Line Sweep Coverage in Wireless Sensor Networks Barun Garain and Partha Sarathi Mandai

Department of Mathematics Indian Institute of Technology Guwahati, Guwahati-781 039, India

Email: {b.gorain,psm}@iitg.ernet.in

Abstract-Traditional coverage in wireless sensor networks requires continuous monitoring of target objects or regions. Unlike traditional coverage, periodic monitoring by a set of mobile sensor nodes is sufficient in sweep coverage. The sweep coverage problem for covering a set of points is NP-hard and it cannot be approximated within a factor of 2 [15]. The sweep coverage problem for a given bounded region is also NP-hard [11]. In this paper, we study sweep coverage for covering a set of line segments on a plane. We prove that the problem is NP-hard and cannot be approximated within a factor 2. Our proposed algorithm achieves the best possible approximation factor 2. As an application of line sweep coverage problem we formulate a data gathering problem, where minimum number of data mules periodically collect data from a set of mobile sensor nodes. The mobile sensor nodes arbitrarily move along their paths, which are line segments. We prove that this problem is NP-hard and propose a 3 approximation algorithm to solve it.

Key words: Sweep Coverage, Approximation Algorithm, Eu­lerian Graph, TSP, Mobile Sensor, Data Mule, Data Gathering, Wireless Networks.

I. INTRODUCTION

Coverage is one of the most important issues in wireless sensor networks (WSNs). Generally, coverage is defined as the quality of surveillance of sensing functions. In practice, sensor nodes are deployed over an area of interest to get desire information from the area. To get complete information from a region, it is necessary that the region must be completely covered by the set of deployed sensor nodes. The quality of data collected from the area of interest depends on how well the area of interest is covered by the deployed sensor nodes. For example in forest monitoring [1] full coverage is required, where every location of the forest must be covered by at least one sensor node such that any unusual activities like forest fire, activities of poachers, etc. can be immediately detected. Similarly, if boundary of the forest is in the coverage range of sensor nodes then it might be possible for eliminating the poaching activities and illegal entry through the boundary. The objective of the coverage problem is to improve the coverage performance when the WSN is unable to satisfy the requirements and if it satisfies the requirements, then how to extend the network lifetime with coverage guarantee. Depending on the subject to be covered or monitored by the sensor nodes, three types of coverage problems are widely studied in literature. Those are point coverage problem, area coverage problem and barrier coverage problem. In point coverage, the objective is to cover a given set of discrete points

978-1-4799-3635-9/14/$31.00 ©2014 IEEE

by the sensor nodes. In application, targets can be represented as the set of discrete points [12], [16]. In area coverage, the objective is to cover every point of a given region by sensor nodes [4], [20], [22]. The objective of the barrier coverage problem is to identify some penetration path [18] over the sensor field with some desired property. Covering a set of line segments [9] is also a type of barrier coverage problem.

There are typical applications where only periodic patrol inspections are sufficient for a certain set of points of interest instead of continuous monitoring like traditional coverage as discussed above. For example continuous monitoring is not requited for some applications like police patrolling, where police patrol officers are assigned to enforce laws and ordi­nances, regulate traffic, control crowds, prevent crime, and arrest violators by periodic visiting the targeted regions. In WSNs this periodic monitoring by mobile sensor nodes is named as sweep coverage. There are many works [7], [10], [11], [15], [24] in literature discussed about the point sweep coverage problem, where a given set of discrete points are monitored by a set of mobile sensor nodes at least once within a given period of time. Finding minimum number of mobile sensor nodes for the sweep coverage of a given set of discrete points on a plane is NP-hard [15]. The area sweep coverage problem is proposed in [11], and proved that the problem is NP-complete. In this paper we formulate sweep coverage problem for a set of line segments.

A. Contribution

In this paper, our contributions on sweep coverage problems are as follows:

• We have introduced sweep coverage concept for covering a set of line segments on a plane. We have proved that finding minimum number of mobile sensor nodes for sweep coverage of a set of line segments is NP-hard and cannot be approximated within a factor of 2. A 2 approximation algorithm is proposed for solving the line sweep coverage problem, which is the best possible approximation factor.

• A data gathering problem by a set of data mules is formulated and proved that the problem is NP-hard. A 3 approximation algorithm is proposed to solve it with the help of line sweep coverage problem.

