Geometric algorithms for sensor networks

We present an implementation of a recent algorithm to compute shortest-path trees in unit disk graphs in O(n log n) worst-case time, where n is the number of disks.In the minimum-separation problem, we are given n unit disks and two points s and t, not contained in any of the disks, and we want to compute the minimum number of disks one needs to retain so that any curve connecting s to t intersects some of the retained disks. We present a new algorithm solving this problem in O(n 2 log 3 n) worst-case time and its implementation.

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“…Analytical and algorithmic tools in graph theory and computational geometry have been widely used in the modeling and analysis of connectivity in ad hoc networks [15], together with geometrical probability, stochastic geometry, and percolation theories in recent years. For example, a connected dominating set is introduced in ad hoc networks to create a virtual backbone for the network [16].…”

Section: Related Workmentioning

confidence: 99%

A New Approach to the Directed Connectivity in Two-Dimensional Lattice Networks

Zhang

Cai

Pan

et al. 2014

IEEE Trans. on Mobile Comput.

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Abstract-The connectivity of ad hoc networks has been extensively studied in the literature. Most recently, researchers model ad hoc networks with two-dimensional lattices and apply percolation theory for connectivity study. On the lattice, given a message source and the bond probability to connect any two neighbor vertices, percolation theory tries to determine the critical bond probability above which a giant connected component appears. This paper studies a related but different problem, directed connectivity: what is the exact probability of the connection from the source to any vertex following certain directions? The existing studies in math and physics only provide approximation or numerical results. In this paper, by proposing a recursive decomposition approach, we can obtain a closed-form polynomial expression of the directed connectivity of square lattice networks as a function of the bond probability. Based on the exact expression, we have explored the impacts of the bond probability and lattice size and ratio on the lattice connectivity, and determined the complexity of our algorithm. Further, we have studied a realistic ad hoc network scenario, i.e., an urban VANET, where we show the capability of our approach on both homogeneous and heterogeneous lattices and how related applications can benefit from our results.

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Geometric algorithms for sensor networks

Cited by 24 publications

References 124 publications

Virtual coordinates in hyperbolic space based on Ricci flow for WLANs

Virtual coordinates in hyperbolic space based on Ricci flow for WLANs

Two optimization problems for unit disks

A New Approach to the Directed Connectivity in Two-Dimensional Lattice Networks

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