Line coverage measures in wireless sensor networks

doi:10.1016/j.jpdc.2014.03.004

Journal of Parallel and Distributed Computing

Volume 74, Issue 7, July 2014, Pages 2596-2614

https://doi.org/10.1016/j.jpdc.2014.03.004 Get rights and content

Highlights

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The paper measures a quality of sensor deployment.
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The smallest $k$ -covered line segment computation problem considers the intruders’ perspective.
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The largest $k$ -uncovered line segment computation problem considers the defenders’ perspective.

Abstract

The coverage problem in wireless sensor networks addresses the problem of covering a region with sensors. Many different definitions of coverage are there in the literature depending on the goal of the coverage. In this paper, we address the problem of determining the quality of a sensor deployment against an intruder who can walk along a straight line. A line segment $ℓ$ is said to be k-covered if it intersects the sensing regions of at least $k$ sensors distributed in $R$ . Similarly, it is said to be k-uncovered if it intersects the sensing regions, that is assumed to be circular, of at most $k - 1$ sensors. We introduce two new metrics, smallest $k$ -covered line segment and longest $k$ -uncovered line segment, for measuring the quality of line coverage achieved by a sensor deployment. The intruder can walk a distance less than the smallest $k$ -covered line segment without ever being detected by $k$ sensors. So, this metric gives an estimate on the distance an intruder can walk in a straight line path before being detected by $k$ sensors. On the other side, the defender would want to deploy sensors so that the length of the longest $k$ -uncovered line segment is minimized. Given a deployment of $n$ sensors, we propose deterministic algorithms to determine the smallest $k$ -covered line segment and longest $k$ -uncovered line segment where the line segments can be of the following types: (i) axis-parallel (horizontal and vertical) line segments, (ii) line segments whose one endpoint is fixed and is of arbitrary orientation and (iii) arbitrary line segments. The time complexities for the first and second types of line segments are $O ((n + χ) log n)$ for both smallest $k$ -covered line segment and longest $k$ -uncovered line segment, where $χ$ is the number of intersections among $n$ circles. For the arbitrary line segment case, the smallest $k$ -covered segment can be determined in $O (χ^{2} log n + n^{\frac{11}{3} + ϵ})$ time, whereas, the longest $k$ -uncovered segment can be determined in $O (χ^{2} log n + n^{2 + β + ϵ})$ time, where $β = {log}_{2} (1 + \sqrt{5}) - 1$ and $ϵ$ is a small value greater than or equal to $0$ . All our algorithms take linear space.

Introduction

The design of wireless sensor networks has become an important subject of research in recent years. A wireless sensor network (WSN) consists of a number of tiny devices equipped with sensors to sense one or more parameters such as temperature and motion. Each sensor node has limited computation capacity and battery power, and they can communicate with other nearby sensors. Each such sensor has a sensing range within which it can sense the parameter(s) and a communication range over which it can communicate with other sensors. The sensors are deployed in the region that needs to be monitored. The sensor nodes are capable of sending the sensed data to base stations for further processing. Sensor networks have been used in different applications such as habitat monitoring [17], intruder detection [18], and target tracking [24], [29].

The coverage problem is an important problem in many wireless sensor network applications. In this problem, a set of sensors deployed in a region is used to cover an area or parts of it. Broadly speaking, coverage is a measure that determines how well a network of sensors monitors an area. Various definitions of coverage of an area have been proposed depending on the target application. Inherent to these coverage measures is a geometric idea that captures how safe the field is. As an example, the $k$ -coverage problem [13] requires that each point in the area be in the sensing range of at least $k$ sensors. This is useful in certain target monitoring applications in which it is necessary to track the movement of a target object by sensing it with at least $k$ sensors at all points. Research in coverage can be broadly classified into two types: (i) given a deployment of sensors over an area, compute the coverage measure and (ii) given a coverage measure, determine how to deploy or place sensors so as to achieve the coverage. Our work in this paper is focused on the first type.

There are other measures of coverage depending on applications at hand. In applications related to monitoring borders and boundaries, a coverage measure called barrier coverage is useful. A barrier is an annular belt-like region surrounding an area. The sensors are spread across the belt-like region. The barrier is said to be $k$ -barrier covered [15], [21] if every path that passes through the barrier cuts the sensing range of at least $k$ sensors. Other definitions of coverage include area coverage [16], breach and support [19]. Several works provide algorithms for achieving various types of coverage in sensor networks by suitable placement of the sensors [13], [26], [15], [19].

