MPI-I-2005-1-003

The all-pairs approximate shortest-paths problem is an interesting variant of the classical all-pairs shortest-paths problem in graphs. The problem aims at building a data-structure for a given graph with the following two features. Firstly, for any two vertices, it should report an {\emph{approximate}} shortest path between them, that is, a path which is longer than the shortest path by some {\emph{small}} factor. Secondly, the data-structure should require less preprocessing time (strictly sub-cubic) and occupy optimal space (sub-quadratic), at the cost of this approximation. In this paper, we present algorithms for computing all-pairs approximate shortest paths in a weighted undirected graph. These algorithms significantly improve the existing results for this problem.