Constant query time (1 + ϵ)-approximate distance oracle for planar graphs

Gu, Qian-Ping; Xu, Gengchun

doi:10.1016/j.tcs.2018.08.024

Cited by 4 publications

(1 citation statement)

References 20 publications

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“…In this subsection, we remove the assumption on the spread using the contraction technique of Kawarabayashi, Sommer, and Thorup [KST13]. The same technique was used in previous results [GX19,CS19]. The idea is to have for each scale r ∈ {2 0 , 2 1 , .…”

Section: Removing the Spread Assumptionmentioning

confidence: 99%

Optimal Approximate Distance Oracle for Planar Graphs

Le¹,

Wulff-Nilsen²

2021

Preprint

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A (1 + )-approximate distance oracle of an edge-weighted graph is a data structure that returns an approximate shortest path distance between any two query vertices up to a (1 + ) factor. Thorup (FOCS 2001, JACM 2004 and Klein (SODA 2002) independently constructed a (1 + )-approximate distance oracle with O(n log n) space, measured in number of words, and O(1) query time when G is an undirected planar graph with n vertices and is a fixed constant. Many follow-up works gave (1 + )-approximate distance oracles with various trade-offs between space and query time. However, improving O(n log n) space bound without sacrificing query time remains an open problem for almost two decades. In this work, we resolve this problem affirmatively by constructing a (1 + )-approximate distance oracle with optimal O(n) space and O(1) query time for undirected planar graphs and fixed .We also make substantial progress for planar digraphs with non-negative edge weights. For fixed > 0, we give a (1 + )-approximate distance oracle with space o(n log(N n)) and O(log log(N n) query time; here N is the ratio between the largest and smallest positive edge weight. This improves Thorup's (FOCS 2001, JACM 2004) O(n log(N n) log n) space bound by more than a logarithmic factor while matching the query time of his structure. This is the first improvement for planar digraphs in two decades, both in the weighted and unweighted setting.

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Section: Removing the Spread Assumptionmentioning

confidence: 99%

Optimal Approximate Distance Oracle for Planar Graphs

Le¹,

Wulff-Nilsen²

2021

Preprint

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Optimal Approximate Distance Oracle for Planar Graphs

Wulff-Nilsen

2022

2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)

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Almost Optimal Exact Distance Oracles for Planar Graphs

Charalampopoulos

Gawrychowski

Long

et al. 2023

J. ACM

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We consider the problem of preprocessing a weighted directed planar graph in order to quickly answer exact distance queries. The main tension in this problem is between space S and query time Q , and since the mid-1990s all results had polynomial time-space tradeoffs, e.g., Q = ~ Θ( n/√ S ) or Q = ~Θ( n 5/2 /S 3/2 ). In this article we show that there is no polynomial tradeoff between time and space and that it is possible to simultaneously achieve almost optimal space n 1+ o (1) and almost optimal query time n o (1) . More precisely, we achieve the following space-time tradeoffs: n 1+ o (1) space and log 2+ o (1) n query time, n log 2+ o (1) n space and n o (1) query time, n 4/3+ o (1) space and log 1+ o (1) n query time. We reduce a distance query to a variety of point location problems in additively weighted Voronoi diagrams and develop new algorithms for the point location problem itself using several partially persistent dynamic tree data structures.

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Constant query time (1 + ϵ)-approximate distance oracle for planar graphs

Cited by 4 publications

References 20 publications

Optimal Approximate Distance Oracle for Planar Graphs

Optimal Approximate Distance Oracle for Planar Graphs

Optimal Approximate Distance Oracle for Planar Graphs

Almost Optimal Exact Distance Oracles for Planar Graphs

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Constant query time (1 + ϵ)-approximate distance oracle for planar graphs

Cited by 4 publications

References 20 publications

Optimal Approximate Distance Oracle for Planar Graphs

Optimal Approximate Distance Oracle for Planar Graphs

Optimal Approximate Distance Oracle for Planar Graphs

Almost Optimal Exact Distance Oracles for Planar Graphs

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Constant query time (1 + ϵ)-approximate distance oracle for planar graphs