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Faster All-Pairs Shortest Paths Via Circuit Complexity

Profile image of Ryan WilliamsRyan Williams

Abstract

We present a new randomized method for computing the min-plus product (a.k.a., tropical product) of two n × n matrices, yielding a faster algorithm for solving the all-pairs shortest path problem (APSP) in dense n-node directed graphs with arbitrary edge weights. On the real RAM, where additions and comparisons of reals are unit cost (but all other operations have typical logarithmic cost), the algorithm runs in time n 3

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Given a simple connected unweighted undirected graph G = (V (G), E(G)) with |V (G)| = n and |E(G)| = m, we present a new algorithm for the all-pairs shortest-path (APSP) problem. The running time of our algorithm is in O(n2 log n). This bound is an improvement over previous best known O(n2.376) time bound of Raimund Seidel (1995) for general graphs. The algorithm presented does not rely on fast matrix multiplication. Our algorithm with slight modifications, enables us to compute the APSP problem for unweighted directed graph in time O(n2 log n), improving a previous best known O(n2.575) time bound of Uri Zwick (2002).

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Speeding up shortest path algorithms
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Given an arbitrary, non-negatively weighted, directed graph G=(V,E) we present an algorithm that computes all pairs shortest paths in time O(m^* n + m n + nT_ψ(m^*, n)), where m^* is the number of different edges contained in shortest paths and T_ψ(m^*, n) is a running time of an algorithm to solve a single-source shortest path problem (SSSP). This is a substantial improvement over a trivial n times application of ψ that runs in O(nT_ψ(m,n)). In our algorithm we use ψ as a black box and hence any improvement on ψ results also in improvement of our algorithm. Furthermore, a combination of our method, Johnson&#39;s reweighting technique and topological sorting results in an O(m^*n + m n) all-pairs shortest path algorithm for arbitrarily-weighted directed acyclic graphs. In addition, we also point out a connection between the complexity of a certain sorting problem defined on shortest paths and SSSP.

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FB-APSP: A new efficient algorithm for computing all-pairs shortest-paths
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We describe a new forward-backward method for an all-pairs shortest-paths (APSP) algorithm. While most APSP algorithms only scan edges forward, the algorithm proposed here also scans all edges backward because it assumes that edges in the outgoing and incoming adjacency lists of the vertices appear with the same importance. The running time of the algorithm on a directed graph with n vertices, and m edges and positive real-valued edge weights in a deterministic way is O(nδ 2 2 log δ (n−1)−1), and the space complexity is O(nδ 2 log δ (n−1)−1), where δ = m/n is the density of the graph. Simulations on graphs with up to 10000 vertices with real, positive weights show that our FB-APSP algorithm, using additional working space less than 75% of space used by Speed-Up Floyd-Warshall algorithm, is faster on large sparse graphs, particularly planar graphs.

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Unified all-pairs shortest path algorithms in the chordal hierarchy
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The objective of this paper is to advance the view that solving the all-pairs shortest path (APSP) problem for a chordal graph G is a two-step process: the first step is determining vertex pairs at distance two (i.c., computing C') and the second step is finding the vcrtcx pairs at distance three or more. The main technical result here is that the APSP problem for a chordal graph can be solved in O(n') time (optimally), if G' is already known. It can be shown that computing G* for chordal graphs is as hard as for general graphs. We then show certain subclasses of chordal graphs for which G' can be computed more efficiently. This leads to optimal APSP algorithms for these classes of graphs in a more natural way than previously known results. Finally, we present an optimal parallel algorithm for the APSP problem on chordal graphs by exploiting new structural properties of shortest paths. Our parallel algorithm uses 0(:24(rz)) operations where M(n) is the time needed for the fastest known algorithm for multiplying two n x II matrices over a ring.

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On the Exponent of the All Pairs Shortest Path Problem
Oded Margalit

Journal of Computer and System Sciences, 1997

The upper bound on the exponent, ω, of matrix multiplication over a ring that was three in 1968 has decreased several times and since 1986 it has been 2.376. On the other hand, the exponent of the algorithms known for the all pairs shortest path problem has stayed at three all these years even for the very special case of directed graphs with uniform edge lengths. In this paper we give an algorithm of time O n ν log 3 n , ν = (3 + ω)/2, for the case of edge lengths in {−1, 0, 1}. Thus, for the current known bound on ω, we get a bound on the exponent, ν < 2.688. In case of integer edge lengths with absolute value bounded above by M , the time bound is O (M n) ν log 3 n and the exponent is less than 3 for M = O(n α ), for α < 0.116 and the current bound on ω.

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Related topics

  • Algorithms
  • Circuit Complexity
  • The shortest path problems
  • Matrix Multiplication
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