Accepted Manuscript

Upper and lower bounds for dynamic cluster assignment formulti-target tracking in heterogeneous WSNs

Marjan Naderan, Mehdi Dehghan, Hossein Pedram

PII: S0743-7315(13)00068-3DOI: http://dx.doi.org/10.1016/j.jpdc.2013.04.007Reference: YJPDC 3176

To appear in: J. Parallel Distrib. Comput.

Received date: 13 March 2012Revised date: 26 October 2012Accepted date: 29 April 2013

Please cite this article as: M. Naderan, M. Dehghan, H. Pedram, Upper and lower bounds fordynamic cluster assignment for multi-target tracking in heterogeneous WSNs, J. ParallelDistrib. Comput. (2013), http://dx.doi.org/10.1016/j.jpdc.2013.04.007

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Highlights

Modeling a cluster-based target tracking in WSNs as an Integer Linear Programming. Lagrangian relaxation upper bound shows a better bound than pure LP relaxation. LP relaxation with Randomized Rounding provides a good lower bound. Results of heuristic clustering methods remain between the upper and lower bounds. Dynamic clustering methods are better than hybrid ones in small numbers of targets.

*Highlights (for review)

1

Upper and Lower Bounds for Dynamic Cluster

Assignment for Multi-Target Tracking in

Heterogeneous WSNs

Marjan Naderan, Mehdi Dehghan*, Hossein Pedram

Computer Engineering and Information Technology Department

Amirkabir University of Technology

424 Hafez Avenue, Tehran, Iran

E-mail: {naderan, dehghan and pedram}@aut.ac.ir

Abstract- In this paper, we consider the problem of cluster task assignment to maximize total utilities of

nodes for target coverage in heterogeneous Wireless Sensor Networks. We define this problem as assigning

the tasks of Cluster Head (CH) and Cluster Members (CM) to nodes for each target and requiring

communication connectivity between every CH with its members. The utility of each node for each target is

defined as a function of its distance to the target and its remaining energy. We propose an upper bound based

on Lagrangian Relaxation (LR) and a lower bound by Linear Programming (LP) relaxation with a

combination of Randomized Rounding (RR) and a greedy-based heuristic. Furthermore, we propose a

distributed heuristic algorithm based on matching and a general assignment problem. Dynamic movements

of targets are taken into account by intra/inter-cluster task reassignments. Simulation results, compared

with optimal values, reveal that the LR upper bound performs better than the bound reached by pure LP

relaxation. The lower bound obtained by LP relaxation combined with the RR technique provides close

results in comparison with the distributed heuristic algorithm. Furthermore, the results of the distributed

heuristic algorithm remain between the upper and lower bounds and close to optimal values.

* Corresponding author: Mehdi Dehghan, Associate professor, e-mail address: dehghan@aut.ac.ir, URL:

http://ceit.aut.ac.ir/~dehghan. Postal address: Mobile Ad hoc and Wireless Sensor Networks lab, Department of Computer Engineering and Information Technology, Amirkabir University of Technology, PO Box 15875-4413, 424 Hafez Avenue, Tehran, Iran. Tel: +98-2164542749, Fax: +98-2166495521

*ManuscriptClick here to view linked References

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Keywords- WSN; task assignment; clustering; task reassignment; LP relaxation; Lagrangian relaxation;

randomized rounding; heterogeneous nodes

1 Introduction

Wireless Sensor Networks (WSNs) consist of small-sized low cost sensor nodes with limited

resources of computation, energy and wireless communications. Traditional applications of these

networks include long time monitoring of environments, event detection and mobile target tracking [1].

From the application layer point of view, a WSN’s mission (or application) can be represented as a set

of tasks (or roles) [2][3]. Task assignment has received a lot of attention, especially in parallel processing,

distributed systems and servers [4][5]. It also stays in close relation with some resource allocation and

scheduling problems [6][7][8][9].

For WSNs, optimal task assignment remains an important issue due to their scarce availability of

resources and their application-dependent nature. For instance, in cluster-based target tracking methods,

dynamic movements of targets require task reassignments which may induce high energy consumption

and considerable amount of message exchanges among nodes [10][11].

In this paper, we consider a target tracking mission from the task assignment point of view in a pre-

deployed configuration of sensors. The mission consists of Cluster Head (CH) and Cluster Member (CM)

tasks which are reassigned to nodes when the targets have dynamic movements. We assume a slow

dynamic (time varying) environment such that each target is assigned to one sensor node holding the CH

task and a number of other nodes which hold the CM tasks. The cluster members send their readings to

the CH which performs a localization algorithm for determining the location of the target. We assume

passive sensors, such as acoustic sensor nodes, and the localization of targets by sensors’ location is

performed via the lateration method which is based on distances [12]. For the special method of

Trilateration, three CM nodes are required while multilateration requires more nodes to improve the

accuracy which degrades due to errors in measurements. Therefore, we define the number of CM nodes

to remain within a minimum and maximum value. In our problem, connectivity constraint is additionally

required between each CH and its CMs, while we assume heterogeneous sensor nodes with different

communication and sensing ranges for different nodes.

The main contributions of this paper are that we provide an Integer Linear Programming (ILP) model

for this problem, and propose an upper bound according to the Lagrangian relaxation. A lower bound is

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also obtained with LP relaxation together with the Randomized Rounding technique and a greedy-based

heuristic algorithm. Furthermore, a distributed heuristic is provided based on matching and assignment

which considers task reassignments for the case of targets movements with intra/inter-cluster

reassignments.

Despite the upper and lower bound algorithms are executed in a centralized manner, they have great

importance in providing an insight on the level of correctness of optimal and heuristic results. To the best

of our knowledge, these approaches have not been taken into account in any of the task assignment

related studies especially for target tracking in WSNs. In addition, the method we present here for

obtaining the upper and lower bounds can be applied to any other complicated and developed clustering

technique as long as the constraints are defined within an LP model (or more generally a convex model).

Results of these methods are compared with optimal values in terms of performance measures, such as

the number of CH and CM nodes, total utilities of nodes and overhead of control packets.

The rest of this paper is as follows: Section 2, reviews some of the most related studies according to

our problem. Section 3, defines the assumptions and the formulation of our problem. In Section 4, we

provide the upper and lower bounds with the distributed heuristic algorithm. In Section 5, simulation

results are presented and finally in Section 6 we conclude the paper.

2 Related work

The clustering concept has been taken into account in the past by many researches in the field of

WSN. In LEACH [13], as the most well-known method for clustering in WSNs, each node is selected as a

cluster head in each round with a probability. In [14], CPEQ is a cluster-based periodic, event-driven and

query-based protocol which improves the routing strategy PEQ through data aggregation when

comparing their energy maps. Due to the fact that the clustering measure is mostly dependent on the

application in WSNs, a number of clustering methods have been presented specifically for object tracking.

