Maintaining Longest Paths in Cyclic Graphs

Abstract

This paper reconsiders the problem of maintaining longest paths in directed graphs, which is at the core of many scheduling applications. It presents bounded incremental algorithms for arc insertion and deletion running in time O(||δ|| + |δ|log|δ|) on Cyclic<0 graphs (i.e., graphs whose cycles have strictly negative lengths), where |δ| and ||δ|| are measures of the change in the input and output. For Cyclic≤0 graphs, maintaining longest paths is unbounded under reasonable computational models; when only arc insertions are allowed, it is shown that the problem can be solved in O(||δ|| + |δ|log|δ|) time even in the presence of zero-length cycles. The algorithms directly apply to shortest paths (by negating the lengths), leading to simpler algorithms than previously known and reducing the worst-case complexity of an operation from Õ(n m) to O(n + m) for Cyclic>0 graphs with n vertices and m arcs.