In some applications, it is more important to cover some parts of a set of line segments rather than the entire line segment. One example of such an application is monitoring intrusions in the corridors of a building or a road network. Here, the objects are represented by a set of line segments (usually horizontal and vertical for corridors, and arbitrarily oriented for roads). While full surveillance of all corridors or all roads is desirable, it may be too costly. Thus, it may be sufficient to ensure that the intruder entering the corridor/road is detected at least a fixed number of times somewhere in the corridor/road. Motivated by such applications, some of the existing works on coverage focus on line coverage in a region rather than points in an area [4], [5], [11], [22]. The coverage measures proposed are track coverage, trap coverage, etc.

In this paper, we propose two different coverage metrics for measuring the line coverage ability of a deployed WSN in a bounded rectangular region $R$ . A line is said to be $k$ -covered if it intersects the sensing range of at least $k$ sensors. Similarly, it is said to be $k$ -uncovered if it intersects the sensing ranges of at most $k - 1$ sensors. The two coverage measures are:

(Smallest $k$ -covered line segment:) It is the minimum length line segment that is $k$ -covered. From an intruder’s perspective, given a deployment of sensors, the intruder would like to find the length of the smallest $k$ -covered line segment so that if it walks a distance less than that along a line, it is sure not to be detected by $k$ or more sensors.
(Longest $k$ -uncovered line segment:) It is the maximum length line segment that is $k$ -uncovered. From a defender’s point of view, it would like to deploy the sensors in such a manner such that the length of the longest $k$ -uncovered line segment is minimized.

We design algorithms to compute the above measures for axis-parallel line segments and line segments with arbitrary orientation.

The rest of the paper is organized as follows. Section 2 presents some related works on coverage in WSNs. Section 3 presents our motivation and formal problem statements. Sections 4 Algorithm for finding smallest, 5 Algorithm for finding longest describe algorithms for finding the smallest $k$ -covered and the longest $k$ -uncovered axis-parallel line segments respectively. Section 6 presents algorithms for smallest $k$ -covered line segment and longest $k$ -uncovered line segment from a given endpoint $p$ inside $R$ . Section 7 presents algorithms for finding smallest $k$ -covered and longest $k$ -uncovered line segment of any arbitrary orientation. Experimental results are presented in Section 8. Finally, Section 9 concludes the paper.

Section snippets

Related work

Given a deployment of sensors in a field, several algorithms have been proposed to compute different types of coverage in wireless sensor networks. Huang and Tseng [13] designed an algorithm to verify whether an area is $k$ -covered. They proved that if the perimeters of all the sensors’ circular sensing region within the bounded region are $k$ -covered by their neighbors then the whole area is also $k$ -covered. To verify whether the perimeter of a sensor is $k$ -covered by its neighbors, they determine

Problem statement and preliminaries

Consider a set of $n$ sensors in a bounded rectangular region $R$ . We assume that the sensors are points in the plane with uniform sensing range. The sensing range of a sensor $s$ is a real number $ρ (s)$ (say), such that it can sense inside a circular region of radius $ρ (s)$ . Without loss of generality, let this sensing range be 1 for all the sensors. Therefore, we can model the total sensing region as a collection of unit circles. Let the circle corresponding to a sensor $i$ be denoted by $C_{i}$ .

Consider an

Algorithm for finding smallest $k$ -covered axis-parallel line segment

We subdivide the problem into two parts. We will find the horizontal smallest $k$ -covered line segment and the vertical smallest $k$ -covered line segment separately, and then find the minimum of the two. A plane sweep algorithm [9] for the case of horizontal line segment is presented next; the algorithm for the case of vertical line segments is exactly symmetrical.

Any horizontal line segment may or may not be $k$ -covered. If a segment is $k$ -covered, we define it to be of minimal length as follows.

Definition 3

Minimal Length $k$ -Covered Horizontal Segment

Algorithm for finding longest $k$ -uncovered axis-parallel line segment

In this section, we sketch the algorithm for finding the longest $k$ -uncovered axis parallel line segment. Overall, the algorithm works in the same manner as that for finding the smallest $k$ -covered axis-parallel line segment, finding first the horizontal and then the vertical such segment, and then taking the maximum of the two.

To find the longest $k$ -uncovered line segment, we make the assumption that if an endpoint of a line segment is on the circumference of a circle $C_{i}$ , then the segment is not

Line coverage ability of a sensor network in $R$ from a given point

In this section, we determine the line coverage ability of a sensor network with respect to an input point $p$ inside $R$ . This problem arises in the case an intruder is dropped inside $R$ . We determine the smallest $k$ -covered line segment and longest $k$ -uncovered line segment where one endpoint of the line segment is fixed at $p$ , and the line segment can be of arbitrary orientation.