For instance, HCMTT, and its predecessor HCTT, [15] use dynamic clusters above a static cluster structure

such that dynamic cluster heads are finally admitted by static ones. While this technique guarantees no

loss of object, other methods have used either static clusters or dynamic ones. Tree-based methods have

been also presented in the literature of object tracking such as DCTC in [16], but their high maintenance

4

overhead turns them unsuitable compared to cluster-based methods. A survey of clustering schemes can

be followed in [17] and on the subject of object tracking methods in [11].

We have used HCTT control message overhead for comparison with our method in Section 5 due to

its resemblance in application scenarios. In addition, the first phase of our clustering scheme inherits that

of DCTC in [16] in its distributed election scheme. It is also worth mentioning that many clustering

schemes have similarities in setting up the structure, including our method. The novelty of this study is

modeling the clustering method from the view point of assignment with a LP representation and

obtaining the upper and lower bounds on the results. The modeling itself depends on the clustering

constraints and network assumptions which may differ from one clustering method to another. Unless

the constraints remain linear and the whole model remains convex, the framework is applicable for

obtaining upper and lower bounds. Small changes are also valid in reaching the upper and lower bounds;

for instance, the rounding technique used for lower bound may be another rounding technique.

Given that our problem is modeled from the view point of assignment, we briefly investigate

researches on this problem at the remainder of this section. The assignment problem consists of a

number of agents and tasks in which, any agent can be assigned to perform any task, incurring some cost

and aims at finding a maximum weight matching in a weighted bipartite graph [18]. In the Generalized

Assignment Problem (GAP) both tasks and agents have a size and the size of each task might vary from

one agent to the other. GAP is NP-hard; and different versions of it have been investigated in [19].

The assignment problem has been also used in some of the WSN-related studies. In [10], the sensor-

mission assignment is modeled as a bipartite graph with correspondence to the semi-matching problem.

The problem is formulated as an ILP model with centralized and distributed algorithms presented for it.

Based on [10], the authors in [20] investigated the same problem with energy harvesting sensor nodes,

leading to a Mixed Integer Programming (MIP) model.

In [4], task assignment is modeled with placement and routing constraints embedded inside the input

matrices. Two MIP problems are introduced with centralized greedy-based heuristic algorithms, in

addition to the multi-path routing versions of the problems. In [21], sensor-mission assignment is

investigated with the assumption that sensors may be shared and reassigned between tasks (in contrast

to [10]) and with a multi-round Knapsack-based algorithm as its solution. In [22], a landmine network is

modeled as a non-linear IP model to destroy the intruding targets using the minimum cost pre-deployed

mines with greedy-based and layering algorithms as its solutions.

5

The market-based approach for the task assignment problem in WSNs has been also taken into

account. For instance, in [23], free sensor nodes are viewed as sellers and allocated sensor nodes are as

buyers. The term role assignment is also used in [2] and [3]. For instance, in [2], access control is

designed such that authorization is given to nodes based on their roles and in [3], roles of coordinator and

collaborator are assigned to nodes to perform routing functionalities. Furthermore, in [24], the target

coverage problem, as a sensor selection scheme, aims to maximize the number of cover sets, subject to

choosing the sensors according to their energy level and sensing range.

Our work differs from previous ones in the sense that we have considered the clustering structure in

our formulation, and we propose upper and lower bounds based on LP and Lagrangian relaxations. In

[10], missions or tasks have a demand that may be fractional, while in our problem, the tasks map to

targets which could not be satisfied fractionally (the demands are 0 or 1). In knapsack-based algorithms

such as in [10] and [21], when a node is assigned to a task, its utility is divided fractionally among them,

while in our target tracking application all targets (tasks) that remain inside the sensing range of a sensor

are covered identically. Moreover, our problem differs from [24] regarding problem objective and

assumptions since clustering is not considered in [24]. In addition, in [4], task assignment with placement

constraints is strongly dependent on the input matrices, which determining them for large numbers of

sensors and targets is difficult.

3 Assumptions and Problem formulation

We assume the tracking WSN consists of a set of static sensor nodes denoted as S = {s1, s2, …, sN} and

the set of targets or objects with O = {o1, o2, …, oM}, such that M < N, i.e., there are more sensor nodes in the

area than the targets. These nodes and targets are distributed randomly across the area. We divide the

total duration of the target tracking network into a finite number of time intervals, in which the targets

are assumed to be static in these intervals but their locations are not necessarily the same in two

consecutive time intervals. Therefore, our mathematical model for obtaining upper and lower bounds is

restricted to static nodes and targets. In fact, a mobility model is required to add to the problem

formulation which may lead to a nonlinear model. However, we have extended the heuristic algorithm to

track mobile targets in Section 4.3.2, as well.

6

Another assumption is on the type of sensor nodes, which we assume to be regular and passive, such

as acoustic sensors. It is assumed that each node knows its location, provided by a localization method at

the beginning phase of network operation, which is not the intention of our paper. Nodes are also aware

of their neighbors’ locations, which is obtained by exchanging packets between neighbor nodes in a

preprocessing phase. Moreover, we do not assume GPS equipment, as it imposes additional cost for nodes

and may not operate properly in some environments. On contrary, if one wishes to use GPS on nodes, the

assumption of minimum number of CM nodes remains acceptable as more than one location information

increases the possibility of correctness for localization.

In addition, we suppose a routing/aggregation strategy is provided by the lower layers to deliver the

tracking data towards the sink. In fact, our algorithms deal with the clustering structure and coverage of

targets by nodes, which are assumed to be executed above the network layer in the layered architecture

of WSNs [11].

Another assumption is on the object detection service by sensor nodes. Nearly in all of the target

tracking methods, which deal with communication and coverage issues, it is assumed that nodes are able

to distinguish targets by some means of signal processing algorithms [10][11][25] and [26]. In fact, it is

specifically mentioned in [26] that that the object detection service is an orthogonal service to object

tracking, which assigns a unique ID to every object detected by nodes in the network.

Table 1 shows the parameters and variables used in our model with their short descriptions.

We select xij as a binary variable which is 1 if sensor node i is the CH of target j and 0 otherwise. In

addition, yij is the binary variable which is 1 if sensor node i is the CM of target j and 0 otherwise. Thus,

the variables xij and yij correspond to the CH and CM tasks, respectively. A graph-based representation of

our problem can be viewed in Figure 1, in which the set of sensor nodes are on the left and the set of

targets are on the right side of the bipartite graph G.

We require the following constraints for our problem:

1- Each target must have exactly one CH node.

2- Every sensor node holds the CH task for at most one target. Accordingly, the number of nodes

with CH tasks is at most equal to the number of targets and there may be sensors with no CH

tasks assigned.

