## Coverage in Wireless Ad-hoc Sensor Networks (2002)

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Citations: | 128 - 10 self |

### BibTeX

@MISC{Li02coveragein,

author = {Xiang-Yang Li and Peng-Jun Wan and Ophir Frieder},

title = {Coverage in Wireless Ad-hoc Sensor Networks},

year = {2002}

}

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### Abstract

Sensor networks pose a number of challenging conceptual and optimization problems such as location, deployment, and tracking [1]. One of the fundamental problems in sensor networks is the calculation of the coverage. In [1], it is assumed that the sensor has the uniform sensing ability. In this paper, we give efficient distributed algorithms to optimally solve the best-coverage problem raised in [1]. Here, we consider the sensing model: the sensing ability diminishes as the distance increases. As energy conservation is a major concern in wireless (or sensor) networks, we also consider how to find an optimum bestcoverage -path with the least energy consumption. We also consider how to find an optimum best-coverage-path that travels a small distance. In addition, we justify the correctness of the method proposed in [1] that uses the Delaunay triangulation to solve the best coverage problem. Moreover, we show that the search space of the best coverage problem can be confined to the relative neighborhood graph, which can be constructed locally.

### Citations

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Citation Context ...and ut. Here, the weight of a path is the maximum weight of all of its edges. A path is the shortest path if it has the minimum weight among all paths connecting us and ut. The Bellman-Ford algorithm =-=[23] can b-=-e modified to solve this shortest path problem. 5. Let ðus;utÞbe a computed path and k ðus;utÞk be the weight of the path. Then, the path concatenating the edge sus, path ðus;utÞ, and the edge u... |

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Citation Context ...paper, we must only study how to find the best-coverage-path, which also achieves the maximum closest-sensor observability. 2.2 Geometry Notations Delaunay triangulation and Voronoi diagram [4], [5], =-=[6]-=- are widely used in many areas. We begin with definitions of the Voronoi diagram and the Delaunay triangulation. We assume that all wireless nodes are given as a set S of n vertices in a two-dimension... |

711 |
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Citation Context ... the paper, we must only study how to find the best-coverage-path, which also achieves the maximum closest-sensor observability. 2.2 Geometry Notations Delaunay triangulation and Voronoi diagram [4], =-=[5]-=-, [6] are widely used in many areas. We begin with definitions of the Voronoi diagram and the Delaunay triangulation. We assume that all wireless nodes are given as a set S of n vertices in a two-dime... |

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Citation Context ...tern recognition. The constrained Gabriel graph of a point set V , denoted by GGðV Þ, consists of all edges uv such that kuvk 1 and the diskðu; vÞ does not contain any node from V . Gabriel and So=-=kal [13]-=- defined the Gabriel graph for the geographic variation analysis. Obviously, the relative neighborhood graph is always a subgraph of the Gabriel graph. See Fig. 2 for an illustration of their definiti... |

254 |
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Citation Context ...nd v. The constrained relative neighborhood graph RNGðV Þ over a point set V has an edge ðu; vÞ if kuvk 1 and the luneðu; vÞ does not contain any point from V in the interior. Toussaint and Jaro=-=mczyk [11], -=-[12] gave the first definition of the relative neighborhood graph to study the pattern recognition. The constrained Gabriel graph of a point set V , denoted by GGðV Þ, consists of all edges uv such ... |

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Citation Context ...ity and, thus, continuouity of network service. In those scenarios, we often have to find the optimum number of base stations required to achieve some system objectives; see [20] for more details. In =-=[21]-=-, Hall studied how many wireless nodes with fixed coverage radius r are needed so that every point of a unit square region is covered by some wireless node with high probability. If the connectivity c... |

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Citation Context ...st of the paper, we must only study how to find the best-coverage-path, which also achieves the maximum closest-sensor observability. 2.2 Geometry Notations Delaunay triangulation and Voronoi diagram =-=[4]-=-, [5], [6] are widely used in many areas. We begin with definitions of the Voronoi diagram and the Delaunay triangulation. We assume that all wireless nodes are given as a set S of n vertices in a two... |

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Citation Context ...The constrained relative neighborhood graph RNGðV Þ over a point set V has an edge ðu; vÞ if kuvk 1 and the luneðu; vÞ does not contain any point from V in the interior. Toussaint and Jaromczyk =-=[11], [12] g-=-ave the first definition of the relative neighborhood graph to study the pattern recognition. The constrained Gabriel graph of a point set V , denoted by GGðV Þ, consists of all edges uv such that k... |

