Dynamic Approximate All-Pairs Shortest Paths: Breaking the O(mn) Barrier and Derandomization

Years and Authors of Summarized Original Work

2013; Henzinger, Krinninger, Nanongkai

Problem Definition

Given an undirected, unweighted graph with n nodes and m edges that is modified by a sequence of edge insertions and deletions, the problem is to maintain a data structure that quickly answers queries that ask for the length d(u, v) of the shortest path between two arbitrary nodes u and v in the graph, called the distance of u and v. The fastest exact algorithm for this problem is randomized and takes amortized $O\left (n^{2}\left (\log n +\log ^{2}((m + n)/n)\right )\right )$ time per update and constant query time [6, 11]. In the decremental case, i.e., if only edge deletions are allowed, there exists a deterministic algorithm with amortized time O(n ²) per deletion [7]. More precisely, its total update time for a sequence of up to m deletions is O(mn ²). Additionally, there is a randomized algorithm with O(n ³log² n) total update time and constant query time [1]. However, in...