Almost optimal distance oracles for planar graphs

Charalampopoulos, Panagiotis; Gawrychowski, Paweł; Mozes, Shay; Weimann, Oren

doi:10.1145/3313276.3316316

Cited by 30 publications

(56 citation statements)

References 46 publications

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“…Unfortunately, this is not true in general. However, non-simple holes do not pose a significant obstacle, and can be avoided by suitably extending the graphs, as discussed numerous times in the past, see e.g., [23,53,56,72].…”

Section: :6mentioning

confidence: 99%

“…There is a dual representation VD * (S, ω) of Voronoi diagram VD(S, ω) as a tree of constant degree with O(|S|) vertices and edges [41]. An efficient FR-Dijkstra based algorithm for computing VD * (S, ω) was presented by Charalampopoulos et al [23].…”

Section: Single-source Shortest Paths and Strong Connectivity In Dynamic Planar Graphsmentioning

confidence: 99%

“…To the best of our knowledge, this result constitutes the first known application of additively weighted Voronoi diagrams machinery (first introduced by Cabello [20]) in dynamic graph algorithms. More specifically, it is obtained by combining fully dynamic maintenance of r-divisions [66,75], the shortest paths algorithm for dense distance graphs [38], the recent efficient construction of dual Voronoi diagrams [23] via FR-Dijkstra [38], and the efficient point location data structure for Voronoi diagrams [41].…”

Section: Introductionmentioning

confidence: 99%

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Single-source shortest paths and strong connectivity in dynamic planar graphs

Charalampopoulos

Karczmarz

2022

Journal of Computer and System Sciences

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Efficient algorithms for computing and processing additively weighted Voronoi diagrams on planar graphs have been instrumental in obtaining several recent breakthrough results, most notably the almost-optimal exact distance oracle for planar graphs [Charalampopoulos et al., STOC'19], and subquadratic algorithms for planar diameter [Cabello, SODA'17, Gawrychowski et al., SODA'18]. In this paper, we show how Voronoi diagrams can be useful in obtaining dynamic planar graph algorithms and apply them to classical problems such as dynamic single-source shortest paths and dynamic strongly connected components.First, we give a fully dynamic single-source shortest paths data structure for planar weighted digraphs with O(n 4/5 ) worst-case update time and O(log 2 n) query time. Here, a single update can either change the graph by inserting or deleting an edge, or reset the source s of interest. All known non-trivial planarity-exploiting exact dynamic single-source shortest paths algorithms to date had polynomial query time. Further, note that a data structure with strongly sublinear update time capable of answering distance queries between all pairs of vertices in polylogarithmic time would refute the APSP conjecture [Abboud and Dahlgaard, FOCS'16].Somewhat surprisingly, the Voronoi diagram based approach we take for single-source shortest paths can also be used in the fully dynamic strongly connected components problem. In particular, we obtain a data structure maintaining a planar digraph under edge insertions and deletions, capable of returning the identifier of the strongly connected component of any query vertex. The worst-case update and query time bounds are the same as for our single-source distance oracle. To the best of our knowledge, this is the first fully dynamic strong-connectivity algorithm achieving both sublinear update time and polylogarithmic query time for an important class of digraphs.

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Section: :6mentioning

confidence: 99%

Section: Single-source Shortest Paths and Strong Connectivity In Dynamic Planar Graphsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Single-source shortest paths and strong connectivity in dynamic planar graphs

Charalampopoulos

Karczmarz

2022

Journal of Computer and System Sciences

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“…There have been recent developments in exact distance oracles for planar graphs [8], not discussed in the introduction. The ideas from [8] can be combined with the ideas of the current paper to yield improvements on the tradeoff for our problem.…”