B. Related Work

Most of the coverage problems presented in the papers [9], [14], [15] are NP-complete. Several heuristics [17], [22]

and approximation algorithms [8], [9] are proposed to solve the problems. The point coverage problems are studied in the papers [8], [12], [16]. In [8], the authors considered a geometric version of the point coverage problem called unit disk cover problem. They discussed about the computational complexity of the problem which is NP-hard. A constant factor approximation algorithm is provided to solve the problem. In paper [23], Wang et al. proposed three movement assisted algorithms for area coverage, which are VEC (vector based algorithm), VOR (voronoi based algorithm) and Minimax. In this paper authors used voronoi diagram to identify coverage holes. These movement strategies give efficient improvements of the coverage with mobile sensor nodes. In the paper [17], Ma et al. proposed a distributed heuristic where in each iteration sensor nodes move such a way that the overall topology becomes closer to an equilateral triangulation of the plane which is the optimal layout for area coverage problem.

The concept of sweep coverage initially came from the context of robotics [3]. There are several papers [7], [10], [11], [15], [24] in literature on point sweep coverage problem. In [10], Du et al. considered two different movement constraints of mobile sensor nodes for sweep coverage. In the first approach, mobile sensor nodes move in the same path in every time period and in the second approach the nodes move in different paths in different time periods. The problem of point sweep coverage is rigorously studied in the paper [15] by Li et al. The authors proved that finding minimum number of mobile sensor nodes for sweep coverage of a set of discrete points is NP-hard. The authors showed that this problem is equivalent to solve the traveling salesman problem (TSP) and also proved that the point sweep coverage problem cannot be approximated less than a factor of 2. One (2 + f) approximation and a 3 approximation algorithms are proposed. The authors remarked on impossibility to design distributed local algorithms, which can guarantee required sweep coverage, i.e. , a mobile sensor node cannot locally determine whether all the points of interest are sweep covered without global information. In [11], a 2 approximation algorithm for point sweep coverage problem is proposed, which is the best possible approximation factor according to the result in [15]. A distributed 2 approximation algorithm is proposed for the point sweep coverage problem, where static sensor nodes are considered instead of points of interest. The area sweep coverage problem is formulated and proved that the problem is NP-complete. A 2V2 ap­proximation algorithm is provided to solve the problem for a square region. The area sweep coverage problem for arbitrary bounded region is also investigated in that paper. To extend lifetime of sweep coverage, Yang et al. in [25] utilized base station as a power source for refueling or replacing battery of the mobile sensor nodes periodically. The authors proposed two heuristics for sweep coverage with one base station and multiple base stations respectively.

II. LINE SWEEP COVERAGE PROBLEM

Let £ = {h, l2, ··· , In} be a set of line segments on a two dimensional plane. Each line li is defined by two end

points (ai, bi), where ai and bi are points on the plane. A line segment li is said to be covered by a set of sensor nodes if and only if each point on li is covered by at least one sensor node. Based on the above coverage metric, we give the respective definition of line sweep coverage as follows:

Definition 1 (Line sweep coverage): A line segment li is said to be t-line sweep covered if and only if all points of li are visited at least once by a mobile sensor node in every t time period.

Definition 2 (Line Sweep Coverage Problem): Let £ {h, l2, ··· , In} be a set of line segments and {SI, S2, ··· , sm} be a set of mobile sensor nodes. For given t > 0 and v > 0, find the minimum number of mobile sensor nodes with uniform speed v such that each line of £ is t-line sweep covered.

A. Complexity result

Theorem i: The line sweep coverage problem is NP-hard and cannot be approximated within a factor of 2 unless P=NP.

Proof Finding minimum number of mobile sensor nodes with uniform velocity to guarantee sweep coverage for a set of points in two dimensional plane is NP-hard and it cannot be approximated within a factor of 2 unless P=NP, as proved in the paper [15] by Li et al. The point sweep coverage problem as proposed in [15] is a special case of line sweep coverage problem when the end points of each line segments are same, i.e, all line segments are points. Therefore, the line sweep coverage problem is NP-hard and cannot be approximated within a factor of 2 unless P=NP. •

B. Basic idea

Let £ = {h, l2, ··· , In} be a set of line segments on a two dimensional plane. Let S be the set of shortest distance line Sij between every pair of line segments (li' 1 j) for i cf j. We define a complete weighted graph G = (V, E), where V = {VI, V2, . . . , Vn} is the set of vertices. The vertex Vi represents line segment li for i = 1 to n. E = V x V is the set of edges, where the edge (Vi, V j) represents Sij E Sand edge weight w ( Vi, V j) = length of Sij. Let T be a minimum spanning tree (MST) of G. Using the above transformation, T can be represented as Te, where Te = £U{ Sij : (Vi, Vj) E T }.

