New Bounds for Old Algorithms: On the Average-Case Behavior of Classic Single-Source Shortest-Paths Approaches

Abstract

Despite disillusioning worst-case behavior, classic algorithms for single-source shortest-paths (SSSP) like Bellman-Ford are still being used in practice, especially due to their simple data structures. However, surprisingly little is known about the average-case complexity of these approaches. We provide new theoretical and experimental results for the performance of classic label-correcting SSSP algorithms on graph classes with non-negative random edge weights. In particular, we prove a tight lower bound of Ω(n ²) for the running times of Bellman-Ford on a class of sparse graphs with O(n) nodes and edges; the best previous bound was Ω(n ^{4/3 − ε}). The same improvements are shown for Pallottino’s algorithm. We also lift a lower bound for the approximate bucket implementation of Dijkstra’s algorithm from Ω(n logn / loglogn) to Ω(n ^{1.2 − ε}). Furthermore, we provide an experimental evaluation of our new graph classes in comparison with previously used test inputs.

Partially supported by the DFG grant ME 3250/1-2, and by MADALGO – Center for Massive Data Algorithmics, a Center of the Danish National Research Foundation.