Faster shortest-path algorithms using the acyclic-connected tree

Abstract

We provide a method to obtain beyond-worst-case time complexity for any single-source-shortest-path (SSSP) algorithm by exploiting modular structures in graphs. The key novelty is a graph decomposition, called the acyclic-connected (A-C) tree, which breaks up a graph into a recursively nested sequence of strongly connected components in topological order. The A-C tree is optimal in the sense that it maximally decomposes the graph, formalised by a parameter called nesting width, measuring the extent to which a graph can be decomposed. We show how to compute the A-C tree in linear time, allowing it to be used as a preprocessing step for SSSP. Indeed, we transform any SSSP algorithm by first computing the A-C tree, and then running the SSSP algorithm in a careful recursive manner on the A-C tree. We illustrate this with two state-of-the-art algorithms: Dijkstra's algorithm and the recent sparse graph algorithm of Duan et al., obtaining improved time complexities of $O(m+n\log(\mathrm{nw}(G)))$ and $O(m\alpha(n)+m\log^{2/3}(\mathrm{nw}(G)))$, respectively, where $\mathrm{nw}(G) \leq n$ is the nesting width of the graph $G$, and $\alpha(n)$ is the extremely slow-growing inverse Ackermann function. Some classes of graphs, such as directed acyclic graphs, have bounded nesting width, and we obtain linear-time SSSP algorithms for these graphs.