Connected k-coverage in two-dimensional wireless sensor networks using hexagonal slicing and area stretching

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Abstract

The problem of coverage in two-dimensional (2D) wireless sensor networks is challenging and is still open. Precisely, determining the minimum sensor density (i.e, minimum number of sensors per unit area) that is required to cover a 2D field of interest (FoI), where every point in the field is covered by at least one sensor, is still under investigation. The problem of 2D k-coverage, which requires that every point in a 2D FoI be covered by at least k sensors, where k1, is more challenging. In this paper, we attempt to address the 2D connected k-coverage problem, where a 2D FoI is k-covered, while the underlying set of sensors k-covering the field forms a connected network. We propose to solve this problem using an approach based on slicing 2D FoI into convex regular hexagons. Our goal is to achieve k-coverage of a 2D FoI with a minimum number of sensors in order to maximize the network lifetime. First, we compute the minimum sensor density for 2D k-coverage using the regular convex hexagon, which is a 2D paver (i.e., covers a 2D field without gaps or overlaps). Indeed, we found that the regular convex hexagon best assimilates the sensors’ sensing disk with respect to our proposed metric, sensing range usage rate. Second, we derive the ratio of the communication range to the sensing range of the sensors to ensure connected k-coverage. Third, we propose an energy-efficient connected k-coverage protocol based on hexagonal slicing and area stretching. To this end, we formulate a multi-objective optimization problem, which computes an optimum solution to the 2D k-coverage problem that meets two requirements: Maximizing the size of the k-covered area, Ck, so as to minimize the sensor density to k-cover a 2D FoI (Requirement 1) and maximizing the area of the sensor locality, Lk, i.e., the region where at least k sensors are located to k-cover Ck, so as to minimize the interference between sensors (Requirement 2). Fourth, we show various simulation results to substantiate our theoretical analysis. We found a close-to-perfect match between our theoretical and simulation results.

Introduction

Coverage is a fundamental step in sensor deployment, which is an important phase in the design and implementation of wireless sensor networks. In general, sensing applications may require different degrees of coverage k, where k 1 is a natural number. In particular, some sensing applications may need a high degree of coverage k, (i.e., k > 1). For instance, triangulation-based sensor localization requires a sensor, which wants to compute its location, to be within the range of three anchor sensors (i.e., sensors already knowing their locations). That is, the degree of coverage required for sensor localization is k=3. The problem of coverage in two-dimensional (2D) wireless sensor networks (i.e., 2D coverage) is challenging, and is still open. More specifically, determining the minimum sensor density (i.e., number of sensors per unit area) that is required to cover a 2D field of interest (FoI), where every point in the field is covered by at least one sensor, has been investigated by several researchers. Likewise, given k1, the problem of 2D k-coverage, which requires that every point in a 2D FoI be covered by at least k sensors simultaneously, is even more challenging. It is worth mentioning that the k-coverage operation is done appropriately if the collected data by the sensors can reach the base station (also, known as the sink) for further analysis and processing. Therefore, in order to accomplish the k-coverage task successfully, it is essential that network connectivity during the entire network operation, be maintained among all of those sensors that are selected to k-cover a 2D FoI.

In this paper, we attempt to address the problem of 2D connected k-coverage, where a 2D FoI is k-covered, while the wireless sensor network being formed is connected. However, there are two major challenges to solve this problem. The first one is due the scarce energy resources of the sensors. Hence, it is essential to minimize the total energy consumption of the sensors, which depends on the number of sensors. Thus, it is important to solve the 2D connected k-coverage problem using a minimum number of sensors in order to maximize the lifetime of the individual sensors. The second challenge is due to the geometric characteristics of the sensing range of the sensors, which is generally supposed to be a disk in 2D wireless sensor networks. It is important that the area of the sensing range actually utilized be maximized in order to minimize the total number of active sensors to k-cover a 2D FoI. Our main goal is to extend the network lifetime, while accomplishing the target mission of the underlying sensing application.

We believe that there exists a tight relationship between coverage and paving. Indeed, the former generally yield gaps, and, thus, allows overlap to remove any gaps. However, the latter does not yield any overlaps or gaps. In order to solve the connected k-overage problem in 2D wireless sensor networks, we propose a regular hexagonal field slicing-based approach, where a 2D FoI is divided into regular convex hexagons. Indeed, the latter has an interesting geometric property in that it can pave a 2D field. Our proposed solution will be discussed in subsequent sections.

Next, we define a few key terms that we use in our analysis. Then, we formulate the problem we want to investigate in this paper, and present our major contributions.

Definition 1 Sensing Range

The sensing range of a sensor si is the area where any occurring event in it will be surely sensed by si.  

