Decremental Data Structures for Connectivity and Dominators in Directed Graphs

TL;DR

This paper introduces a decremental data structure for efficiently maintaining strongly connected components and dominator trees in directed graphs under edge deletions, enabling fast connectivity queries and improving dynamic graph algorithms.

Contribution

The paper presents the first decremental data structures supporting complex connectivity and dominator queries in directed graphs with optimal or near-optimal time complexity.

Findings

Supports constant-time connectivity queries after deletions.

Processes edge deletions in $O(m n \,\log n)$ total time.

Provides the first algorithms for maintaining dominator trees under edge deletions.

Abstract

We introduce a new dynamic data structure for maintaining the strongly connected components (SCCs) of a directed graph (digraph) under edge deletions, so as to answer a rich repertoire of connectivity queries. Our main technical contribution is a decremental data structure that supports sensitivity queries of the form "are $ u $ and $ v $ strongly connected in the graph $ G \setminus w $?", for any triple of vertices $ u, v, w $, while $ G $ undergoes deletions of edges. Our data structure processes a sequence of edge deletions in a digraph with $n$ vertices in $O(m n \log{n})$ total time and $O(n^2 \log{n})$ space, where $m$ is the number of edges before any deletion, and answers the above queries in constant time. We can leverage our data structure to obtain decremental data structures for many more types of queries within the same time and space complexity. For instance for edge-related queries, such as testing whether two query vertices $u$ and $v$ are strongly connected in $G \setminus e$, for some query edge $e$. As another important application of our decremental data structure, we provide the first nontrivial algorithm for maintaining the dominator tree of a flow graph under edge deletions. We present an algorithm that processes a sequence of edge deletions in a flow graph in $O(m n \log{n})$ total time and $O(n^2 \log{n})$ space. For reducible flow graphs we provide an $O(mn)$-time and $O(m + n)$-space algorithm. We give a conditional lower bound that provides evidence that these running times may be tight up to subpolynomial factors.