A simple and practical $o(\sqrt{n})$-time algorithm for shortest paths in power law graphs

Abstract

Computing shortest paths in large graphs is, and remains, a fundamental and practically motivated problem. While many algorithms were proposed to calculate shortest path between pairs of vertices efficiently, many of them (index-based methods) require substantial preprocessing, while others (traversal-based methods) have higher time complexity. In this paper, we propose and analyze Pruned Bidirectional Search (PBS), a simple sublinear approximation algorithm for power-law graphs with parameter $\beta\in[2,3)$: our algorithm does not require any preprocessing, yet exhibits performance comparable to light index-based algorithms (of linear or sublinear index size): that is, PBS runs in time $O(n^{(1-1/\log\log n)/2})$ and, with high probability, returns a path with length within $\frac{41}{32}$ of the shortest path. Moreover, if one does allow a $n^{\Theta(2-1/\log\log n)}$-time preprocessing step, its query time improves to $n^{\Theta(1/\log\log n})$. We complement our theoretical results by experiments on both real-world and synthetic power-law graphs, which show that PBS is typically $1.84\times$-$7.76\times$ times faster than existing alternatives, while achieving an approximation ratio at most 1.05.