Connectivity maintenance and coverage preservation in ...yzchen/papers/papers/deployment... · their work in order to support general random wireless sensor networks by providing

Ad Hoc Networks 3 (2005) 744–761

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Connectivity maintenance and coverage preservationin wireless sensor networks

Di Tian *, Nicolas D. Georganas *

Distributed and Collaborative Virtual Environments Research Laboratory (DISCOVER), School of Information Technology

and Engineering, University of Ottawa, 800 King Edward Avenue, 550 Cumberland, Ottawa, ON, Canada K1N 6N5

Received 1 December 2003; accepted 1 March 2004Available online 9 April 2004

Abstract

In wireless sensor networks, one of the main design challenges is to save severely constrained energy resources andobtain long system lifetime. Low cost of sensors enables us to randomly deploy a large number of sensor nodes. Thus, apotential approach to solve lifetime problem arises. That is to let sensors work alternatively by identifying redundantnodes in high-density networks and assigning them an off-duty operation mode that has lower energy consumption thanthe normal on-duty mode. In a single wireless sensor network, sensors are performing two operations: sensing and com-munication. Therefore, there might exist two kinds of redundancy in the network. Most of the previous work addressedonly one kind of redundancy: sensing or communication alone. Wang et al. [Intergrated Coverage and ConnectivityConfiguration in Wireless Sensor Networks, in: Proceedings of the First ACM Conference on Embedded NetworkedSensor Systems (SenSys 2003), Los Angeles, November 2003] and Zhang and Hou [Maintaining Sensing Coverageand Connectivity in Large Sensor Networks. Technical report UIUCDCS-R-2003-2351, June 2003] first discussedhow to combine consideration of coverage and connectivity maintenance in a single activity scheduling. They provideda sufficient condition for safe scheduling integration in those fully covered networks. However, random node deploy-ment often makes initial sensing holes inside the deployed area inevitable even in an extremely high-density network.Therefore, in this paper, we enhance their work to support general wireless sensor networks by proving another con-clusion: ‘‘the communication range is twice of the sensing range’’ is the sufficient condition and the tight lower bound toensure that complete coverage preservation implies connectivity among active nodes if the original network topology(consisting of all the deployed nodes) is connected. Also, we extend the result to k-degree network connectivity andk-degree coverage preservation.� 2004 Elsevier B.V. All rights reserved.

Keywords: Coverage; Network connectivity; Scheduling integration; Redundancy; Wireless sensor networks

1570-8705/$ - see front matter � 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.adhoc.2004.03.001

* Corresponding authors. Address: 1504-25 Leith Hill Rd., Toronto, Canada M2J 1Z1. Tel./fax: +1 416 4918631.E-mail addresses: [email protected] (D. Tian), [email protected] (N.D. Georganas).

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D. Tian, N.D. Georganas / Ad Hoc Networks 3 (2005) 744–761 745

1. Introduction

Recently, the concept of wireless sensor networks has attracted a great deal of research attention due to awide-range of potential applications that will be enabled by such networks, such as battlefield surveillance,machine failure diagnosis, biological detection, home security, smart spaces, inventory tracking, etc. [1–4].A wireless sensor network consists of tiny sensing devices, deployed in a region of interest. Each device hasprocessing and wireless communication capabilities, which enable it to gather information about the envi-ronment and to generate and deliver report messages to the remote base station (remote user). The basestation aggregates and analyzes the report messages received and decides whether there is an unusual orconcerned event occurrence in the monitored area.

In wireless sensor networks, the energy source provided for sensors is usually battery power, which hasnot yet reached the stage for sensors to operate for a long time without recharging. Moreover, since sensorsare often intended to work in remote or hostile environments, such as a battlefield or desert, it is undesir-able or impossible to recharge or replace the battery power of all the sensors. However, long system lifetimeis always expected by many monitoring applications. The system lifetime, which is measured by the time tillall nodes have been drained out of their battery power or the network no longer provides an acceptableevent detection ratio, directly affects network usefulness. Therefore, conserving energy resource and pro-longing system lifetime is an important issue in design of large-scale wireless sensor networks.

Due to low cost of sensors, it is well accepted that wireless sensor networks can be deployed with a veryhigh node density (even up to 20 nodes/m3 [5]). In such high-density networks with energy efficient designrequirement, it is not desirable to have all nodes continuously monitoring the environment all the time. Ifall the sensors were vigilant at the same time, a single event would trigger many sensors to generate reportmessages. When these sensors intend to send their highly redundant report messages simultaneously, thenetwork would suffer excessive energy consumption and packet collision. Recent research advances haveshown that energy saving can be achieved by assigning a subset of nodes to provide monitoring serviceat any instant in a high-density network. In the literature, some algorithms [6–8] have been proposed fordynamically scheduling node activity to minimize the number of active nodes while maintaining the originalsystem sensing coverage and the quality of monitoring service.

Monitoring is just one duty assumed by sensors. Due to limited communication capability, each sensornode has to act as router to help others to forward data packets to remote base stations. Similarly, redun-dancy also exists in communication domain in high-density networks. Excessive amount of energy would bewasted by having all nodes to continuously listen to the media, if adaptively turning off radio of some nodeshas no influence on network connectivity and communication performance. In the literature, several pro-tocols [9–11] have been proposed for dynamic management of node activity in communication domain.

In high-density wireless sensor networks, energy waste always exists if only one kind of redundancy isidentified and removed from the network. Minimizing active nodes has to maintain both network connec-tivity and sensing coverage. Without sufficient coverage, the network cannot guarantee the quality of mon-itoring service. Without network connectivity, active nodes may not be able to send data back to remotebase stations. Most of the previous work addressed only one kind of redundancy: sensing or communica-tion alone. [13,14] are the earliest papers to address how to combine consideration of coverage and connec-tivity maintenance in a single activity scheduling. They both proved that ‘‘the communication range is atleast twice of the sensing range’’ is the sufficient condition to ensure that a full coverage of a convex areaimplies connectivity among active nodes inside the convex area. However, we argue that their proofs onlyguarantee safe integration of scheduling in those networks where all the deployed nodes can cover a convexhull without sensing holes inside. Random sensor deployment (such as throwing sensors from a plane) isalways believed to be one of the major advantages of wireless sensor networks over traditional wired sensornetworks. In such randomly deployed networks, it is hard or impossible to 100% guarantee complete cov-erage of the monitored region even if the node density is very high. Therefore, in this paper, we first enhance

746 D. Tian, N.D. Georganas / Ad Hoc Networks 3 (2005) 744–761

their work in order to support general random wireless sensor networks by providing the proof of anotherconclusion: ‘‘the communication range is twice of the sensing range’’ is the sufficient condition and the tightlower bound to ensure that complete coverage preservation implies connectivity among active nodes if theoriginal network topology (consisting of all the deployed nodes) is connected. It is worth to notice thatnode-scheduling algorithms with the capability of complete coverage preservation have existed in the liter-ature [6,7]. Also, we extend the result to figure out the relationship between k-degree network connectivityand k-degree coverage preservation to support different applications that may require different toleranceagainst node failures. Finally, we verify our analytical results through experiments.