Definition 2 Communication Range

The communication range of a sensor si is the area where si can directly communicate with any sensor sj in it.  

Definition 3

A point in a 2D FoI is said to be k-covered if it is located within the sensing range of at least k sensors. A 2D FoI is said to be k-covered if every point in this field is k-covered. k is called the degree of coverage.  

Definition 4

A set of at least k sensors in a 2D FoI form a connected k-covered network if this field is k-covered and these sensors are guaranteed to be connected to each other.  

Definition 5 Sensor Density

The sensor density is the number of sensors per unit area.  

Definition 6 2D Paver

A 2D regular convex polygon is said to be a 2D paver if it can pave a 2D FoI through replication of its congruent copies without any overlaps or gaps between any pair of congruent copies.  

The major questions that are related to the connected k-coverage problem in 2D wireless sensor network, and which we attempt to answer in this paper, are listed below:

  • What is the best convex polygonal shape that can be used to model the sensing range of the sensors, while maximizing its actual area being utilized to pave a 2D FoI?

  • What is the minimum sensor density that is required to k-cover a 2D FoI, where k1?

  • What is the ratio of the communication range to the sensing range of the sensors to guarantee connected k-coverage of a 2D FoI, where k1?

  • What is the most energy-efficient sensor selection approach that enables connected k-coverage of a 2D FoI using a minimum number of sensors, with k1?

Our major contributions in this paper to solve the connected k-coverage problem in 2D wireless sensor networks can be summarized as follows:

  • We investigate well-known regular 2D pavers, namely triangle, square, and convex regular hexagon, with a goal to find the best convex polygon to model the sensors’ sensing range and maximize its area being utilized for paving a 2D FoI. Our study is based on our proposed metric, called sensing area usage rate. This metric shows that the convex regular hexagon is the best polygonal paver of a 2D field.

  • We model the sensors’ sensing range using the convex regular hexagon. Our analysis shows that there is a close relationship between the area of the k-covered region, denoted by Ck (i.e., region to be k-covered by at least k sensors) and the area of the sensor locality, denoted by Lk (i.e., region where the set of at least k sensors are located to k-cover Ck). We find that to compute the optimum sensor density to k-cover a 2D FoI, we need to solve a multi-objective optimization problem with a goal to maximize the areas of Ck and Lk.

  • Based on our solution to the above-mentioned multi-objective optimization problem, we determine the relationship that exists between the communication range and the sensing range of the sensors, which is required to ensure connected k-coverage of a 2D field.

  • We propose an energy-efficient k-coverage protocol based on hexagonal slicing of a 2D FoI and area stretching. More precisely, this protocol exploits our solution to this multi-objective optimization problem to select a minimum number of sensors, where k1.

  • We generalize our proposed solution to the connected k-coverage problem in 2D wireless sensor networks by accounting for a probabilistic sensing model, where the sensors’ sensing range is not necessarily a disk, as well as sensor heterogeneity, where the sensors may not have the same sensing range, communications range, and initial energy reserve.

  • We support our theoretical analysis with various simulation results.

The remainder of this paper is organized as follows. In Section 2, we present our network model along with the energy model that are used in this work. In Section 3, we review existing approaches that were proposed to address the problem of coverage in 2D wireless sensor networks. In Section 4, we study regular convex pavers, and determine the best regular convex polygonal shape that paves a 2D field, using our proposed metric, namely sensing area usage rate. In Section 5, we investigate the problem of connected k-coverage in 2D wireless sensor networks. More precisely, we compute the sensor density that is required to k-cover a 2D field. Then, we derive the relationship that should exist between the communication range and the sensing range of the sensors in order to maintain network connectivity in 2D k-covered wireless sensor networks. In Section 6, we provide a generalization of our solution to the connected k-coverage in 2D wireless sensor networks by considering a more general sensing model, namely a probabilistic sensing model, and sensor heterogeneity. In Section 7, we present our simulation results. In Section 8, we conclude, and discuss our future work.

Section snippets

Network and energy models

First, we specify the network model used in our study of the connected k-coverage problem in 2D wireless sensor networks. Then, we describe our energy model.

Related work

In this section, we review existing approaches for coverage and k-coverage in 2D wireless sensor networks. In addition, we present the unique features of our proposed study in this paper, compared to our previous work [4].

