Directed Single–Source Shortest–Paths in Linear Average–Case Time

The quest for a linear-time single-source shortest-path (SSSP) algorithm on directed graphs with positive edge weights is an ongoing hot research topic. While Thorup recently found an O(n +m) time RAM algorithm for undirected graphs with n nodes, m edges and integer edge weights in f0; : : : ; 2w 1g where w denotes the word length, the currently best time bound for directed sparse graphs on a RAM is O(n+m log logn). In the present paper we study the average-case complexity of SSSP. We give simple label-setting and label-correcting algorithms for arbitrary directed graphs with random real edge weights uniformly distributed in [0; 1℄ and show that they need linear time O(n+m) with high probability. A variant of the label-correcting approach also supports parallelization. Furthermore, we propose a general method to construct graphs with random edge weights which incur large non-linear expected running times on many traditional shortest-path algorithms.