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We consider the all pairs all shortest paths (APASP) problem, which maintainsthe shortest path dag rooted at every vertex in a directed graph G=(V,E) withpositive edge weights. For this problem we present a decremental algorithm(that supports the deletion of a vertex, or weight increases on edges incidentto a vertex). Our algorithm runs in amortized O(\vstar^2 \cdot \log n) time perupdate, where n=|V|, and \vstar bounds the number of edges that lie on shortestpaths through any given vertex. Our APASP algorithm can be used for thedecremental computation of betweenness centrality (BC), a graph parameter thatis widely used in the analysis of large complex networks. No nontrivialdecremental algorithm for either problem was known prior to our work. Ourmethod is a generalization of the decremental algorithm of Demetrescu andItaliano [DI04] for unique shortest paths, and for graphs with \vstar =O(n), wematch the bound in [DI04]. Thus for graphs with a constant number of shortestpaths between any pair of vertices, our algorithm maintains APASP and BC scoresin amortized time O(n^2 \log n) under decremental updates, regardless of thenumber of edges in the graph.

Decremental All-Pairs ALL Shortest Paths and Betweenness Centrality