The rest of this paper is organized as follows. Section 2 presents a review of the related work in the lit-erature. In Section 3, we provide definition for all the terms and notations used in the next sections. In Sec-tion 4, we figure out the relationship between connectivity maintenance and coverage preservation. InSection 5, we extend the result in Section 4 to higher degree. In Section 6, we present some experimentalresults. In Section 7, we conclude the paper.

2. Literature review

Minimizing energy consumption and prolonging the system lifetime is an important issue for wireless adhoc networks. Controlling network topology and scheduling node activity as a promising approach havebeen studying actively and broadly recently. There are several ways for topology control in the literature.

The first approach tries to reduce energy consumption in overhearing by restricting relay stations to asubset of the nodes in a network graph. [9–12 and etc.] fall into this category. In GAF [9], each node knowsits geographic location and uses this information to divide a given area into fixed square grids. The size ofgrid is designed to ensure that any pair of nodes within adjacent grids can directly communicate with eachother. Thus, the algorithm selects only one node in each grid to relay packets at any instant and makesother nodes go into sleep mode to save energy. Unlike GAF, [10,11] decide if a node should serve as a domi-nating node (relay station) based on connectivity among its neighbors. They differ in how to remove redun-dant dominating nodes caused by simultaneously marking. In Ascent [12], each node self-determines itsparticipation into multi-hop routing as a relay station, based on its contribution to network connectivity.Different from the above algorithms that are based on position information or neighbor connectivity, eachnode in Ascent evaluates the contribution by checking its local node density and measured message lossrate.

The second approach is to let each node adaptively self-configure its radio status while minimizing theimpact on global connectivity and communication performance. For instance, Xu et al. [15] proposed ascheme in which energy is conserved by letting nodes turn off their radio when they are not involved intodata communication (transmitting and receiving). They also present how to leverage neighbor size to in-crease the duration of the sleeping time.

The third approach restricts which neighbors can communicate with each concerned node, thereby form-ing an underlying subnetwork owning a lower node degree than the original one. In this way, transmissionpower and traffic interference in routing and searching can be reduced. [16–18] presented distributed andlocalized algorithms to form a minimum-power topology, which contains the minimum-power paths be-tween any pair of nodes. The difference between them is that [16,17] rely on location information, while[18] uses direction information. Unlike them, other algorithms try to take advantage of properties of somewell-known graphs. For instance, [19] proposed routing algorithms based on prior constructed Gabrielgraph, relative neighborhood graph and Delaunay triangulation. It also presented a localized algorithmto form a planar Delaunay triangulation.

All the above topology control techniques are based on a communication network graph. They do notaddress the issue of sensing redundancy.


In the recent years, some research efforts have been made on sensing coverage in wireless sensor network.[20] presented how to use ILP (Integer Linear Program) technique to solve several optimal 0/1 node sched-uling problems in wireless sensor networks. For example, what is the minimal active node set that can com-pletely cover a sensor field? [21] designed a heuristic approach to find k-sets of nodes so that each node setcan cover the whole monitored area and the cardinality of node sets is maximal. The approaches presentedin the above two papers are both centralized, therefore requiring collection of global information. [6–8] pro-vided alternative, suboptimal, distributed algorithms to solve same or similar problems. In [8], a subset ofsensor nodes is selected out initially and operates in active mode until they run out of their energy or aredestroyed. Other nodes fall asleep and wake up occasionally to probe their local neighborhood. If there isno working node within its probing range, a sleeping node starts to work, otherwise it sleeps again. In [8],probing range can be adjusted to control sensing redundancy. However, the original sensing coverage maybe reduced after node scheduling. Tian and Georganas [6] proposed a distributed node-scheduling algo-rithm that can provide complete coverage preservation. In their algorithm, each node arithmetically calcu-lates the union of all the sectors covered by its neighbors and determines its working status according to thecalculation result. Yan et al. [7] presented another distributed node-scheduling algorithm, wherein eachnode divides its sensing area into small grids. For a grid, each sensor node sorts those neighbors thatcan cover this grid, in the ascending order of their associated reference points. Based on the sequence ofthe reference points, the nodes in the sequence can coordinate their on-duty time in order to continuouslymonitor that concerned grid. By merging the results of all the covered grids, each node can finally determineits working time. Their algorithm can support k-degree coverage preservation by extending each node�sworkspace proportionally along the sequence of reference points.

Like those topology control techniques introduced before, the above algorithms still remove one kind ofredundancy without addressing the other (i.e. in communication domain).

To the best of our knowledge, [13,14] are the earliest papers discussing how to integrate activity sched-uling in both domains: communication and sensing. Both of them independently proved the same conclu-sion: if a convex region is completely covered by a set of nodes, the communication graph consisting ofthese nodes is connected when RP 2r. In this paper, we enhance their work by providing the proof: ifan original network is connected and the identified active nodes can cover the same region as all the originalnodes, then the active nodes are connected when R P 2r. The difference between their conclusion and ourscan be better understood through a simple scenario as illustrated in Fig. 1. In this scenario, the sensing fieldcan be divided into four subfields 1–4, with each subfield completely covered by a subset of nodes. We callthe node subset as subset 1–4 correspondingly. There are two narrow gaps in the middle of the sensing fieldthat cannot be monitored by any node. Furthermore, the communication graph consisting of all the nodesare connected. After node scheduling, redundant nodes is removed from each subset without changing thecoverage graph. According to the work in [13,14], we can only know that the active nodes within each sub-set are connected with each other, but cannot determine if the active nodes among different subsets are con-nected. However, the latter can be confirmed to be affirmative, according to the proofs in the next sections.Besides theoretical result, [13,14] proposed a distributed protocol to identify redundant nodes in sensingdomain, respectively. Both protocols were designed based on the same theorem: a convex area is completelycovered by a set of nodes if all the intersection points inside the area are covered. Moreover, [13] presentedhow to integrate the proposed protocol with the existing SPAN protocol [11] for R < 2r cases.

