Optimal Approximate Distance Oracle for Planar Graphs

Le, Hung; Wulff-Nilsen, Christian

doi:10.1109/focs52979.2021.00044

Cited by 6 publications

(2 citation statements)

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“…Thorup [63] proved that on nonnegatively weighted planar graphs, (1 + 𝜖)-approximate distances can be reported in 𝑂 (𝜖 −1 + log log 𝑛) time by an oracle of space 𝑂 (𝑛𝜖 −1 log 2 𝑛). Refer to [13, 34, 42-44, 48, 68] for other spacetime-approximation tradeoffs on undirected planar graphs and to [48] for an improved tradeoff on directed planar graphs. See Table 1.…”

Section: History Of Planar Distance Oraclesmentioning

confidence: 99%

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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Section: History Of Planar Distance Oraclesmentioning

confidence: 99%

Almost Optimal Exact Distance Oracles for Planar Graphs

Charalampopoulos

Gawrychowski

Long

et al. 2023

J. ACM

Self Cite

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“…Directed planar graphs, on the other hand, admit very efficient reachability oracles: almost three decades of research [4,5,7,8] eventually resulted in the optimal solution by Holm, Rotenberg and Thorup [13], who presented an oracle that has O(n) storage and O(1) query time. Moreover, there are various (approximate) distance oracles for directed planar graphs; see for example [11,16,21] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

A Note on Reachability and Distance Oracles for Transmission Graphs

Berg¹

2022

Preprint

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Let P be a set of n points in the plane, where each point p ∈ P has a transmission radius r(p) > 0. The transmission graph defined by P and the given radii, denoted by Gtr(P ), is the directed graph whose nodes are the points in P and that contains the arcs (p, q) such that |pq| ⩽ r(p).An and Oh [Algorithmica 2022] presented a reachability oracle for transmission graphs. Their oracle uses O(n 5/3 ) storage and, given two query points s, t ∈ P , can decide in O(n 2/3 ) time if there is a path from s to t in Gtr(P ). We show that the clique-based separators introduced by De Berg et al. [SICOMP 2020] can be used to improve the storage of the oracle to O(n √ n) and the query time to O( √ n). Our oracle can be extended to approximate distance queries: we can construct, for a given parameter ε > 0, an oracle that uses O((n/ε) √ n log n) storage and that can report in O((, where d hop (s, t) is the hop-distance from s to t. We also show how to extend the oracle to so-called continuous queries, where the target point t can be any point in the plane.To obtain an efficient preprocessing algorithm, we show that a clique-based separator of a set F of convex fat objects in R d can be constructed in O(n log n) time.

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