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Table 1. Parameters and variables used in this paper

Description Parameter/variable

The set of sensor nodes S

Cardinality of set S or the number of sensor nodes N

Index used for sensor nodes as sS or s=1, …, N i, s

The set of targets (or objects) O

Cardinality of set O or the number of targets M

Index used for targets as j=1, …, M j, k

Binary variable indicating the assignment of CH task to node i for target j xij

Binary variable indicating the assignment of CM task to node i for target j yij

Minimum number of nodes with CM task for each target CMmin

Maximum number of nodes with CM task for each target CMmax

Sensing range of node i, between minimum value SRimin and maximum value SRi

max SRi

Communication range of node i, between minimum value CRimin and maximum value

CRimax

CRi

Utility (function) of node i for target j wij

Euclidean distance between sensor node i and target j d(si, oj)

The weight parameter in utility function wij ω

Remaining energy of node i eirem

The bipartite graph including sensor nodes and targets G

The set of vertices of G: V=S⋃O V

The set of weights of edges of G: W= S xO W

A weighted matching F ⊆ W F

An assignment where H ⊆ W-F H

The set of determined CH nodes CH

A constant parameter showing the priority of CH task over CM c

The number of memberships of a node as CM for different targets b

Lagrangian of problem 1-CTA L(x, y;λp, γq)

p=1,2,q=1,2

Matrix of Lagrangian multipliers for inequality constraints λp

Matrix of Lagrangian multipliers for equality constraints γq

The dual function of problem 1-CTA D(λp, γq)

Index used for iterations t

Positive scalar step sizes for Lagrange multipliers α1t, α2t, α3t, α4t

Constant between 0 and 2 δt

Average number of neighbors of each node Δ

Average number of nodes that detect a target m

Number of MYW messages NMYW

Number of CHC messages NCHC

Number of IMCH messages NIMCH

Number of URCM messages NURCM

3- For each target, the sensor which has the CH task has the highest utility for it. This utility is

defined in terms of the Euclidean distance between each node and target and the remaining

energy of each node as (extended from [10]):

8

21 if ,

1 ,

0 otherwise

rem

i i j i

i jij

e d s o SRd s ow

ωω

(1)

where 0 ≤ ω ≤ 1 is a weight parameter, d(si, oj) is the Euclidean distance between sensor i and

target j, SRi is the sensing range of sensor i and eirem is the remaining energy of node i. According

to (1), the utility of node i for target j corresponds to the weight of the edge relating node i and

target j in graph G. Furthermore, if two sensors have the same distance to a target, the one with

more energy is assigned the CH task.

4- For each target, at least CMmin sensor nodes, which have the most utilities for it at the next levels,

are assigned the CM task for that target. We define the maximum number of nodes having the CM

task as CMmax, which behaves as a measure of coverage quality depending on the applications

needs.

5- The CM nodes must stay in the communication range (CR) of the CH, so that they can send their

data to the CH node (connectivity constraint).

6- The CH and CM nodes must have the target inside their sensing range.

7- If a sensor node is assigned the CH task for one target, it could not be assigned the CM task for

any other target.

Figure 1. Model of our problem as a bipartite graph.

Each target is assigned to one node as a CH and three other nodes as CMs.

Sensors Targets

CH

CM

Set S Set O

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8- If a sensor is assigned a CM task for one target, it could be assigned the CM task of other targets

as long as its number of memberships is less than b. For instance, in Fig. 1, each sensor node

covers at most 2 targets.

9- Finally, it is desirable that the CH tasks have precedence on CM tasks, i.e., first all CH tasks must

be assigned and next the CM tasks (precedence constraint). According to this, each target may

have at least one CH node, such that if we could not assign CMmin nodes for it, its location can be

estimated by the location of its CH node.

Additionally, we have the following relations for the sensing and communication ranges (CR) for

heterogeneous nodes:

SRimin ≤ SRi ≤ SRi

max

CRimin ≤ CRi ≤ CRi

max

SRi ≤ CRi/2 , SRimax ≤ CRi

max/2 ∀i=1,…,N

(2)

(3)

(4)

where (2) and (3) imply that sensing and communication ranges of nodes are within an interval of

minimum and maximum values. In addition, each node may have a different minimum and maximum

value from other nodes which encompasses heterogeneous nodes. Condition (4) is the necessary and

sufficient condition to ensure that complete coverage in a convex region implies also connectivity [27]. It

is also applied in some target tracking studies as in [25]. Despite we used connected scenarios to obtain

optimal results, this condition guarantees connectivity in case of complete coverage in general for our

work.

What we aim is first a matching of sensors to targets, so that (ideally) each target is covered by a CH

node (task) and second, the assignment of at least CMmin nodes to each CH node. In fact, covering all

targets may not be possible; hence, we want to maximize the weighted sum of the utilities, i.e., weighted

sum of weights of edges. It is also worth mentioning that our current problem is a strict version of

coverage in which a target is covered if it is located inside the sensing range of a sensor, and otherwise it

is not covered. In fact, we do not have partially covered targets, the situation which may exist in camera-

related object tracking problems. We can now define our problem formally as follows:

Problem 1-CTA (One Cluster-per-Target Allocation): Assume a weighted bipartite graph G=(V, W),

where V=S⋃O and S = {s1, s2, …, sN} is the set of sensor nodes, O = {o1, o2, …, oM} is the set of objects

(targets) and W= SxO = {wij | si∊S ∧ oj∊O} is the set of edges with weights connecting a sensor vertex si to

an object vertex oj. We first require to find a weighted matching F ⊆ W (no two chosen edges share the

10

same sensor, nor the same target), such that i j

ij ij

s S o O

w x

, where {(i, j) ∊F | si ∊ S, oj ∊ O}, is maximized. At

the second step, we need an assignment H ⊆ W-F (every edge might share the same sensor or might share

the same target), such that i j

ij ij

s S CH o O

w y

, where {(i, j) ∊ H | si ∊ S-CH, oj ∊ O}, is maximized, and CH = {si

| (i, j) ∊F } is the set of CH nodes, min max

1,

N

kj

k k i

CM y CM

and d(si, sk) ≤ min(CRi, CRk).

If the number of memberships of CM nodes to clusters is restricted by one, the second problem is

named a semi-matching. The ILP representation of the problem is:

Problem 1-CTA:

1 1

MaximizeN M

ij ij ij

i j

w c x y

s.t.

(5)

1

1N

ij

i

x

For each j=1, …, M (5-1)

1

1M

ij

j

x

For each i=1, …, N (5-2)

min max

1

N

ij

i

CM y CM

For each j=1, …, M (5-3)

, min ,ij kj i k i kx y d s s CR CR For each j=1, …, M,

i,k=1, …, N, i≠k

(5-4)

1 1

1M M

ij ij

j j

y x b

For each i=1, …, N (5-5)

, , ,ij i j i ij i j ix d s o SR y d s o SR

For each i=1,…, N (5-6)

xij , yij ∊ {0, 1}, c > b

Also subject to equations (1)-(4)

(5-7)

in which, c, inside the objective function, is a constant which shows priority of selecting the sensor with

the highest value of wij as the CH. b represents the number of memberships of nodes for different clusters

(targets) as CMs. We require c>b to guarantee priority of selecting CH nodes over the CM ones. Constraint

(5-4), also referred as connectivity constraint, requires that for a target j, its CH and CM nodes, with

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indices i and k, must be in the communication range of each other. In fact, if xij=1 and yjk=1, then d(si, sk)

min(CRi, CRk). Note that problem 1-CTA is linear, since the connectivity constraint (5-4) can be turned to

linear terms according to the technique presented in [4]. Table 2 shows the numbered constraints defined

in this section with their corresponding LP representations in (5). For instance, in constraint number 3, it

was required that the CH node of a target holds the highest utility value; thus, we associate this

requirement with (1), which is the definition of the utility function, and maximizing the objective function

in (5) with the constant value c as a multiplier for xij variable. In the next section, we present the lower

and upper bounds to problem 1-CTA and a distributed heuristic with its extension for dynamic

movements of targets.