It is worth to mention that there are some other papers discussing coverage problem for wireless sensornetworks. However, the definition of their coverage is different from the one addressed above. In the abovepapers, coverage is the geographical area that can be covered by at least one sensor node. While in [22–25],coverage is used to indicate with how much confidence, the sensors can detect an intruded object penetrat-ing through the network. [24] presented a centralized method using Vornonoi diagram and Delaunay tri-angulation to find minimal exposure path (a path connecting two points which maximizes the distanceof the path to all sensor nodes) and maximal exposure path (a path connecting two points which maximizes

1

3 4

2

Fig. 1. Illustration of difference between the proof in [13,14] and that in this paper.


the smallest observability of all the points on the path) in a network graph, respectively. In [23], anothercentralized approach was designed to find a path connecting two points that minimizes the integral observ-ability along the path. Clouqueur et al. [22] introduced energy and noise into sensing model. Based on theirsensing model, they proposed a centralized solution to find a minimum exposure path. Li [25] revisited themaximal exposure path problem and proposed a localized, distributed solution. Also, they extended theiralgorithm to find a maximal exposure path with least energy consumption and smallest traveling distance.

Besides activity scheduling, node deployment strategies are directly related with sensing coverage, aswell. [26] addressed how to use ILP (Integer Linear Programming) technique to solve such problems: (i)given a grid-based region and types of a sensor set, determine where to deploy these sensors to minimizethe overall sensor cost under a certain coverage constraint (for example, each grid is covered at least byk nodes). (ii) how to deploy sensor nodes to ensure that each grid can be covered by an unique node subsetso that target position can be determined by identifying the set of observers. [27] presented a sequentialnode deployment strategy that, instead of deploying all the nodes at one time, adds the nodes step by stepuntil the desired minimum exposure is achieved.

3. Preliminaries

To facilitate the future description, we first define some terms and notations used in the future. We viewa wireless sensor network from two perspectives: communication and sensing.

3.1. Communication model

We assume that two sensors can directly exchange messages if their Euclidean distance is not larger thana communication range R and the communication range is homogeneous among all the sensors in a net-work. We characterize a wireless sensor network as a communication graph G(V,E), where each elementin the node set V(G) represents a sensor in the network. For any pair of nodes v and u in V(G), the edge(v,u) is in the edge set E(G) if and only if the Euclidean distance between v and u, denoted as d(v,u), isnot larger than the communication range R. We call two nodes are adjacent if they are incident to a com-mon edge. The neighbor set of node v, denoted as N(v), is the set of those nodes that are adjacent to node v.In other words, N(v) contains all the other nodes that are located within the communication range R awayfrom node v. Each node in N(v) is called a neighbor of node v. We define a path P in G(V,E) as an alter-


nating sequence of nodes in V(G) and edges in E(G), with each edge being incident to the nodes immediatelyproceeding and succeeding it in the sequence. P can be represented by its complete edge sequence as

P = (n1, n2)(n2, n3) � � � (nl�2, nl�1)(nl�1,nl) or simplified as n1 $Pnl. We call two nodes v and u in V(G) con-

nected if there is a path P in G(V,E), whose ends are the given nodes, and the whole graph G(V,E) con-nected if every pair of nodes in V(G) is connected.

3.2. Sensing model

Next, we introduce the concept of sensing coverage into a network graph G(V,E). We define A as theregion where sensors are initially deployed in and are supposed to monitor afterwards. We assume eachsensor can do 360� observation. We denote the maximal circular area centered at a sensor that can be cov-ered by that sensor as its sensing area S(v). The radius of S(v) is called sensor v�s sensing range. We assumethat all sensors have the same sensing range r. We also define the coverage C(X) of a node set X, as theunion of sensing area covered by each node in X i.e. C(X) =["v2XS(v). The overall sensing coverage ofa network graph G(V,E) is C(V(G)) =["v2V(G)S(v).

3.3. Sensing scheduling with coverage preservation

We do not restrict our discussion on those networks whose initial sensing coverage is always the same orlarger than the expected monitoring region A. Instead, we consider general networks that may have small‘‘sensing holes’’ geographically scattered throughout the deployed area. We argue that the latter is more real-istic situation because wireless sensor networks use a random node deployment strategy. Such a strategy im-plies that it is difficult or impossible to completely ensure that each point in the region A can be covered by atleast one sensor node even at a very high node density. However, it is possible that after removing some sens-ing redundant nodes, the remaining nodes can cover the same area as initially, without introducing any cov-erage loss. There have been such node-scheduling algorithms in Refs. [6,7]. Such algorithms assign eachsensor node a sensing status, active or inactive. Accordingly, the node set V(G) in the initial network graphG(V,E) is divided into two parts: Vactive(G) and Vinactive(G). Vactive(G) consists of all the nodes obtaining anactive status after scheduling. Vinactive(G) contains the others, i.e. Vinactive(G) = V(G) � Vactive(G). We areinterested in a subgraph of G(V,E), denoted as G(Vactive), which is induced by the node set Vactive(G), withthe edge set as the subset of E(G) consisting of those edges whose both ends are in Vactive(G). We also definean active path as a path inG(V,E), whose all the immediate nodes are active ones. The counterpart is inactivepath owning at least one inactive immediate node. If nodes u and v in V(G) can be communicated through anactive path, we call these two nodes are actively connected, denoted as u $active v.

4. Network connectivity on coverage preservation

As mentioned above, some existing node-scheduling algorithms have the capability of complete coveragepreservation, i.e. C(Vactive) = C(V). Next, we will prove that RP 2r is the sufficient condition and the tightlower bound to ensure connectivity of any subgraph G(Vactive) induced from a connected network graphG(V,E), here the induced subgraph G(Vactive) satisfied that the condition: C(Vactive) = C(V).

