Worst-case time decremental connectivity and k-edge witness

TL;DR

This paper presents new algorithms for decremental graph connectivity and k-edge witness problems, achieving improved worst-case time bounds and space efficiency, along with a distributed data structure for k-edge witness.

Contribution

It introduces a simple decremental connectivity algorithm with worst-case bounds and a novel distributed data structure for k-edge witness queries.

Findings

Handles edge deletions in O(k log n) worst-case time

Provides a k-edge witness algorithm with O(k^2 log n) worst-case time

Achieves space complexity improvements for certain graph parameters

Abstract

We give a simple algorithm for decremental graph connectivity that handles edge deletions in worst-case time $O(k \log n)$ and connectivity queries in $O(\log k)$, where $k$ is the number of edges deleted so far, and uses worst-case space $O(m^2)$. We use this to give an algorithm for $k$-edge witness (``does the removal of a given set of $k$ edges disconnect two vertices $u,v$?'') with worst-case time $O(k^2 \log n)$ and space $O(k^2 n^2)$. For $k = o(\sqrt{n})$ these improve the worst-case $O(\sqrt{n})$ bound for deletion due to Eppstein et al. We also give a decremental connectivity algorithm using $O(n^2 \log n / \log \log n)$ space, whose time complexity depends on the toughness and independence number of the input graph. Finally, we show how to construct a distributed data structure for \kvw by giving a labeling scheme. This is the first data structure for \kvw that can efficiently distributed without just giving each vertex a copy of the whole structure. Its complexity depends on being able to construct a linear layout with good properties.