Near-Optimal Decremental Hopsets with Applications

TL;DR

This paper introduces the first efficient decremental algorithm for maintaining hopsets with polylogarithmic hopbound, enabling near-optimal dynamic approximate shortest path computations in graphs.

Contribution

It presents the first decremental hopset construction with polylogarithmic hopbound and applies it to improve dynamic approximate shortest path algorithms.

Findings

Decremental hopset algorithm with polylogarithmic hopbound

Improved bounds for dynamic approximate all-pairs shortest paths

Near-optimal bounds for multi-source shortest paths and distance sketches

Abstract

Given a weighted undirected graph $G=(V,E,w)$, a hopset $H$ of hopbound $\beta$ and stretch $(1+\epsilon)$ is a set of edges such that for any pair of nodes $u, v \in V$, there is a path in $G \cup H$ of at most $\beta$ hops, whose length is within a $(1+\epsilon)$ factor from the distance between $u$ and $v$ in $G$. We show the first efficient decremental algorithm for maintaining hopsets with a polylogarithmic hopbound. The update time of our algorithm matches the best known static algorithm up to polylogarithmic factors. All the previous decremental hopset constructions had a superpolylogarithmic (but subpolynomial) hopbound of $2^{\log^{\Omega(1)} n}$ [Bernstein, FOCS'09; HKN, FOCS'14; Chechik, FOCS'18]. By applying our decremental hopset construction, we get improved or near optimal bounds for several distance problems. Most importantly, we show how to decrementally maintain $(2k-1)(1+\epsilon)$-approximate all-pairs shortest paths (for any constant $k \geq 2)$, in $\tilde{O}(n^{1/k})$ amortized update time and $O(k)$ query time. This improves (by a polynomial factor) over the update-time of the best previously known decremental algorithm in the constant query time regime. Moreover, it improves over the result of [Chechik, FOCS'18] that has a query time of $O(\log \log(nW))$, where $W$ is the aspect ratio, and the amortized update time is $n^{1/k}\cdot(\frac{1}{\epsilon})^{\tilde{O}(\sqrt{\log n})}$. For sparse graphs our construction nearly matches the best known static running time / query time tradeoff. We also obtain near-optimal bounds for maintaining approximate multi-source shortest paths and distance sketches, and get improved bounds for approximate single-source shortest paths. Our algorithms are randomized and our bounds hold with high probability against an oblivious adversary.