On the Difficulty of Some Shortest Path Problems

Abstract

We prove super-linear lower bounds for some shortest path problems in directed graphs, where no such bounds were previously known. The central problem in our study is the replacement paths problem: Given a directed graph G with non-negative edge weights, and a shortest path P = e 1, e 2, . . . , e p between two nodes s and t, compute the shortest path distances from s to t in each of the p graphs obtained from G by deleting one of the edges e i . We show that the replacement paths problem requires ω(m√n) time in the worst case whenever m = O(n√n). Our construction also implies a similar lower bound for the k shortest paths problem for a broad class of algorithms that includes all known algorithms for the problem. To put our lower bound in perspective, we note that both these problems (replacement paths and k shortest paths) can be solved in near linear time for undirected graphs.