Fast and Compact Exact Distance Oracle for Planar Graphs

Cohen-Addad, Vincent; Dahlgaard, Søren; Wulff-Nilsen, Christian

doi:10.1109/focs.2017.93

Cited by 34 publications

(41 citation statements)

References 30 publications

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“…This approach is also the one taken in [7]. However, our construction is arguably simpler and more elegant than that of [7].…”

Section: Introductionmentioning

confidence: 99%

“…This approach is also the one taken in [7]. However, our construction is arguably simpler and more elegant than that of [7]. Our main technical contribution is a novel point location data structure for Voronoi diagrams (see below).…”

Section: Introductionmentioning

confidence: 99%

“…WulffNilsen [31] showed how to achieve constant query-time with O(n 2 (log log n) 4 / log n) space, improving the above tradeoff for close to quadratic space. Very recently, Cohen-Addad, Dahlgaard, and Wulff-Nilsen [7], inspired by the ideas of Cabello [4] made significant progress by presenting an oracle with O(n 5/3 ) space and O(log n) query-time. This is the first oracle for planar graphs that achieves truly subquadratic space and subpolynomial query-time.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Better Tradeoffs for Exact Distance Oracles in Planar Graphs

Gawrychowski¹,

Mozes²,

Weimann³

et al. 2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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We present an O(n 1.5 )-space distance oracle for directed planar graphs that answers distance queries in O(log n) time. Our oracle both significantly simplifies and significantly improves the recent oracle of Cohen-Addad, Dahlgaard and Wulff-Nilsen [FOCS 2017], which uses O(n 5/3 )-space and answers queries in O(log n) time. We achieve this by designing an elegant and efficient point location data structure for Voronoi diagrams on planar graphs.We further show a smooth tradeoff between space and query-time. For any S ∈ [n, n 2 ], we show an oracle of size S that answers queries inÕ(max{1, n 1.5 /S}) time. This new tradeoff is currently the best (up to polylogarithmic factors) for the entire range of S and improves by polynomial factors over all previously known tradeoffs for the range S ∈ [n, n 5/3 ].

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“…This approach is also the one taken in [7]. However, our construction is arguably simpler and more elegant than that of [7].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Better Tradeoffs for Exact Distance Oracles in Planar Graphs

Gawrychowski¹,

Mozes²,

Weimann³

et al. 2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

Self Cite

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show abstract

“…They also improve the stateof-the-art tradeoff between space and query time (see also [33]). Using Theorem 1.1 yields an optimal preprocessing time for the oracles in [23], matching their space requirements up to polylogarithmic factors.…”

Section: The Aboveõ(|smentioning

confidence: 99%

“…Our detailed study of Voronoi diagrams for planar graphs, and the resulting algorithm of Theorem 1.1 are of independent interest, and we believe it will find additional uses. For example, in a very recent work, CohenAddad, Dahlgaard, and Wulff-Nilsen [23] show that the Voronoi diagram approach can be used to construct exact distance oracles for planar graphs with subquadratic space and polylogarithmic query time. They show an oracle of sizeÕ(n 5/3 ), query time O(log n), and preprocessing time O(n 2 ).…”

Section: The Aboveõ(|smentioning

confidence: 99%

Voronoi Diagrams on Planar Graphs, and Computing the Diameter in Deterministic Õ(n^5/3) Time

Gawrychowski

Kaplan

Mozes

et al. 2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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We present an efficient construction of additively weighted Voronoi diagrams on planar graphs. Let G be a planar graph with n vertices and b sites that lie on a constant number of faces. We show how to preprocess G inÕ(nb 2 ) time 1 so that one can compute any additively weighted Voronoi diagram for these sites inÕ(b) time.We use this construction to compute the diameter of a directed planar graph with real arc lengths inÕ(n 5/3 ) time. This improves the recent breakthrough result of Cabello (SODA'17), both by improving the running time (from O(n 11/6 )), and by providing a deterministic algorithm. It is in fact the first truly subquadratic deterministic algorithm for this problem. Our use of Voronoi diagrams to compute the diameter follows that of Cabello, but he used abstract Voronoi diagrams, which makes his diameter algorithm more involved, more expensive, and randomized.As in Cabello's work, our algorithm can also compute the Wiener index of a planar graph (i.e., the sum of all pairwise distances) within the same bound. Our construction of Voronoi diagrams for planar graphs is of independent interest. It has already been used to obtain fast exact distance oracles for planar graphs FOCS'17].

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Balancing graph Voronoi diagrams with one more vertex

Ducoffe

2023

Networks

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Let be a graph with unit‐length edges and nonnegative costs assigned to its vertices. Given a list of pairwise different vertices , the prioritized Voronoi diagram of with respect to is the partition of in subsets so that, for every with , a vertex is in if and only if is a closest vertex to in and there is no closest vertex to in within the subset . For every with , the load of vertex equals the sum of the costs of all vertices in . The load of equals the maximum load of a vertex in . We study the problem of adding one more vertex at the end of in order to minimize the load. This problem occurs in the context of optimally locating a new service facility (e.g., a school or a hospital) while taking into account already existing facilities, and with the goal of minimizing the maximum congestion at a site. There is a brute‐force algorithm for solving this problem in time on ‐vertex ‐edge graphs. We prove a matching time lower bound–up to sub‐polynomial factors–for the special case where and , assuming the so called Hitting Set Conjecture of Abboud et al. On the positive side, we present simple linear‐time algorithms for this problem on cliques, paths and cycles, and almost linear‐time algorithms for trees, proper interval graphs and (assuming to be a constant) bounded‐treewidth graphs.

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Fast and Compact Exact Distance Oracle for Planar Graphs

Cited by 34 publications

References 30 publications

Better Tradeoffs for Exact Distance Oracles in Planar Graphs

Better Tradeoffs for Exact Distance Oracles in Planar Graphs

Voronoi Diagrams on Planar Graphs, and Computing the Diameter in Deterministic Õ(n^5/3) Time

Balancing graph Voronoi diagrams with one more vertex

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Fast and Compact Exact Distance Oracle for Planar Graphs

Cited by 34 publications

References 30 publications

Better Tradeoffs for Exact Distance Oracles in Planar Graphs

Better Tradeoffs for Exact Distance Oracles in Planar Graphs

Voronoi Diagrams on Planar Graphs, and Computing the Diameter in Deterministic Õ(n5/3) Time

Balancing graph Voronoi diagrams with one more vertex

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Product

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Voronoi Diagrams on Planar Graphs, and Computing the Diameter in Deterministic Õ(n^5/3) Time