Dynamic DFS Tree in Undirected Graphs: breaking the $O(m)$ barrier

TL;DR

This paper introduces new algorithms for maintaining DFS trees in undirected graphs under dynamic updates, achieving sublinear worst-case times and enabling efficient solutions for related connectivity problems.

Contribution

It presents the first sublinear worst-case time algorithms for dynamic DFS trees and deterministic solutions for dynamic connectivity problems.

Findings

Fault tolerant DFS structure with polylogarithmic size and update time.

Fully dynamic DFS algorithm with worst-case O(√mn) update time.

Incremental DFS with worst-case O(n polylog n) per insertion.

Abstract

Depth first search (DFS) tree is a fundamental data structure for solving various problems in graphs. It is well known that it takes $O(m+n)$ time to build a DFS tree for a given undirected graph $G=(V,E)$ on $n$ vertices and $m$ edges. We address the problem of maintaining a DFS tree when the graph is undergoing {\em updates} (insertion and deletion of vertices or edges). We present the following results for this problem. (a) Fault tolerant DFS tree: There exists a data structure of size ${O}(m ~polylog~ n)$ such that given any set ${\cal F}$ of failed vertices or edges, a DFS tree of the graph $G\setminus {\cal F}$ can be reported in ${O}(n|{\cal F}| ~polylog~ n)$ time. (b) Fully dynamic DFS tree: There exists a fully dynamic algorithm for maintaining a DFS tree that takes worst case ${O}(\sqrt{mn} ~polylog~ n)$ time per update for any arbitrary online sequence of updates. (c) Incremental DFS tree: Given any arbitrary online sequence of edge insertions, we can maintain a DFS tree in ${O}(n ~polylog~ n)$ worst case time per edge insertion. These are the first $o(m)$ worst case time results for maintaining a DFS tree in a dynamic environment. Moreover, our fully dynamic algorithm provides, in a seamless manner, the first deterministic algorithm with $O(1)$ query time and $o(m)$ worst case update time for the dynamic subgraph connectivity, biconnectivity, and 2-edge connectivity.