An illustration is shown from Fig. 1 to Fig. 4, where a set of line segments is shown in Fig. 1, corresponding complete graph G is shown in Fig. 2, an MST T of G is shown in Fig. 3 and the representation Te of T is shown in Fig. 4.

We construct a graph Ge from Te by introducing vertices at the end points of each line segment in Te, which may split l;'s into several smaller line segments. According to Fig. 5, vertices of GI: are {al, p, bl, a2, q, b2, a3, b3, a4", b4}. The vertex P splits line segment (aI, bd into two smaller line segments (aI, p) and (p, bl). Similarly, vertices q and I split (a2, b2) and (a4, b4) into (a2, q), (q, b2) and (a4, I), (I, b4) respectively, whereas the line segment (a3, b3) remains same. Each of these line segments and the lines corresponding to the edges of T together are the edges of the graph G 1:. According to Fig. 5, edges of GI: are {(al, p), (p, bl), (a2, q),

Fig. I: Set of line segments £

V I

�,- --, ' -, " , -,-�

-', " ',V3 --","',,/

" , \

'.",

,, , , , , '

/ /

//

'"II' /

V4

Fig. 2: Complete graph G

, • V 4

Fig. 3: MST, T of G

The graph G £, is a tree and the sum of the edge weight of G £, is w(T) + L�=I Ii, where w(T) is the sum of the edge weights of T. Following Algorithm 1 LINESWEEPCOV­

ERAGE computes a tour on G £, and find number of mobile sensor nodes and their movement paths for solving line sweep coverage problem.

C. Analysis

Lemma 1: According to the Algorithm: LrNESWEEPCOV­

ERAGE each point on Ii can be visited by at least one mobile sensor node in every time period t for i = 1,2, . . . n.

Proof" Since the mobile sensor nodes are moving along the Eulerian tour E, each edges of G £, are visited. Let us consider any point p on a line segment Ii and let tf be the time when a mobile sensor node visited p last time. Now we have to prove that the point p must be visited by at least one mobile sensor node in tf + t time. According the deployment

b4

Fig. 4: T£,

b4

Fig. 5: G£,

strategy of mobile sensor nodes any two consecutive mobile sensor nodes are within the distance of vt at any time. So, when a mobile sensor node visited p at tf another mobile sensor node is on the way to p and within the distance of vt along E. Hence p will be again visited by another mobile sensor node within next t time. •

Lemma 2: If Lopt is the length of the optimal TSP tour for visiting all points of every line segment in £, then w(T) + L�=l Ii .-::: Lopt.

Proof" The optimal TSP tour Lopt contains two types

Algorithm 1 LrNESWEEPCOVERAGE

1: Construct complete weighted graph G from the given set of line segments £.

2: Find an MST T of G. 3: Construct G£,. 4: Find Eulerian graph after doubling each edge of G£,. 5: Find an Eulerian tour E on the Eulerian graph. Let I E I

be the length of E. 6: Partition E into [I:tll parts and deploy i':t'l mobile

sensor nodes at al partition points, one for each. 7: Each mobile sensor node then starts moving at the same

time along E in same direction.

of movement paths; movement paths along the line segments of £ and movement paths between the line segments. Let total length of the movement paths along the line segments be Lalong. Since Lopt is the optimal tour for visiting all points of each line segment Ii E £, therefore,

n Lalong :;:, L Ii

i=l (1)

Let Le be the optimal TSP tour on G. Then w(T) s: Le. Let total length of the movement paths between the line segments be Lbetween. Since Le is the optimal TSP tour on G and the weights of all edges of G are taken to be the shortest distance between respective line segments, therefore,

Lbetween :;:, Le (2)

Now, from equation 1 and equation 2, Le + L�=l Ii s: Lopt. Hence w(T) + L�=l Ii s: Lopt. •

Theorem 2: The approximation factor of the Algorithm LrNESWEEPCOVERAGE is 2.

