Improved algorithms for the k simple shortest paths and the replacement paths problems

Gotthilf, Zvi; Lewenstein, Moshe

doi:10.1016/j.ipl.2008.12.015

Cited by 62 publications

(56 citation statements)

References 14 publications

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“…We call this problem single-source replacement paths (SSRP). SSRP is a natural generalization of the well-studied replacement paths problem (RP) [2], [8], [9], [14], [15], [16], [20], [22], where both the source s and the target t are fixed. Somehow surprisingly, SSRP has not received much attention in the literature.…”

Section: Theorem 2 For Any Integer 1 ≤ L ≤ N There Is a Randomized mentioning

confidence: 99%

“…For weighted planar digraphs, the runtime can be reduced toÕ(n) as shown by Emek, Peleg and Roditty [8]. For arbitrary directed graphs with arbitrary edge weights, the fastest known algorithm for RP is by Gotthilf and Lewenstein [9] and runs in O(mn + n 2 log log n) time. For dense graphs with arbitrary edge weights, nothing much better than cubic time is known.…”

Section: B Related Workmentioning

confidence: 99%

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Improved Distance Sensitivity Oracles via Fast Single-Source Replacement Paths

Grandoni

Williams

2012

2012 IEEE 53rd Annual Symposium on Foundations of Computer Science

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Abstract-A distance sensitivity oracle is a data structure which, given two nodes s and t in a directed edge-weighted graph G and an edge e, returns the shortest length of an s-t path not containing e, a so called replacement path for the triple (s, t, e). Such oracles are used to quickly recover from edge failures.In this paper we consider the case of integer weights in the interval [−M, M ], and present the first distance sensitivity oracle that achieves simultaneously subcubic preprocessing time and sublinear query time. More precisely, for a given parameter α ∈ [0, 1], our oracle has preprocessing timeÕ(M n

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Section: Theorem 2 For Any Integer 1 ≤ L ≤ N There Is a Randomized mentioning

confidence: 99%

Section: B Related Workmentioning

confidence: 99%

Improved Distance Sensitivity Oracles via Fast Single-Source Replacement Paths

Grandoni

Williams

2012

2012 IEEE 53rd Annual Symposium on Foundations of Computer Science

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“…For this problem -k shortest paths without loops -an influential algorithm was proposed by Yen [12], which was the basis for many of the currently known algorithms (e.g. Hershberger et al [6], Gotthilf and Lewenstein [4]). However, as mentioned in the introduction, routing applications are very time-critical and the existing algorithms tend to be too slow for this purpose.…”

Section: Alternative Routes 31 K Shortest Pathsmentioning

confidence: 99%

Applications of graph algorithms in GIS

Vanhove

Fack

2010

SIGSPATIAL Special

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This paper describes ongoing PhD research on applications of graph algorithms in Geographical Information Systems. Many GIS problems can be translated into a graph problem, especially in the domain of routing in road networks. Our research aims to evaluate and develop efficient methods for different variants of the routing problem. Standard existing shortest path algorithms are not always suited for use in road networks, e.g. in a realistic situation forbidden turns and turn penalties need to be taken into account. An experimental evaluation of different methods for this purpose is presented. Another interesting problem is the generation of alternative routes. This can be modelled as a k shortest paths problem, where a ranking of k paths is desired rather than only the shortest path itself. A new heuristic approach for generating alternative routes is presented and evaluated.

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“…Gotthilf and Lewenstein [8] recently improved this to O(mn+n 2 log log n), but no o(mn) algorithms are known.…”

Section: Introductionmentioning

confidence: 99%

“…Gotthilf and Lewenstein [8] recently improved this to O(mn+n 2 log log n), but no o(mn) algorithms are known.We present the first approximation algorithm for replacement paths in directed graphs with positive edge weights. Given any ∈ [0, 1), our algorithm returns (1 + )-approximate replacement paths in O( −1 log 2 n log(nC/c)(m+n log n)) = O(m log(nC/c)/ ) time, where C is the largest edge weight in the graph and c is the smallest weight.…”

mentioning

confidence: 99%

A Nearly Optimal Algorithm for Approximating Replacement Paths and k Shortest Simple Paths in General Graphs

Bernstein

2010

Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms

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Let G = (V, E) be a directed graph with positive edge weights, let s, t be two specified vertices in this graph, and let π(s, t) be the shortest path between them.In the replacement paths problem we want to compute, for every edge e on π(s, t), the shortest path from s to t that avoids e. The naive solution to this problem would be to remove each edge e, one at a time, and compute the shortest s − t path each time; this yields a running time of O(mn + n 2 log n). Gotthilf and Lewenstein [8] recently improved this to O(mn+n 2 log log n), but no o(mn) algorithms are known.We present the first approximation algorithm for replacement paths in directed graphs with positive edge weights. Given any ∈ [0, 1), our algorithm returns (1 + )-approximate replacement paths in O( −1 log 2 n log(nC/c)(m+n log n)) = O(m log(nC/c)/ ) time, where C is the largest edge weight in the graph and c is the smallest weight.We also present an even faster (1 + ) approximate algorithm for the simpler problem of approximating the k shortest simple s − t paths in a directed graph with positive edge weights. That is, our algorithm outputs k different simple s − t paths, where the kth path we output is a (1 + ) approximation to the actual kth shortest simple s − t path. The running time of our algorithm is O(k −1 log 2 n(m + n log n)) = O(km/ ). The fastest exact algorithm for this problem has a running time of O(k(mn + n 2 log log n)) = O(kmn) [8]. The previous best approximation algorithm was developed by Roditty [15]; it has a stretch of 3/2 and a running time of O(km √ n) (it does not work for replacement paths).Note that all of our running times are nearly optimal except for the O(log(nC/c)) factor in the replacements paths algorithm. Also, our algorithm can solve the variant of approximate replacement paths where we avoid vertices instead of edges.

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Improved algorithms for the k simple shortest paths and the replacement paths problems

Cited by 62 publications

References 14 publications

Improved Distance Sensitivity Oracles via Fast Single-Source Replacement Paths

Improved Distance Sensitivity Oracles via Fast Single-Source Replacement Paths

Applications of graph algorithms in GIS

A Nearly Optimal Algorithm for Approximating Replacement Paths and k Shortest Simple Paths in General Graphs

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