Near Optimal Dual Fault Tolerant Distance Oracle

TL;DR

This paper introduces a near-optimal dual fault-tolerant distance oracle for undirected, unweighted graphs that provides instant shortest path queries avoiding two edges, with improved simplicity and comparable space complexity.

Contribution

It presents a new dual fault-tolerant distance oracle with constant query time and nearly optimal space, simplifying previous complex solutions.

Findings

Query time is O(1) with high probability.

Space complexity is nearly optimal at ~O(n^2).

Improves upon prior work by Pettie and Duan in both simplicity and performance.

Abstract

We present a dual fault-tolerant distance oracle for undirected and unweighted graphs. Given a set $F$ of two edges, as well as a source node $s$ and a destination node $t$, our oracle returns the length of the shortest path from $s$ to $t$ that avoids $F$ in $O(1)$ time with a high probability. The space complexity of our oracle is $\Tilde{O}(n^2)$ \footnote{$\Tilde{O}$ hides poly$\log n$ factor }, making it nearly optimal in terms of both space and query time. Prior to our work, Pettie and Duan [SODA 2009] designed a dual fault-tolerant distance oracle that required $\Tilde{O}(n^2)$ space and $O(\log n)$ query time. In addition to improving the query time, our oracle is much simpler than the previous approach.