Xing et al. [34] addressed the k-coverage problem in wireless sensor networks, and proposed a protocol, named connected coverage protocol. Huang et al. [17] proposed distributed protocols for coverage and connectivity. Bai et al. [5] developed an optimal deployment strategy to

Investigating two-dimensional pavers

It is well known that the only 2D regular convex shapes, which can pave the Euclidean plane without overlaps or gaps, are the equilateral triangle, square, and regular hexagon. Kershner [19], indeed, showed that a regular n-gon can pave the plane only if n=3, 4, or 6. Indeed, the vertex angle of a regular n-gon is n2n×π. Thus, an n-gon can pave the plane only if m of these vertices can meet at a point to fill 2π. That is, we should have m×(n2)n×π=2π, or mn=2(n+m). The only positive integer (

Slicing-based connected k-coverage

In this section, we investigate the problem of connected k-coverage of a 2D FoI. Precisely, we consider two sensor placement and selection approaches to achieve k-coverage of a 2D field. First, we slice a 2D field randomly into tangential congruent regular convex hexagons to produce a 2D regular hexagonal grid, as shown in Fig. 2. For each k-coverage approach, we compute the sensor density that is required to k-cover a 2D field. Then, we determine the corresponding relationship that should

Generalization

In this section, we generalize our proposed solution to the connected k-coverage problem in two-dimensional wireless sensor networks. Precisely, first, we extend our deterministic sensing model to a stochastic sensing model. Second, we take into consideration network heterogeneity, where we account for heterogeneous sensors. Next, we discuss both extensions in order to make our proposed approach more realistic.

Performance evaluation

In this section, we specify the simulation setup. Then, we discuss the simulation results of our proposed approaches to the problem of connected k-coverage in wireless sensor networks, i.e., cone-based and perimeter-based approaches.

Summary

In this paper, we investigated the problem of connected k-coverage in 2D wireless sensor networks using regular convex hexagons. First, we studied all 2D regular convex pavers to determine the best paver that maximizes the area of the sensors’ sensing range utilized based on our proposed sensing range usage rate metric. We found that the regular convex hexagon is the “best” paver as it yields the maximum sensing range usage rate. Second, we computed the sensor density for 2D k-coverage. Our

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The author gratefully acknowledges the insightful comments of the anonymous reviewers which helped improve the quality and presentation of the paper significantly. This work is partially supported by the US National Science Foundation (NSF) grants CNS-0917089 and CNS-1054935.

Habib M. Ammari Habib M. Ammari is an Associate Professor, the Founding Director of Wireless Sensor and Mobile Ad-hoc Networks, Internet of Things, and Applied Cryptography Engineering (WiSeMAN-IoT-ACE) Research Lab, and the Graduate Computer Science Program Director in the Department of Electrical Engineering and Computer Science, Frank H. Dotterweich College of Engineering, at Texas A&M University Kingsville (TAMUK). He received his tenure in May 2014 in the Department of Computer and

