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Faster Replacement Paths [article]

Virginia Vassilevska Williams
2010 arXiv   pre-print
The replacement paths problem for directed graphs is to find for given nodes s and t and every edge e on the shortest path between them, the shortest path between s and t which avoids e.  ...  Our result shows that, at least for small integer weights, the replacement paths problem in directed graphs may be easier than the related all pairs shortest paths problem in directed graphs, as the current  ...  For unweighted graphs, both the replacement paths problem and the second shortest simple path problem are closely related to Boolean matrix multiplication; for general directed graphs with arbitrary weights  ... 
arXiv:1007.2216v1 fatcat:pvxa43lspfavjmviiii7ry6ga4

Near Optimal Algorithm for the Directed Single Source Replacement Paths Problem

Shiri Chechik, Ofer Magen, Artur Czumaj, Anuj Dawar, Emanuela Merelli
2020 International Colloquium on Automata, Languages and Programming  
In the Single Source Replacement Paths (SSRP) problem we are given a graph G = (V, E), and a shortest paths tree rooted at a node s, and the goal is to output for every node t ∈ V and for every edge  ...  e in the length of the shortest path from s to t avoiding e.  ...  The k shortest simple paths problem can be solved by invoking the replacement paths algorithm k times and adding a very small weight to the path found in each invocation.  ... 
doi:10.4230/lipics.icalp.2020.81 dblp:conf/icalp/ChechikM20 fatcat:ncfyhk47c5crvfhopgluvgrpoi

Faster replacement paths [chapter]

Virginia Vassilevska Williams
2011 Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms  
The replacement paths problem for directed graphs is to find for given nodes s and t and every edge e on the shortest path between them, the shortest path between s and t which avoids e.  ...  Our result shows that, at least for small integer weights, the replacement paths problem in directed graphs may be easier than the related all pairs shortest paths problem in directed graphs, as the current  ...  The author would like to thank Ryan Williams for his moral support and the anonymous reviewers for their valuable input.  ... 
doi:10.1137/1.9781611973082.102 dblp:conf/soda/Williams11 fatcat:aanila5k6fdjzk4n2lu6ztguye

Minimum Cuts and Shortest Cycles in Directed Planar Graphs via Noncrossing Shortest Paths

Hung-Chun Liang, Hsueh-I Lu
2017 SIAM Journal on Discrete Mathematics  
Let $G$ be an $n$-node simple directed planar graph with nonnegative edge weights.  ...  The kernel of our result is an $O(n\log\log n)$-time algorithm for computing noncrossing shortest paths among nodes well ordered on a common face of a directed plane graph, which is extended from the algorithm  ...  a shortest cycle in unweighted directed planar G in o(n 3/2 ) time.  ... 
doi:10.1137/16m1057152 fatcat:s6osvpbcwjeydpmcmfmuncq7k4

Fine-Grained Complexity and Conditional Hardness for Sparse Graphs [article]

Udit Agarwal, Vijaya Ramachandran
2017 arXiv   pre-print
Our sparse reductions for directed path problems in the $\tilde{O}(mn)$ class establish that several problems in this class, including 2-SiSP (second simple shortest path), Radius, and Eccentricities,  ...  in the graph, the length of a longest shortest path starting at that vertex.  ...  Hence, the k-th shortest path in the collection of k-SiSPs from x 0,i to x 1,i in log n G i , 1 ≤ i ≤ ⌈log n⌉ (after removing duplicates), corresponds to the k-th SiSC passing through x.  ... 
arXiv:1611.07008v3 fatcat:vuwsyquqn5bwlp2kyx7iks5wxi

On Dynamic Shortest Paths Problems [chapter]

Liam Roditty, Uri Zwick
2004 Lecture Notes in Computer Science  
directed or undirected graphs, are, in a strong sense, at least as hard as the static all-pairs shortest-paths problem.  ...  (ii) A randomized fully-dynamic algorithm for the all-pairs shortestpaths problem in directed unweighted graphs with an amortized update time ofÕ(m √ n) and a worst case query time is O(n 3/4 ).  ...  There is a decremental algorithm for maintaining the first k levels of a single-source shortest-paths tree, in a directed or undirected unweighted graph, whose total running time, over all deletions, is  ... 
doi:10.1007/978-3-540-30140-0_52 fatcat:ekwwih5dcfe6hjfntk7r24mnou

On Dynamic Shortest Paths Problems

Liam Roditty, Uri Zwick
2010 Algorithmica  
directed or undirected graphs, are, in a strong sense, at least as hard as the static all-pairs shortest-paths problem.  ...  (ii) A randomized fully-dynamic algorithm for the all-pairs shortestpaths problem in directed unweighted graphs with an amortized update time ofÕ(m √ n) and a worst case query time is O(n 3/4 ).  ...  There is a decremental algorithm for maintaining the first k levels of a single-source shortest-paths tree, in a directed or undirected unweighted graph, whose total running time, over all deletions, is  ... 
doi:10.1007/s00453-010-9401-5 fatcat:2pf45vg5zrg4pmtlxp5ewidsua

Finding k Simple Shortest Paths and Cycles [article]

Udit Agarwal, Vijaya Ramachandran
2016 arXiv   pre-print
The problem of finding multiple simple shortest paths in a weighted directed graph $G=(V,E)$ has many applications, and is considerably more difficult than the corresponding problem when cycles are allowed  ...  We also give hardness results for sparse graphs, relative to the complexity of computing a minimum weight cycle in a graph, for several variants of problems related to finding $k$ simple paths and cycles  ...  For unweighted directed graphs, Roditty and Zwick [29] gave anÕ(km √ n) randomized algorithm for k-SiSP.  ... 
arXiv:1512.02157v2 fatcat:nxxmrpjor5hilbx5p66ggmuzu4

