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Optimal Cluster Association in Two-TieredWireless Sensor Networks

WeiZhao Wang∗ Wen-Zhan Song† Xiang-Yang Li∗ Kousha Moaveni-Nejad∗

Abstract—In this paper, we study the two-tiered wirelesssensor network (WSN) architecture and propose the optimalcluster association algorithm for it to maximize the overallnetwork lifetime. A two-tiered WSN is formed by number ofsmall sensor nodes (SNs), powerful application nodes (ANs),and base-stations (BSs, or gateways). SNs capture, encode,and transmit relevant information to ANs, which then sendthe combined information to BSs. Assuming the locationsof the SNs, ANs, and BSs are fixed, we consider how toassociate the SNs to ANs such that the network lifetime ismaximized while every node meets its bandwidth require-ment. When the SNs are homogeneous (e.g., same band-width requirement), we give optimal algorithms to maximizethe lifetime of the WSNs; when the SNs are heterogeneous,we give a2-approximation algorithm that produces a net-work whose lifetime is within 1/2 of the optimum. We alsopresent algorithms to dynamically update the cluster associ-ation when the network topology changes. Numerical resultsare given to demonstrate the efficiency and optimality of theproposed approaches. In simulation study, comparing net-work lifetime, our algorithm outperforms other heuristicsalmost twice.

Keywords— Network lifetime, wireless sensor networks,two-tiered, clustering.

I. I NTRODUCTION

The advances in Micro Electro-Mechanical Sys-tems (MEMS) technology, digital circuit design andwireless communication have enabled the built ofsmall, cheap and low power sensors,e.g., the Mica2and Mica2Dot motes [1]. With the introduction ofthese components, the new systems which are com-posed of thousands even millions of tiny computingdevices interacting with the environment and com-municating with each other becomes possible. Inthe meanwhile, several possible applications based onwireless sensor networks composed of thousands ofsuch tiny device are expected to come into practice inthe near future. These applications may include but

∗Illinois Institute of Technology, Chicago, IL, USA, Email:wangwei4@iit.edu ,xli@cs.iit.edu , moavkoo@iit.edu .The research of Xiang-Yang Li was supported in part by NSF CCR-0311174.∗Washington State University, Vancouver, WA 98686, Email:

songwz@wsu.edu . The research of Wen-Zhan Song is supportedby NASA ESTO 05-AIST05-0082

are not restricted to: (1)military applicationsin thebattle field, i.e., enemy surveillance, target tracking[2], [3] and countersniper systems [4]; (2)environ-mental monitoringin the countryside, i.e., microcli-mate monitoring on Great Duck Island, Maine [5],[6]; (3) structural monitoring and emergency rescue,i.e., structural health monitoring of the Golden GateBridge in San Francisco.

The low power of these tiny sensors raises a uniquechallenge for the large scale wireless sensor net-works. Even these small sensors are able to act as arouter to store and relay the data for other small sen-sors, it may take tens or even hundreds of hops for theinformation to reach the base station which is clearlynot affordable in most circumstances. Another con-cern is that these small sensors that are one-hop awayfrom the base station would have to route nearly everypiece of information generated by the sensors in thenetwork, which makes them run-out of power (calleddie) very quickly. Thus, the architecture of WSNsshould be revisited. Due to its possible massive size,it is natural to build the large scale WSNs in a hierar-chical model. In fact, several works have already ad-dressed different issues regarding this hierarchical ar-chitecture, including minimizing the number of clus-ters [7], [8], minimizing the total energy consumption[9] and maximizing the lifetime [10], [11]. Followingthe work in [10], we consider how to maximize thelifetime of a wireless sensor network with thousandsof tiny and simple sensor nodes (SNs) deployed ina region. In addition, there are several applicationnodes (ANs) who collect the information from smallsensors, process the collected information to producea local-view and send the information back to thebase-station(BS) if necessary. Every application nodecan communicate with the base station either by somedirect radio communication or routing through the re-lay of other application nodes. The base station oftencan connect to the Internet and is in charge of build-ing the global view of the WSN, controls the applica-tion nodes, sends the data to remote site and reports

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emergency events if necessary.

Usually the small sensors sleep for most of thetime, only periodically sample the data and send toapplication nodes. Thus, once deployed, they usu-ally could last several months even years before theirbattery needed to be replaced, if it is feasible. Al-though the application node has much more powercompared with the small sensor nodes, they also con-sume much more energy due to the following reasons:(1) the ANs need to wake up or stay idle for muchlonger time, which is usually proportional to the num-ber of small sensors in the cluster; (2) the ANs usuallyhave transmission of bitstreams over much greaterdistances; (3) the ANs need to do more extensive lo-cal computations. If not carefully designed, the appli-cation nodes usually can only survive a shorter periodof time than the small sensors can. On the other hand,if one or two small sensors die, the entire WSN willgenerally function properly while one or two appli-cation nodes running out of power could result in thecomplete lost of the coverage of certain area. Thus,here we focus on how to maximize the lifetime of ap-plication nodes instead of small sensors. In a WSN,usually a global view should be maintained with cer-tain quality in order to work properly. Rememberthat the global view is built from the local-views sentby those application nodes, thus, the lifetime of theWSN heavily depends on the lifetime of certain ANs.Therefore, we define the lifetime of a wireless sensornetwork as a certain function of the lifetime of appli-cation nodes. In [11], the authors study the effect ofthe position of the base station on the lifetime of theWSN. In their model, they assumed that there are al-ready some clusters and each cluster is composed ofone application node and several small sensors. Eachapplication node in a cluster can receive the informa-tion from all small sensors in the same cluster andsend it to the base station either by direct connectionor intermediate routing. However, how to form thecluster and what effect the cluster formation has onthe lifetime of the WSNs has not been answered. Inthis paper, we mainly focus on how to form the clus-ters efficiently and study the effect of cluster forma-tion on the lifetime of the WSN.

We categorize the WSN into four different cases:whether the SNs are homogeneous or heterogeneous,whether the ANs are homogeneous or heterogeneous.Here SNs are said to be homogeneous if the averageproduced data rates of SNs are same; otherwise, they

are said to be heterogeneous. The ANs are said tobe homogeneous if (1) the energy consumption func-tion of every AN (based on total amount of data pro-duced by the SNs in its cluster) is the same, and (2)the initial power of ANs are same; otherwise, theseANs are said to be heterogeneous. The main contri-butions of this paper are as follows. First, we presenta series of max-flow-based methods to form the clus-ters that optimally maximize the lifetime of the WSNwhen the small sensors are homogenous. We sepa-rately study the cases when the ANs are homogenousor heterogeneous. Since these methods are central-ized, we then present a new approach (called smooth-ing) with a low time complexity that is suitable fordynamic updating. When the small sensors are het-erogeneous we present an algorithm to form clusterswhose lifetime is no less than1

2of the optimum. We

then conduct extensive simulations to study the prac-tical performances of our method compared with theperformances of some simple heuristics. Our theoret-ical results are corroborated by our simulation stud-ies.

The remainder of the paper is organized as follows.In Section II, we present some preliminaries and pre-vious works. In Section III, we study the scenario inwhich the small sensors are the same. We then studythe generalized case when the small sensors could bedifferent in Section IV. We also discuss some otherissues in Section V. Extensive simulations have beenconducted to study the practical performances of ourproposed solutions. We conclude our paper in Sec-tion VII with some possible future directions.