125 |
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Citation Context ...path among all optimum best-coverage-paths. It is well-known that the relative neighborhood graph and the Gabriel graph are not spanners [24], [10]. Although Delaunay triangulation is a spanner [25], =-=[26]-=-, we know that it cannot always be constructed efficiently in a distributed manner. Thus, we have to use some other geometry structure that is a spanner and can be constructed efficiently. Recently, L... |

118 | Distributed construction of planar spanner and routing for ad hoc wireless networks
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Citation Context ...aunay triangulation. Let UDelðSÞ be the graph by removing all edges of DelðSÞ that are longer than one unit, i.e., UDelðSÞ DelðSÞ\UDGðSÞ. Call UDelðSÞ the unit Delaunay triangulation. Li e=-=t al. [7], [8] pro-=-vided an efficient localized algorithm that constructs a planar graph, called localized Delaunay triangulation P LDeðSÞ, which contains UDelðSÞ as a subgraph. Thus, the constructed graph can be us... |

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Citation Context ...rtest path among all optimum best-coverage-paths. It is well-known that the relative neighborhood graph and the Gabriel graph are not spanners [24], [10]. Although Delaunay triangulation is a spanner =-=[25]-=-, [26], we know that it cannot always be constructed efficiently in a distributed manner. Thus, we have to use some other geometry structure that is a spanner and can be constructed efficiently. Recen... |

102 | Critical Power for Asymptotic Connectivity
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Citation Context ... the connectivity coverage is of concern, then we need to find how many wireless nodes are needed in a unit area square such that the resulting UDG is connected with high probability. Gupta and Kumar =-=[22]-=- studied the dual of this problem: If the number of nodes is fixed, then what is the smallest r such that the resulting UDG is connected with high probability. In [1], Meguerdichian et al. developed c... |

64 | Power efficient and sparse spanner for wireless ad hoc networks
- Li, Wan, et al.
- 2001
(Show Context)
Citation Context ...t require use of the structure UDelðSÞ or even DelðSÞ. We review how to construct the localized Delaunay triangulation P LDelðSÞ later. Various proximity subgraphs of the unit disk graph were st=-=udied [9], [10]-=-. For convenience, let diskðu; vÞ be the closed disk with diameter uv; let diskðu; v; wÞ be the circumcircle defined by the triangle 4uvw; let Bðu; rÞ be the circle centered at u with radius r. ... |

57 |
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Citation Context ...es edges from the Delaunay triangulation in Oðn log nÞ time. Several methods [15], [16] were then proposed for constructing RNG efficiently. The construction of the Gabriel graph is a tad easier; se=-=e [17]-=-. The algorithm by Matula and Sokal [17] is based on the observation that the Gabriel graph only contains those edges in the Delaunay triangulation that do not intersect their Voronoi edges. Fig. 3 il... |

49 | Sparse power efficient topology for wireless networks
- LI, WAN, et al.
- 2002
(Show Context)
Citation Context ...uire use of the structure UDelðSÞ or even DelðSÞ. We review how to construct the localized Delaunay triangulation P LDelðSÞ later. Various proximity subgraphs of the unit disk graph were studied=-= [9], [10]. For -=-convenience, let diskðu; vÞ be the closed disk with diameter uv; let diskðu; v; wÞ be the circumcircle defined by the triangle 4uvw; let Bðu; rÞ be the circle centered at u with radius r. Call t... |

45 |
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Citation Context ...nay triangulation, a simple method with Oðn 2 Þ time complexity is to check whether the lune of each edge in the Delaunay triangulation is empty. If it is empty, then we add that edge to RNG. Supowi=-=t [14] ga-=-ve the first Oðn log nÞ timecomplexity algorithm to construct the relative neighborhood graph in two dimensions using the l2 metric, which eliminates edges from the Delaunay triangulation in Oðn lo... |

41 | On the spanning ratio of gabriel graphs and beta-skeletons - Bose, Devroye, et al. |

25 | GPS-free Positioning - Capkun, Hamdi, et al. - 2001 |

20 | System To Place Observers On A Polyhedral Terrain - Marengoni, Draper, et al. - 1996 |

17 |
Frieder O. Sparse power efficient topology for wireless networks
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Citation Context ...n to . Node keeps triangle � if both and have triangle � remained. The collection of edges kept by all nodes is called planar �� . Various proximity subgraphs of the unit disk graph were studied [5], =-=[6]-=-. For convenience, let �� � � be the closed disk with diameter ; let �� � � � be the circumcircle defined by the triangle � ; let � � be the circle centered at with radius . Call the interior of the i... |

16 |
Coverage Opportunities For Global Ocean Color
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- 1998
(Show Context)
Citation Context ...ch that every point of the room is watched by at least one observer. A sensor network was also used to detect the global ocean color by assembling and merging data from satellites at different orbits =-=[19]-=-. Notice that the coverage problem we study here is different from the coverage problem studied in cellular network. Traditionally, in cellular networks, coverage studies the maintainence of connectiv... |