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    Habib M. Ammari Habib M. Ammari is an Associate Professor, the Founding Director of Wireless Sensor and Mobile Ad-hoc Networks, Internet of Things, and Applied Cryptography Engineering (WiSeMAN-IoT-ACE) Research Lab, and the Graduate Computer Science Program Director in the Department of Electrical Engineering and Computer Science, Frank H. Dotterweich College of Engineering, at Texas A&M University Kingsville (TAMUK). He received his tenure in May 2014 in the Department of Computer and Information Science, College of Engineering and Computer Science, at the University of MichiganDearborn, where he served on the Distinguished Research Award Committee in 2015. Also, he received tenure at the Higher School of Communications in Tunis, Tunisia (Sup’Com Tunis) in 1998. He is the recipient of the 2021 Outstanding Graduate Instructor Teaching Award presented by the Center for Teaching Effectiveness at TAMUK. Also, he received the 2020 Professor of the Year Award, Computer Science major, presented by the College of Engineering at TAMUK. He is the recipient of the 2018 Albert Nelson Marquis Lifetime Achievement Award. He was selected as instructor at Stanford University in the Stanford Summer College Academy 2016 program, where he taught “Discrete Mathematical Structures: Foundational Concepts in Computer Science, Engineering, and Mathematics”. He obtained his second Ph.D. degree in Computer Science and Engineering from the University of Texas at Arlington, in May 2008, and his first Ph.D. in Computer Science from the Faculty of Sciences of Tunis, in December 1996. His main research interests lay in the area of wireless sensor and mobile ad hoc networks, including connected k-coverage, geographic forwarding, physical and information security, applied cryptography, and computational geometry in wireless sensor networks. He has a strong publication record in top-quality journals, such as ACM TOSN, ACM TAAS, IEEE TPDS, IEEE TC, Elsevier Ad Hoc Networks, Elsevier COMNET, Elsevier PMC, Elsevier JPDC, Elsevier COMCOM, and high-quality conferences, such as IEEE SECON, IEEE ICDCS, IEEE MASS, and EWSN. He published his first Springer book, “Challenges and Opportunities of Connected k-Covered Wireless Sensor Networks: From Sensor Deployment to Data Gathering” in August 2009. Also, he is the author and editor of two Springer books, “The Art of Wireless Sensor Networks: Fundamentals” and “The Art of Wireless Sensor Networks: Advanced Topics and Applications”, which have been published in January 2014. In addition, he published these two current Springer books,“Mission-oriented sensor networks and systems: Art and science - Foundations” and “Mission-oriented sensor networks and systems: Art and science - Advances” in January 2019. He is the recipient of the US National Science Foundation (NSF) CAREER Award in January 2011, a three-year US NSF Research Grant Award in June 2009, the National Security Agency (NSA) Award in 2017, and the Faculty Research and Development Grant Award from Hofstra College of Liberal Arts and Sciences in May 2009. In March 2014, he was recognized with the Distinguished Research Award at the University of Michigan-Dearborn. Furthermore, in May 2010, he was recognized with the Lawrence A. Stessin Prize for Outstanding Scholarly Publication (i.e., Distinguished Research Award) at Hofstra University. He is the recipient of the Nortel Outstanding CSE Doctoral Dissertation Award in February 2009, and the John Steven Schuchman Award for 2006–2007 Outstanding Research by a Ph.D. Student in February 2008. He received the Best Paper Award at EWSN in 2008, and the Best Paper Award at the IEEE PerCom 2008 Google Ph.D. Forum. He received several other prestigious awards, including the Best Graduate Student Paper Award (Nokia Budding Wireless Innovators Awards First Prize) in May 2004, the Best Graduate Student Presentation Award (Ericsson Award First Prize) in February 2004, and Laureate in Physics and Chemistry for academic years 1987 and 1988. Also, he was selected as the ACM Student Research Competition Finalist at the ACM MobiCom in September 2005. Also, he was selected for inclusion in the Marquis Who’s Who in the World in 2019 and 2018, AcademicKeys Who’s Who in Sciences Higher Education in 2017, Who’s Who in America in 2017, AcademicKeys Who’s Who in Engineering Higher Education in 2012, the AcademicKeys Who’s Who in Sciences Higher Education in 2011, Feature Alumnus in the University of Texas at Arlington CSE Department’ Newsletter in Spring 2011, Who’s Who in America in 2010, and the 2008–2009 Honors Edition of Madison Who’s Who Among Executives and Professionals. He received several service awards, including the Certificate of Appreciation Award at MiSeNet 2014, the Certificate of Appreciation Award at ACM MiSeNet 2013, the Certificate of Appreciation Award at the IEEE DCoSS 2013, the Certificate of Appreciation Award at the ACM MobiCom 2011, the Outstanding Leadership Award at the IEEE ICCCN 2011, and the Best Symposium Award at the IEEE IWCMC 2011. He serves as the Founding Coordinator of the CIS Distinguished Lecture Series, and as Coordinator of the CIS Faculty Research Talk Series since 2017. In addition, he was the Founding Coordinator of both of the Distinguished Lecture Series, and the Research Colloquium Series, in the College of Engineering and Computer Science at the University of MichiganDearborn from 2011–2015. He was successful to invite ACM Turing Award Winners to his distinguished lecture series, such as Dr. Manuel Blum from Carnegie Mellon University (CMU), and Dr. Shafi Goldwasser from (MIT), who gave talks at the University of MichiganDearborn on January 25, 2013, and October 25, 2013, respectively, and Dr. Martin E. Hellman from Stanford University, who gave a talk at Fordham University on October 22, 2018. He was invited to give several invited talks at reputed universities. Indeed, he was invited to give a talk at the Third Arab–American Frontiers of Sensor Science Symposium, which was organized by the US National Academy of Sciences on December 5–7, 2015. Also, he served as external examiner of several Ph.D. Dissertations. He is the Founder of the Annual International Workshop on Mission-Oriented Wireless Sensor Networking (MiSeNet), which has been co-located with ACM MobiCom, IEEE INFOCOM, and IEEE MASS conferences since 2012. He served as Associate Editor of several prestigious journals, such as ACM TOSN, IEEE TC, IEEE Access, and Elsevier PMC. He serves on the Steering Committee of MiSeNet, the Annual International Conference on Distributed Computing in Sensor Systems (DCOSS), and the International Workshop on Wireless Mesh and Ad-hoc Networking (WiMAN). Moreover, he served as General Chair, Program Chair, Track Chair, Session Chair, Publicity Chair, Web Chair, and Technical Program Committee member of numerous ACM and IEEE conferences, symposia, and workshops. He is an IEEE Senior Member.

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