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The minimum linear arrangement problem is widely used and studied in many practical and theoretical applications. In this paper we present a linear-time algorithm for the problem inspired by the algebraic multigrid approach which is based on weighted edge contraction rather than simple contraction. Our results turned out to be better than every known result in almost all cases, while the short running time of the algorithm enabled experiments with very large graphs.

This paper deals with the Minimum Linear Arrangement problem from an experimental point of view. Using a test-suite of sparse graphs, we experimentally compare several algorithms to obtain upper and lower bounds for this problem. The algorithms considered include Successive Augmentation heuristics, Local Search heuristics and Spectral Sequencing. The test-suite is based on two random models and "real life" graphs. As a consequence of this study, two main conclusions can be drawn: On one hand, the best approximations are usually obtained using Simulated Annealing, which involves a large amount of computation time. However, solutions found with Spectral Sequencing are close to the ones found with Simulated Annealing and can be obtained in significantly less time. On the other hand, we notice that there exists a big gap between the best obtained upper bounds and the best obtained lower bounds. These two facts together show that, in practice, finding lower and upper bounds for the Minimum ...

In this paper, we introduce a new model of random graph, that we call random sector graph. This model aims to provide a tool for studying communication problems in networks of sensors using laser communication such as the ones addressed in the Smart Dust project. Current technology allows steering the laser cannon along a contigous sector, providing undirectional communication. Thus, random sector graphs are a generalization of random geometric graphs, in which this restricted communication is taken into account. We provide tight estimations of the maximum and minimum degree and show that random sector graphs are connected for an adequate selection of the sector radius.

In this paper we analyze the computational power of random geometric networks in the presence of random (edge or node) faults considering several important network parameters. We rst analyze how to emulate an original random geometric network G on a faulty network F . Our results state that, under the presence of some natural assumptions, random geometric networks can tolerate a constant node failure probability with a constant slowdown. In the case of constant edge failure probability the slowdown is an arbitrarily small constant times the logarithm of the graph order. Then we consider several network measures, stated as linear layout problems (Bisection, Minimum Linear Arrangement and Minimum Cut Width). Our results show that random geometric networks can tolerate a constant edge (or node) failure probability while maintaining the order of magnitude of the measures considered here. Finally we show that, with high probability, random geometric networks with (edge or node) faults do h...

In this paper we analyze the Hamiltonian properties of faulty random networks. This consideration is of interest when considering wireless broadcast networks. A random geometric network is a graph whose vertices correspond to points uniformly and independently distributed in the unit square, and whose edges connect any pair of vertices if their distance is below some specified bound. A faulty random geometric network is a random geometric network whose vertices or edges fail at random. Algorithms to find Hamiltonian cycles in faulty random geometric networks are presented. 1 Introduction The use of distributed computing in wireless networks is a computational model that is gaining increasing importance in computer science and telecommunication. In this setting, the processors, scattered geographically, communicate through transmitters, e#ectively forming a wireless broadcast network. The following setting arises in applications of wireless broadcast networks: A set of stations are loc...

In this paper we present two randomized algorithms to compute the bisection width of random cubic graphs with n vertices, giving an asymptotic upper bound for the bisection width respectively of 0.174039n and 0.174501n. We also obtain an asymptotic lower bound for the size of the max bisection of a random cubic graph with n vertices of 1.325961n and 1.325499n. The analysis is based on the differential equation method.

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Abstract — In this paper, we study network coding capacity for random wireless networks. Previous work on network coding capacity for wired and wireless networks have focused on the case where the capacities of links in the network are independent. In this paper, we consider a more realistic model, where wireless networks are modeled by random geometric graphs with interference and noise. In this model, the capacities of links are not independent. We consider two scenarios, single source multiple destinations and multiple sources multiple destinations. In the first scenario, employing coupling and martingale methods, we show that the network coding capacity for random wireless networks still exhibits a concentration behavior around the mean value of the minimum cut under some mild conditions. Furthermore, we establish upper and lower bounds on the network coding capacity for dependent and independent nodes. In the second one, we also show that the network coding capacity still follows a concentration behavior. Our simulation results confirm our theoretical predictions. I.

Abstract. In this work we consider the problem of finding Hamilton Cycles in graphs derived from the uniform random intersection graphs model Gn,m,p. In particular, (a) for the case m = n α, α> 1 we give a result that allows us to apply (with the same probability of success) any algorithm that finds a Hamilton cycle with high probability in a Gn,k graph (i.e. a graph chosen equiprobably form the space of all graphs with k edges), (b) we give an expected polynomial time algorithm for the n case p = constant and m ≤ α for some constant α, and (c) we show log n that the greedy approach still works well even in the case m = o ( n log n) and p just above the connectivity threshold of Gn,m,p (found in [21]) by giving a greedy algorithm that finds a Hamilton cycle in those ranges of m, p with high probability. 1