More Algorithms for All-Pairs Shortest Paths in Weighted Graphs

Chan, Timothy M.

doi:10.1137/08071990x

Cited by 150 publications

(171 citation statements)

References 37 publications

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“…Slightly subcubic algorithms for this problem have also been obtained. The fastest currently known algorithm is by Chan [15], running in time O( n 3 (log log n) 3 log 2 n ). For Monge matrices, distance multiplication can easily be performed in time O(n 2 ), using the standard row-minima searching technique of Aggarwal et al [1] to perform matrix-vector multiplication in linear time (see also [6,57]).…”

Section: Fast Implicit Distance Multiplicationmentioning

confidence: 99%

Fast Distance Multiplication of Unit-Monge Matrices

Tiskin

2013

Algorithmica

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Monge matrices play a fundamental role in optimisation theory, graph and string algorithms. Distance multiplication of two Monge matrices of size n can be performed in time O(n 2 ). Motivated by applications to string algorithms, we introduced in previous works a subclass of Monge matrices, that we call simple unit-Monge matrices. We also gave a distance multiplication algorithm for such matrices, running in time O(n 1.5 ). Landau asked whether this problem can be solved in linear time. In the current work, we give an algorithm running in time O(n log n), thus approaching an answer to Landau's question within a logarithmic factor. The new algorithm implies immediate improvements in running time for a number of algorithms on strings and graphs. In particular, we obtain an algorithm for finding a maximum clique in a circle graph in time O(n log 2 n), and a surprisingly efficient algorithm for comparing compressed strings. We also point to potential applications in group theory, by making a connection between unit-Monge matrices and Coxeter monoids. We conclude that unit-Monge matrices are a fascinating object and a powerful tool, that deserve further study from both the mathematical and the algorithmic viewpoints.

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Section: Fast Implicit Distance Multiplicationmentioning

confidence: 99%

Fast Distance Multiplication of Unit-Monge Matrices

Tiskin

2013

Algorithmica

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“…Let M σ be a k × k matrix with elements M σ i,j = e i (σ) · P i,j . We can now express v n as: Faster methods for max-times matrix multiplication [11,12] and standard matrix multiplication [14,32] can be used to reduce the k 3 term. However, for small values of k this is not profitable.…”

Section: Preliminariesmentioning

confidence: 99%

Speeding Up HMM Decoding and Training by Exploiting Sequence Repetitions

Mozes

Weimann

Ziv-Ukelson

Combinatorial Pattern Matching

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Abstract. We present a method to speed up the dynamic program algorithms used for solving the HMM decoding and training problems for discrete time-independent HMMs. We discuss the application of our method to Viterbi's decoding and training algorithms [33], as well as to the forward-backward and BaumWelch [6] algorithms. Our approach is based on identifying repeated substrings in the observed input sequence. Initially, we show how to exploit repetitions of all sufficiently small substrings (this is similar to the Four Russians method). Then, we describe four algorithms based alternatively on run length encoding (RLE), Lempel-Ziv (LZ78) parsing, grammar-based compression (SLP), and byte pair encoding (BPE). Compared to Viterbi's algorithm, we achieve speedups of Θ(log n) using the Four Russians method, Ω( r log r ) using RLE, Ω( log n k ) using LZ78, Ω( r k ) using SLP, and Ω(r) using BPE, where k is the number of hidden states, n is the length of the observed sequence and r is its compression ratio (under each compression scheme). Our experimental results demonstrate that our new algorithms are indeed faster in practice. Furthermore, unlike Viterbi's algorithm, our algorithms are highly parallelizable.

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“…For dense graphs with arbitrary edge weights, nothing much better than cubic time is known. 5 Vassilevska Williams and Williams [23] showed that the RP problem in directed graphs is equivalent to APSP, under subcubic reductions, i.e. essentially either both problems admit truly subcubic algorithms, or neither of them does.…”

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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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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More Algorithms for All-Pairs Shortest Paths in Weighted Graphs

Cited by 150 publications

References 37 publications

Fast Distance Multiplication of Unit-Monge Matrices

Fast Distance Multiplication of Unit-Monge Matrices

Speeding Up HMM Decoding and Training by Exploiting Sequence Repetitions

Improved Distance Sensitivity Oracles via Fast Single-Source Replacement Paths

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