All-pairs almost shortest paths

Dorit Dor, Shay Halperin, Uri Zwick

Research output: Contribution to journalArticlepeer-review

204 Scopus citations

Abstract

Let G = (V, E) be an unweighted undirected graph on n vertices. A simple argument shows that computing all distances in G with an additive one-sided error of at most 1 is as hard as Boolean matrix multiplication. Building on recent work of Aingworth et al. we describe an O(min{n3/2m1/2, n7/3})-time algorithm APASP2 for computing all distances in G with an additive one-sided error of at most 2. Algorithm APASP2 is simple, easy to implement, and faster than the fastest known matrix-multiplication algorithm. Furthermore, for every even k>2, we describe an O(min{n2-2/(k+2)m2/(k+2), n2+2/(3k-2)})-time algorithm APASPk for computing all distances in G with an additive one-sided error of at most k. We also give an O(n2)-time algorithm APASP for producing stretch 3 estimated distances in an unweighted and undirected graph on n vertices. No constant stretch factor was previously achieved in O(n2) time. We say that a weighted graph F = (V, E′) k-emulates an unweighted graph G = (V, E) if for every u, v∈V we have δG(u, v)≤δF(u, v)≤δG(u, v)+k. We show that every unweighted graph on n vertices has a 2-emulator with O(n3/2) edges and a 4-emulator with O(n4/3) edges. These results are asymptotically tight. Finally, we show that any weighted undirected graph on n vertices has a 3-spanner with O(n3/2) edges and that such a 3-spanner can be built in O(mnHLF) time. We also describe an O(n(m2/3+n))-time algorithm for estimating all distances in a weighted undirected graph on n vertices with a stretch factor of at most 3.

Original languageEnglish
Pages (from-to)1740-1759
Number of pages20
JournalSIAM Journal on Computing
Volume29
Issue number5
DOIs
StatePublished - Mar 2000

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