The total communication cost of our networking protocol is Oðn log nÞ bits, which is within a constant factor of the optimum to construct any structure in a distributed manner. It contains all edges that are both in the unit-disk graph and the Delaunay triangulation of all nodes. In this paper, we present a novel localized networking protocol that constructs a planar 2.5-spanner of UDG, called the localized Delaunay triangulation (LDEL), as network topology. Given a set of wireless nodes, we model the network as a unit-disk graph (UDG), in which a link uv exists only if the distance kuvk is at most the maximum transmission range. However, it is expensive to construct the Delaunay triangulation in a distributed manner. recently developed a localized routing protocol that guarantees that the distance traveled by the packets is within a constant factor of the minimum if Delaunay triangulation of all wireless nodes is used, in addition, to guarantee the delivery of the packets. However, it is well-known that the spanning ratios of these two graphs are not bounded by any constant (even for uniform randomly distributed points). Typically, relative neighborhood graph (RNG) or Gabriel graph (GG) is used as such planar structure. Several localized routing protocols guarantee the delivery of the packets when the underlying network topology is a planar graph. 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. In addition, we justify the correctness of the method proposed in that uses the Delaunay triangulation to solve the best coverage problem.
#Planar disk graph proof citesee how to#
We also consider how to find an optimum best-coverage-path that travels a small distance. 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. Here, we consider the sensing model: the sensing ability diminishes as the distance increases. In this paper, we give efficient distributed algorithms to optimally solve the best-coverage problem raised in. In, it is assumed that the sensor has the uniform sensing ability. One of the fundamental problems in sensor networks is the calculation of the coverage. We then prove some structural properties of such graphs which allows us to design polynomial-time algorithms to decide whether a bipartite graph, resp., a planar graph, has treebreadth one.Sensor networks pose a number of challenging conceptual and optimization problems such as location, deployment, and tracking. We show that it is NP-complete to decide whether a graph belongs to this class. We further investigate graphs with treebreadth one, i.e., graphs that admit a tree-decomposition where each bag has a dominating vertex. Namely, we prove that computing these graph invariants is NP-hard. In this paper, we answer open questions of and about the computational complexity of treebreadth, pathbreadth and pathlength.
Pathlength and pathbreadth are defined similarly for path-decompositions. The treelength and the treebreadth of a graph are the minimum length and breadth of its tree-decompositions respectively. Roughly, the length and the breadth of a tree-decomposition are the maximum diameter and radius of its bags respectively. We then prove some structural properties of such graphs which allows us to design polynomial-time algorithms to decide whether a bipartite graph, resp., a planar graph (or more generally, a triangle-free graph, resp., a K3,3-minor-free graph), has treebreadth one.ĭuring the last decade, metric properties of the bags of tree-decompositions of graphs have been studied. We further investigate graphs with treebreadth one, i.e., graphs that admit a tree decomposition where each bag has a dominating vertex. (Algorithm theory-SWAT 2014, Springer, pp 158–169, 2014) about the computational complexity of treebreadth, pathbreadth and pathlength. In this paper, we answer open questions of Dragan and Köhler (Algorithmica 69(4):884–905, 2014) and Dragan et al. Pathlength and pathbreadth are defined similarly for path decompositions. The treelength and the treebreadth of a graph are the minimum length and breadth of its tree decompositions respectively. Roughly, the length and the breadth of a tree decomposition are the maximum diameter and radius of its bags respectively. During the last decade, metric properties of the bags of tree decompositions of graphs have been studied.
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