Fully dynamic shortest paths and negative cycles detection on digraphs with arbitrary arc weights

Abstract

We study the problem of maintaining the distances and the shortest paths from a source node in a directed graph with arbitrary arc weights, when weight updates of arcs are performed. We propose algorithms that work for any graph and require linear space and optimal query time. If a negative-length cycle is added during weight-decreaseop-erations it is detected by the algorithms. The algorithms explicitly deal with zero-length cycles. We show that, if the graph has a k-bounded accounting function (as in the case of graphs with genus, arboricity, degree, tree width or pagenumber bounded by k, and k-inductive graphs) the algorithms require O(k · n · logn) worst case time. In the case of graphs with n nodes and m arcs k = O(√m); this gives O(√m · n · logn) worst case time per operation, which is better for a factor of O(√mlogn) than recomputing everything from scratch after each update. If we perform also insertions and deletions of arcs, then the above bounds become amortized. © Springer-Verlag Berlin Heidelberg 1998.