II. PRELIMINARIES

A. Two-Tiered Wireless Sensor Networks

A two-tiered wireless sensor network (WSN) con-sists of a set of small sensor nodes (SN), denoted asSM = {s1, s2, · · · , sm}, a set of application nodes(AN), denoted asVN = {v1, v2, · · · , vn}, and at leastone base station (BS). The ANs and SNs formclus-ters, and in each cluster there are many SNs and oneAN. For simplicity, we assume that the applicationnodevi is in clusterCi and the set of small sensors inclusterCi is Si ⊆ SM . A small sensor, once triggeredby the internal timer or some external signals, startsto capture and encode the environmental phenomena(such as temperature, moisture, motion measure, etc)and broadcast the data directly to all ANs within itstransmission range and to certain ANs via the relay

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of some other neighboring sensors. Here, if ANvi

can receive the data from the small sensorsj, then wecall vi is a neighbor ofsj. Here sensorsj may have toreach ANvi via relay of other sensors. For notationalsimplicity, we useN(vi) to denote the neighboringsmall sensors of ANvi. Remember that although sev-eral ANs can receive the data packets from the smallsensorsj, only the AN in the same cluster assj pro-cesses the information. Here, we assume that onceformed, the cluster formation does not change overthe time. We also letri be the data-rate of the smallsensorsi generates andr(S) =

∑si∈S ri be the total

data-rates produced by a set of small sensorsS. Usu-ally, the data-rateri(t) is a function over the timetinstead of a constant. However, if we average the rateover a period of timeT , e.g., one day or one week,most often it is a constant. Thus, we can define therateri as the the average rate over a period of time,

i.e., ri =∫ T+T0

T0ri(t)

T. When receiving the raw data

from SNs from its cluster, an AN might create an ap-plication specific local-view for the whole cluster byexploring some correlations among the data sent bydifferent SNs. In the meanwhile, some data fusioncan be conducted by ANs to alleviate the redundancein the raw data sent by SNs. After an AN creates alocal-view of the data, it then forwards the informa-tion to a BS that generates a comprehensive global-view for the entire WSN. Notice that here an AN cancommunicate directly with a BS, or optionally, ANscan be involved in inter-AN relaying if such activitiesare needed and applicable.

B. Energy Model and Notations

In this paper, we assume that SNs wake up period-ically to collect, process and transmit the data to theANs. This assumption is natural and used often inpractice. For instance, the small sensors are config-ured to transmit once every5 mins in the Great DuckIsland project [6]. After the formation of the clusters,the ANs are able to decide the time slot in which theyneed to wake up to receive the data sent by SNs fromits own cluster. It is critical to reduce the wake uptime due to the high power consumption for idle lis-tening, i.e., sometimes it is as large as the receivingpower and about1/2 of the transmit power. Recentlyminimizing the wake up time has been addressed atdifferent layers [12], [13], [14]. In this paper, our fo-cus is on how to form clusters properly so that thenetwork lifetime is maximized. How to schedule the

wake-up time for SNs has been addressed extensively[15], [16], [17] and our formation could be coupledwith any of these schemes. It is reasonable to expectthat the live time of an ANdecreaseswhen the num-ber of small sensors in its clusterincreases. Thus, foran AN, its energy consumption mainly are composedof three parts: (1) the energy consumed to receive theinformation from the small sensors of its own clus-ter, (2) the overhearing cost incurred by those sensorsnot belonging its cluster and can reach this AN; (3)the energy consumed to process the information andthe energy consumed to send the information to thebase station; and (4) the energy consumed when theANs are idle listening. We implicitly assume that thepower consumption of (2) - (4) is a fixed value. En-ergy consumption for ANs in different applicationsand scenario may be different. Thus, we do not relyon any special assumptions about energy consump-tion. Given an ANvi, let Si ∈ SM be the set of smallsensors in its logical cluster. The power consumptionof the AN vi is a general functionpi(r(Si), N(vi)),wherer(Si) is the total data-rate of the small sensorsin Si. SinceN(vi) does not depend on the clusterformation and can be taken as a constant for a givenapplication nodevi, we can simplify the power con-sumption function aspi(r(Si)). This model takes intoaccount all other power consumption that is a fixedvalue. The only assumption in this paper is that func-tion pi(x) should satisfy thatpi(x) > pi(x

′) whenx > x′, i.e., the more information the small sensors inthe cluster generate, the more energy the applicationnode consumes. Notice that, the above monotone in-creasing property is only assumed to be true for eachAN. For two different ANsvi andvj, it is possiblethatpi(x) < pj(x

′) whenx > x′.

C. Lifetime of a Two-Tiered WSN

In this paper, we assume thatPi is the initialbattery power level of the application nodevi andpi(r(Si)) is its average energy consumption ratewhen the set of small sensorsSi is in the clusterCi. The lifetime of an individual ANvi is define asli = Pi

pi(r(Si)). According to thecriticality of a mis-

sion, several different definitions of sensor networklifetime has been defined in the literature.• CRITICAL APPLICATION NODE L IFETIME (CANLT):The mission fails when any AN runs out of energy,i.e., the lifetimeLN is LN = minN

i=1{li}. The firstAN that run out of energy are denoted as the critical

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AN. If mission fails whenα percentage ANs are runout of energy, then it is calledα-Critical ApplicationNode Lifetime (α-CANLT).• FULL COVERAGE L IFETIME (FCLT): A small sen-sor is called acoveredsensor if it has at least one aliveAN neighbor. The total sensing area of allcoveredsensors is called the covered area of the WSN here.The mission fails when the covered area of the WSNis smaller than the originally covered area. If missionfails when the ratio of the covered area over the orig-inally covered area is smaller than percentageβ, thenit is calledβ-Full Coverage Lifetime (FCLT).

In the literature, the definition of network lifetimeimplicitly assumes that small sensors have longerlifetime than application nodes. The reasons include,but are not limited to, several practical considera-tions: (1) In many applications, small sensors aredensely deployed to provide better tolerance. Hence,the small sensors could schedule themselves to sev-eral independent groups, and when one group is onduty, other groups can sleep. Thus, the lifetime ofsmall sensors in a group can be prolonged as a func-tional unit. (2) Application nodes need collect datafrom small sensors and send to base station whichmight be far away, which often costs more energy.

In this paper, we adopt the above definitions whenconducting theoretical analysis and simulations. No-tice that, there are also several other different defini-tions of lifetime of a wireless sensor network, e.g.,see [11]. Our analysis can be extended to those sce-narios similarly.

D. Previous Works

Numerous literatures have discussed efficient clus-ter formation for wireless ad hoc and sensor net-works. Although almost all works assumed that thereare some nodes acting asclusterheadswho are incharge of gathering the information from other nodesand sending back to some base stations, the criteriaof forming the clusters vary from case to case. Onefundemental difference between the cluster formationproblem studied in this paper and the traditional clus-ter formation problems is thateverynode could be aclusterhead in the traditional methods, while only theAN can be the clusterhead for the problems studiedhere.

In the Linked Cluster Algorithm (LCA) [7], a nodebecomes the clusterhead if it has the highest identityamong all nodes within one hop of itself or among all

nodes within one hop of one of its neighbors. This al-gorithm was improved by the LCA2 algorithm [18],which generates a smaller number of clusters. TheLCA2 algorithm elects the node, with the lowest IDamong all nodes which are not within1-hop of anychosen clusterheads, as a new clusterhead. The algo-rithm proposed in [19], chooses the node with high-est degree among its1−hop neighbors as a cluster-head. In [20], the authors propose a distributed algo-rithm that is similar to the LCA2 algorithm. The Dis-tributed Clustering Algorithm (DCA) uses weightsassociated with nodes to elect clusterheads [21]. Itelects the node that has the highest weight among its1-hop neighbors as the clusterhead. The DCA al-gorithm is suitable for networks in which nodes arestatic or moving at a very low speed. The Max-Min d−cluster Algorithm proposed in [22] generatesd−hop clusters with a run-time ofO(d) rounds. Allabove approaches are aiming to minimize the numberof clusters such that any node in any cluster is at mostd hops away from the clusterhead.

