Deterministic Algorithms for Decremental Shortest Paths via Layered Core Decomposition

In the decremental single-source shortest paths (SSSP) problem, the input is an undirected graph G = (V, E) with n vertices and m edges undergoing edge deletions, together with a fixed source vertex s ∈ V . The goal is to maintain a data structure that supports shortest-path queries: given a vertex v ∈ V , quickly return an (approximate) shortest path from s to v. The decremental all-pairs shortest paths (APSP) problem is defined similarly, but now the shortest-path queries are allowed between any pair of vertices of V .

Both problems have been studied extensively since the 80's, and algorithms with near-optimal total update time and query time have been discovered for them. Unfortunately, all these algorithms are randomized and, more importantly, they need to assume an oblivious adversary -a drawback that prevents them from being used as subroutines in several known algorithms for classical static problems. In this paper, we provide new deterministic algorithms for both problems, which, by definition, can handle an adaptive adversary.

Our first result is a deterministic algorithm for the decremental SSSP problem on weighted graphs with O(n 2+o(1) ) total update time, that supports (1 + ǫ)-approximate shortest-path queries, with query time O(|P | • n o(1) ), where P is the returned path. This is the first (1 + ǫ)-approximation adaptive-update algorithm supporting shortest-path queries in time below O(n), that breaks the O(mn) total update time bound of the classical algorithm of Even and Shiloah from 1981. Previously, Bernstein and Chechik [STOC'16, ICALP'17] provided a Õ(n 2 )-time deterministic algorithm that supports approximate distance queries, but unfortunately the algorithm cannot return the approximate shortest paths. Chuzhoy and Khanna [STOC'19] showed an O(n 2+o(1) )-time randomized algorithm for SSSP that supports approximate shortest-path queries in the adaptive adversary regime, but their algorithm only works in the restricted setting where only vertex deletions, and not edge deletions are allowed, and it requires Ω(n) time to respond to shortest-path queries.

Our second result is a deterministic algorithm for the decremental APSP problem on unweighted graphs that achieves total update time O(n 2.5+δ ), for any constant δ > 0, supports approximate distance queries in O(log log n) time, and supports approximate shortest-path queries in time O(|E(P )| • n o(1) ), where P is the returned path; the algorithm achieves an O(1)-multiplicative and n o(1) -additive approximation on the path length. All previous algorithms for APSP either assume an oblivious adversary or have an Ω(n 3 ) total update time when m = Ω(n 2 ), even if an o(n)-multiplicative approximation is allowed.

To obtain both our results, we improve and generalize the layered core decomposition data structure introduced by Chuzhoy and Khanna to be nearly optimal in terms of various parameters, and introduce a new generic approach of rooting Even-Shiloach trees at expander sub-graphs of the given graph. We believe both these technical tools to be interesting in their own right and anticipate them to be useful for designing future dynamic algorithms that work against an adaptive adversary.