Our proof is based on the following assumptions:

1. There may be a large number of sensor nodes in the network. However, the number is always finite.2. There are no two nodes located at the exactly same position.

First, we prove the sufficient condition.


Lemma 1. If R P 2r and C(Vactive) = C(V), let node u and node v be any pair of nodes in V(G) satisfying (i)

u 2 Vinactive(G), (ii) 2r < d(u,v) 6 R, then there must exist another node w satisfying (i) w 2 Vactive(G), (ii)

w 2 N(v), (iii) w 2 N(u), (iv) d(u,w) 6 2r.

Proof. As illustrated in Fig. 2, suppose segment uv interacts with the boundary of S(u) at the point P, theremust exist an active node, assumed w, whose distance to point P is not larger than sensing range r. Other-wise, the system overall sensing coverage would be reduced after node scheduling (because point P can becovered by the inactive node u, but cannot be reached by any active node). The position of node w is deter-mined by its distance to point P and the angle a with 0 6 d(w,P) 6 r and 0 6 a 6 p. According totrigonometry,

dðv;wÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidðw; P Þ2 þ dðv; PÞ2 � 2� dðw; P Þ � dðv; P Þ � cos a

q;

here d(v,P) = d(u,v) � r.The above expression gets its maximal value d(u,v) when and only when d(w,P) = r and a = p. In other

words, when and only when node w is located at the same place as node u. Since there are no two nodes inthe network located at the same position, d(v,w) must be smaller than d(u,v). Because d(u,v) 6 R, we haved(v,w) 6 R.

Furthermore,

dðu;wÞ ¼ dðu; P Þ2 þ dðw; P Þ2 � 2� dðu; PÞ � dðw; P Þ � cosðp� aÞ;
where d(u,P) = r . d(u,w) gets its maximal value 2r when and only when d(w,P) = r and a = 0, i.e. node w islocated at point Q. { RP 2r, ) node w is adjacent to node u.
Therefore, we conclude that for any neighboring node located farther than twice of the sensing rangeaway from an inactive node, there must exist another active node in the network that is adjacent to boththem and is located within twice of the sensing range from the inactive node, if the communicate range islarger than twice of the sensing range and the overall sensing coverage is completely preserved after nodescheduling. h

Lemma 2. If C(Vactive) = C(V), let node u and node v be any pair of nodes in V(G) satisfying (i) u 2Vinactive(G), (ii) d(u,v) 6 2r, then there must exist another node w satisfying (i) w 2 Vactive(G), (ii) Q 2 S(w),

where point Q is the intersection point between segment uv and the boundary of S(v) as illustrated in Fig. 3.

Proof. We split out consideration into two cases.Case i: The first case is that node v is an inactive node. Because coverage is completely preserved after

node scheduling, there must be an active node that can cover point Q.Case ii: We prove the lemma for the case that node v is an active node. Since in the network, there are no

two nodes whose positions are exactly the same, there must be other nodes besides node v in the active node

u vP Q

w

α

Fig. 2. Illustration for Lemma 1.

uv

w

PQ

O

x

uv

w

P

QO

x

R

ββ



set Vactive(G). Otherwise, the sensing area of node u cannot be completely preserved after node scheduling.Suppose in Vactive � {v}, node w has the minimal distance to point Q. Assume d(w,Q) = r + g, g > 0. Asillustrated in Fig. 3, the segment wQ interacts with the boundary of S(w) at point P. The distance betweennode w and point P is d(w,P) = r and the distance between point P and point Q is d(P,Q) =d(w,Q) � d(w,P) = g. We consider the following two sub-cases.

For the sub-case shown on the left of Fig. 3, i.e. 90� 6 angle \PQv 6 180�, we can always find a pointO, located in segment PQ and satisfying (i) 0 < d(O,Q) = e < g, (ii) 0 < d(O,P) = g � e < g, (iii) O 2 S(u).{ d(w,O) = r + g � e > r, ) node w cannot cover point O,

* dðv;OÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidðv;QÞ2 þ dðO;QÞ2 � 2� dðv;QÞ � dðO;QÞ � cos\PQv

qP

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidðv;QÞ2 þ dðO;QÞ2

q

> dðv;QÞ ¼ r;

where 90� 6 \PQv 6 180� ) node v cannot cover point O, either. If Vactive � {v,w} is not empty, let node xbe any node in it, whose distance to point Q is d(x,Q) = r + g + d, d P 0 because w is the node inVactive � {v} whose distance to point Q is minimal.

dðx;OÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidðx;QÞ2 þ dðO;QÞ2 � 2� dðx;QÞ � dðO;QÞ � cos b

qP dðx;QÞ � dðO;QÞ ¼ r þ gþ d� e

P r þ g� e ¼ dðw;OÞ;

where 0 6 b 6 p. This implies that if node w cannot cover point O, other active nodes in Vactive � {v,w}cannot, either. Obviously, it contradicts with the condition C(Vactive) = C(V).

For the sub-case shown on the right of Fig. 3, i.e. 0� 6 angle\PQv < 90�, we always find a point O that islocated in line PQ, satisfying (i) 0 < d(O,Q) = e 0 < g, (ii) 0 < d(O,P) = g + e 0 > g, (iii) O 2 S(u). { d(w,O) =d(w,Q) + d(Q,O) = r + g + e 0 > r, ) node w cannot cover point O.

* dðv;OÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidðv;QÞ2 þ dðO;QÞ2 � 2� dðv;QÞ � dðO;QÞ � cos\OQv

q>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidðv;QÞ2 þ dðO;QÞ2

q

> dðv;QÞ ¼ r;

where 90� < \OQv = (180� � \PQv) 6 180�.


) node v cannot cover point O, either. If Vactive � {v,w} is not empty, let node x be any node in it, whosedistance to point Q is d(x,Q) = r + g + d 0, d P 0.

dðx;OÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidðx;QÞ2 þ dðO;QÞ2 � 2� dðx;QÞ � dðO;QÞ � cos b

qP dðx;QÞ � dðO;QÞ ¼ r þ gþ d0 � e0

P r þ g0 � e0 > r;

where 0 6 b 6 p. This implies that there is no active node that can cover point O. So it contradicts with thecondition C(Vactive) = C(V), as well.