In [23], the authors proposed a clustering algorithmthat aims at maximizing the lifetime of the networkby determining optimal cluster size and optimal as-signment of nodes to clusterheads. They assumedthat the clusterhead consumes power to send the datato the nodes in its cluster via broadcast. Thus, thepower consumption of the cluterhead depends on itstransmission range, not on the number of nodes inthis cluster. This is fundamentally different from ourmodel in which the ANs consume power to processand relay the collected information that is closely re-lated to the number of small sensors in the cluster.

Results reported in [10], [11] are closest to this pa-per in spirit. In [10], Panet al. studied the problem ofmaximizing lifetime of a two-tiered WSN with focuson the top-tier. By assuming theprior knownfixedcluster formation, the authors mainly studied how toplace the base-station in the network such that thelifetime of the WSN is maximized. The ANs are as-sumed to be homogenous in [10] and generalized tobe heterogenous in [11]. The authors also discussedhow to relay the packets via ANs to some fixed basedstations. In this paper, we will focus on the lower-tier of the two-tiered WSN: how to form the cluster(associate small sensors to application nodes) so thenetwork lifetime is maximized.

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III. H OMOGENEOUSSMALL SENSORS

In this section, we study the case when the smallsensors are homogeneous,i.e., all small sensors havethe same data rate, sayr. Thusr(S) = r · |S|, where|S| is the number of small sensors in the setS. Wespecifically discuss how to maximize the lifetime un-der the critical application node lifetime (CANLT)definition when the application nodes are homoge-nous in subsection III-A and heterogenous in subsec-tion III-B.

A. Homogeneous Application Nodes

In this subsection, we discuss how to maximizethe lifetime of the WSN when all application nodesare homogeneous, i.e., their initial on-board energyare the same, sayP and the energy consumptionfunctions are the same, sayp(x). Remember thatLN = minn

i=1{li} = minni=1

Pp(r·|Si|)) andp(x) is in-

creasing. Thus maximizing the lifetimeLN of theWSN is equivalent to minimizing the maximum clus-ter size. For simplicity, we denotexi,j = 1 if thesensorsj belongs to clusterCi, andxi,j = 0 other-wise. LetN(vi) be the set of sensors who arevi’sneighbors. We formalize the problem of maximizingLN as the following Integer Programming (IP).

min maxvi∈VN

∑sj∈SM

xi,j (1)

Subject to constraints

xi,j = 0,∀vi,∀sj 6∈ N(vi); (2)

xi,j ∈ {0, 1},∀sj,∀vi; (3)∑vi

xi,j = 1,∀sj (4)

Obviously, a feasible solution of the above IP prob-lem is afeasiblecluster formation. For simplicity, theset of small sensors in the clusterCi is denoted asSi

in this paper, when no confusion is caused. Next wepresent two different approaches to solve the aboveIP exactly.

A.1 Efficient Centralized Approach

First, we use a Max-Flow approach to solve the IP(1). By adopting the traditional techniques and re-laxing the constraintsxi,j ∈ {0, 1} to 0 ≤ xi,j, wetransform the IP (1) to a linear programming as fol-lows:

min T (5)

subject to constraints

xi,j = 0,∀vi,∀sj 6∈ N(vi); (6)

xi,j ≥ 0, ∀vi, ∀sj; (7)∑sj∈SM

xi,j ≤ T, ∀vi; (8)

∑vi

xi,j = 1,∀sj (9)

Given the linear programming, we first constructa flow network as shown in Figure 1 withs as thesource andt as the sink. There is a directional link−→svi, 1 ≤ i ≤ n with capacityk, a directional linkbetween−−→visj with capacity1 if sj ∈ N(vi) and a di-rectional link

−→sjt with capacity1. If T ≤ k, then a

solutionx of the above linear programming implies afeasible flow for the corresponding flow network de-fined in Figure 1:xi,j is the flow fromvi to sj. On theother hand, whenk ≤ T , a feasible flowf with totalflow m for flow network defined in Figure 1 implies afeasible solution for the linear programming also:xi,j

is the flowf(vi, sj). In the following Lemma 1 weshow the relation between a feasible solution to thefeasible system (6) and the maximum network flowin the network shown in Figure (1).

Lemma 1:If we fix T = k, then there is a feasiblesolution to the FS (6) if and only if the maximum flowof the network illustrated by Figure 1 ism, wheremis the cardinality ofSM .

Proof: The proof that if there is a feasible solu-tion to the FS (6) then the maximum flow of the net-work illustrated by Figure 1 ism is trivial and omit-ted here. We only prove that if the maximum flowof the network illustrated by Figure 1 ism then thereis a feasible solution to the FS (6). Without loss ofgenerality, letfi,j denote the flow on linkvisj andobviously, fi,j = 0 for all vi and sj 6∈ N(vi) be-cause the link−−→visj does not exist. Notice that for ev-ery nodevi, all the inflow comes from link−→svi, thus∑

sj∈SMfi,j ≤ k for each nodevi. If the cardinality

of the flow ism, then the flow on each link−→sjt is 1

which implies that∑

vixi,j = 1. This proves thatfi,j

is a feasible solution to FS (6).Usually, for a maximum flow problem, the flow on

each directional link could be any real number. For-tunately, the solution generated by the flow-network,illustrated by Figure 1, has a well-known property.

Lemma 2:If the capacity function takes on onlyintegral values, then the maximum flowf has the

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1

v

2v

iv

nv

1s

2s

js

ms

S T

k

k

k

k

1

1

1

1

1

1

1

1

1

1

1

1

v

2v

iv

nv

1s

2s

js

ms

k 1

k 2

k n−1

k n

S T1

1

1

1

1

1

1

1

1

1

1

(a) homogeneous SNs (b) heterogenous SNs

Fig. 1. A flow network for two-tiered WSN.

property that|f | is integer-valued. Moreover, for allverticesu andv, the flow on edgeuv is an integer.

Lemma 2 immediately shows that the flow on eachlink −−→visj is either0 or 1. Remember that the flowon link −−→visj corresponds toxi,j, which implies thatxi,j ∈ {0, 1} for every vi and sj ∈ N(vi). Thus,a feasible solution to FS (6) also satisfies the con-straints (2). Letxmin be the solution to IP (1) andTmin = min maxvi∈VN

∑sj∈SM

xmini,j .

If we know the exact value ofTmin, then we canfind xmin by solving the maximum flow problem innetwork (1). Remember thatTmin is a non-negativeinteger and is at mostm, thus by performing a binarysearch onTmin we can find the exact value.

We can find the solution to IP (1) by solvinglog m max-flow problems for different values ofT .Thus, the time complexity for Max-Flow approach ism · log m · (n+m)3, which is very expensive and im-practical. Notice that the cluster formation problemwith minimum cluster size becomes the MaximumCardinality Matching problem in a bipartite graph[24]. In [24], Hopcroft and Karp presents the bestknown algorithm that achieves the time complexity√

m · nm. This reduces the time complexity fromO((n + m)3) to O(nm · (n + m)1/2 log(n + m)) fora fix valueT . Therefore, we can solve the IP (1) intimeO(n ·m3/2 log2(m)).