8 | Farinaz Koushanfar, Gang Qu, and Miodrag Potkonjak. Exposure in wireless ad-hoc sensor networks - Meguerdichian |

8 | The relative neighborhood graph of a planar set - Toussaint - 1980 |

7 | Miodrag Potkonjak. Exposure in wireless ad hoc sensor networks - Meguerdichian, Koushanfar, et al. - 2001 |

5 |
Localized delaunay triangulation is as good as unit disk graph
- Li
(Show Context)
Citation Context ...is not appropriate, thus, to require the construction of the Delaunay triangulation in the wireless communication environment because of the possible massive communications it requires. Therefore, Li =-=[7] studied a -=-subset of the Delaunay triangulation. Let UDelðSÞ be the graph by removing all edges of DelðSÞ that are longer than one unit, i.e., UDelðSÞ DelðSÞ\UDGðSÞ. Call UDelðSÞ the unit Delaunay tr... |

5 |
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Citation Context ...mplexity algorithm to construct the relative neighborhood graph in two dimensions using the l2 metric, which eliminates edges from the Delaunay triangulation in Oðn log nÞ time. Several methods [15]=-=, [16]-=- were then proposed for constructing RNG efficiently. The construction of the Gabriel graph is a tad easier; see [17]. The algorithm by Matula and Sokal [17] is based on the observation that the Gabri... |

4 |
System to Place Observers on a
- Marengoni, Draper, et al.
- 1996
(Show Context)
Citation Context ...e among the first several researchers to identify the importance of using Delaunay triangulation and Voronoi diagram in sensor network coverage. A related problem is the so-called art gallery problem =-=[18]-=-, in which one must determine the number of observers necessary to cover an art gallery room such that every point of the room is watched by at least one observer. A sensor network was also used to de... |

4 |
Farinaz Koushanfar, Miodrag Potkonjak, and Mani Srivastava, “Coverage problems in wireless ad-hoc sensor network
- Meguerdichian
- 2001
(Show Context)
Citation Context ...d-hoc Sensor Networks Xiang-Yang Li, Peng-Jun Wan, Ophir Frieder Abstract— Sensor networks pose a number of challenging conceptual and optimization problems such as location, deployment, and tracking =-=[1]-=-. One of the fundamental problems in sensor networks is the calculation of the coverage. In [1], it is assumed that the sensor has the uniform sensing ability. In this paper, we give efficient distrib... |

4 | Power ecient and sparse spanner for wireless ad hoc networks - Li, Wan, et al. |

3 |
On the Spanning Ratio
- Bose, Devroye, et al.
- 2002
(Show Context)
Citation Context ...the edges of this path of not more than 5=2 times the shortest path among all optimum best-coverage-paths. It is well-known that the relative neighborhood graph and the Gabriel graph are not spanners =-=[24]-=-, [10]. Although Delaunay triangulation is a spanner [25], [26], we know that it cannot always be constructed efficiently in a distributed manner. Thus, we have to use some other geometry structure th... |

3 | Ophir Frieder, \Sparse power ecient topology for wireless networks - Li, Wan, et al. - 2002 |

2 |
Constructing the Relative Neighborhood Graph in Three-Dimensional Euclidean
- Jaromczyk, Kowaluk
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(Show Context)
Citation Context ...timecomplexity algorithm to construct the relative neighborhood graph in two dimensions using the l2 metric, which eliminates edges from the Delaunay triangulation in Oðn log nÞ time. Several method=-=s [15]-=-, [16] were then proposed for constructing RNG efficiently. The construction of the Gabriel graph is a tad easier; see [17]. The algorithm by Matula and Sokal [17] is based on the observation that the... |

2 |
The Automatic Location of Base Stations for Optimized Cellular Coverage: A New Combinatorial Approach
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Citation Context ... maintainence of connectivity and, thus, continuouity of network service. In those scenarios, we often have to find the optimum number of base stations required to achieve some system objectives; see =-=[20]-=- for more details. In [21], Hall studied how many wireless nodes with fixed coverage radius r are needed so that every point of a unit square region is covered by some wireless node with high probabil... |

1 | is possible that there is some other sensor node inside the circle diskðu; wÞ. Thus, we can only claim that it is at most 1 - It |

1 | An optimal algorithm for constructing in l p metric - Jaromczyk, Kowaluk, et al. - 1991 |

1 | 4, 2002 DRAFT \Coverage opportunities for global ocean color in a multimission era - Gregg, Esaias, et al. |