Therefore, we conclude Lemma 2 by contradiction.

Lemma 3. If R P 2r and C(Vactive) = C(V), let node u and node v be any pair of nodes in V(G) satisfying (i)

u 2 Vinactive(G), (ii) r < d(u,v) 6 2r, then there must exist another node w satisfying (i) w 2 Vactive(G), (ii)

w 2 N(v), (iii) w 2 N(u), (iv) d(u,w) < d(u,v).

Proof. As illustrated in Fig. 4, segment uv interacts with the boundary of S(u) at point P. According toLemma 2, there must exist another node, supposing node w, which is an active one and whose distanceto point P does not exceed the sensing range, i.e. 0 6 d(w,P) 6 r. Let the angle of \wPv be b, we have0 6 b 6 p. According to trigonometry,

dðv;wÞ2 ¼ dðv; pÞ2 þ dðw; P Þ2 � 2� dðv; pÞ � dðw; P Þ � cos b;

where d(v,P) = r. Obviously, d(v,w) gets its maximal value 2r when and only when d(w,P) = r and b = p, i.e.w is located at point Q. { RP 2r, ) node w is adjacent to node v. Furthermore,

dðu;wÞ2 ¼ dðu; pÞ2 þ dðw; P Þ2 � 2� dðu; pÞ � dðw; P Þ � cosðp� bÞ;

with d(u,w) reaching its maximal value r + d(u,P) when and only when d(w,P) = r and b = 0, i.e. nodew is located at the same place as node v. Since there are no two nodes in the network located at thesame position, d(u,w) must be smaller than d(u,v). Also, { v 2 N(u) ) w 2 N(u). Therefore, we concludethat for any neighbor node located within twice of the sensing range and farther than the sensing rangeaway from an inactive node, there must exist another active node in the network that is adjacent toboth them and whose distance to that inactive node is smaller than the neighbor node, when the com-municate range is at least twice of the sensing range and coverage is completely preserved after nodescheduling. h

Lemma 4. If RP 2r, let node u, node v and node w be any three nodes in V(G) satisfying (i) 0 < d(u,v) 6 r,

(ii) 0 < d(u,w) 6 r, then node v and node w are adjacent to each other.

u vP

w

Qβ



Proof. As illustrated in Fig. 5, the distance between node v and node w can be expressed as

dðv;wÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidðu; vÞ2 þ dðu;wÞ2 � 2� dðu; vÞ � dðu;wÞ � cos/

q;

where 0 < d(u,v) 6 r, 0 < d(u,w) 6 r and 0 6 / 6 p. So d(v,w) gets its maximal value 2r when and only when/ = p and d(u,v) = d(u,w) = r. Since RP 2r, we conclude that any pair of nodes, which are located withinthe sensing range away from a same node, can communicate with each other directly. h

Lemma 5. If RP 2r and C(Vactive) = C(V), let node u, node v and node w be any three nodes in V(G) satis-

fying (i) u 2 Vinactive(G), (ii) v 2 N(u), (iii) w 2 N(u), then node v and node w can communicate with each other

directly or through an active path in G(V,E).

Proof. According to the distance of node v and nodew to node u, we split our consideration into several cases.Case i: 0 < d(u,v) 6 r and <d(u,w) 6 r.According to Lemma 4, node v and node w can communicate directly with each other.Case ii: In {v,w}, one node has a distance to node u not larger than the sensing range. The other�s

distance to node u is larger than the sensing range and does not exceed twice of the sensing range. Withoutloss of generality, we assume that 0 < d(u,v) 6 r and r < d(u,w) 6 2r.

If d(v,w) 6 R, then node v and node w can communicate with each other directly. Otherwise (i.e.d(v,w) > R), according to Lemma 3, there must exist another active node x in Vactive(G) satisfying (i)x 2 (N(u)\N(w)), (ii) d(u,x) < d(u,w). It is easy to know v 5 x because x 2 N(w) and v 62 N(w). If we can findnode x further satisfying 0 < d(u,x) 6 r, then referring to the conclusion for case i, node v and x are adjacentwith each other. Thus, node v and w can communicate with each other via the active path (v,x)(x,w).Otherwise (i.e. r < d(u,x) < d(u,w) 6 2r), according to Lemma 3, there must exist another active node y with(y 5 x 5 w) satisfying that y 2 (N(x)\N(u)) and d(u,y) < d(u,x). If the selected node y is node v, then nodev and node w can communicate with each other via the active path (v = y,x)(x,w). Otherwise (i.e.v 5 y 5 x 5 w), if we can find such an active node y further satisfying 0 < d(u,y) 6 r, according to theconclusion for case i, node v and node y are adjacent with each other. Thus, node v and node w cancommunicate with each other via the active path (v,y)(y,x)(x,w). However, if r < d(u,y) < d(u,x) <d(u,w) 6 2r, there must be another active node z satisfying that z 2 (N(u)\N(y)) and d(u,z) < d(u,y).Similarly, it can be concluded that there is an active path (v = z,y)(y,x)(x,w) or (v,z)(z,y)(y,x)(x,w) unlessr < d(u,z) 6 2r. The proof is going on if the new identified active node has a distance to node u larger thanthe sensing range. Since, the number of nodes in the graph G(V,E) is finite, the process can always end at anactive node that equals to node v or whose distance to node u is equal to or smaller than the sensing range.

u

v

w

P

Q

φ

Fig. 5. Illustrate for Lemma 4.


In any case, there exists a network path between node v and node w whose immediate nodes are all activeones. So we can conclude the Lemma 5 for case ii.

Case iii: In {v,w}, one node has a distance to node u equal to or smaller than the sensing range. The otherhas a distance to node u exceeding twice of the sensing range. Without loss of generality, we assume0 < d(u,v) 6 r and 2r < d(u,w).

If d(v,w) 6 R, then node v and node w can communicate with each other directly. Otherwise (i.e.d(v,w) > R), according to Lemma 1, there must exist another active node x in Vactive(G) satisfying thatx 2 (N(u)\N(w)) and d(u,x) 6 2r. According to the conclusions for cases i and ii, node v and node x areadjacent to each other or can communicate through an active path v $active x. Correspondingly, the activepath between node v and node w is (v,x)(x,w) or v $active xþ ðx;wÞ. Therefore, we conclude the Lemma 5 forcase iii.