4 Lower and upper bounds and algorithms

In this section, we provide three solutions to the problem 1-CTA in centralized and distributed

fashions. We have used two approaches in this section in addition to the heuristic method:

By LP relaxation of the integer variables and rounding the fractional solutions by the

Randomized Rounding technique [28] and a greedy-based heuristic.

By Lagrangian Relaxation (LR) of the original ILP problem

Despite the LR method does not necessarily reach feasible solutions for xij and yij variables, its upper

bound is better than the LP relaxation according to [29]. Furthermore, we introduce a simpler version of

the 1-CTA problem which is used in the rest of this paper as:

Table 2. The correspondence between constraints defined and their LP representation in (5)

Constraint number LP representation equation

1 (5-1)

2 (5-2)

3 (1) and maximizing the objective function in (5)

4 (1), (5-3) and maximizing the objective function in (5)

5 (5-4)

6 (5-6)

7, 8 (5-5)

9 (5-7)

12

1 1

MaximizeN M

ij ij ij

i j

w c x y

s.t.

(6)

1

1N

ij

i

x

For each j=1, …, M (6-1)

1

1M

ij

j

x

For each i=1, …, N (6-2)

1

3N

ij

i

y

For each j=1, …, M (6-3)

1 1

1M M

ij ij

j j

y x

For each i=1, …, N (6-4)

xij , yij ∊ {0, 1}, c>1 (6-5)

in which for simplicity, we have removed the connectivity constraint (5-4) as we can investigate scenarios

with connected sensor nodes. In addition, we set CMmin=CMmax=3 in (5-3) which is sufficient for the

Trilateration technique, and reduce the number of clusters each node can attend as a CM in (5-5) to one

(b=1). We have used problem 1-CTA and problem (6) as the same in the rest.

4.1 Lower bound by LP relaxation and rounding

A typical solution to ILP problems is to relax the integer variables, solve the relaxed LP problem by

any polynomial time algorithm, and finally round the fractional solutions by a heuristic algorithm to

integer results. Since this method mostly results in solutions not better than the pure heuristic method,

the results of this method are viewed as a lower bound.

It is also worth mentioning that as solving the relaxed LP problem is done centrally, the whole

algorithm is centralized. This remains acceptable since the results are used as a lower bound and in

practice, the distributed heuristic algorithm in section 4.3 may be used in real-life implementations.

However, there are also distributed methods which can also provide solutions to the relaxed LP, which

we have left as future work in Section 6.

Currently, we use the Randomized Rounding technique [28] to round the fractional solutions of the

relaxed LP model. In this technique, a binary value xij is 1 if a random variable value (using the rand

13

function in C++) is greater than the fractional solution ˆijx . Furthermore, in our method a node may be as

a CH if it has not been a CH or CM for another target. Thus, we have to check the previous assignments of a

special node i, and if it has not been assigned to any other target the rounded value of ˆijx is xij=1. A similar

situation holds for rounding the yij variables.

The complete algorithm is presented in Algorithm 1. Suppose 1-CTAR is the rational relaxation of 1-

CTA; i.e., xij and yij∊ [0,1] in (6). This new problem is given to a solver which returns fractional results

namely, ˆijx and ˆ

ijy . The algorithms is continued with selecting first the CH tasks and next the CM ones.

Furthermore, the fractional solutions ˆijx are sorted in decreasing order such that node results with more

ˆijx are treated first, the same as in greedy algorithms. To round the fractional solutions, not only the

probability is taken into account, but also the previous assignments must be considered (statement at line

6). Hence, if a node is allocated as CH for a target, it cannot be allocated as a CH or CM for any other target,

even if its probability is higher than ˆijx .

The same procedure holds for the CM task with the difference that at least CMmin and at most CMmax

number of CM nodes must be selected. Statement 13 implies this and since we used CMmin=CMmax=3 in (6),

we can repeat statements 14-19 three times.

4.2 Upper bound based on Lagrangian Relaxation

To obtain the Lagrangian upper bound, we must have the Lagrangian of 1-CTA which is:

1

1,2, 1,21 1 1 1

, , , 1N M M N

p q

ij ij ij j ijp qi j j i

L x y w cx y xλ γ γ

1 2 2

1 1 1 1 1 1 1

1 3 1N M M N M M M

i ij j ij i ij ij

i j j i i j j

x y x yλ γ λ

(7)

where we have used the notation in [30] which applies the vector γ for equality and λ for inequality

constraints. x and y are matrices containing xij and yij, γq and λp are the matrices γ=(γijq, i=1,..,N, j=1,…,M,

q=1,2), λ=(λpij, p=1,2, i=1,…,N, j=1,…,M). Accordingly, the dual function is:

1,2, 1,2, Maximize , , ,

. . , 0,1

p q p q

p q

ij ij

D L x y

s t x y

λ γ λ γ

(8)

and the master dual problem is:

14

Algorithm 1: LP Relaxation based on Randomized Rounding (LPR3)

1- Solve 1-CTAR, the results are as ˆ ˆ, [0,1]ij ijx y

2- For each target j do

3- Sort the vector ˆijx for i=1, …, N in decreasing order.

4- For i=1, …, N do

5- Choose ˆijx

6- If ( Prob.[xij=1]

ˆijx ) and (xik=0 for all k j)

7- Set the variable xij =1; Break

8- Else continue

9- If ( i >N )

10- Set xij=0

11- For each target j do

12- Sort the vector ˆijy for i=1, …, N in decreasing order.

13- Do statements 14-19 at least CMmin and at most CMmax times (for problem (6) three times)

14- For i=1, …, N do

15- Choose ˆijy

16- If ( Prob.[yij=1] ˆijy ) and (xij=0) and (yik=0 for all k j )

17- Set the variable yij =1; Break

18- Else continue

19- If ( i > N )

20- Set yij=0

21- If ( less than CMmin CMs could not be selected )

Announce “could not assign CMmin CMs for this target” and release back the sensors

1,2, 1,2

Minimize ,

. . 0

p q

p q

p

D

s t

λ γ

λ

(9)

The function D(λp, γq) is the maximum of a finite set of linear functions of λp and γq and therefore, it is

convex and piecewise linear where each piece corresponds to a point in x and y. In addition, D(λp, γq) is

differentiable except at points where the dual problem has multiple optimal solutions. At differentiable

points the derivatives of D(λp, γq) with respect to λp and γq for p=1,2 and q=1,2 are given by its gradients.

