Improved Shortest Path Algorithms For Nearly Acyclic Directed Graphs

This paper presents new algorithms for computing single source shortest paths (SSSPs) in a nearly acyclic directed graph G. The first part introduces higher-order decomposition. This decomposition is an extension of the technique of strongly connected component (sc-component) decomposition. The second part presents a new method for measuring acyclicity based on modifications to two existing methods. In the new method, we decompose the graph into a 1-dominator set, which is a set of acyclic subgraphs where each subgraph is dominated by one trigger vertex. Meanwhile we compute sc-components of a degenerated graph derived from triggers. Using this preprocessing, a new SSSP algorithm has O(m + r logl) time complexity, where r is the size of the 1-dominator set, and l is the size of the largest sccomponent. In the third part, we modify the concept of a 1-dominator set to that of a 1-2-dominator set. Each of acyclic subgraphs obtained by the 12-dominator decomposition are dominated by one or two trigger vertices cooperatively. Such subgraphs are potentially larger than those decomposed by the 1dominator set. Thus fewer trigger vertices are needed to cover the graph.