A.2 Efficient Distributed Algorithm by Smoothing

Although the previous approach computes a clus-tering quickly in centralized manner, it may be tooexpensive to collect the necessary information. In thissubsection, we propose a different approach that canbe implemented efficiently in a distributed manner.The basic idea of this approach is to construct a vir-tual directed graph on ANs and iteratively move thesensors from those clusters who have the largest num-ber of small sensors to smaller clusters. In the virtualdirected graph, there is an edge−−→vivk from AN vi to

vk if there is a sensorsj that can be moved from thecluster ofvi to the cluster ofvk. The weight of theedge is the number of such small sensors that can bemoved from the cluster ofvi to the cluster ofvk. Fol-lowing algorithm presents the method constructing avirtual graph based on a feasible solutionx to FS (2).

Algorithm 1 Constructing the virtual graphInput: A set of ANsVN , a set of small sensorsSM

and a feasible solutionx, e.g., assigningsj randomlyto avi wheresj ∈ N(vi).Output: A directed virtual graphV G(x).1: SetVN as the vertices for virtual graphV G.2: for every pair ofvi andsj such thatxi,j = 1 do3: for everyvk such thatsj ∈ N(vk) do4: if there is no directed edge−−→vivk from vi to

vk then5: Add a directed edge−−→vivk from vi to vk.

Set the weight of the edge toc(vivk) = 1.6: else7: Update the weight asc(vivk) = c(vivk)+

1.

In the directed virtual graphV G(x), if there is apath fromvi to vj, then we sayvi reachesvj. Allvertices thatvi can reach forms a setRi(x), calledthecliquecentered at the ANvi. Given a solutionx ofFS (2) and its corresponding virtual graphV G(x), wehave the following property about cliques (its proof isomitted due to space limit).

Lemma 3:Given a feasible assignmentx of smallsensors to ANs and its corresponding virtual graphV G(x), for any ANvi and its cliqueRi(x) in V G(x),if x′ is also a feasible assignment of SNs to ANs, wehave

∑vj∈Ri(x) |Si(x)| ≤ ∑

vj∈Ri(x′) |Si(x′)|

The Algorithm relies on the relation betweenωi(x), ωj(x) andTmin wherevi is the AN with thelargest weight andvj is the AN with the smallestweight inRi(x).

Lemma 4:Let vi be the AN with the largest weightandvj be the AN with the smallest weight inRi(x)under any feasible assignmentx, then |Sj(x)| ≤Tmin ≤ |Si(x)|.

Proof: Remember thatvi has the maximumweight among all ANs, thusTmin ≤ |Si(x)| triviallyholds. Therefore, we only need to prove|Sj(x)| ≤Tmin. We prove this by contradiction. For the sakeof contradiction, we assume all ANs reachable byvi

has a weight greater thanT . From the assumption

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that |Sk(x)| > Tmin for everyvk ∈ Ri(x), we haveS(Ri(x)) =

∑vk∈Ri(x) |Sk(x)| > Tmin · |Ri(x)|,

where|Ri(x)| denotes the number of ANs reachableby vi.

Let xmin be the solution to IP (1). FromLemma 3, we obtain that

∑vk∈Ri(x) |Sk(x

min)| ≥∑vk∈Ri(x) |Sk(x)| > Tmin · |Ri(x)|. Remember

that Tmin · |Ri(x)| ≥ ∑vk∈Ri(x) |Sk(x

min)|. ThusTmin · |Ri(x)| > Tmin · |Ri(x)|, which is a contradic-tion. This finishes our proof.

Given a virtual graph constructed by Algorithm 1based on a feasible assignment of SNs to ANs, ourapproach to find a better solution is to iteratively ap-ply a process calledSMOOTH to reduce the maximumweight of the application nodes if possible. Here, theweight of an application nodevi under a feasible as-signmentx is the number of small sensors assignedto the clusterCi, denoted asωi(x).

Algorithm 2 Smooth AlgorithmInput: A feasible assignmentx.Output: A solution to IP (1).1: Construct virtual graphV G(x) based on the fea-

sible assignmentx using Algorithm 1.2: repeat3: Find the AN with the largest weight, sayvi. If

there are more than one such ANs, choose onerandomly.

4: Find the AN with the smallest weight inRi(x), sayvj. If there are more than one suchANs, choose one randomly.

5: Apply procedureSMOOTH(vi, vj, V G(x),x).6: until ωi(x) ≤ ωj(x) + 1

Theorem 5:Algorithm 2 terminates after at mostm iterations, with an solution to IP (1).

Proof: From Lemma 4, we haveωj(x) ≤Tmin ≤ ωi(x). If Algorithm 2 does not stop at thisiteration, we haveωi(x) > ωj(x) + 1, which impliesthatTmin > ωj(x) + 1. For a feasible solutionx, wedefineδi(x) = |Si(x)| − |Si(x

min)| if ωi(x) > Tmin

and0 otherwise. Let∆(x) =∑

vi∈VNδi(x), it is not

difficult to observe that∆(x) will be decreased by1for each iteration. Thus, Algorithm 2 terminates afterat mostm iterations.

Remember when Algorithm 2 terminates, we haveωi(x) ≤ ωj(x) + 1. Combining with the rela-tion ωj(x) ≤ Tmin ≤ ωi(x), we haveωj(x) ≤Tmin ≤ ωi(x) ≤ ωj(x) + 1. This implies that

Algorithm 3 SMOOTH(vi, vj, V G(x),x)

Input: A feasible assignmentx and its correspondingdirected virtual graphV G(x), a pair of nodesvi andvj.1: Let vi0vi1 · · · vik be the path connectingvi andvj

with the minimum number of hop. Here,vi0 = vi

andvik = vj.2: for t = 0 to k − 1 do3: Assume thatxt,l = 1 for some SNs` with

s` ∈ N(vt) ands` ∈ N(vt+1). Setxt,l = 0andxt+1,l = 1, i.e., moves` from clusterCt toclusterCt+1.

4: for everyva such thats` ∈ N(va) do5: Updatec(−−→vitva) = c(−−→vitva) − 1. Remove

directed link−−→vitva if c(−−→vitva) = 0.

6: Update−−−−−→c(vit+1va) = c(−−−−→vit+1va) + 1. Add a

directed link−−−−→vit+1va if c(−−−−→vit+1va) = 1.7: Setωj(x) = ωj(x) + 1 andωi(x) = ωi(x)− 1.

Tmin = ωi(x) − 1 or Tmin = ωi(x). First, weconsider the case whenTmin = ωi(x) − 1. Inthis case, we haveωj(x) ≥ Tmin for every vj ∈Ri(x) which implies

∑vj∈Ri(x) |Si(x)| ≥ Tmin ·

|Ri(x) − 1| + Tmin + 1. For the solutionxmin

of IP (1), every AN’s weight is not greater thanTmin. Thus,

∑vj∈Ri(x) |Si(x

min)| ≤ Tmin · |Ri(x)|.From Lemma 3, we have

∑vj∈Ri(x) |Si(x

min)| ≥∑vj∈Ri(x) |Si(x)|. This implies thatTmin · |Ri(x)| ≥

Tmin · |Ri(x) − 1| + Tmin + 1, which is a contra-diction. Thus,Tmin = ωi(x). Remember thatx is asolution to the FS (2). Therefore,x is a solution to IP(1). This finishes our proof.