For instance, for λi1:

1,2, 1,2

1

,1

p q

p q

ij

ii

Dx

λ γ

λ

(10)

which is the coefficient of λi1. It is also possible to apply this gradient method to minimization of D(λp, γq)

with some adaptation at the points where D(λp, γq) is non-differentiable by a procedure called sub-

gradient method [29][31]. At points where D(λp, γq) is non-differentiable, the sub-gradient method

15

chooses arbitrarily from the set of alternative optimal Lagrangian solutions and uses the coefficient

vectors of Lagrange multipliers as though they were the gradient of D(λp, γq). The result is a procedure

that determines a sequence of values for Lagrange multipliers by beginning at initial points and applying:

1, 1 1,

1 1t t t t

j j ij

i

xγ γ α

(11)

2, 1 2,

2 3t t t t

j j ij

i

yγ γ α

(12)

1, 1 1,

3 1t t t t

i i ij

j

xλ λ α

(13)

2, 1 2,

4 1t t t t t

i i ij ij

j j

x yλ λ α

(14)

where α1t, α2

t, α3t and α4

t are the scalar step sizes, xijt and yij

t are optimal solutions to D(λp, γq) with dual

variables set to their values in the t-th iteration and [.]+ denotes projection onto the set R+ of nonnegative

real numbers. Values of scalar step sizes are updated according to [29]:

*

, 1,...,4 2

Solution to , ,

Sum of subgradients

p q p q

tt

k k

D Dδ λ γ λ γα

(15)

in which, D*(λp, γq) is the best solution found for D(λp, γq) until the t-th iteration. tδ is a constant between

0 and 2, starting at 2, and is reduced by a factor of ½ if we fail to find a better solution at the t-th iteration.

The sum of sub-gradients is also computed as:

Sum of subgradients= 1 3 1 1t t t t t

ij ij ij ij ij

i i j j j

x y x x y

(16)

Finally, the iterative algorithm is as presented in Algorithm 2. Note that solving D(λp, γq) is equal to

solving an unconstrained problem since, we have relaxed the constraints by Lagrange multipliers into the

objective function. It is obvious that the above algorithm is executed centrally and provides an upper

bound on the problem.

16

Algorithm 2: Lagrangian Relaxation Upper Bound (LRUB)

Choose γj1,0=0, γj

2,0=0, λi1,0=0, λi

2,0=0, t=0, bestValue=∞

While ( t < iterationLimit ) do

1- Solve D(λp, γq) to find a value on the objective function, namely currentObjective.

2- Compute values of sub-gradients as in (16) and value of step size according to (15)

3- Update Lagrange multipliers as in (11)-(14)

4- If ( currentObjective < bestValue ) then bestValue=currentObjective

5- Else counter=counter+1

6- If (counter=4) then 1 / 2t tδ δ and counter=0

7- Increment t and go to 1.

4.3 Distributed algorithm based on matching and assignment

In this section, a greedy-based approach is chosen to assign each target to the best sensor to hold the

CH task. We start with a two-stage procedure for the “cluster construction” which consists of solving the

matching problem for determining the CH tasks and solving the general assignment problem to

determine the CM tasks. Next, we add the actions of nodes in case of target movements. Our algorithm in

this section is named Distributed Algorithm based on Matching and Assignment (DAMA).

4.3.1 Cluster construction

The CH matching stage consists of two phases to determine sensor i with maximum wij for each target

j (inspired from [16] for a tree-based method). In the first phase, each sensor node i which senses target j

computes the wij value according to (1) and sends this value together with its ID, target ID and its distance

to target* to its one-hop neighbors in a MYW message. To avoid conflict of packets each node sets a timer

based on its wij value and upon expiration of this timer, it sends its MYW message. In the second phase,

each node which receives MYW messages from neighbors compares these values with its own. If its own

value is greater than the received ones, it sends a CHC message (CH candidate) to its one hop neighbors

including its wij value, ID and target ID. A one hop transmission is sufficient in this phase since reception

of the message by all nodes surrounding the target is guaranteed according to (4). If a node which

receives a CHC message checks that its own value is less than the received value, it will no longer be a CH

candidate and it gives up. On the other hand, if it detects that its value is greater than all the received

values, sets itself as a CH and informs its neighbors by sending an IMCH message.

* d(si, oj) parameter in (1)

17

After the CH matching stage, one hop nodes from the CH node, send their data on target j together

with a periodic MYW message. The CH node selects the best three of them to hold the CM tasks by sending

a URCM message. Each CH node preserves a table for its members to include their wij values for each

target it is tracking. In addition, each CM node saves the CH node which belongs to it and the distance to

the corresponding target. Note that according to constraint (6-4), each CM node belongs to at most one

cluster. To localize the target, the CH node, which has the locations of its CM nodes and their distances

from the target (sent by the MYW message), performs the lateration algorithm. Figure 2 shows a

flowchart for the cluster construction procedure.

Figure 2. Flowchart of DAMA algorithm, the cluster construction phase.

Node i detects a new target j, it

sends wij message to its one-hop

neighbors in a MYW message.

No

Node i is no longer a CH

candidate and it gives up.

Yes

No

Yes

Is the w value of node i

greater than all received

values?

Does node i receive any

CHC message?

Node i sets itself as a CH for

target j and informs neighbors

with a IMCH message.

Send a CHC message to one-hop

neighbors and wait for some time.

Wait for other messages.

Nodes that receive IMCH message

set node i as the CH for target j and

send their w values to i.

Node i selects the best three members

by sending URCM message.

CH matching

stage

CM assignment

stage

18

As an analysis for the overhead of the cluster construction phase, we can operate as follows: assume

M targets exist in the network, the average number of neighbors of each node is Δ and m nodes detect

each target on average. Accordingly, we have:

M m MYW packets are sent at the first stage.

If m/2 nodes around each target send CHC messages to their neighbors, overall, / 2M m

CHC messages are sent.

Finally, one node sends IMCH message to its neighbors, hence, M IMCH messages are sent.

In the CM assignment stage, for each target, neighbor nodes send MYW packets to their CH

nodes and next CH nodes select 3 nodes as CM by sending URCM packets. Therefore, M Δ

MYW and 3 M URCM packets are sent.

Therefore, the overall number of packets sent during the cluster construction phase is the sum of

number of MYW, CHC, IMCH and URCM packets as NMYW, NCHC, NIMCH, NURCM, respectively:

Total overhead= MYW CHC IMCH URCMN N N N

= / 2 3M m M m M M MΔ

3 / 2 4M m Δ

(17)

4.3.2 Dynamic movements

Movements of targets affect the dynamic clusters and require reassignment of tasks. A reassignment

is needed when:

a target, that is surrounded by a cluster, is moving inside its cluster (intra-cluster

reassignment) and

a new target is detected by some nodes which probably hold tasks for other targets (inter-

cluster reassignment)

We design our algorithm based on the tasks that the two sending and receiving nodes of MYW

message have. In all cases we first check the node to see whether it has a CH task or not, and next we

check the CM task. In addition, the number of CM assignments to nodes in each cluster is controlled by the

CH of that cluster to satisfy constraints (6-1)-(6-4).

