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Undirected 3-Fault Replacement Path in Nearly Cubic Time

Shucheng Chi, Ran Duan, Benyu Wang, Tianle Xie

TL;DR

A 3FRP algorithm in deterministic $\tilde{O}(n^3)$ time for undirected weighted graphs, which almost matches the size of the output and shows that a truly subcubic algorithm for 2FRP in undirected graphs does not exist under APSP-hardness conjecture.

Abstract

Given a graph $G=(V,E)$ and two vertices $s,t\in V$, the $f$-fault replacement path ($f$FRP) problem computes for every set of edges $F$ where $|F|\leq f$, the distance from $s$ to $t$ when edges in $F$ fail. A recent result shows that 2FRP in directed graphs can be solved in $\tilde{O}(n^3)$ time [arXiv:2209.07016]. In this paper, we show a 3FRP algorithm in deterministic $\tilde{O}(n^3)$ time for undirected weighted graphs, which almost matches the size of the output. This implies that $f$FRP in undirected graphs can be solved in almost optimal $\tilde{O}(n^f)$ time for all $f\geq 3$. To construct our 3FRP algorithm, we introduce an incremental distance sensitivity oracle (DSO) with $\tilde{O}(n^2)$ worst-case update time, while preprocessing time, space, and query time are still $\tilde{O}(n^3)$, $\tilde{O}(n^2)$ and $\tilde{O}(1)$, respectively, which match the static DSO [Bernstein and Karger 2009]. Here in a DSO, we can preprocess a graph so that the distance between any pair of vertices given any failed edge can be answered efficiently. From the recent result in [arXiv:2211.05178], we can obtain an offline dynamic DSO from the incremental worst-case DSO, which makes the construction of our 3FRP algorithm more convenient. By the offline dynamic DSO, we can also construct a 2-fault single-source replacement path (2-fault SSRP) algorithm in $\tilde{O}(n^3)$ time, that is, from a given vertex $s$, we want to find the distance to any vertex $t$ when any pair of edges fail. Thus the $\tilde{O}(n^3)$ time complexity for 2-fault SSRP is also almost optimal. Now we know that in undirected graphs 1FRP can be solved in $\tilde{O}(m)$ time [Nardelli, Proietti, Widmayer 2001], and 2FRP and 3FRP in undirected graphs can be solved in $\tilde{O}(n^3)$ time. In this paper, we also show that a truly subcubic algorithm for 2FRP in undirected graphs does not exist under APSP-hardness conjecture.

Undirected 3-Fault Replacement Path in Nearly Cubic Time

TL;DR

A 3FRP algorithm in deterministic time for undirected weighted graphs, which almost matches the size of the output and shows that a truly subcubic algorithm for 2FRP in undirected graphs does not exist under APSP-hardness conjecture.

Abstract

Given a graph and two vertices , the -fault replacement path (FRP) problem computes for every set of edges where , the distance from to when edges in fail. A recent result shows that 2FRP in directed graphs can be solved in time [arXiv:2209.07016]. In this paper, we show a 3FRP algorithm in deterministic time for undirected weighted graphs, which almost matches the size of the output. This implies that FRP in undirected graphs can be solved in almost optimal time for all . To construct our 3FRP algorithm, we introduce an incremental distance sensitivity oracle (DSO) with worst-case update time, while preprocessing time, space, and query time are still , and , respectively, which match the static DSO [Bernstein and Karger 2009]. Here in a DSO, we can preprocess a graph so that the distance between any pair of vertices given any failed edge can be answered efficiently. From the recent result in [arXiv:2211.05178], we can obtain an offline dynamic DSO from the incremental worst-case DSO, which makes the construction of our 3FRP algorithm more convenient. By the offline dynamic DSO, we can also construct a 2-fault single-source replacement path (2-fault SSRP) algorithm in time, that is, from a given vertex , we want to find the distance to any vertex when any pair of edges fail. Thus the time complexity for 2-fault SSRP is also almost optimal. Now we know that in undirected graphs 1FRP can be solved in time [Nardelli, Proietti, Widmayer 2001], and 2FRP and 3FRP in undirected graphs can be solved in time. In this paper, we also show that a truly subcubic algorithm for 2FRP in undirected graphs does not exist under APSP-hardness conjecture.

Paper Structure

This paper contains 38 sections, 46 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Other related work.
  3. Organization.
  4. Preliminaries
  5. Notations and Assumptions
  6. Replacement Paths
  7. DSOs
  8. Technical Overview
  9. 2FRP algorithm between $s,t$
  10. Only $d_1$ is on $st$
  11. Only $d_1,d_2$ are on $st$
  12. Ideas of 3FRP algorithm
  13. Only $d_1$ is on $st$.
  14. Only $d_1,d_2$ are on $st$.
  15. All of $d_1,d_2,d_3$ are on $st$.
  16. ...and 23 more sections

Key Result

Theorem 1.1

The 3FRP problem in undirected real-weighted graphs can be solved in $\tilde{O}(n^3)$ time.

Theorems & Definitions (78)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • ...and 68 more