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Near Optimal Dual Fault Tolerant Distance Oracle

Dipan Dey, Manoj Gupta

TL;DR

This work presents a near-optimal dual fault-tolerant distance oracle for undirected and unweighted graphs with two faults. It achieves $ ilde{O}(n^2)$ space and $O(1)$ query time with high probability by introducing landmark-based detours, the concept of D-close vertices, and a refined set of maximisers partitioned into two groups to bound space. The method builds on and simplifies prior Do$(2)$ approaches, notably reducing space from $O(n^4)$ while maintaining constant-time queries, albeit with probabilistic guarantees and current applicability to unweighted graphs. The resulting oracle offers fast exact distance queries under two-edge failures and provides structural insights (detours, clean/ intermediate vertices, and trapezoids) that may inform broader fault-tolerant graph algorithms.

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.

Near Optimal Dual Fault Tolerant Distance Oracle

TL;DR

This work presents a near-optimal dual fault-tolerant distance oracle for undirected and unweighted graphs with two faults. It achieves space and query time with high probability by introducing landmark-based detours, the concept of D-close vertices, and a refined set of maximisers partitioned into two groups to bound space. The method builds on and simplifies prior Do approaches, notably reducing space from while maintaining constant-time queries, albeit with probabilistic guarantees and current applicability to unweighted graphs. The resulting oracle offers fast exact distance queries under two-edge failures and provides structural insights (detours, clean/ intermediate vertices, and trapezoids) that may inform broader fault-tolerant graph algorithms.

Abstract

We present a dual fault-tolerant distance oracle for undirected and unweighted graphs. Given a set of two edges, as well as a source node and a destination node , our oracle returns the length of the shortest path from to that avoids in time with a high probability. The space complexity of our oracle is \footnote{ hides poly 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 space and query time. In addition to improving the query time, our oracle is much simpler than the previous approach.
Paper Structure (18 sections, 12 theorems, 14 equations, 10 figures, 1 table, 1 algorithm)

This paper contains 18 sections, 12 theorems, 14 equations, 10 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Overview
  3. Overview of the approach in DuanR22
  4. Overview of our approach
  5. Preliminaries
  6. Oracle for two faults
  7. Finding a $D$-close vertex from $s$
  8. The detour starts on the primary path
  9. The detour starts on the secondary path
  10. From $D$-close vertex to a clean vertex
  11. The detour starts on the primary path
  12. The detour starts on the secondary path
  13. From a clean vertex to an intermediate vertex
  14. Space required by our data-structures
  15. Group 1 maximisers
  16. ...and 3 more sections

Key Result

Theorem 1.1

For an undirected and unweighted graph, there is a dual fault-tolerant oracle that takes $\tilde{O}(n^2)$ space and answers each query in $O(1)$ time with high probability.

Figures (10)

  • Figure 1: After at most three calls to the maximizer, we are sure to find an intermediate vertex.
  • Figure 2: The primary path contains $e_1$ and the secondary path contains $e_2$.
  • Figure 3: Start of the detour
  • Figure 4: After at most five consecutive calls to the maximiser, we are sure to find an intermediate vertex.
  • Figure 5: If $d \in T_s(x)$, the paths $sd \diamond e_1$ is not a prefix of $st \diamond e_1$
  • ...and 5 more figures

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 14 more