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First passage percolation on random graphs with finite mean degrees
, 2009
"... We study first passage percolation on the configuration model. Assuming that each edge has an independent exponentially distributed edge weight, we derive explicit distributional asymptotics for the minimum weight between two randomly chosen connected vertices in the network, as well as for the numb ..."
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We study first passage percolation on the configuration model. Assuming that each edge has an independent exponentially distributed edge weight, we derive explicit distributional asymptotics for the minimum weight between two randomly chosen connected vertices in the network, as well as for the number of edges on the least weight path, the socalled hopcount. We analyze the configuration model with degree powerlaw exponent τ> 2, in which the degrees are assumed to be i.i.d. with a tail distribution which is either of powerlaw form with exponent τ − 1> 1, or has even thinner tails (τ = ∞). In this model, the degrees have a finite first moment, while the variance is finite for τ> 3, but infinite for τ ∈ (2, 3). We prove a central limit theorem for the hopcount, with asymptotically equal means and variances equal to α log n, where α ∈ (0, 1) for τ ∈ (2, 3), while α> 1 for τ> 3. Here n denotes the size of the graph. For τ ∈ (2, 3), it is known that the graph distance between two randomly chosen connected vertices is proportional to log log n [25], i.e., distances are ultra small. Thus, the addition of edge weights causes a marked change in the geometry of the network. We further study the weight of the least weight path, and prove convergence in distribution of an appropriately centered version. This study continues the program initiated in [5] of showing that log n is the correct scaling for the hopcount under i.i.d. edge disorder, even if the graph distance between two randomly chosen vertices is of much smaller order. The case of infinite mean degrees (τ ∈ [1, 2)) is studied in [6], where it is proved that the hopcount remains uniformly bounded and converges in distribution.
Shortestweight paths in random regular graphs, arXiv preprint arXiv:1210.2657
, 2012
"... Abstract Consider a random regular graph with degree d and of size n. Assign to each edge an i.i.d. exponential random variable with mean one. In this paper we establish a precise asymptotic expression for the maximum number of edges on the shortestweight paths between a fixed vertex and all the o ..."
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Abstract Consider a random regular graph with degree d and of size n. Assign to each edge an i.i.d. exponential random variable with mean one. In this paper we establish a precise asymptotic expression for the maximum number of edges on the shortestweight paths between a fixed vertex and all the other vertices, as well as between any pair of vertices. Namely, for any fixed d ≥ 3, we show that the longest of these shortestweight paths has about α log n edges where α is the unique solution of the equation α log
Random Shortest Paths: NonEuclidean Instances for Metric Optimization Problems
"... Abstract. Probabilistic analysis for metric optimization problems has mostly been conducted on random Euclidean instances, but little is known about metric instances drawn from distributions other than the Euclidean. This motivates our study of random metric instances for optimization problems obtai ..."
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Abstract. Probabilistic analysis for metric optimization problems has mostly been conducted on random Euclidean instances, but little is known about metric instances drawn from distributions other than the Euclidean. This motivates our study of random metric instances for optimization problems obtained as follows: Every edge of a complete graph gets a weight drawn independently at random. The length of an edge is then the length of a shortest path (with respect to the weights drawn) that connects its two endpoints. We prove structural properties of the random shortest path metrics generated in this way. Our main structural contribution is the construction of a good clustering. Then we apply these findings to analyze the approximation ratios of heuristics for matching, the traveling salesman problem (TSP), and the kcenter problem, as well as the runningtime of the 2opt heuristic for the TSP. The bounds that we obtain are considerably better than the respective worstcase bounds. This suggests that random shortest path metrics are easy instances, similar to random Euclidean instances, albeit for completely different structural reasons. 1
AllPairs Shortest Paths in O(n 2 ) time with high probability
"... Abstract We present an allpairs shortest path algorithm whose running time on a complete directed graph on n vertices whose edge weights are chosen independently and uniformly at random from [0, 1] is O(n 2 ), in expectation and with high probability. This resolves a long standing open problem. Th ..."
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Abstract We present an allpairs shortest path algorithm whose running time on a complete directed graph on n vertices whose edge weights are chosen independently and uniformly at random from [0, 1] is O(n 2 ), in expectation and with high probability. This resolves a long standing open problem. The algorithm is a variant of the dynamic allpairs shortest paths algorithm of Demetrescu and Italiano. The analysis relies on a proof that the number of locally shortest paths in such randomly weighted graphs is O(n 2 ), in expectation and with high probability. We also present a dynamic version of the algorithm that recomputes all shortest paths after a random edge update in O(log 2 n) expected time.
9. Bibliography............................................................................101. Team
"... c t i v it y e p o r t 2008 Table of contents 1. Team.................................................................................... 1 ..."
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c t i v it y e p o r t 2008 Table of contents 1. Team.................................................................................... 1
9. Bibliography............................................................................111. Team Research Scientist
"... c t i v it y e p o r t 2009 Table of contents 1. Team.................................................................................... 1 ..."
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c t i v it y e p o r t 2009 Table of contents 1. Team.................................................................................... 1