An intra-cluster reassignment occurs when a target moves slowly such that it still remains inside its

cluster. The procedure is as follows: Since, each CM periodically sends its w value in a MYW message to

19

the CH node, the CH node (assume node i) on receiving this message from all its members, compares the

values with its own. If CH node i detects that some member node (e.g., node s) has a greater value, it starts

the handover procedure with sending a CHD (CH Dismiss) message to its one hop neighbors to inform

them it is not a CH node anymore. On the other hand, it signals a cluster construction procedure to create

a new cluster.

For the case of detecting a new target, assume node i detects a new target j and it sends MYW

message to its one-hop neighbors, say node s as one of the receiving nodes. We denote the previous target

of nodes i and s (if they track) as target k. Note that according to (4) if two nodes i and s sense a new

target j, they are certainly neighbors of each other and they can cooperatively track the target. Three

cases may exist depending on the task of nodes i and s:

1. None of them hold any task

2. One of them holds a task

3. Both of them hold a task

In case 1, it is required that the nodes run the cluster construction procedure. This is due to the fact

that the target has entered a new area since, nodes do not hold any task.

In case 2, assume node i holds a task (the same is true for node s). If node i is a CH and wij>wik, it sends

a CHD message to its members and start the cluster construction procedure. This is due to the constraint

that the CH task has precedence on CM task. If node i is a CM for current target k and wij>wik, it sends a

CMD (CM Dismiss) message to its neighbors to end its membership and starts the cluster construction

procedure, since the new target j does not have any clusters yet. The CH node which receives a CMD

message, must find a new member to reach the limit number 3 for its CMs.

Case 3, is also similar to case 2. Depending on the nodes tasks, they send a CHD or CMD if they detect

wij>wik or wsj>wsk, and a cluster construction procedure is initiated.

Table 3 represents the messages in our Dynamic DAMA algorithm together with their descriptions.

Note that all messages are transmitted to only one hop neighbors.

20

Table 3. Messages in the Dynamic DAMA algorithm

Message name Message description

MYW Each node that detects a target periodically sends this message to its one-

hop neighbors.

CHC The node that detects a higher wij value than its neighbors sends this

message to its neighbors.

IMCH The node that is a CH sends this message to its neighbors.

URCM The CH sends this message to its neighbors to announce them as CMs.

CHD Each node on receiving a CHD message removes the current CH and starts

the cluster construction procedure.

CMD The CH node for the corresponding CM, removes it from its list of CMs and

sends URCM message to another node in its neighbor list.

5 Simulation Results

To evaluate and compare the performance of our algorithms, we implemented DAMA and LPR3, as

application layer algorithms, with the C++ programming language, the LRUB method and the fractional

solutions of LPR3 by AIMMS 3.11 [32], and the results of HCTT algorithm from our previous work [15].

The latter is chosen as a hybrid clustering target tracking method implemented by the Castalia 3.2

framework [33]. We evaluated the measures of number of CH nodes, number of CM nodes, total utility of

nodes, and average number of control packets. Since these are application layer parameters and are not

dependent on lower layer implementations, we are able to use the results of HCTT for comparison.

While the amount of code for implementation with C++ and AIMMS were not so large, the runtime of

DAMA and LPR3 for 600 nodes and above were rather time-consuming. In fact, scenarios given as input to

AIMMS (for optimal solutions, LRUB and fractional solutions to LPR3) require configurations with

feasible solutions. Thus, as mentioned for problem (6-4), to obtain optimal results, we had to investigate

scenarios with communication connectivity between CH and CM nodes and also scenarios with exactly

one CH and three CM nodes for each target. Otherwise, the linear model returns infeasibility error.

Henceforth, for configurations with large numbers of nodes (600-1000 nodes) we run DAMA at a number

of 50 runs to find configurations which also have optimal values in AIMMS. This process consumes nearly

one to two hours for runtime.

Consequently, for scenarios with feasible solutions AIMMS is able to find it in few seconds, as the

model is linear. It is also worth mentioning that for scenarios of 600 nodes and above, HCTT

21

implementation takes nearly half an hour to complete the simulation time (which was 3600 seconds)

while the amount of C++ code in Castalia was quit considerable. Finally, the simulations were performed

on a system with two cores Intel T7200 running at 2 GHz and 4 GB of memory. The C++ compiler used for

implementing DAMA and LPR3 was the Microsoft Visual studio 2010 and for the Castalia framework was

the g++ compiler executed on the Ubuntu operating system.

Scenarios of simulations contain random placements of nodes and targets in different areas according

to Table 4. It can be seen from the table, that with increase in the number of nodes, we also increase the

number of targets. Therefore, in the rest of this paper, increase in the number of nodes also means

increase in the number of targets. Every point in the graphs is the average of 10 runs. The communication

and sensing ranges of all nodes were set as 20m and 10m, respectively, to agree with constraint (4).

5.1 Convergence of the LRUB method

We first evaluated the convergence of the LRUB method since it is executed iteratively. Furthermore,

this method only gives optimal upper bounds for the objective value and not feasible solutions for the

assignment variables, hence, we evaluated the value of objective function which is the total utilities of

nodes for targets as shown in Fig. 3. The algorithm was tested for 50 nodes, 5 targets, 100 iterations and

with the first two constraints relaxed (constraints (6-1) and (6-2)). To obtain the positive step sizes of

(15), we had to use a constant parameter LB instead of the optimal value of the dual function. Different

values of Fig. 3 correspond to different values of LB.

Table 4. Number of targets in grid networks with different number of nodes

Number of nodes Size of the network (m2) Number of targets

100 30 x 30 10

200 50 x 50 20

300 70 x 70 30

400 90 x 90 40

500 100 x 100 50

600 120 x 120 60

700 140 x140 70

800 150 x 150 80

900 170 x 170 90

1000 190 x 190 100

22

a

b

Figure 3. Convergence of the objective function in LRUB, for 50 nodes, 5 targets and

a) LB=0, LB=10, b) LB=100, LB=1000, LB=10000.

It can be seen from Fig. 3.a that for LB=0 and LB=10, LRUB converges in less than 20 iterations, thus,

the behavior of the algorithm is only depicted for these iterations. On the other hand, in Fig. 3.b, for

LB=100, 1000, 10000, the algorithm converges after 30 iterations, and therefore, 50 iterations are shown

in Fig. 3.b.

12

13

14

15

16

17

18

19

0 2 4 6 8 10 12 14 16 18 20

UB(k), LB=0

UB(k), LB=10

Iterations

Ob

ject

ive

valu

e

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10 12 14 16 18 20

x 10000

UB(k), LB=100

UB(k), LB=1000

UB(k), LB=10000

Iterations

Ob

ject

ive

valu

e

23

5.2 The number of CH and CM nodes, value of total utilities

Figures 4 and 5 depict the number of CH and CM nodes for the three algorithms DAMA, LPR3 and

HCTT compared to the optimal value. As seen from Fig. 4.a, the three algorithms have very near optimal

values in selecting the CH nodes. When the number of nodes (targets) increases, especially from 600

nodes and above, a slight difference can be seen which is shown in a closer view in Fig. 4.b. It can be

viewed that the values of DAMA and HCTT remain between the optimal and LPR3 values, as was expected.