Now we analyze the time complexity of Algorithm2. In procedureSMOOTH(vi, vj, V G(x),x), there areat mostn nodes on the path betweenvi, vj and up toniterations in the “FOR” loop between line4-7. Thus,the time complexity ofSMOOTH(vi, vj, V G(x),x) isO(n2). From Theorem 5, it takes at mostO(m · n2)for Algorithm 2 to terminate. Constructing the virtualgraph based on a feasible solutionx could take timeO(m·n2). Thus, the total time complexity of smooth-ing algorithm is alsoO(m · n2). If n = o(

√m), then

Algorithm 2 outperforms the best known max-flowalgorithm bylog2 m; whenn is a constant the timecomplexity becomesO(m) which is optimal.

8

A.3 Efficient Distributed Implementation

So far we have illustrated the basic idea of theSmoothing algorithm, which clearly can be imple-ment in a distributed manner. In the remainder of thesection, we will describe how this method can be im-plemented efficiently. Given an ANvi, we sayvi isadjacent to ANvj if there is a small sensorsk in theclusterCi

⋂N(vj). If vi andvj are not adjacent, then

we define the distance betweenvi andvj as the small-est number of hops between them if we consider theadjacent graph of the ANs. For an ANvi that is adja-centvj, let ` be the largest non-negative integer such

thatpj(r·

∑sk∈SM

xj,k+`·r)Pj

< ωi(x). We define thedif-

ferenceof vi andvj asdifi,j(x) = `. Based on thenotation of difference, we have following localizedalgorithm.

Algorithm 4 Distributed Smoothing algorithm forAN vi

Input: An initial assignmentx, γi,j for every adja-cent AN vj that is the number of sensors that are inN(vi)

⋂ Cj.1: When vi receive an UPDATE-LEAVE or

UPDATE-JOIN message from an adjacent ANvj, it updatesγi,j if necessary.

2: Let vj be one ofvi’s adjacent AN with the maxi-mum difference. Here, we break the tie arbitrar-ily.

3: if difi,j(x) ≥ 1 then4: Send a REQUEST message to ANvj.5: When vj receives all REQUEST messages the

ANs that adjacent to it, it sends out an ACK mes-sage to the AN that has the maximum weight andREJECT messages to all other ANs.

6: if vi receives an ACK message fromvj then7: Choose one SN, saysk, in Ci

⋂N(vj). Set

xi,k = 0 and send SUCC message with the IDk to vj.

8: Upateγi,j = γi,j − 1 and send the UPDATE-LEAVE message with IDk to all adjacentANs.

9: When vj receives the SUCC message fromvi

with ID k, it first setsxj,k = 0 and γ(j, i) =γ(j, i) + 1. After that it also sends UPDATE-JOIN message with IDk to all adjacent ANs.Remark: Afterward, we also say that the smallsensorsk is migratingfrom clusterCi to Cj.

Regarding the distributed Algorithm 4, we have thefollowing theorem.

Theorem 6:Algorithm 4 converges in at mostm·nrounds and total message complexity isO(n2 ·m) ifthe ANs are homogeneous.

Proof: Given an assignmentx, we denoteκi(x)as the number of small sensors inith largest cluster.LetΓi(x) =

∑ij=1 κj(x), andxk be the assignment of

sensors in roundk. ConsideringΓk =∑n

i=1 Γi(xk).

If there is a small sensor joiningCik and leavingCjk

in roundk, then|Sik | > |Sjk | + 1 andik < ik. No-tice that after the small sensor migrating from clus-ter Cik to Cjk , Γ`(x

k) decreases by1 if j < ` ≤ iand does not change otherwise. Thus,Γk decreasesby 1 for every small sensor migrating. It is not diffi-cult to observe that if there is no small sensor migrat-ing in roundk, then Algorithm 4 terminates. SinceΓ1 < n ·m, Algorithm 4 terminates in at mostn ·mrounds.

In every round, every AN sends only one RE-QUEST message and receives at most one REJECTmessage. Thus, there is at mostO(n) REQUESTand REJECT messages. It is also not difficult to ob-serve that every AN sends at most one ACK mes-sages. Thus, there are at mostO(n2 · m) RE-QUEST, ACK and REJECT messages in total. Onthe other hand, there is exact one UPDATE-LEAVEand UPDATE-JOIN message for every small sensormigrating. Thus, there are at mostO(n·m) UPDATE-LEAVE and UPDATE-JOIN messages. Therefore,the overall message complexity isO(n2 ·m).

Notice that the message complexity analysis isvery pessimistic. In simulations, it is much smallerthan the worst case analysis. Observe that when Al-gorithm 4 terminates, it not necessarily gives an op-timal solution. However, Algorithm 4 gives the bestsolution among all localized algorithms in which ev-ery AN can only know the information of its adja-cent ANs. Furthermore, if we define the diameter ofthe network as the largest distance of the ANs, wehave the following theorem (its proof is omitted dueto space limit).

Theorem 7:When Algorithm 4 terminates, it givesan assignment with maximum cluster size at mostT ≤ Tmin + D whereD is the diameter of the net-work.

9

A.4 Dynamic updating

In wireless sensor networks, some old sensors mayrun out of battery and new sensors may be deployedfrom time to time. Furthermore, in certain appli-cations, some sensors are able to move. Thus, itis necessary to consider how to dynamically updatethe cluster when the WSN changes. Here, we focuson the case when single small sensor leaves or joinsthe WSN. If there are more multiple sensors joiningand/or leaving, it could be easily be reduced to thesingle sensor case. When an AN node runs out ofpower, the small sensors in its cluster has to be re-assigned,i.e., it is treated as the case that a group ofsmall sensors joins the networks. We discuss howto dynamically update the cluster formation after theformation of the cluster according to the localized Al-gorithm 4 by cases.

Small sensor leaving. In this case, we assumethe sensorsk originally belongs to clusterCj. Theweight of vj is decreased by1. Assumevi is theAN with the largest weight that is adjacentvj. Ifωj(x) ≥ ωi(x) then vj sends an UPDATE-LEAVEmessage with IDk to every adjacentva with ID k.Otherwise, we first choose any arbitrary small sensors` ∈ Ci

⋂N(vj). Removes` from Ci and assign it to

Cj. vj sends an UPDATE-LEAVE message with IDk and an UPDATE-JOIN message with ID̀to everyadjacent AN;vj sends an UPDATE-LEAVE messagewith ID ` to every adjacent AN.

Small sensor joining. In this case, we assume thesensorsk joins the wireless sensor network. Find theAN vi with the minimum weight such thatsk ∈ N(vi)and assignsk to clusterCi. vi sends out an UPDATE-JOIN with ID k to every adjacent AN.

For any individual small sensor leaving or joining,the clustering terminates in at mostD rounds and theoverall message complexity is at mostO(n·D) whereD is the diameter of the network. However, in sim-ulations, we found that the message complexity andconvergence rounds are bothO(1) most of the time.

B. Heterogeneous Application Nodes

In subsection III-A, we discuss how to form theclusters when both the small sensors and applicationnodes are homogeneous. However, in practice, suchnode homogeneity cannot always be guaranteed. Forexample, the initial onboard energy of ANs built bydifferent vendors may not be proportional to the bit-rate at which they generate, or the application nodes

could be redeployed (e.g., new ANs join the systemlong after old ANs have been activated). Further-more, two different application nodes may consumedifferent energy to receive, process and send the in-formation to the base station even given the same setof small sensors. Thus, it is more practical to assumethe application nodes are heterogenous. In this paper,we consider the heterogeneity in two ways: the initialon board energyP and energy consumption functionp(x) wherex is the sum of the rate of the small sen-sors in the cluster.

