Average update times for fully-dynamic all-pairs shortest paths

Abstract

We study the fully-dynamic all pairs shortest path problem for graphs with arbitrary non-negative edge weights. It is known for digraphs that an update of the distance matrix costs O ( n 2.75 polylog ( n ) ) worst-case time (Thorup, 2005 [20]) and O ( n 2 log 3 ( n ) ) amortized time (Demetrescu and Italiano, 2004 [4]) where n is the number of vertices. We present the first average-case analysis of the undirected problem. For a random update we show that the expected time per update is bounded by O ( n 4 / 3 + ε ) for all ε > 0 . If the graph is outside the critical window, we prove even smaller bounds.