On the other hand, form Fig. 5, selecting three CM nodes is not as good as selecting the CH ones,

especially when the number of nodes increases to 600 nodes and above. This is due to the fact that when

the number of targets increases, it is harder to find exactly CMmin=3 nodes as CM, since, we have restricted

the number of memberships of nodes in different clusters to one. This not only holds for DAMA, but also

for LPR3 and HCTT, as well. Once more, in this figure, the values of DAMA and HCTT remain between the

optimal and LPR3 values.

Figure 6 shows the average of total utilities of nodes with upper bounds LRUB and LPUB and the

lower bound as LPR3. The LPUB (Linear Programming Upper Bound) is obtained by simply converting

the integer variables to real ones and solving it via the AIMMS software. As expected, the LRUB reaches

better upper bound than LPUB and it is closer to optimal values. This closeness, reduces when the

number of nodes increases, especially for 900 and 1000 nodes. On the other hand, LPR3 lower bound

finds acceptable results which are near the values from heuristic algorithms, i.e., DAMA and HCTT.

a b

Figure 4. Average number of CH nodes a) for 100-1000 nodes and b) for 600-1000 nodes from a closer view.

0

10

20

30

40

50

60

70

80

90

100

0 100 200 300 400 500 600 700 800 900 1000

Optimal

LPR3

DAMA

HCTT

Avg

. nu

mb

er o

f C

Hs

Number of nodes

55

60

65

70

75

80

85

90

95

100

600 700 800 900 1000

Optimal

LPR3

DAMA

HCTT

Avg

. nu

mb

er o

f C

Hs

Number of nodes

24

Figure 5. Average number of CM nodes

Figure 6. Average of total utility functions with upper bounds LRUB and LPUB, and lower bound LPR3

5.3 Overhead of control packets for distributed methods

In the final series of simulations, we tested the two distributed heuristic methods, DAMA and HCTT,

in terms of the number of control packets. The result is shown in Fig. 7, in which the locations of objects

were changed randomly in five consecutive time slots. HCTT is based on a hybrid method of constructing

both static and dynamic clusters, while DAMA serves only as a dynamic clustering method. Static clusters

0

50

100

150

200

250

300

0 100 200 300 400 500 600 700 800 900 1000

Optimal

LPR3

DAMA

HCTT

Avg

. nu

mb

er o

f C

Ms

Number of nodes

20

40

60

80

100

120

140

160

180

0 100 200 300 400 500 600 700 800 900 1000

Optimal

LPUB

LRUB

LPR3

DAMA

HCTT

Number of nodes

Avg

. su

m o

f u

tilit

ies

25

in HCTT are constructed before entering the targets, while dynamic clusters in HCTT and DAMA are based

on targets locations. To accompany with HCTT, the number of targets and number of nodes are reduced

as in Table 5. Scenarios of target movements for DAMA are intra- and inter-cluster movements but no

new target is added and only the targets move from their current cluster towards nodes that construct

new clusters. It can be seen from Fig. 7 that for small numbers of targets, control packets overhead of

HCTT is larger than DAMA, since it includes static clusters as well. In contrast, when the number of targets

increases, the overhead of DAMA (which is a dynamic clustering method) increases since, these methods

are dependent on the number of targets.

Table 5. Number of nodes and targets for

scenarios of control packet overhead for DAMA

and HCTT

Number of targets Number of nodes

2 100

4 200

6 300

8 400

10 500

12 600

14 700

16 800

Figure 7. Overhead of control packets for DAMA and HCTT

6 Conclusion and Future Work

In this paper, we investigated the possibility of obtaining upper and lower bounds for a dynamic

cluster assignment scenario. We assumed the tasks of CH and CM for nodes and a utility function for each

node based on its distance to the targets and its remaining energy. An upper bound is reached with

applying the Lagrangian relaxation which shows near optimal values and is much better than LP

relaxation upper bound. The lower bound is obtained by LP relaxation and rounding the results by

Randomized rounding and a greedy-based heuristic. Finally, a distributed greedy-based heuristic

algorithm is presented which shows good results between the upper and lower bounds in terms of

number CH and CM nodes and the total usefulness of nodes.

0

200

400

600

800

1000

1200

1400

1600

0 2 4 6 8 10 12 14 16 18

DAMA

HCTT

Number of targets

Avg

. nu

mb

er o

f co

ntr

ol p

acke

ts

26

A number of future works remain for this problem which includes seeking distributed algorithms for

LPR3 and LRUB. The optimization model can be extended to adapt for tree-based structures in target

tracking. Furthermore, combining parameters of other layers such as transmission power and rate leads

to interesting cross layer problems as joint optimization of task assignment and power control or rate

control, respectively. Network layer routing may be also taken into account as a challenging problem,

directing to joint optimization of task assignment and routing.

Acknowledgements

This work is supported in part by Iran Telecommunications Research Center (ITRC) under grant

#1042/500.

References

[1] J. Yick, B. Mukherjee, D. Ghosal, Wireless Sensor Network survey, Computer Networks, 52 (2008) 2292-2330.

[2] S. Misra, A. Vaish, Reputation-based role assignment for role-based access control in wireless sensor networks, Computer

Communication, 34 (2011) 281-294.

[3] E. F. Nakmura, H. S. Ramos, L. A. Villas, H. A. B. F. de Oliveira, A. L.L. de Aquino, A. A.F. Loureiro, A reactive role assignment

for data routing in event-based wireless sensor networks, Computer Networks, 53 (2009) 1980-1996.

[4] A. Pathak, V. K. Prasanna, Energy-Efficient Task Mapping for Data-Driven Sensor Network Macroprogramming, IEEE Trans.

on Computers, 59(7) (2010) 955-968.

[5] F. Semchedine, L. Bouallouche-Medjkoune, D. Aissani, Task assignment policies in distributed server systems: a survey,

Journal of Network and Computer Applications, 34(4) (2011) 1123-1130.

[6] G. Mainland, D. C. Parkes, M. Welsh, Decentralized, Adaptive Resource Allocation for Sensor Networks, in: Proc. Second

USENIX/ACM Symp. on Networked Systems Design and Implementations (NSDI 2005), Symp. A Quarterly J. in Modern Foreign

Literatures, USENIX Association, 2 (2005), pp. 315-328.

[7] Y. Tian, E. Ekici, Cross-Layer Collaborative In-Network Processing in Multihop Wireless Sensor Networks, IEEE Trans. on

Mobile Computing, 6(3) (2007) 297-310.

[8] T. Xie, X. Qin, An Energy-Delay Tunable Task Allocation Strategy for Collaborative Applications in Networked Embedded

Systems, IEEE Trans. on Computers, 57(3) (2008) 329-343.

[9] W. Chen, H. Miao, K. Wada, Autonomous Market-Based Approach for Resource Allocation in a Cluster-Based Sensor

Network, in: Proc. IEEE Symp. on Computational Intelligence in Multi-Criteria Decision Making (MCDM 2009), Nashville, TN, March-

April (2009), pp. 1-8.