With respect to a point $p$ lying outside a circle $C_{i}$ , we divide the circumference of $C_{i}$ into two parts. The portion of the

Line coverage ability for any arbitrary line segment in a given region

In this section, we determine the smallest $k$ -covered segment $ℓ_{β}$ and longest $k$ -uncovered segment $ℓ_{α}$ in $R$ . There is no restriction on the orientation as well as position of the segment. ${[p, q]}_{S_{k}}$ denotes a line segment which has endpoints at $p$ and $q$ and intersects a set of $k$ circles $S_{k}$ .

Definition 25

Minimal Length $k$ -Covered Segment

A $k$ -covered line segment of type ${[p, q]}_{S_{k}}$ is said to be of minimal length if (a) it does not contain any other $k$ -covered segment and (b) its length is less than any $k$ -covered segment of type ${[p^{'}, q^{'}]}_{S_{k}}$ where $p^{'}$ and $q^{'}$

Actual time and sensor distribution

We report the actual running time for all the variations of smallest $k$ -covered and longest $k$ -uncovered line segments (axis-parallel line segments as in Sections 4 Algorithm for finding smallest, 5 Algorithm for finding longest, line segment with one endpoint fixed as in Section 6 and line segment of arbitrary orientation as in Section 7).

Table 1, Table 2 report the time in milliseconds for all the above cases for smallest $k$ -covered and longest $k$ -uncovered line segment respectively. We have

Conclusion

In this paper, we have introduced two new coverage measures—smallest $k$ -covered segment and longest $k$ -uncovered segment to measure the quality of coverage of a sensor network. These measures indicate the quality of coverage a deployment of sensors can provide against an intruder walking in a straight line path. We view the problem both from the intruder’s and defender’s perspective using the smallest $k$ -covered segment and longest $k$ -uncovered segment respectively. We determine the above two

Acknowledgments

We want to thank the three anonymous reviewers for their incisive comments that helped us to improve the work.

Dinesh Dash received M.Sc. degree in Computer & Information Science from University of Calcutta, Kolkata in 2002, and M.Tech. degree in Computer Science from the same university in 2004. From July 2004 to June 2007, he worked in Asansol Engineering college, Asansol under the West Bengal University of Technologies, India as a Lecturer. Since January 2008, he has been a research scholar in the Department of Computer Science & Engineering in IIT Kharagpur. His research interests are in the areas

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Arobinda Gupta received his Ph.D. in Computer Science from the University of Iowa, Iowa City, in 1997. From 1997 to 1999, he was with the Windows 2000 Distributed Infrastructure group in Microsoft Corp., Redmond, Washington, USA. Since Oct. 1999, he is a faculty in Indian Institute of Technology Kharagpur, where he is currently a Professor in the Department of Computer Science & Engineering and School of IT. His current research interests are in distributed algorithms, ad hoc and sensor networks, and delay tolerant networks.

Arijit Bishnu received the B.E. degree in electrical engineering from Burdwan University, India, in 1995, the M.Tech. degree in computer science and the Ph.D. degree, both from the Indian Statistical Institute, Kolkata, India, in 1998 and 2003, respectively. Currently he is an Associate Professor of the Advanced Computing and Microelectronics Unit, Indian Statistical Institute, Kolkata, India. His research interests include algorithms, discrete computational geometry and its applications.

Subhas C. Nandy received his M.Sc. in Statistics from the University of Calcutta, M.Tech. in Computer Science from Indian Statistical Institute and Ph.D. in Computer Science from the University of Calcutta in 1982, 1985 and 1996 respectively. He joined Indian Statistical Institute, Kolkata in 1986 and now he is a professor in the Advanced Computing and Microelectronics Unit of Indian Statistical Institute.

His current research interest includes the algorithmic aspects of graph theory and computational geometry.

^☆: A most preliminary version of the paper appeared as a poster presentation at ICDCN 2012.

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Line coverage measures in wireless sensor networks☆

Highlights

Abstract

Introduction

Section snippets

Related work

Problem statement and preliminaries

Algorithm for finding smallest k-covered axis-parallel line segment

Minimal Length k-Covered Horizontal Segment

Algorithm for finding longest k-uncovered axis-parallel line segment

Line coverage ability of a sensor network in R from a given point

Line coverage ability for any arbitrary line segment in a given region

Minimal Length k-Covered Segment

Actual time and sensor distribution

Conclusion

Acknowledgments

J. Algorithms

Comput. Oper. Res.

Comput. Netw.

Ad Hoc Networks

A geometric transversal approach to analyzing track coverage in sensor networks

IEEE Trans. Comput.

Grid coverage for surveillance and target location in distributed sensor networks