Proof' The total edge weights of Ge is w(T) + L�=11i. Now, 1[; 1= 2(w(T) + L�=11i)' since Eulerian tour [; found by the Algorithm 1: LrNESWEEPCOVERAGE after doubling each edges of Ge. By Lemma 2, I [; IS: 2Lopt. Let Nopt be t�e number of mobile sensor n�des required fZ�,}.ptimal

solutIOn. Then Nopt x vt :;:, Lopt, I.e. , Nopt :;:, l----;;t I. The

number of mobile sensor nodes calculated by the Algorithm

1 is II:n (=N, say). Therefore, the approximation factor of

the Algorithm 1 is equal to � s: 12Lopt l / 1 Lopt l s: 2 . • Nopt I vt I vt

III. DATA GATHERING BY DATA MULES

In this section we consider a data gathering problem by a set of data mules [2], [5], [13], [21], which we have formulated as a variation of line sweep coverage problem explained below. A set of mobile sensor nodes are moving on a plane to collect coverage information. Each mobile sensor node is moving along a path, which is a line segment. The movement of the mobile sensor nodes are arbitrary along their respective paths i. e. , a node moves in any direction along the straight line with arbitrary speed and it can stop its movement for arbitrary time period. But at any point of time the mobile sensor nodes never leave their trajectories and which is the only constraint of their movements. A set of data mules are moving with uniform speed v in the plane for collecting data from the mobile sensor nodes. A data mule can collect data from a mobile sensor node whenever it meets the mobile sensor node on its trajectory. It is possible to collect data instantaneously whenever the data mule meets a mobile sensor node. The definition of the problem is given below.

Definition 3: (Minimum number of data mule for data gath­

ering (MDMDG)) A set of mobile sensor nodes are moving arbitrarily on a plane along line segments. Find minimum number of data mules, which are moving with uniform speed v, such that data can be collected from each of the mobile sensor nodes at least once in every t time period.

The points sweep coverage problem [15] is a special in­stance of the MDMDG when two end points of every line segment are same, i.e. the mobile sensor nodes are behaving like a static sensor node. Therefore we can state the following theorem.

Theorem 3: The MDMDG problem is NP-hard and cannot be approximated within a factor of 2 unless P=NP. The proof of the above theorem directly follows from the Theorem 1.

The following Lemma 3 shows that to visit the mobile sensor nodes, each point of all paths must be visited by the data mules.

Lemma 3: To solve the MDMDG problem, each and every point of all line segments must be visited by the set of data mules.

Proof' We will prove the lemma by the method of contradiction. Let I be a line segment for which all points of I are not visited by the data mules. Therefore, there exist one point p on I such that p is not visited by any data mule. One mobile node can stop its movement for sometime and which can be allowed for the arbitrary nature of their movements. Now, if the mobile sensor node on I remains static at p for more than t time then the mobile sensor node is not visited by any data mule and which contradict the condition of t sweep coverage. Hence, each and every point of all line segments must be visited by the set of data mules. •

We now find the minimum path traveled by a single data mule to visit all the mobile sensor nodes. According to the problem definition and by Lemma 3, within any time period [t', t' + t], all points of each line segment are visited by a data mule. Therefore, the total length of the tour traversed by all the data mules within that time period is greater or equals to the optimal tour traversed by a single data mule in order to visit all mobile sensor nodes. The following lemma gives the nature of the optimal tour traversed by a single data mule for visiting all the mobile sensor nodes.

Lemma 4: It may not be possible to visit a mobile sensor node by a data mule unless it visits the whole line segment, which is movement path of the mobile sensor node, from one end to the other end continuously.

Proof' If a line segment is not visited continuously then in the following scenario the data mule cannot visit a mobile sensor node. Let I' and I" be the two parts of a line segment I. The data mule continuously visits I' and during that time period the mobile sensor node remains on I". After sometime, when the mobile sensor node remains on I', the data mule visits I" continuously. So, in this scenario the data mule cannot visit mobile sensor node. •

Let Lopt be the optimal tour for visiting all mobile sensor nodes by one data mule. Now, the optimal tour Lopt contains two types of movement paths: the movement paths along the line segments and the paths between pair of line segments. The paths between pair of line segments are the lines which connect end points of the pair of line segments.

We construct euclidian complete graph G2n with 2n vertices ai, bi, 'i = 1,2"" n, where ai and bi are the two end points of

the line Ii. The edge set E( G2n) of G2n is given by E( G2n) = {Ii : i = 1,2" " , n}U{(ai, aj): i "l- j}U{(bi, bj): i "l­j} U {(ai, bj) : i "I- j} U {(bi, aj) : i "I- j}. The weight of each of the edges is equal to the euclidian distance between the two vertices.

