An expected-case sub-cubic solution to the all-pairs shortest path problem in R

It has been shown by Alon et al. that the so-called ‘all-pairs shortest-path’ problem can be solved in O((MV ) log(V )) for graphs with V vertices, with integer distances bounded by M . We solve the more general problem for graphs in R (assuming no negative cycles), with expected-case running time O(V 2.5 log(V )). While our result appears to violate the Ω(V ) requirement of “Funny Matrix Multiplication” (due to Kerr), we find that it has a sub-cubic expected time solution subject to reasonable conditions on the data distribution. The expected time solution arises when certain sub-problems are uncorrelated, though we can do better/worse than the expected-case under positive/negative correlation (respectively). Whether we observe positive/negative correlation depends on the statistics of the graph in question. In practice, our algorithm is significantly faster than Floyd-Warshall, even for dense graphs. 1 Problem Definition The all-pairs shortest path problem [Dijkstra, 1959] consists of solving d(v, v) = min p∈Pv,v′ f(p) (1) for all vertices v, v ∈ V , where Pv,v′ is the space of all paths connecting v to v in V , and f(p) is the path length, i.e., f(p) = ∑|p|−1 i=1 e(pi, pi+1) where e(pi, pj) is the weight of the edge connecting pi to pj, or ∞ if no such edge exists. A simple divide-and-conquer solution to (eq. 1) can be obtained by defining d(u, v, k) to be the shortest path between u and v containing at most k edges. This solution exploits the fact that d(u, v, k) = { e(u, v) if k = 1 minx (d(u, x, k/2) + d(x, v, k/2)) otherwise (2) This allows us to solve the all-pairs shortest path problem via Algorithm 1, which we requires Θ(V 3 log(V )) time (this is by no means the optimal solution, though it is this version to which our improvements apply). The authors are with the Statistical Machine Learning Program at NICTA, and the Research School of Information Sciences and Engineering, Australian National University. Queries should be addressed to [email protected].