Connectivity Oracles for Graphs Subject to Vertex Failures

TL;DR

This paper presents new deterministic and randomized data structures, called connectivity oracles, that efficiently answer connectivity queries in graphs after vertex failures, handling unbounded failures with improved time and space complexities.

Contribution

Introduction of the first efficient connectivity oracles for general graphs capable of managing an unbounded number of vertex failures, with novel decomposition techniques.

Findings

Deterministic structure processes batched vertex failures in  O(d^3) time.

Randomized Monte Carlo structure processes failures in  O(d^2) time with efficient query handling.

Edge-failure oracle answers connectivity queries in  (\log\log n) time using  O(n \\log^2 n) space.

Abstract

We introduce new data structures for answering connectivity queries in graphs subject to batched vertex failures. A deterministic structure processes a batch of $d\leq d_{\star}$ failed vertices in $\tilde{O}(d^3)$ time and thereafter answers connectivity queries in $O(d)$ time. It occupies space $O(d_{\star} m\log n)$. We develop a randomized Monte Carlo version of our data structure with update time $\tilde{O}(d^2)$, query time $O(d)$, and space $\tilde{O}(m)$ for any failure bound $d\le n$. This is the first connectivity oracle for general graphs that can efficiently deal with an unbounded number of vertex failures. We also develop a more efficient Monte Carlo edge-failure connectivity oracle. Using space $O(n\log^2 n)$, $d$ edge failures are processed in $O(d\log d\log\log n)$ time and thereafter, connectivity queries are answered in $O(\log\log n)$ time, which are correct w.h.p. Our data structures are based on a new decomposition theorem for an undirected graph $G=(V,E)$, which is of independent interest. It states that for any terminal set $U\subseteq V$ we can remove a set $B$ of $|U|/(s-2)$ vertices such that the remaining graph contains a Steiner forest for $U-B$ with maximum degree $s$.