Let T2n be a MST of G2n containing all edges Ii E E(T2n), i = 1,2" " , n. We compute T2n using Kruskal's algorithm after including all edges Ii, i = 1, 2, . . . , n in the initial edge set of T2n. Until the spanning tree is formed we ap­ply Kruskal's algorithm on the remaining edges, E( G2n)\ {Ii: i = 1,2" " , n} of G2n. An Eulerian graph is formed from T2n as described in Christofides algorithm [6].

Based on the above discussions, we propose Algorithm 2 to solve MDMDG problem where each line segments are of length less than or equal to ¥. Algorithm 2 MDMDG

I: Use Kruskal algorithm to find an MST T2n of G2n with the initial set of edges containing all edges Ii for i = 1,2" " , no

2: Construct an Eulerian graph from T2n using Christofides algorithm [6].

3: Find an Eulerian tour £2n from the Eulerian graph. Let I £2n I be the length of £2n

4: Partition £2n into 121��n Il parts of length ¥ and deploy

1 21��nll data mules at all partition points, one for each.

5: Bach data mule then starts moving at the same time along

£2n in same direction.

A. Analysis

Theorem 4: According to the Algorithm:MDMDG, each mobile node is visited by a data mule at least once in every t time period.

Proof" Let at time t', a mobile node is at the position p on its path when it is visited by a data mule. Another data mule will reach p at time t' + �, since at any instance distance between two consecutive data mules is ¥. Since the length of the path of the mobile node is less than or equal to ¥, the maximum displacement of the mobile node from p in the direction of movement of the data mules is less than or equal to ¥. Hence the next data mule will reach the current position of the mobile node in at most � time after reaching p. Therefore within t' + t time, the mobile node will be again visited by another data mule. •

Theorem 5: The approximation factor of the Algo­rithm:MDMDG is 3.

Proof" According to the Christofides algorithm [6] we can write I £2n IS: � Lopt. Let Nopt be the number of data mules required for optimal solution. Then Nopt x vt :;0. Lopt, i.e. , Nopt :;0. I L;t l· The number of data mule calculated

by the Algorithm 2 is 121��nll C=N, say). Therefore, the

approximation factor for the Algorithm 2 is equal to NN <

12X ��OT'i l / I L;;t l s: 3. opt

•

IV. CONCLUSION

Unlike traditional coverage, in sweep coverage periodic monitoring is maintained by mobile sensor nodes instead of continuous monitoring. There are many applications in indus­try, where periodic monitoring is required for identification of specific preventive maintenance. For example, periodic monitoring of electrical equipments like motors and generators is required to check their partial discharges [19].

In this paper we have introduced sweep coverage concept to cover a set of line segments on a plane. In line sweep coverage, mobile sensor nodes periodically visit all points of each line segment. We have proved that the problem is NP-hard and cannot be approximated within a factor of 2. We have proposed 2 approximation algorithm for solving line sweep coverage problem. As an application of line sweep coverage problem, we have defined a data gathering problem: MDMDG, where the concept of line sweep coverage is applied for gathering data by utilizing minimum number of data mules. A 3 approximation algorithm is proposed to solve the problem. In future we want to investigate the sweep coverage problems in presence of obstacles.

REFERENCES

[I] Fadi M. AI-Turjman, Hossam S. Hassanein, and Mohamed A. Ibnkahla. Connectivity optimization for wireless sensor networks applied to forest monitoring. In Proceedings of the 2009 IEEE international conference

on Communications, ICC'09, pages 285-290, Piscataway, NJ, USA, 2009. IEEE Press.

[2] Giuseppe Anastasi, Marco Conti, and Mario Di Francesco. Data collection in sensor networks with data mules: an integrated simulation analysis. In IN: PROC. OF IEEE ISCC, 2008.

[3] Maxim A. Batalin and Gaurav S. Sukhatme. Multi-robot dynamic coverage of a planar bounded environment. In in IEEEIRSJ International Conference on Intelligent Robots and Systems (Submitted, 2002.

[4] Mihaela Cardei and Ding-Zhu Du. Improving wireless sensor net­work lifetime through power aware organization. Wireless Networks, 11(3):333-340, 2005.

[5] Giiner D. <,;:elik and Eytan Modiano. Dynamic vehicle routing for data gathering in wireless networks. In CDC, pages 2372-2377, 2010.

[6] N. Christofides. Worst-case analysis of a new heuristic for the travelling salesman problem. Technical Report 388, Graduate School of Industrial Administration, Carnegie Mellon University, 1976.

[7] Hung-Chi Chu, Wei-Kai Wang, and Yong-Hsun Lai. Sweep coverage mechanism for wireless sensor networks with approximate patrol times. Ubiquitous, Autonomic and Trusted Computing, Symposia and Work­shops on, 0:82-87. 2010.