In this subsection, we redefineweightof a AN vi

for assignmentx asωi(x) =pi(r·

∑sj∈SM

xi,j)

Pi, where

Pi is the initial onboard energy andpi(x) is energyconsumption function. Here, the lifetime of the net-work is defined as

L = max minvi∈VN

Pi

pi(r ·∑

sj∈SMxi,j)

= min maxvi∈VN

ωi(x).

Thus maximizing the lifetime is equivalent to min-imizing the maximum weight over all ANs. Simi-lar to the approach for the homogenous applicationnode case, we formalize the problem as an IntegerProgramming as follows.

min maxvi∈VN

pi(r ·∑

sj∈SMxi,j)

Pi

(10)

Subject to constraint

xi,j = 0,∀vi,∀sj 6∈ N(vi); (11)

xi,j ∈ {0, 1}, ∀sj,∀vi; (12)

and∑vi

xi,j = 1,∀sj (13)

B.1 Max-Flow Approach

Similar to the homogenous case, by adopting thetraditional techniques, we transform the IP (1) into anew IP as follows:

min T (14)

Subject to constraint

xi,j = 0,∀vi,∀sj 6∈ N(vi); (15)

xi,j ∈ {0, 1}, ∀sj,∀vi; (16)∑vi

xi,j = 1, ∀sj; (17)

∑sj∈SM

xi,j ≤ ki,∀vi (18)

10

Hereki =p−1

i (T ·Pi)

r. Then

∑sj∈SM

xi,j ≤ ki is equiv-

alent topi(r·

∑sj∈SM

xi,j)

Pi≤ T sincepi() is assumed to

be monotone non-decreasing.Let xmin be the solution to IP (14) andTmin be the

minimum weight of the ANs. Unlike in the homoge-nous SN case, the valueTmin could be any positivereal number here. Thus, the simple binary search onTmin does not work. However, since there must existan indexi such that

∑sj∈SM

xi,j = ki, we can guesssuch index fromi = 1 to n then perform a binarysearch on

∑sj∈SM

xi,j. Therefore, we only need todecide whether there is a solution for the followingFeasible System for a givenT .

xi,j = 0,∀vi,∀sj 6∈ N(vi); (19)

xi,j ∈ {0, 1},∀sj,∀vi; (20)∑vi

xi,j = 1,∀sj; (21)

∑sj∈SM

xi,j ≤ ki, ∀vi (22)

Here, we construct a network withs as the sourceand t as the sink as shown in Figure 1. There isa directional link−→svi (1 ≤ i ≤ n) with capacityki, a directional link between−−→visj with capacity1 ifsj ∈ N(vi) and a directional link

−→sjt with capacity1.

Similar to Lemma 1 for homogenous case, we havethe following lemma. The proof is straightforwardand is omitted here.

Lemma 8:There is a solution to FS (19) for a givenT if and only if the maximum flow of the network (1)is m, wherem is the cardinality ofSM .

By guessing theT for log m · n time, we can findthe solution to Integer Programming (10) by solv-ing log m · n bipartite max-flow problems. Thus, thetime complexity for Max-Flow approach isO(n2 ·m3/2 log2(m)) which is very expensive and imprac-tical.

B.2 Smoothing Algorithm

In this subsection we continue to show that oursmoothing Algorithm 2 also applies to the heteroge-nous case with only minor modification.

Lemma 9:Let vi be the AN with the largest weightand vj be the AN with the lowest weight that isreachable byvi in a feasible assignmentx. Thenωj(x) ≤ Tmin ≤ ωi(x).

Algorithm 5 Smoothing algorithm for heterogenousANsInput: An Integer Programming (10).Output: The solution to Integer Programming (10).1: Find a feasible solutionx, e.g., randomly assign

every SN to a neighboring AN.2: Construct a virtual graphV G(x) based onx by

applying Algorithm 1.3: repeat4: Choose any one of AN with the largest weight

randomly, sayvi.

5: Defineω+k (x) =

pk(r·∑sj∈SMxk,j+r)

Pk.

6: Find the AN vj with the smallestω+j (x) in

Ri(x). If there are more than one such ANs,choose one randomly.

7: Apply procedureSMOOTH HETE(vi, vj, V G(x),x) ifωi(x) > ω+

j (x)

8: until ωi(x) ≤ ω+j (x)

Algorithm 6 ProcedureSMOOTH HETE(vi, vj, V G(x),x)

Input: A feasible solutionx of FS (11) and its cor-responding directed virtual graphV G(x), a pair ofnodesvi andvj.1: Letvi0(vi)vi1 · · · vik(vj) be the path connectingvi

andvj with the minimum number of hop. Here,vi0 = vi andvik = vj.

2: for t = 0 to k − 1 do3: Assumext,l = 1 ands` ∈ N(vt+1). Setxt,l =

0 andxt+1,l = 1.4: for everyva such thats` ∈ N(va) do5: Updatec(vitva) = c(vitva) − 1. Remove

directed linkvit(va) if c(vitva) = 0.6: for everyvb such thats` ∈ N(vb) do7: c(vit+1vb) = c(vit+1vb) + 1. Add a directed

link vit+1vb if c(vit+1vb) = 1.8: Update ωj(x) = ω+

j (x) and ωi(x) =pi(r·

∑sj∈SM

xi,j−r)

Pi.

11

Theorem 10:Algorithm 5 outputs a solution of IP(14) and terminates afterm iterations.

The proof of this theorem is omitted here due tospace limit. Surprisingly, the time complexity of Al-gorithm 5 is alsoO(m ·n2), which is exactly the sameas in the homogenous case. This reduces the timecomplexity by an order of

√m log2 m and more im-

portantly, Algorithm 4 also works for the heteroge-nous case with only modification of the definition ofdifference. However, we only have the following con-jecture for the convergence and message complexityof localized smoothing algorithm. It is an open andinteresting problem to either prove or disprove thefollowing conjecture.

Conjecture 1:Algorithm 4 terminates after at mostn ·m rounds and the total message complexityO(n2 ·m) when the ANs are heterogenous.

IV. H ETEROGENEOUSSMALL SENSORS

Usually in WSNs, several different kinds of sen-sors cooperate together to fulfill some certain goals.Some sensors may generate data at a higher rate thanothers do,e.g., the visual sensors have a bit-rate that ismuch higher than the bit-rate generated by a temper-ature sensor. Even in scenarios when all small sen-sors are of same type, sometimes sensors located atdifferent locations may need to sample the data at adifferent time interval. Thus, it is more reasonable toassume that in a WSN different type of sensors pro-duce different bit-rates.

By assuming that every small sensor has its owndata rateri, we formalize the problem of maximizingthe lifetime as an Integer Programming as follows:

min maxvi∈VN

pi(∑

sj∈SMrj · xi,j)

Pi

(23)

Subject to constraint

xi,j = 0,∀vi,∀sj 6∈ N(vi); (24)

xi,j ∈ {0, 1},∀sj,∀vi; (25)∑vi

xi,j = 1,∀sj (26)

Unlike the case for homogenous SNs in which wecan find the solution that maximizes the lifetime ex-actly, Theorem 11 shows that it is NP-Hard to find thesolution to IP (23).

Theorem 11:We can not find the solution of IP(23) in polynomial time ifP 6= NP .

Proof: We consider the special case when ap-plication nodes are homogeneous. In this case, sincepi(x) = p(x) is increasing, it is equivalent to mini-mizing the maximum

∑sj∈SM

rj · xi,j subject to con-straints (24). If every ANvi satisfies thatN(vi) =SM − vi, then the problem becomes the traditionaljob scheduling problem [25], [26], which is known tobe NP-Hard. This finishes our proof.

