Connectivity Oracle Under Vertex Failures by Shortcutting Unbreakable Decomposition

2026-05-07

AI Summary

This work addresses the problem of efficiently answering connectivity queries in graphs following vertex failures. It proposes a novel connectivity oracle that, for the first time, achieves near-linear space and preprocessing time, with both update and query complexities depending solely on the number of failures $k$ and independent of the graph size $n$. Built upon an inseparable decomposition framework, the approach introduces tree-decomposition shortcutting, internally guided oracle construction, and a new patch-set mechanism. For any constant $k$, the oracle supports updates in $O(k^6)$ time and answers pairwise connectivity queries in optimal $O(k)$ time, thereby significantly overcoming the exponential bottlenecks in space or time that plague existing methods.

Abstract

We give an improved connectivity oracle under vertex failures. After a set of $k$ vertices fails, our oracle performs an $O(k^{6})$-time update independent of the graph size $n$, and then answers pairwise connectivity queries in optimal $O(k)$ time. For constant $k$, it uses near-linear space and can be built in near-linear preprocessing time. In contrast, all prior oracles with $n$-independent update time[PSS+22, vdBS19] either require $Ω(n^{2})$ space or incur $2^{2^{O(k)}}$ update and query time. Moreover, their preprocessing time is polynomially large in $n$, far from near-linear. Our oracle builds on the unbreakable decomposition framework of[PSS+22], but introduces three new ingredients: (i) shortcutting over the tree decomposition to reduce space from quadratic to near-linear, (ii) bootstrapping that leverages $n$-dependent oracles internally to obtain near-linear preprocessing, and (iii) a new patch set mechanism that yields conditionally optimal $O(k)$ query time.

Problem

• connectivity oracle
• vertex failures
• graph connectivity
• near-linear space
• update time

Innovation

• connectivity oracle
• vertex failures
• unbreakable decomposition
• shortcutting
• near-linear preprocessing