DeepAI AI Chat
Log In Sign Up

Near-Optimal Deterministic Single-Source Distance Sensitivity Oracles

06/29/2021
by   Davide Bilò, et al.
Hasso Plattner Institute
0

Given a graph with a source vertex s, the Single Source Replacement Paths (SSRP) problem is to compute, for every vertex t and edge e, the length d(s,t,e) of a shortest path from s to t that avoids e. A Single-Source Distance Sensitivity Oracle (Single-Source DSO) is a data structure that answers queries of the form (t,e) by returning the distance d(s,t,e). We show how to deterministically compress the output of the SSRP problem on n-vertex, m-edge graphs with integer edge weights in the range [1,M] into a Single-Source DSO of size O(M^1/2n^3/2) with query time O(1). The space requirement is optimal (up to the word size) and our techniques can also handle vertex failures. Chechik and Cohen [SODA 2019] presented a combinatorial, randomized O(m√(n)+n^2) time SSRP algorithm for undirected and unweighted graphs. Grandoni and Vassilevska Williams [FOCS 2012, TALG 2020] gave an algebraic, randomized O(Mn^ω) time SSRP algorithm for graphs with integer edge weights in the range [1,M], where ω<2.373 is the matrix multiplication exponent. We derandomize both algorithms for undirected graphs in the same asymptotic running time and apply our compression to obtain deterministic Single-Source DSOs. The O(m√(n)+n^2) and O(Mn^ω) preprocessing times are polynomial improvements over previous o(n^2)-space oracles. On sparse graphs with m=O(n^5/4-ε/M^7/4) edges, for any constant ε > 0, we reduce the preprocessing to randomized O(M^7/8m^1/2n^11/8)=O(n^2-ε/2) time. This is the first truly subquadratic time algorithm for building Single-Source DSOs on sparse graphs.

READ FULL TEXT

page 3

page 9

page 13

page 15

page 19

page 23

page 27

page 29

05/17/2019

Deterministic Combinatorial Replacement Paths and Distance Sensitivity Oracles

In this work we derandomize two central results in graph algorithms, rep...
06/30/2022

Near Optimal Algorithm for Fault Tolerant Distance Oracle and Single Source Replacement Path problem

In a graph G with a source s, we design a distance oracle that can answe...
04/12/2020

A Fast Algorithm for Source-Wise Round-Trip Spanners

In this paper, we study the problem of efficiently constructing source-w...
02/17/2021

Constructing a Distance Sensitivity Oracle in O(n^2.5794M) Time

We continue the study of distance sensitivity oracles (DSOs). Given a di...
04/22/2022

Parameterized Sensitivity Oracles and Dynamic Algorithms using Exterior Algebras

We design the first efficient sensitivity oracles and dynamic algorithms...
09/15/2022

Algorithms and Lower Bounds for Replacement Paths under Multiple Edge Failures

This paper considers a natural fault-tolerant shortest paths problem: fo...
05/08/2022

Near-Optimal Deterministic Vertex-Failure Connectivity Oracles

We revisit the vertex-failure connectivity oracle problem. This is one o...