Greedy spanners are optimal in doubling metrics

Borradaile, Glencora; Le, Hung; Wulff-Nilsen, Christian

doi:10.1137/1.9781611975482.145

Cited by 31 publications

(63 citation statements)

References 19 publications

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“…We now show how to construct level-0 clusters and prove their invariant (I1). The construction is similar to the construction of BLW in [8]. Level-0 clusters are constructed by greedily breaking the MST into subtrees of diameter at least L 0 and at most 6L 0 .…”

Section: Light Greedy Spanners In the Euclidean Spacementioning

confidence: 99%

“…Let K be the cluster graph where each vertex of K corresponds to an -cluster and each edge of K corresponds to a spanner edge connecting the two corresponding -clusters. Let ∆ K be the degree of K. BLW showed (Lemma 3.1 in [8]) that ∆ K ≤ −O(d) when the point set is in the metric of doubling dimension d using the packing argument. By adapting their proof to the point set in R d , we can derive the upper bound ∆ K ≤ O( −d ).…”

Section: Light Greedy Spanners In the Euclidean Spacementioning

confidence: 99%

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Truly Optimal Euclidean Spanners

Solomon

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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Euclidean spanners are important geometric structures, having found numerous applications over the years. Cornerstone results in this area from the late 80s and early 90s state that for any ddimensional n-point Euclidean space, there exists a (1 + )-spanner with n · O( −d+1 ) edges and lightness (normalized weight) O( −2d ). 1 Surprisingly, the fundamental question of whether or not these dependencies on and d for small d can be improved has remained elusive, even for d = 2. This question naturally arises in any application of Euclidean spanners where precision is a necessity (thus is tiny). In the most extreme case is inverse polynomial in n, and then one could potentially improve the size and lightness bounds by factors that are polynomial in n.The state-of-the-art bounds n · O( −d+1 ) and O( −2d ) on the size and lightness of spanners are realized by the greedy spanner. In 2016, Filtser and Solomon [25] proved that, in low dimensional spaces, the greedy spanner is "near-optimal"; informally, their result states that the greedy spanner for dimension d is just as sparse and light as any other spanner but for dimension larger by a constant factor. Hence the question of whether the greedy spanner is truly optimal remained open to date.The contribution of this paper is two-fold.1 Introduction Background and motivationSparse spanners. Let P be a set of n points in R d , d ≥ 2, and consider the complete weighted graph G P = (P, P 2 ) induced by P , where the weight of any edge (x, y) ∈ P 2 is the Euclidean distance |xy| between its endpoints. Let H = (P, E) be a spanning subgraph of G P , with E ⊆ P 2 , where, as in G P , the weight function is given by the Euclidean distances. For any t ≥ 1, H is called a t-spanner for P if for every x, y ∈ P , the distance d G (x, y) between x and y in G is at most t|xy|; the parameter t is called the stretch of the spanner and the most basic goal is to get it down to 1 + , for arbitrarily small > 0, 1 The lightness of a spanner is the ratio of its weight and the MST weight. without using too many edges. Euclidean spanners were introduced in the pioneering SoCG'86 paper of Chew [16], who showed that O(n) edges can be achieved with stretch √ 10, and later improved the stretch bound to 2 [17]. The first Euclidean spanners with stretch 1 + , for an arbitrarily small > 0, were presented independently in the seminal works of Clarkson [18] (FOCS'87) and Keil [38] (see also [39]), which introduced the Θ-graph in R 2 and R 3 , and soon afterwards was generalized for any R d in [45,2]. The Θ-graph is a natural variant of the Yao graph, introduced by Yao [51] in 1982, where, roughly speaking, the space R d around each point p ∈ P is partitioned into cones of angle Θ each, and then edges are added between each point p ∈ P and its closest points in each of the cones centered around it. The Θ-graph is defined similarly, where, instead of connecting p to its closest point in each cone, we connect it to a point whose orthogonal projection to some fixed ray contained in the cone is closest to p....

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Section: Light Greedy Spanners In the Euclidean Spacementioning

confidence: 99%

Section: Light Greedy Spanners In the Euclidean Spacementioning

confidence: 99%

Truly Optimal Euclidean Spanners

Solomon

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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“…In metrics of constant doubling dimension d, Borradaile, Le and Wulff-Nilsen [15] showed that greedy spanners have lightness −O(d) , improving upon previous lightness bounds by Smid [52] and Gottlieb [28]. In Section 6, we construct a spanner oracle with strong sparsity O( −d ).…”

Section: Other Applications Of Our Techniquesmentioning

confidence: 74%

“…In the lightness analysis of greedy spanners for Euclidean and doubling metrics [15,42], the authors implicitly used sparsity of the greedy spanner to bound the weight. We observe that their analysis can be turned into a non-greedy algorithm that explicitly uses sparse spanners to construct light spanners.…”

Section: Our Techniquesmentioning

confidence: 99%

“…Our sparse spanner oracles are inspired by previous (implicit) light spanner constructions in Euclidean and doubling metrics [15,42]. Specifically, these constructions repeatedly select a specific subset of points Q in the input metric and construct a sparse spanner S for Q with two properties: (a) it preserves, up to (1 + ) factor, the distance between every pair of points of Q in range [ /2, ] for some positive real number and (b) w(S) = O(|Q| ).…”

Section: Our Techniquesmentioning

confidence: 99%

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A PTAS for subset TSP in minor-free graphs

Le¹

2020

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

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We give the first PTAS for the subset Traveling Salesperson Problem (TSP) in H-minorfree graphs. This resolves a long standing open problem in a long line of work on designing PTASes for TSP in minor-closed families initiated by Grigni, Koutsoupias and Papadimitriou in FOCS'95. The main technical ingredient in our PTAS is a construction of a nearly light subset (1 + )-spanner for any given edge-weighted H-minor-free graph. This construction is based on a necessary and sufficient condition given by sparse spanner oracles: light subset spanners exist if and only if sparse spanner oracles exist. This relationship allows us to obtain two new results: • An (1 + )-spanner with lightness O( −d+2 ) for any doubling metric of constant dimension d. This improves the earlier lightness bound −O(d) obtained by Borradaile, Le and Wulff-Nilsen [15].

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Pattern Matching in Doubling Spaces

Allair

Vigneron

2021

Lecture Notes in Computer Science

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Greedy spanners are optimal in doubling metrics

Cited by 31 publications

References 19 publications

Truly Optimal Euclidean Spanners

Truly Optimal Euclidean Spanners

A PTAS for subset TSP in minor-free graphs

Pattern Matching in Doubling Spaces

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