Deterministic Incremental APSP with Polylogarithmic Update Time and Stretch

TL;DR

This paper introduces a deterministic data structure for dynamic graphs that efficiently updates and approximates all-pairs shortest paths with polylogarithmic time per update and query, advancing the state of the art in partially dynamic graph algorithms.

Contribution

It presents the first deterministic partially dynamic APSP data structure with polylogarithmic update and query times, improving over previous randomized or dense graph solutions.

Findings

Supports edge insertions with polylogarithmic amortized update time.

Provides polylogarithmic approximation for all-pairs shortest paths.

Operates efficiently on sparse graphs with guaranteed performance.

Abstract

We provide the first deterministic data structure that given a weighted undirected graph undergoing edge insertions, processes each update with polylogarithmic amortized update time and answers queries for the distance between any pair of vertices in the current graph with a polylogarithmic approximation in $O(\log \log n)$ time. Prior to this work, no data structure was known for partially dynamic graphs, i.e., graphs undergoing either edge insertions or deletions, with less than $n^{o(1)}$ update time except for dense graphs, even when allowing randomization against oblivious adversaries or considering only single-source distances.