[10] H. Rowaihy, M. P. Johnson, O. Liu, A. Bar-Noy, T. Brown, T. La Porta, Sensor-Mission Assignment in Wireless Sensor

Networks, ACM Trans. on Sensor Networks, 6(4) (2010) 1-33.

27

[11] M. Naderan, M. Dehghan, H. Pedram, V. Hakami, Survey of Mobile Object Tracking Protocols in Wireless Sensor Networks:

A Network-Centric Perspective, Int. J. Ad Hoc and Ubiquitous Computing, 11(1) (2012) 34–63.

[12] A. Boukerche, H. A. B. F. Oliveira, E. F. Nakamura, A. A. F. Loureiro, Localization Systems for Wireless Sensor Networks,

IEEE Wireless Communications Mag., 14(6) (2007) 6-12.

[13] W. B. Heinzelman, A. P. Chandrakasan, H. Balakrishnan, An Application-Specific Protocol Architecture for Wireless

Microsensor Networks, IEEE Trans. on Wireless Communications, 1(4) (2002) 660-670.

[14] A. Boukerche, R. Werner Nelem Pazzi, R. Borges Araujo, Fault-Tolerant Wireless Sensor Network Routing Protocols for the

Supervision of Context-Aware Physical Environments, J. of Parallel and Distributed Computing, 66(2006) 586-599.

[15] F. Hajiaghajani, M. Naderan , H. Pedram, M. Dehghan, HCMTT: Hybrid Clustering for Multi-Target Tracking in Wireless

Sensor Networks, Accepted in PerCom 2012, SeNAmI Workshop, 19-23 March (2012), Lugano, Switzerland.

[16] W. Zhang, G. Cao, DCTC: Dynamic Convoy Tree-based Collaboration for Target Tracking in Sensor Networks, IEEE Tran. on

Wireless Comm., 3(5), (2004), 1689-1701.

[17] A. A. Abbasi, M. Younis, A Survey on Clustering Algorithms for Wireless Sensor Networks, Computer Communications, 30

(2007) 2826-2841.

[18] R. Ahuja, T. Magnanti, J. Orlin, Network Flows. Prentice Hall, 1993.

[19] D. W. Pentico, Assignment problems: A golden anniversary survey, European J. of Operational Research, 176(2) (2007)

774-793.

[20] T. La Porta, C. Petrioli, D. Spenza, Sensor-Mission Assignment in Wireless Sensor Networks with Energy Harvesting, in:

Proc. of IEEE SECON, June (2011), Salt Lake City, Utah, USA.

[21] T. Le, T. J. Norman, W. Vasconcelos, Agent-based Sensor-Mission Assignment for Tasks Sharing Assets, in: Proc. of 3rd Int.

W. on Agent Technology for Sensor Networks (ATSN), (2009), pp. 33-40.

[22] C. Liu, G. Cao, Minimizing the cost of mine selection via sensor networks, in: Proc. of the IEEE INFOCOM, April (2009), Rio

de Janeiro, Brazil.

[23] A. T. Zimmerman, J. P. Lynch, F. T. Ferrese, Market-based Computational Task Assignment within Autonomous Wireless

Sensor Networks, in: Proc. of the IEEE Int. Conf. on Electro/Information Technology (EIT’09), Windsor, ON, June (2009), pp. 23-28.

[24] M. Cardei, J. Wu, M. Lu, Improving Network Lifetime using Sensors with Adjustable Sensing Ranges, Int. J. of Sensor

Networks, 1(1/2) (2006) 41-49.

[25] H.-W. Tsai, C.-P. Chu, T.-S. Chen, Dynamic Object Tracking in Wireless Sensor Networks, Computer Communications, 30

(2007), 1811-1825.

[26] V. Kulathumani, A. Arora, M. Demirbas, M. Sridharan, Trail: A Distance Snesitive WSN Service for Distributed Object

Tracking, Proc. of EWSN (2007), Delft, The Netherlands.

[27] H. Zhang, J. C. Hou, Maintaining Sensing Coverage and Connectivity in Large Sensor Networks, Ad Hoc and Sensor Wireless

Networks (AHSWN), 1 (2005), 89-124.

[28] P. Raghavan, C. D. Thompson, Randomized Rounding: A Technique for Provably Good Algorithms and Algorithmic Proofs,

Combinatorica, 7(4), (1987) 365-374.

[29] M. L. Fisher, An Applications Oriented Guide to Lagrangian Relaxation, INTERFACES, 15(2), (1985) 10-21.

[30] Boyd, S., Vandenberghe, L., (2004). Convex Optimization. New York, USA, Cambridge University press.

28

[31] D. P. Palomar, M. Chiang, A Tutorial on Decomposition Methods for Network Utility Maximization. IEEE J. on Sel. Areas in

Comm., 24(8) (2006), 1439-1451.

[32] AIMMS, Optimization Software for Mathematical Programming, http://www.aimms.com/, Nov. 2011.

[33] The Castalia Website. [Online]. Available: http://castalia.npc.nicta.com.au/

Marjan Naderan received her B.Sc. degree in Computer Engineering in 2004 and

the M.Sc. degree in Information Technology in 2006 both from Sharif University

of Technology (SUT), Tehran, Iran. She received the Ph.D. degree in Computer

Engineering, major in computer networks in Feb. 2012, from Amirkabir

University of Technology (AUT), Tehran, Iran. Dr. Naderan has currently joined

the Computer Engineering department of Shahid Chamran University (SCU) in

Ahwaz, Iran. She has reviewed papers in several journals and conferences such

as VTC, ICC, JNCA, JPDC, J. of Supercomputing and Trans. on Communications.

Her research interests include computer networks, wireless and mobile

networks, object tracking, network optimization and simulation of network

protocols.

Mehdi Dehghan received his B.Sc. degree in Computer Engineering from Iran

University of Science and Technology (IUST), Tehran, Iran in 1992, and his M.Sc.

and Ph.D. degrees from Amirkabir University of Technology (AUT), Tehran, Iran

in 1995, and 2001, respectively. He joined the Computer Engineering and

Information Technology (CEIT) Department of Amirkabir University of

Technology in 2004. Currently, as an associate professor, Dr. Dehghan is the

director of the Mobile Ad hoc and Wireless Sensor Lab at AUT. His research

interests include high speed networks, network management, mobile ad hoc

networks and fault-tolerant computing.

Hossein Pedram received his B.Sc. degree in Electrical Engineering in 1977

from Sharif University of Technology, Tehran, Iran. He continued with the M.Sc.

degree in Electrical Engineering in 1980 from Ohio State University, Columbus,

Ohio, USA and the Ph.D. degree in Computer Engineering in 1992 from

Washington State University, Pullman, Washington, USA. Dr. Pedram is

currently an associate professor at the Computer Engineering and Information

Technology (CEIT) department in Amirkabir University of Technology (AUT),

Tehran, Iran, and he is also the director of the Asynchronous lab at AUT. His

research interests include computer architecture, asynchronous design,

network on chip, innovations in computer architecture, distributed systems and

networking.

*Author Biography & Photograph