Case iv: Both nodes v and w have a distance to node u larger than the sensing range and equal to orsmaller than twice of the sensing range, i.e. r < d(u,v) 6 2r and r < d(u,w) 6 2r.

If d(v,w) 6 R, then node v and node w can communicate with each other directly. Otherwise (i.e.d(v,w) > R), similar to the proof process for case ii, we can always get a sequence of active nodes, denoted asx1, x2,. . .,xi, i P 1, satisfying (i) d(u,xi) < d(u,xi � 1) < � � � < d(u,x2) < d(u,x1) < d(u,w), (ii) xj 2 N(u),1 6 j 6 i, (iii) x1 2 N(w), (iv) xj 2 N(xj � 1), 2 6 j 6 i, (v) {r < d(u,xj)P 2r and xj 5 v, 1 6 j < i}, (vi){0 < d(u,xi) 6 r or xi = v}. If xi = v, we have got an active path between node v and node w as (w, x1)(x1,x2 � � � (xi�1, xi = v). Otherwise (i.e. xi 5 v and 0 < d(u,xi) 6 r), since 0 < d(u,xi) 6 r and r < d(u,v) 6 2r,according to the conclusion for case ii, we know that node v and node xi must be adjacent to each other orconnected through an active path v $active xi. Because we have got an active path between node w and node xias (w, x1)(x1, x2)� � �(xi�1, xi), after concatenating the above two paths, we get an active path between node vand node w as (w, x1)(x1, x2)� � �(xi�1, xi)(xi, v) or v $active xi þ ðxi; xi�1Þ � � � ðx2; x1Þðx1;wÞ. Therefore, weconclude the Lemma 5 for case iv.

Case v: In {v,w}, one node has the distance to node u larger than the sensing range and equal to orsmaller than twice of the sensing range. The other has the distance exceeding twice of the sensing range.Without loss of generality, we assume r < d(u,v) 6 2r and 2r < d(u,w).

If d(v,w) 6 R, then node v and node w can communicate with each other directly. Otherwise (i.e.d(v,w) > R), according to Lemma 1, there must exist another active node x in Vactive(G) satisfying thatx 2 (N(u)\N(w)) and d(u,x) 6 2r. According to the conclusion for case iv, node v and node x are adjacent toeach other or can communicate through an active path v $active x. So, we can get an active path between nodev and node w as (v,x)(x,w) or v $active xþ ðx;wÞ. Therefore, we conclude the Lemma 5 for case v.

Case vi: Both nodes v and w have the distance to node u larger than twice of the sensing range, i.e.2r < d(u,v) and 2r < d(u,w).

If d(v,w) 6 R, then node v and node w can communicate with each other directly. Otherwise (i.e.d(v,w) > R), according to Lemma 1, there must exist another active node x satisfying that x 2 (N(u)\N(w))and d(u,x) 6 2r. According to the conclusions for case v and case iii, node v and node x are adjacent to eachother or can communicate through an active path v $active x. Thus, we can get an active path between node vand node w as (v,x)(x,w) or v $active xþ ðx;wÞ. Therefore, we conclude the Lemma 5 for case vi. h

Lemma 6. If R P 2r and C(Vactive) = C(V), for any inactive path in G(V,E), there is a corresponding active

path in G(V,E), which ends at the same pair of nodes.

Proof. Let P0 be an inactive path ending at node u and node v: P0 = (u = w1, w2)(w2, w3)� � �(wi�2,wi�1)(wi�1, wi = v). In {wj, 2 6 j 6 i � 1}, the number of inactive nodes is l with l P 1. Suppose wk is thefirst inactive immediate node in path P0. We have all wj, 2 6 j 6 k � 1 are active ones and the numberof inactive nodes in {wj, k + 1 6 j 6 i} is l � 1. According to Lemma 5, node wk�1 and node wk+1 are adja-cent to each other or can communicate through an active path wk�1 !activewkþ1. Thus, we get a new path


P 1 ¼ ðu ¼ w1;w2Þðw2;w3Þ � � � ðwk�2;wk�1Þðwk�1;wkþ1Þðwkþ1;wkþ2Þ � � � ðwi�1;wi ¼ vÞ
or
P 1 ¼ ðu ¼ w1;w2Þ � � � ðwk�2;wk�1Þ þ wk�1 $activewkþ1 þ ðwkþ1;wkþ2Þ � � � ðwi�1;wi ¼ vÞ:
In either case, the number of inactive immediate nodes in the new path P1 is one less than that in path P0. Inthe same way, we can find another new path P2 whose inactive immediate node number is l � 2. The processis going on. In each step, the number of inactive immediate nodes is decremented by one. Since the nodenumber in the graph G(V,E) is finite, the number of node in the old path P0 is finite as well. This impliesthat we can finally get a path between node u and node v in G(V,E), where all immediate nodes are activeones. h
Theorem 1. When RP 2r and the system sensing coverage is completely preserved after node scheduling, if a

network graph G(V,E) is originally connected, then the induced subgraph G(Vactive) must be connected.

Proof. Let node u and node v be any pair of nodes in Vactive(G). { V(G) � Vactive(G) and the network graphG(V,E) is connected, there must exist a path between these two nodes in G(V,E). Let path P be such a path.If P is active, according to the definition of G(Vactive), path P is in G(Vactive). If path P is inactive, accordingto Lemma 6, we can always find a corresponding path Q between node u and node v in G(V,E), whose allimmediate nodes as active ones, i.e. path Q is in G(Vactive). That is, for any pair of nodes in Vactive(G) we canalways find a path in G(Vactive) when the original network graph G(V,E) is connected. The proof ends. h

Theorem 1 just tells us that R P 2r guarantees connectivity maintenance after complete coverage pres-ervation. It does not imply disconnectivity among active nodes for networks with R < 2r. Consider the sim-ple scenario illustrated in Fig. 6 with R = 1.8r. Node u is the only node assigned an inactive status afternode scheduling. The distance between any pair of node v, node w and node x is

ffiffiffi3

pr < R. Therefore,

the subgraph consisting of the three inactive nodes is still connected. So R P 2r is not the necessary con-dition to ensure that coverage preservation implies connectivity maintenance. However, we can prove thatR = 2r is the tight lower bound for this conclusion.

