Shortest Path Queries in Digraphs of Small Treewidth

We consider the problem of preprocessing an n-vertex digraph with real edge weights so that subsequent queries for the shortest path or distance between any two vertices can be efficiently answered. We give algorithms that depend on the treewidth of the input graph. When the treewidth is a constant, our algorithms can answer distance queries in O(α(n)) time after O(n) preprocessing. This improves upon previously known results for the same problem. We also give a dynamic algorithm which, after a change in an edge weight, updates the data structure in time O(n), for any constant 0 < β < 1. The above two algorithms are based on an algorithm of independent interest: computing a shortest path tree, or finding a negative cycle in linear time.