Sublinear-Time Decremental Algorithms for Single-Source Reachability and Shortest Paths on Directed Graphs

We consider dynamic algorithms for maintaining Single-Source Reachability (SSR) and approximate Single-Source Shortest Paths (SSSP) on $n$-node $m$-edge directed graphs under edge deletions (decremental algorithms). The previous fastest algorithm for SSR and SSSP goes back three decades to Even and Shiloach [JACM 1981]; it has $O(1)$ query time and $O (mn)$ total update time (i.e., linear amortized update time if all edges are deleted). This algorithm serves as a building block for several other dynamic algorithms. The question whether its total update time can be improved is a major, long standing, open problem. In this paper, we answer this question affirmatively. We obtain a randomized algorithm with an expected total update time of $O(\min (m^{7/6} n^{2/3 + o(1)}, m^{3/4} n^{5/4 + o(1)}) ) = O (m n^{9/10 + o(1)})$ for SSR and $(1+\epsilon)$-approximate SSSP if the edge weights are integers from $1$ to $W \leq 2^{\log^c{n}}$ and $\epsilon \geq 1 / \log^c{n}$ for some constant $c$. We also extend our algorithm to achieve roughly the same running time for Strongly Connected Components (SCC), improving the algorithm of Roditty and Zwick [FOCS 2002]. Our algorithm is most efficient for sparse and dense graphs. When $m = \Theta(n)$ its running time is $O (n^{1 + 5/6 + o(1)})$ and when $m = \Theta(n^2)$ its running time is $O (n^{2 + 3/4 + o(1)})$. For SSR we also obtain an algorithm that is faster for dense graphs and has a total update time of $O ( m^{2/3} n^{4/3 + o(1)} + m^{3/7} n^{12/7 + o(1)})$ which is $O (n^{2 + 2/3})$ when $m = \Theta(n^2)$. All our algorithms have constant query time in the worst case and are correct with high probability against an oblivious adversary.