TY - UNPB

T1 - Almost Shortest Paths and PRAM Distance Oracles in Weighted Graphs.

AU - Elkin, Michael

AU - Gitlitz, Yuval

AU - Neiman, Ofer

N1 - DBLP's bibliographic metadata records provided through http://dblp.org/search/publ/api are distributed under a Creative Commons CC0 1.0 Universal Public Domain Dedication. Although the bibliographic metadata records are provided consistent with CC0 1.0 Dedication, the content described by the metadata records is not. Content may be subject to copyright, rights of privacy, rights of publicity and other restrictions.

PY - 2019

Y1 - 2019

N2 - Let G=(V,E,w) be a weighted undirected graph with n vertices and m edges, and fix a set of s sources S⊆V. We study the problem of computing {\em almost shortest paths} (ASP) for all pairs in S×V in both classical centralized and parallel (PRAM) models of computation. Consider the regime of multiplicative approximation of 1+ϵ, for an arbitrarily small constant ϵ>0 . In this regime existing centralized algorithms require Ω(min{|E|s,nω}) time, where ω<2.372 is the matrix multiplication exponent. Existing PRAM algorithms with polylogarithmic depth (aka time) require work Ω(min{|E|s,nω}).Our centralized algorithm has running time O((m+ns)nρ), and its PRAM counterpart has polylogarithmic depth and work O((m+ns)nρ), for an arbitrarily small constant ρ>0. For a pair (s,v)∈S×V, it provides a path of length d^(s,v) that satisfies d^(s,v)≤(1+ϵ)dG(s,v)+β⋅W(s,v), where W(s,v) is the weight of the heaviest edge on some shortest s−v path. Hence our additive term depends linearly on a {\em local} maximum edge weight, as opposed to the global maximum edge weight in previous works. Finally, our β=(1/ρ)O(1/ρ).We also extend a centralized algorithm of Dor et al. \cite{DHZ00}. For a parameter κ=1,2,…, this algorithm provides for {\em unweighted} graphs a purely additive approximation of 2(κ−1) for {\em all pairs shortest paths} (APASP) in time O~(n2+1/κ). Within the same running time, our algorithm for {\em weighted} graphs provides a purely additive error of 2(κ−1)W(u,v), for every vertex pair (u,v)∈(V2), with W(u,v) defined as above. On the way to these results we devise a suit of novel constructions of spanners, emulators and hopsets.

AB - Let G=(V,E,w) be a weighted undirected graph with n vertices and m edges, and fix a set of s sources S⊆V. We study the problem of computing {\em almost shortest paths} (ASP) for all pairs in S×V in both classical centralized and parallel (PRAM) models of computation. Consider the regime of multiplicative approximation of 1+ϵ, for an arbitrarily small constant ϵ>0 . In this regime existing centralized algorithms require Ω(min{|E|s,nω}) time, where ω<2.372 is the matrix multiplication exponent. Existing PRAM algorithms with polylogarithmic depth (aka time) require work Ω(min{|E|s,nω}).Our centralized algorithm has running time O((m+ns)nρ), and its PRAM counterpart has polylogarithmic depth and work O((m+ns)nρ), for an arbitrarily small constant ρ>0. For a pair (s,v)∈S×V, it provides a path of length d^(s,v) that satisfies d^(s,v)≤(1+ϵ)dG(s,v)+β⋅W(s,v), where W(s,v) is the weight of the heaviest edge on some shortest s−v path. Hence our additive term depends linearly on a {\em local} maximum edge weight, as opposed to the global maximum edge weight in previous works. Finally, our β=(1/ρ)O(1/ρ).We also extend a centralized algorithm of Dor et al. \cite{DHZ00}. For a parameter κ=1,2,…, this algorithm provides for {\em unweighted} graphs a purely additive approximation of 2(κ−1) for {\em all pairs shortest paths} (APASP) in time O~(n2+1/κ). Within the same running time, our algorithm for {\em weighted} graphs provides a purely additive error of 2(κ−1)W(u,v), for every vertex pair (u,v)∈(V2), with W(u,v) defined as above. On the way to these results we devise a suit of novel constructions of spanners, emulators and hopsets.

U2 - 10.48550/arXiv.1907.11422

DO - 10.48550/arXiv.1907.11422

M3 - Preprint

BT - Almost Shortest Paths and PRAM Distance Oracles in Weighted Graphs.

ER -