Since solving IP (23) is NP-hard, we will presentan algorithm approximating the optimal solution byborrowing some ideas from job scheduling [27], [28].Again we transform IP (23) into Integer Program-ming (27) as follows.

min T (27)

Subject to constraints

xi,j = 0,∀vi,∀sj 6∈ N(vi); (28)

xi,j ∈ {0, 1}, ∀sj,∀vi; (29)∑vi

xi,j = 1, ∀sj; (30)

∑sj∈SM

rj · xi,j ≤ ki,∀vi (31)

Hereki = p−1i (Pi · T ). Let xmin be the solution to

IP (27) andTmin be themin T under solutionxmin.It is easy to observe thatxmin

ij satisfies the followingconstraint.

xi,j = 0 ∀vi, ∀sj rj > ki (32)

If we relax the constraintxi,j ∈ {0, 1} we obtain aLinear Programming as follows.

min T (33)

Subject to constraints

xi,j = 0,∀vi,∀sj 6∈ N(vi); (34)

xi,j ≥ 0, ∀vi, ∀sj; (35)∑vi

xi,j = 1, ∀sj; (36)

∑sj∈SM

rj · xi,j ≤ ki;∀vi (37)

Let x? be the solution to LP (33) plus constraint 32andT ? be the value ofmin T under solutionx?. ThenT ? ≤ Tmin. By binary search onT ? we can find the

12

solutionx? to LP (33) plus constraint 32 in polyno-mial time. Furthermore, we can find a solutionx? thathas some special properties. For a small sensorsj, ifthere exists an ANvi such that0 < xi,j < 1, we callsj is fractionally assigned to clusterCi. We constructa graph with vertexVN

⋃SM and add an edgesjvi if

and only if0 < xi,j < 1. Obviously, it is a bipartitegraph and it is generally known [28], [29] that we cantransform the solutionx? to another solutionx∗ suchthat its corresponding bipartite graph is composed offorests with(or without) a line. Remember that ev-ery node inSM connects to at least two nodes inAN ,thus there is a matching such that every node inSM

can connect to a distinct node inAN . The final solu-tion is to assignsj to cluster with headvi if one of thefollowing two conditions holds:• x∗ij = 1• sj is connected withvi in the matching.

In this section, to make sure that we can guar-antee the performance of the above job-schedulingbased approach, we add one more requirement forthe power consumption functionpi. We assume thatthe marginal cost ofpi(x) is not increasing,i.e., forx1 ≥ x2, pi(x1 + δ)− pi(x1) ≤ pi(x2 + δ)− pi(x2).This assumption is almost universally satisfied. Ifthis assumption is not satisfied, we can construct ex-amples to show that the above approach (based onjob scheduling) cannot provide any theoretical perfor-mance guarantees, although its practical performancemay still be good.

Theorem 12:Our job scheduling based methodproduces a cluster formation such that the lifetime ofthe WSN is at least1

2of the maximum lifetime of the

WSN.Proof: For any ANvi, there is at most ones`

connecting tovi in the matching andr` ≤ ki. Thus,∑sj∈SM

rj · xi,j =∑

sj∈SM\r`rj · xi,j + r` · xil ≤

2ki. Therefore,pi(∑

sj∈SMrj · xi,j) ≤ pi(2ki) ≤

2pi(ki) = 2Pi · T ? ≤ 2Pi · Tmin. This finishes theproof.

V. OTHER ISSUES

In previous sections, we proposed several methodsfor clustering formation for two-tiered wireless sen-sor networks. Although we have addressed the het-erogeneous application nodes and heterogeneous sen-sor nodes, there are still lots of interesting questionsleft for further research.

Multihop Network of SNs: In previous sections,

we assumed that every SN can reach at least one ANdirectly. However, this assumption may not be truein practice because SN’s transmission range is usu-ally smaller than the AN’s. Thus, the SNs formeda routing tree that could be reached by some ANs,which complicated the case. However, if we defineCi as all the SNs that belong to the tree that is rootedat vi andN(vi) as all the SNs that can communicateor belong to some SNs inCi, our localized smooth-ing algorithm still has a pretty good performance inenhancing the life time although it does not achievethe optimal. Our conjecture is that it is impossible tofind the optimal solution in this case unlessP = NPand it could be an interesting open question to findthe some algorithms with good performance guaran-tee that integrates both the routing tree formation ofSNs and the clustering of these SNs.

Multihop Network of ANs : Remember that in or-der to simply our analysis, we assumed that the ap-plication nodes send the packet directly to the basestation. In the real world, it is possible that the appli-cation nodes form a multi-hop relay network to relaythe packets to the base station. Different applicationnodes may be in different positions in this relay net-work according to the base station. This makes theproblem even complicated,i.e., all these ANs that canreach base station directly are expected to relayallthe packets generated in the network. This makes theANs closer to BS easier to be run out of power thanfar-away ANs. Notice that we did not pose any spe-cial restriction on the energy consumption functionof an application node. In other words, we assumeda generic energy consumption function for ANs. Ifwe can defineCi as all the SNs whose traffic need tobe relayed viavi andN(vi) as all the SNs that cancommunicate or belong to some SNs inCi, our local-ized smoothing algorithm still works when the relaynetwork of the ANs are known in advance. However,some algorithms other than smoothing algorithms areneeded to decide the ANs relay network formationwhen it is not known in advance and we list it as oneof our possible future research directions.

Static vs. Dynamic Cluster: In previous sections,we discussed how to maximize the lifetime of theWSNs by forming some logical clusters. We requiredthat the cluster formation is permanent,i.e., the clus-ters do not change during the whole lifetime of WSN.Obviously, the lifetime of the WSN could be furtherimproved if we allow a dynamic cluster formation,

13

i.e., the clusters will adapt to the remaining energylevel of the ANs. Notice that the improvement oflifetime is not guaranteed since there is always over-head to dynamically update the cluster. For example,when the power consumption function satisfies a cer-tain property,e.g., it is linear, the lifetime is improvedby at most a very small fraction. Thus, we only con-sider the static case in our simulation afterwards.

VI. PERFORMANCESTUDIES

We conducted extensive simulations to study theperformance of different algorithms and approachesintroduced in this paper. As mentioned earlier in thispaper the network is composed of application nodesand sensor nodes. Sensor nodes communicate withapplication nodes only and application nodes com-municate with sensor nodes and other applicationsnodes. Here we mainly study the case with hetero-geneous application nodes and homogeneous sensornodes. Each sensor node has a transmission rangeand is able to communicate with application nodeswithin its transmission range and also each sensor hasa sensing range and is able to monitor the area withinits sensing range. We assume that the ANs have thesame properties but are different in the initial powerand the power consumption rate for sending a unitamount of data to the base-station. In addition, weassume that the transmission range and the sensingrange of all sensor nodes are the same. Each appli-cation node consumesone unitof battery power toserve one sensor node for one day.

A. Simulation Environment

We randomly placed2000 sensor nodes in a800feet × 800feet square region, the transmissionrange of each sensor node is set to50feet and thesensing range is set to10feet. Notice that, the smallsensors are typically randomly placed without anystrategic locations. To guarantee that the area of aa × a square feet region is covered with high prob-ability by n small sensors with sensing ranges, weshould have the following relationnπ( s

a)2 ' ln n.

We found that the small sensors with these settingsin our simulations will cover the majority part of theregion. Then we put a different number of applica-tion nodes, from150 to300 (with incremental25) andmeasured the network lifetime based on several dif-ferent definitions of lifetimes. In addition, the initialbattery power of each sensor node is a random value

between100 units and200 units. Hereafter we call asmall sensor node that has power remaining and hasat least one alive application node in their transmis-sion region as analive sensor.

