornetworks.

Errol L. Lloyd received the undergraduatedegrees in both computer science and mathe-matics from Penn State University and the PhDdegree in computer science from the Massa-chusetts Institute of Technology. He is aprofessor of computer and information sciencesat the University of Delaware. Previously, heserved as a faculty member at the University ofPittsburgh and as the program director forComputer and Computation Theory at the US

National Science Foundation (NSF). His research expertise is in thedesign and analysis of algorithms, with a particular concentration onapproximation algorithms. In 1989, He received the NSF OutstandingPerformance Award, and in 1994, he received the University ofDelaware Faculty Excellence in Teaching Award.

Chien-Chung Shen received the BS and MSdegrees from National Chiao Tung University,Taiwan, and the PhD degree from the Universityof California, Los Angeles, all in computerscience. He was a senior research scientist atBellcore Applied Research. He is now anassociate professor in the Department of Com-puter and Information Sciences, University ofDelaware, and a recipient of the US NationalScience Foundation CAREER Award. His re-

search interests include ad hoc and sensor networks, peer-to-peer anddistributed object computing, control and management of broadbandnetworks, and simulation. He is a member of the IEEE and ACM.

. For more information on this or any other computing topic,please visit our Digital Library at www.computer.org/publications/dlib.



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