[8] Gautam K. Das, Robert Fraser, Alejandro L6pez-Ortiz, and Bradford G. Nickerson. On the discrete unit disk cover problem. In WALCOM, pages 146-157,2011.

[9] Dinesh Dash, Arijit Bishnu. Arobinda Gupta. and Subhas C. Nandy. Approximation algorithms for deployment of sensors for line segment coverage in wireless sensor networks. In COMSNETS, pages 1-10,2012.

[10] lunzhao Du, Yawei Li. Hui Liu. and Kewei Sha. On sweep coverage with minimum mobile sensors. In Proceedings of the 2010 IEEE 16th International Conference on Parallel and Distributed Systems. ICPADS '10, pages 283-290, Washington, DC, USA, 2010. IEEE Computer Society.

[11] Barun Gorain and Partha Sarathi Mandal. Point and area sweep coverage in wireless sensor networks. In WiOpt, pages 140-145. IEEE, 2013.

[12] Yu Gu, Yusheng li, lie Li. and Baohua Zhao. Fundamental results on tar­get coverage problem in wireless sensor networks. In GLOBECOM'09, pages 1-6. 2009.

[13] Liron Levin, Michael Segal, and Hanan Shpungin. Optimizing perfor­mance of ad-hoc networks under energy and scheduling constraints. In WiOpt. pages 11-20. IEEE, 2010.

[14] Jing Li, Ruchuan Wang, Haiping Huang, and Lijuan Sun. Voronoi-based coverage optimization for directional sensor networks. Wireless Sensor Network, 1(5):417-424, 2009.

[15] Mo Li, Wei-Fang Cheng, Kebin Liu, Yunhao Liu, Xiang-Yang Li, and Xiangke Liao. Sweep coverage with mobile sensors. IEEE Trans. Mob. Comput., 10(11):1534-1545, 2011.

[16] Mingming Lu, Jie Wu, Mihaela Cardei, and Minglu Li. Energy-efficient connected coverage of discrete targets in wireless sensor networks. In ICCNMC'05, pages 43-52, 2005.

[17] Ming Ma and Yuanyuan Yang. Adaptive triangular deployment algo­rithm for unattended mobile sensor networks. IEEE Trans. Computers,

56(7), 2007. [18] Seapahn Meguerdichian, Farinaz Koushanfar, Miodrag Potkonjak, and

Mani B. Srivastava. Coverage problems in wireless ad-hoc sensor networks. In INFOCOM'OI, pages 1380-1387, 2001.

[19] G. Paoletti and A. Golubev. Partial discharge theory and applications to electrical systems. In Pulp and Paper, 1999. Industry Technical

Conference Record of 1999 Annual, pages 124-138, 1999. [20] Mana Saravi and Bahareh J. Farahani. Distance constrained deployment

(DCD) algorithm in mobile sensor networks. In Proceedings of the 2009 Third International Conference on Next Generation Mobile Applications,

Services and Technologies, NGMAST '09, pages 435-440, Washington, USA, 2009. IEEE Computer Society.

[21] Rahul C. Shah, Sumit Roy, Sushant Jain, and Waylon Brunette. Data mules: modeling and analysis of a three-tier architecture for sparse sensor networks. Ad Hoc Networks, 1(2-3):215-233, 2003.

[22] Guiling Wang, Guohong Cao, and T. LaPorta. A bidding protocol for deploying mobile sensors. In Proceedings. 11th IEEE International Conference on Network Protocols (ICNP 2003), pages 315 - 324, Nov. 2003.

[23] Guiling Wang, Guohong Cao, and Thomas F. LaPorta. Movement­assisted sensor deployment. IEEE Trans. Mob. Comput., 5(6):640-652, 2006.

[24] Min Xi, Kui Wu, Yong Qi, Jizhong Zhao, Yunhao Liu, and Mo Li. Run to potential: Sweep coverage in wireless sensor networks. In ICPP,

pages 50-57, 2009. [25] Meng Yang, Donghyun Kim, Deying Li, Wenping Chen, Hongwei Du,

and Alade O. Tokuta. Sweep-coverage with energy-restricted mobile wireless sensor nodes. In Kui Ren, Xue Liu, Weifa Liang, Ming Xu, Xiaohua Jia, and Kai Xing, editors, WASA, volume 7992 of Lecture Notes in Computer Science, pages 486-497. Springer, 2013.