Let G = (V; E) be an unweighted undirected graph on n vertices. A simple argument shows that computing all distances in G with an additive one-sided error of at most 1 is as hard as Boolean matrix multiplication. Building on recent work of Aingworth, Chekuri and Motwani, we describe g) time algorithm APASP 2 for computing all distances in G with an additive one-sided error of at most 2. The algorithm APASP 2 is simple, easy to implement, and faster than the fastest known matrix multiplication algorithm. Furthermore, for every even k ? 2, we describe an g) time algorithm APASP k for computing all distances in G with an additive one-sided error of at most k.

We give algorithms for finding the k shortest paths (not required to be simple) connecting a pair of vertices in a digraph. Our algorithms output an implicit representation of these paths in a digraph with n vertices and m edges, in time O(m + n log n + k). We can also find the k shortest

In this paper we show that a very simple and efficient approach can be used to solve the all pairs almost shortest path problem on the class of weakly chordal graphs and its different subclasses. Moreover, this approach works well also on graphs with small size of largest induced cycle and gives a

in the problem graph: ( 1) O(n log n + m) for the single-source shortest path problem with nonnegative edge lengths, improved from O(m logfmh+2)n); (2) O(n*log n + nm) for the all-pairs shortest path problem, improved from O(nm lo&,,,+2,n); (3) O(n*logn + nm) for the assignment problem (weighted bipartite

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The essential subgraph H of a weighted graph or digraph G contains an edge (v, w) if that edge is uniquely the least-cost path between its vertices. Let s denote the number of edges of H. This paper presents an algorithm for solving all-pairs shortest paths on G that requires O(ns + n 2 log n

of interest is how almost blind animals manage to establish shortest route paths from their colony to feeding sources and back. In the case of ants, the media used to communicate among individuals information regarding paths and used to decide where to go consists of pheromone trails. A moving ant lays some

are specified? One need consider only paths with at most 2 cusps or reversals. We give a set of paths which is sufficient in the sense that it always contains a shortest path and small in the sense that there are at most 68, but usually many fewer paths in the set for any pair of endpoints and directions. We

We present two new algorithms for solving the All Pairs Shortest Paths (APSP) problem for weighted directed graphs. Both algorithms use fast matrix multiplication algorithms. The first algorithm solves the APSP problem for weighted directed graphs in which the edge weights are integers of small

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