Section: Final Remarksmentioning

confidence: 99%

Exact Distance Oracles for Planar Graphs with Failing Vertices

Charalampopoulos¹,

Mozes²,

Tebeka³

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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We consider exact distance oracles for directed weighted planar graphs in the presence of failing vertices. Given a source vertex u, a target vertex v and a set X of k failed vertices, such an oracle returns the length of a shortest u-to-v path that avoids all vertices in X. We propose oracles that can handle any number k of failures. More specifically, for a directed weighted planar graph with n vertices, any constant k, and for any q ∈ [1,√ n], we propose an oracle of sizeÕ( n k+3/2 q 2k+1 ) that answers queries inÕ(q) time. 1 In particular, we show anÕ(n)-size, O( √ n)-query-time oracle for any constant k. This matches, up to polylogarithmic factors, the fastest failure-free distance oracles with nearly linear space. For single vertex failures (k = 1), ourÕ( n 5/2 q 3 )-size,Õ(q)-query-time oracle improves over the previously best known tradeoff of Baswana et al. [SODA 2012] by polynomial factors for q = Ω(n t ), t ∈ (1/4, 1/2]. For multiple failures, no planarity exploiting results were previously known. * This work was partially supported by Israel Science Foundation (ISF) grants 794/13 and 592/17. 1 TheÕ(·) notation hides polylogarithmic factors.

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“…Indeed, the above question has been raised repeatedly, even after [6], including in the best paper talk of Cabello at SODA 2017 [22]. The search for an answer to this question has been remarkably fruitful from the viewpoint of upper bounds; Cabello's breakthrough (a subquadratic time algorithm for computing the diameter of a planar graph) came following attempts at proving a quadratic lower bound (such a lower bound holds in sparse but non-planar graphs [75]), and the techniques introduced in his work (mainly Abstract Voronoi Diagrams) have led to major breakthroughs in planar graph algorithms (see [25,28,40,41]).…”

Section: Introductionmentioning

confidence: 99%

New hardness results for planar graph problems in p and an algorithm for sparsest cut

Abboud

Cohen-Addad

Klein

2020

Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing

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The Sparsest Cut is a fundamental optimization problem that has been extensively studied. For planar inputs the problem is in P and can be solved inÕ(n 3) time if all vertex weights are 1. Despite a significant amount of effort, the best algorithms date back to the early 90's and can only achieve O(log n)-approximation inÕ(n) time or a constant factor approximation inÕ(n 2) time [Rao, STOC92]. Our main result is an Ω(n 2−ε) lower bound for Sparsest Cut even in planar graphs with unit vertex weights, under the (min, +)-Convolution conjecture, showing that approximations are inevitable in the near-linear time regime. To complement the lower bound, we provide a constant factor approximation in near-linear time, improving upon the 25-year old result of Rao in both time and accuracy. Our lower bound accomplishes a repeatedly raised challenge by being the first fine-grained lower bound for a natural planar graph problem in P. Moreover, we prove near-quadratic lower bounds under SETH for variants of the closest pair problem in planar graphs, and use them to show that the popular Average-Linkage procedure for Hierarchical Clustering cannot be simulated in truly subquadratic time. At the core of our constructions is a diamond-like gadget that also settles the complexity of Diameter in distributed planar networks. We prove an Ω(n/ log n) lower bound on the number of communication rounds required to compute the weighted diameter of a network in the CONGEST model, even when the underlying graph is planar and all nodes are D = 4 hops away from each other. This is the first poly(n) + ω(D) lower bound in the planar-distributed setting, and it complements the recent poly(D, log n) upper bounds of Li and Parter [STOC 2019] for (exact) unweighted diameter and for (1 + ε) approximate weighted diameter.

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Almost optimal distance oracles for planar graphs

Cited by 30 publications

References 46 publications

Single-source shortest paths and strong connectivity in dynamic planar graphs

Single-source shortest paths and strong connectivity in dynamic planar graphs

Exact Distance Oracles for Planar Graphs with Failing Vertices

New hardness results for planar graph problems in p and an algorithm for sparsest cut

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