A.1 Other Heuristics

To study the performance of our smoothing Algo-rithm 2, we compare it with other heuristics listed be-low: (1) [-Nearest] Each sensor node is assigned tothe nearest AN. (2)[-Arbitrary] Each sensor node israndomlyassigned to one of the application nodes in-side its transmission range. (3)[-Smart-Arbitrary] Inthis method, each sensor node israndomlyassignedto one of the application nodes that is inside the sen-sor node’s transmission range. The probability of aSN sj assigned to a neighboring ANvi is the ratioof the remaining power ofvi over the total remainingpower of all neighboring ANs of this SNsj. (4) [-All]Here, each sensor node is assigned toall the applica-tion nodes that are inside the sensor node’s transmis-sion range. This is clearly the worst method. Thuswe will not compare with this method in most simu-lations.

B. Simulation Results

B.1 Lifetime

In this subsection, we compare the lifetime offour different methods under two different definitionsof lifetimes: CANLT, FCLT. Forα-CANLT and β-FCLT lifetime, whenα andβ are smaller,i.e., α =30%; our smoothing algorithm performances muchworst than most the other methods. The reason is thatour smoothing algorithm makes its best effort to max-imize the lifetime whenα = 100% andβ = 100%.Thus, forα or β that is different from100%, we canapply the following simple technique: send1 − α or1−β percentage of the ANs to sleep in a round-robinmanner. This technique can apply to all five methodsmentioned above for the sake of fairness. It is not dif-ficult to observe that the lifetime of all four methodsincrease by a fix percentage (1

1−αor 1

1−β). Thus, it

suffices to use the CANLT and FCLT only. See Sec-tion II for definitions. Figure 2 (a), (b) show the life-time of different assignment methods under lifetimedefinition CANLT, FCLT respectively. We generate100 random WSNs and all results are the average overthe performance of these100 WSNs.

As can be seen, the network lifetime increases al-most linearly with the number of application nodes

14

150 200 250 3000

5

10

15

20

25

30

35

40

Number of application nodes

Num

ber

Net

wor

k Li

fetim

e

Network lifetime befor the first application node dies

Our MethodSmart Arbitrary ANArbitrary ANNearest ANAll AN

150 200 250 3005

10

15

20

25

30

35

40

45

Number of ANs

Net

wor

k Li

fetim

e

Network lifetime (All sensor nodes are covered)

Our MethodSmart Arbitrary ANArbitrary ANNearest AN

1 2 3 4 5 6 70

5

10

15

20

25

30

Number of ANs

Life

time

(CA

NLT

)

Distributed Smoothing AlgorithmCentralized Smoothing Algorithm

(a) CANLT (b) FCLT Localized and Centralized Algorithm

Fig. 2. Comparison of lifetime for different methods

available initially for all methods, except the simplestALL approach that does not perform any logic clusterat all. A striking observation is that, as we expected,our smoothing based method outperforms all othertree methods under all four definitions of lifetimes re-gardless of the density of the application nodes. In allsimulations, we found that our method generally out-performs the other methods by almost100%. In otherwords, the network lifetime is almostdoubledwhenour method is used to form the cluster.

We also compare the performance of the Central-ized Smoothing Algorithm 2 (CSA) and LocalizedSmoothing Algorithm 4 (LSA). We fixed the num-ber of the ANs to50 and varies the number of SNsfrom 200 to 500. Figure 3 (c) shows difference of thelifetime (CANLT) between CSA and LSA, and it isnot difficult to observe that the lifetime of LSA andCSA only differs about5% to 8%. This corroboratesour theoretical analysis and we will only compare thelifetime of CSA with other four methods afterwards.

B.2 Load Balancing

As mentioned in Section III-B, for heterogeneousapplication nodes case, application nodes have dif-ferent initial battery powers, and the objective of theAlgorithm 2 is to assign less sensor nodes to applica-tion nodes that have lower remaining battery powerand more sensor nodes to application nodes that havehigher battery power. To see how good the load bal-ancing of our algorithm is, we run simulation for thenetworks with150 application nodes till all applica-tion nodes die. As can be seen in Figure 3, our algo-rithm achieves a very good load balancing meaningthat all application nodes consume energy at a rateproportional to their initial battery power and thenthey all die together. The result for number of alive

sensor nodes and also the percentage of coverage areaare basically the same as shown in Figure 3 (b) and(c).

B.3 Area Coverage

To further study the area covered by sensor nodeswe build a two tiered sensor network with50 applica-tion nodes,1000 sensor nodes randomly placed in a250feet×250feet region. The transmission range ofall sensor nodes is set to50feet and the sensing rangeis set to10feet. To serve1000 small sensor nodesfor one month,30, 000 units of energy is needed. Weset the initial battery power of each application nodeto a random value between400 and800 units. Eachapplication node has on average600 units of batterypower. Figure 4 depicts the sensor node assignmentand coverage of the network at various date by vari-ous methods.

In the method (calledALL ) that do not performlogic cluster all application nodes die after9 days.As can be seen in Figure 4, after20 days, the ”Near-est AN” and ”Arbitrary AN” methods fail to keep allthe application nodes alive and hence the area is notfully covered. Till the end of day29, our methodmaintains to keep all the application nodes alive andhence the full coverage. Our algorithm guarantees abalanced power consumption of the application nodesand maintain full coverage. For the lifetime definedbased on coverage area, our method improves thelifetime of the network by almost20% for this net-works (composed of1000 sensors and50 applicationnodes in a250feet × 250feet area). Our previoussimulations (see Figure 2 (c) and (d)) reported a largerimprovement if we had more ANs.

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Fig. 3. Comparison for different methods

(a) Our Methods After29 days (b) Nearest Method After15 days (c) Arbitrary method after15 days

(a) Nearest Methods After20 days (b) Arbitrary Method After20 days (c) Nearest method after25 days

(d) Closest Methods After25 days (e) Nearest Method After29 days (f) Arbitrary method after29 days

Fig. 4. Coverage in two-tiered networks.

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VII. C ONCLUSION

In this paper, we studied a generic two-tiered wire-less sensor networks (WSN) composed of small sen-sor nodes (SNs), more powerful application nodes(ANs), and base-stations (BSs). We especially stud-ied how to organize the WSN to form logic clustersto maximize the lifetime of the networks. By us-ing CANLT as the definition of the lifetime for theWSN, we considered the scenarios when the appli-cation nodes are homogeneous or heterogeneous, thesensor nodes are homogeneous or heterogeneous re-spectively. When the sensors are homogeneous, wegive optimal algorithms to maximize the lifetime ofthe networks; when the sensors are heterogeneous,we give a2-approximation algorithm that producesa network whose lifetime is within1/2 of the op-timum. We also showed that it is NP-hard to findthe optimum cluster formation. Our theoretical re-sults are corroborated by extensive simulation stud-ies. Our simulations show that our algorithms actu-ally perform much better not only theoretically butalso for randomly generated WSNs.

There are some interesting important questions thathave not been addressed in this paper. For exam-ple, maximizing the lifetime under other definitionsdeserves further study. It is also important to studyhow the schematically improve the lifetime by inte-grating the routing tree formation of small sensors,the routing tree formation of application nodes andthe attachment of sensor routing trees to applicationnodes. Once again, we point out that our work is onlyan opening but not concluding work of this directionwhich deserve more research efforts in the future.

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