Theorem 2. When R < 2r and the system sensing coverage is completely preserved after node scheduling, the

induced network subgraph G(Vactive) may be disconnected even when the original network graph G(V,E) is

connected.

v w

x

u

Fig. 6. Connectivity maintenance for R < 2r.

O

vu P Q w

Fig. 7. Illustration for Theorem 2.


Proof. Assume R = 2r � n, 0 < n < 2r. Let us consider the scenario shown in Fig. 7, node u and node v areadjacent to each other and has a distance between them as 0 < l = d(u,v) < min(r,n,R). Besides node u andnode v, there are a sufficient number of nodes located on the circle centered at node u with a radius as R + e,as illustrated in Fig. 7. e satisfies that 0 < e < l. The union of the sensing areas of these nodes forms a‘‘ring’’, whose inner radius is R + e � r and whose outer radius is R + e + r. From the figure, we can seethat node u can cover part of node v�s sensing area. The remaining part is a ‘‘crescent’’ that is containedin the ‘‘ring’’. That is because (i) the closed distance from node u to any point in the ‘‘crescent’’ is r thatsatisfies r > R + e � r ({ r = 2r � r = R + n � r > R + l � r > R + e � r), and (ii) the farthest distancefrom node u to any point in the crescent, is d(u,Q) = r + l that satisfies r + l < r + R + e ({r + l = r + R + l � R < r + R + 0 < r + R + e). This implies that node v can take an inactive status duringnode scheduling. Before node scheduling, the network is connected because d(u,v) = l < R andd(v,w) = R + e � l < R. However, after removing node v, node u becomes an ‘‘isolated’’ node because itsdistance to any other active node is R + e > R. Therefore, the proof ends. h

Combining the proofs for Theorems 1 and 2, we can conclude that:

Theorem 3. RP 2r is the sufficient condition and tight lower bound to ensure that complete sensing coveragepreservation implies network connectivity among active nodes in an original connected network graph.

The significance of the above conclusions is that it establishes the relationship between connectivitymaintenance and coverage preservation, enables us to focus on sensing domain when connectivity and cov-erage are both required, therefore greatly simplifies the safe integration of activity scheduling in bothdomains.

5. k-degree connectivity on k-degree coverage preservation

The result in previous section can be extended to the relationship between k-degree connectivity andk-degree coverage preservation with k > 1. That is, RP 2r is the sufficient condition to ensure that k-degreecoverage preservation implies k-degree connectivity maintenance.

Before proving it, we first formulate definitions of k-degree connectivity and k-degree coverage preser-vation. We call a graph G(V,E) is k-connected if for every subset X of V(G) with fewer than k elements,

C(Y)

C(X)

C(Vinactive )P

Fig. 8. Illustration of Theorem 4.


V(G) � X is connected. We call a node-scheduling algorithm can preserve k-degree coverage if for everyinactive node identified by that algorithm, any point in its sensing area can be covered by at least k activenodes. Similar to 1-degree coverage preservation explained in the previous section, this definition acceptsthe situation that part of expected monitoring area A is weakly covered initially due to random nodedeployment, i.e. the following discussion is not restricted to those networks where every point in the ex-pected monitoring area A must be covered by at least k nodes before node scheduling. It just requires thatnode-scheduling algorithms can guarantee that the initially weak coverage would not be exacerbated.

Theorem 4. When RP 2r and the system sensing coverage is completely k-degree preserved after node

scheduling, if a network graph G(V,E) is originally k-connected, then the induced subgraph G(Vactive) must be

k-connected.

Proof. Let node set X be any subset of Vactive fewer than k elements. We denote the node subset Vactive � X

as Y, consisting of all the other elements in Vactive. Based on coverage definition, C(X)[C(Y) = C(Vactive).Since the original system coverage is preserved after node scheduling, we have C(V) = C(Vactive).

* CðV Þ ¼ CðV active þ V inactiveÞ ¼ CðV activeÞ [ CðV inactiveÞ ¼ CðV activeÞ;

* CðV activeÞ � CðV inactiveÞ:
We can prove that C(Vinactive) � C(Y) by contradiction. Suppose there is a point P in C(Vinactive) satisfyingthat P 62 C(Y), as shown in Fig. 8. Point P must be in C(X) because C(X)[C(Y) = C(Vactive) � C(Vinactive).According to the definition of k-degree coverage preservation, point P must be covered by at least k activenodes. Since node set X has fewer than k elements, there must be a node in Y, covering point P. It is con-tradictory with the assumption P 62 C(Y). Therefore, we have C(Vinactive) � C(Y). We can further getC(Y + Vinactive) = C(Y). Based on the definition of k connectivity, the subgraph G(Y + Vinactive) is con-nected. Also, after removing the node subset Vinactive from the network graph G(Y + Vinactive), the systemcoverage is not reduced. Therefore, according to Theorem 1, the subgraph G(Y) is also connected. Remov-ing any subset X with fewer than k elements from G(Vactive), the subgraph G(Vactive � X) is still connected.The means G(Vactive) is k-connected. Therefore, the proof ends. h
6. Experimental results

The experiments are performed based on the light-weighed sensing simulator implemented for the algo-rithm in [6]. However, a small modification is made to increase the size of active node set by dividing eachnode�s sensing area into small grids and determining each node�s status by checking if each grids inside its


sensing area is covered by a node succeeding the concerned node in the scheduling sequence. It is justified inthis context because we concern the connectivity maintenance among minimal active nodes in originalconnected network graphs instead of simplicity and efficiency of scheduling algorithm. Fig. 9 illustrates

Fig. 9. (a) Overall sensing coverage of a network with 350 nodes in a 100 · 100 m2 region, (b) scheduling result with r as 6 m (blackcircles representing inactive nodes), (c) network is connected before node scheduling with R = 9 m, (d) active nodes are disconnectedwith R = 9 m, (e) network is connected before node scheduling with R = 12 m and (f) active nodes are disconnected with R = 12 m.