Tight Hardness for Shortest Cycles and Paths in Sparse Graphs [chapter]

Andrea Lincoln, Virginia Vassilevska Williams, Ryan Williams
2018 Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms  
Replacement Paths, Second Shortest Paths, and so on.  ...  graph, • Second Shortest Path in a directed weighted graph, • Betweenness Centrality of a given node in a directed weighted graph.  ...  Landsberg and Mateusz Michalek for valuable discussions about tensor rank lower bounds. We would like to thank the anonymous reviewers whose suggestions we implemented.  ... 
doi:10.1137/1.9781611975031.80 dblp:conf/soda/LincolnWW18 fatcat:7boftjghqrbsxe4baz736vkjty

Restoration by path concatenation: fast recovery of MPLS paths

Yehuda Afek, Anat Bremler-Barr, Haim Kaplan, Edith Cohen, Michael Merritt
2002 Distributed computing  
The theory pertains to restoration of shortest paths in a network following failure, e.g., we prove that a shortest path in a network after removing k edges is the concatenation of at most k + 1 shortest  ...  paths in the original network.  ...  We thank Noga Alon for simplifying the proof of the existence of a set B of bypass paths in the proof of Theorem 1.  ... 
doi:10.1007/s00446-002-0080-6 fatcat:tts3nnjkxnh6jer6r3bumd5wb4

Near Optimal Algorithm for the Directed Single Source Replacement Paths Problem [article]

Shiri Chechik, Ofer Magen
2020 arXiv   pre-print
In the Single Source Replacement Paths (SSRP) problem we are given a graph $G = (V, E)$, and a shortest paths tree $\widehat{K}$ rooted at a node $s$, and the goal is to output for every node $t \in V$  ...  and for every edge $e$ in $\widehat{K}$ the length of the shortest path from $s$ to $t$ avoiding $e$.  ...  Also, note that any path from goes from x 1 to a i and then preforms a 3 vertex path: If α i,j < 8L − 7i + 5 then k must be a index which minimizes the length of this 3 vertex path.  ... 
arXiv:2004.13673v1 fatcat:4iv3ojzcl5gx5b44w3joyd7p3a

Tight Hardness for Shortest Cycles and Paths in Sparse Graphs [article]

Andrea Lincoln, Virginia Vassilevska Williams, Ryan Williams
2020 arXiv   pre-print
), Radius, Replacement Paths, Second Shortest Paths, and so on.  ...  , Replacement Paths in a directed weighted graph, Second Shortest Path in a directed weighted graph, Betweenness Centrality of a given node in a directed weighted graph.  ...  We thank Pawel Gawrychowski for pointing out a typo in a previous version of the paper.  ... 
arXiv:1712.08147v4 fatcat:4kmi3ia3d5cmriclwvyj6oynwa

Deterministic Combinatorial Replacement Paths and Distance Sensitivity Oracles [article]

Noga Alon, Shiri Chechik, Sarel Cohen
2019 arXiv   pre-print
For the replacement paths problem, let G = (V,E) be a directed unweighted graph with n vertices and m edges and let P be a shortest path from s to t in G.  ...  The {\sl replacement paths} problem is to find for every edge e \in P the shortest path from s to t avoiding e.  ...  For directed unweighted graphs, the randomized replacement paths algorithm of Roditty and Zwick [37] implies that the k simple shortest paths has a randomized O(km √ n) time algorithm.  ... 
arXiv:1905.07483v1 fatcat:bk4olmgm6fdpdn3tk2jgmaelpm

Subcubic Equivalences between Path, Matrix and Triangle Problems

Virginia Vassilevska Williams, Ryan Williams
2010 2010 IEEE 51st Annual Symposium on Foundations of Computer Science  
Finding the second shortest simple path between two nodes in a weighted digraph. Note the only previously known equivalence in the above was that of (1) and (2) .  ...  Finding a minimum weight cycle in a graph of non-negative edge weights. 8. The replacement paths problem on weighted digraphs. 9.  ...  In ESA, pages 580-591, 2004. [RZ05] L. Roditty and U. Zwick. Replacement paths and k simple shortest paths in unweighted directed graphs. In Proc. ICALP, volume 32, pages 249-260, 2005. [Spi03] J. P.  ... 
doi:10.1109/focs.2010.67 dblp:conf/focs/WilliamsW10 fatcat:3h7a5os4vjbk3hqktxahtbe4bm

Efficiently listing bounded length st-paths [article]

Romeo Rizzi, Gustavo Sacomoto, Marie-France Sagot
2014 arXiv   pre-print
The problem of listing the $K$ shortest simple (loopless) $st$-paths in a graph has been studied since the early 1960s.  ...  For a non-negatively weighted graph with $n$ vertices and $m$ edges, the most efficient solution is an $O(K(mn + n^2 \log n))$ algorithm for directed graphs by Yen and Lawler [Management Science, 1971  ...  Algorithm 4 using a binary heap outputs all α-bounded st-paths in increasing order of their lengths in O((nt(n, m) + log γ)γ) total time, using O(mγ) space.  ... 
arXiv:1411.6852v1 fatcat:k3jivx57szhmfmx6hpnwpb5nea
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