Almost Optimal Multiple Source Shortest Paths and Reachability

Given a graph, computing distances and reachabilities from a small set of vertices to the whole graph is an important primitive both in theory and in practice. In undirected unweighted graphs, while computing single-source shortest path (SSSP) requires $O(n^2)$ time in dense graphs, all-pairs shortest paths (APSP) can be computed in $\hat{O}(n^\omega) = O(n^{2.372})$ time [Seidel '95] providing significant savings over running $n$ SSSP instances separately. However, if one needs to compute multiple-source shortest paths (MSSP) from a set of $n^\sigma$ vertices, the previously best known running time was $\hat{O}(\min\{n^\omega, n^{2 + \sigma}\})$: either compute APSP or run SSSP from each source. On the other hand, MSSP is only as hard as computing Boolean matrix product (BMM) between an $n^\sigma \times n$ matrix and $n \times n$ matrix, leaving a significant gap. Our first main result is an almost optimal algorithm for MSSP on undirected unweighted graphs running in $\hat{O}(n^{\omega(\sigma, 1, 1)})$ time, which gives a smooth interpolation between the SSSP and APSP algorithms. The main technical tool behind our result is a novel graph decomposition, which may be of independent interest. Next, we study the multiple-source reachability problem, where we need to determine whether a given set of $n^\sigma$ vertices can reach each of the vertices in a given directed graph. Multiple-source reachability can also be solved in $\hat{O}(\min\{n^\omega, n^{2 + \sigma}\})$ time, with the same lower bound from rectangular BMM. We give an optimal algorithm that runs in $\hat{O}(n^{\omega(\sigma, 1, 1)})$ time, again matching the running time for BMM. Our algorithm for multiple-source reachability can be generalized to MSSP on DAGs. As an application, we provide an $O(n^{2.084})$ time algorithm for computing an $\widetilde{O}(n)$-size shortcut set that reduces diameter to $O(n^{1/3})$.