Table 1Thenumber of network topologieswith connectivitymaintenance after coveragepreserving sensing-scheduling (Nnodes in a 100 · 100m2

region, communication range R = 10 m)

N R/r

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2

200 670 813 879 955 982 997 1000 999 1000 1000 1000 1000 1000250 679 794 901 971 984 994 995 999 1000 1000 1000 1000 1000300 648 794 896 957 980 996 996 1000 1000 1000 1000 1000 1000350 699 812 885 954 985 997 1000 1000 1000 1000 1000 1000 1000400 634 784 890 952 989 997 998 1000 1000 1000 1000 1000 1000450 656 803 887 960 982 995 999 1000 1000 1000 1000 1000 1000500 666 785 888 953 978 991 999 1000 1000 1000 1000 1000 1000550 674 801 885 955 987 998 998 1000 1000 1000 1000 1000 1000


a network topology with 350 nodes uniform randomly placed in a 100-by-100-m region. When the sensingrange is 6 m and the original system sensing coverage is shown as Fig. 9(a). The modified coverage-preserv-ing node scheduling algorithm in [6] identifies some inactive nodes as marked in Fig. 9(b) while maintainingthe system sensing coverage is unchanged. Fig. 9(d) shows that the connectivity among active nodes is des-troyed with the communication range is 1.5 times of the sensing range. While in the same sensing config-uration, if the communication range is twice the size of the sensing range, the connectivity is still maintainedas illustrated in Fig. 9(f). Energy saving can be achieved by communicating upon the simplified networkgraph with a lower node density and lower link density as shown in Fig. 9(f) and by monitoring the envi-ronments with less active nodes as shown in Fig. 9(b), compared with the original network graph.

We also investigate the impact of the ratio of communication range to sensing range on the relationshipbetween connectivity maintenance and coverage preservation. The networks we study have a sensing den-sity from average 1.6 nodes per sensing area to 17.3 nodes per sensing area, which are obtained by varyingthe deployed node number from 200 to 550 and varying the size of the sensing range based on a certainratio to a fixed communication range as 10 m. For each configuration, we generate 1000 random networktopologies that are original connected. Among these 1000 network topologies, those maintaining connec-tivity after coverage preserving sensing-scheduling are counted. The results are listed in Table 1. Consistentwith our expectation, the smaller the ratio of the communication range to the sensing range is, the lower theconnectivity possibility is. However, as shown the data, the lower bound of the ratio for connectivity main-tenance seems to be 1.8 instead of 2.0. That is because we just run 1000 network topologies for each con-figuration and the scenario illustrated in Fig. 7 is a rare and extreme situation in those networks with a veryhigh node density. Furthermore, from the table, we observe that node density seems to have no significantimpact on connectivity possibility when the ratio of communication range to sensing range is fixed.Approximating the connectivity possibility based on a given ratio of communication range to sensing rangecan be one of our future works.

7. Conclusion

In this paper, we provided complete proof: ‘‘the communication range is twice of the sensing range’’ isthe sufficient condition and the tight lower bound to ensure that complete coverage preservation impliesconnectivity among active nodes if the original network topology (consisting of all the deployed nodes)is connected. The significance of the above conclusion is that it enables us to focus on sensing domain whenconnectivity and coverage are both required, therefore greatly simplifies the safe integration of activityscheduling in both domains. We also extend the result for k-degree by proving that: ‘‘the communication


range is twice of the sensing range’’ is the sufficient condition to ensure that k-degree coverage preservationimplies k-degree connectivity among active nodes if the original network topology (consisting of all the de-ployed nodes) is k-connected.

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Di Tian received the B.S. in Computer Science from Nankai University, Tianjin, China, in 1996 and the M.S.in Computer Engineering from Beijing University of Posts & Telecommunication, Beijing, China in 1999.From April 1999 to December 2000, she worked at Bell Lab Innovations, Lucent China Co. Ltd. as a memberof technical staff. From July 1999 to December 1999, she was an intern at Kenan Systems Co., Boston, MA.She is currently pursuing the Ph.D., degree at School of Information Technology and Engineering, Universityof Ottawa. Her research interests include wireless sensor network and mobile ad hoc network.

Nicolas D. Georganas, OOnt, FIEEE, FRSC, FCAE, FEIC is Distinguished University Professor and CanadaResearch Chair in Information Technology at the School of Information Technology and Engineering,University of Ottawa.

He received the Dipl. Ing. degree in Electrical Engineering from the National Technical University ofAthens, Greece, in 1966 and the Ph.D., in Electrical Engineering (Summa cum Laude) from the University ofOttawa in 1970.

He has published over 300 technical papers and is co-author of the book ‘‘Queueing Networks––ExactComputational Algorithms: A Unified Theory by Decomposition and Aggregation’’, MIT Press, 1989. He hasreceived research grants and contracts totaling more than $51 million and has supervised more than 175researchers, among which 90 graduate students (25 Ph.D., 65 MASc) and 18 PostDocs.

In 1990, he was elected Fellow of IEEE. In 1994, he was elected Fellow of the Engineering Institute ofCanada. In 1995, he was co-recipient of the IEEE INFOCOM�95 Prize Paper Award. In 1997, he was inducted as Fellow in the
Canadian Academy of Engineering and Fellow of the Royal Society of Canada. In 1998, he was selected as the University of OttawaResearcher of the Year and also received the University 150th Anniversary Medal for Research. In 1999, he was awarded the ThomasW. Eadie Medal of the Royal Society of Canada, funded by Bell Canada, for his contributions to Canadian and Internationaltelecommunications. In 2000, he received the A.G.L. McNaughton Gold Medal and Award for 1999–2000, the highest distinction ofIEEE Canada; the Julian C. Smith Medal of the Engineering Institute of Canada; the OCRI President�s Award (jointly with Dr. SamyMahmoud) for the creation of the National Capital Institute of Telecommunications (NCIT); the Bell Canada Forum Award from theCorporate-Higher Education Forum, the Researcher Achievement Award, from the TeleLearning Network of Centres of Excellenceand a Canada Research Chair in Information Technology. In 2001, he was appointed Distinguished University Professor of theUniversity of Ottawa and he was also received the Order of Ontario, the province�s highest and most prestigious honour. In 2002, hereceived the Killam Prize for Engineering, Canada�s most distinguished award for outstanding career achievements.

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Connectivity maintenance and coverage preservation in ...yzchen/papers/papers/deployment... · their work in order to support general random wireless sensor networks by providing