November 17, Wednesday 14:15, Room 303, Jacobs Building
Title
Lecturer : Oren Weimann
Lecturer homepage : http://www.wisdom.weizmann.ac.il/~oweimann/
Affiliation : Faculty of Mathematics and Computer Science, Weizmann Institute
Let G = (V,E) be a directed edge-weighted graph and let P be a shortest path from s to t in G. The "replacement paths" problem
asks to compute, for every edge e on P, the shortest s-to-t path that avoids e.
Apart from approximation algorithms and algorithms for special graph classes, the naive solution to this problem -- removing each edge e on P one at a time and computing the shortest s-to-t path each time -- is surprisingly the only known solution for directed weighted graphs, even when the weights are integrals. In particular, although the related shortest paths problem has benefited from fast matrix multiplication, the replacement paths problem has not, and still required cubic time.
For an n-vertex graph with integral edge-lengths in {-M,...,M}, we give an O(Mn^{2.584}) time algorithm that uses fast matrix multiplication and is sub-cubic for appropriate values of M. We also show how to construct a distance sensitivity oracle in the same time bounds. A query (u,v,e) to this oracle requires sub-quadratic time and returns the length of the shortest u-to-v path that avoids the edge e. Our results also apply for avoiding multiple edges and for avoiding vertices rather than edges.