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arXiv · 2605.07168

Connectivity Oracle Under Vertex Failures by Shortcutting Unbreakable Decomposition

Xizhe Li·Yaowei Long·David Pidugu·Thatchaphol Saranurak·Benyu Wang
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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 $\Omega(n^{2})$ 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.

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Xizhe Li, Yaowei Long, David Pidugu, Thatchaphol Saranurak, Benyu Wang. 2026-05-08. Connectivity Oracle Under Vertex Failures by Shortcutting Unbreakable Decomposition. https://arxiv.org/abs/2605.07168

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