The Greedy Spanner is Existentially Optimal

Filtser, Arnold; Solomon, Shay

doi:10.1145/2933057.2933114

Cited by 32 publications

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“…This loss in the dimension becomes more significant if we restrict the attention to Euclidean spaces, as then the comparison is between Euclidean dimension d and doubling dimension 2d, but spanners for metrics with doubling dimension 2d (or even d) tend to admit significantly weaker guarantees (as a function of and d) than the corresponding ones for d-dimensional Euclidean spaces. Consequently, this result by [25] does not resolve Questions 1 and 2 for two reasons. First, it only implies near-optimality of the greedy spanner, which, as mentioned, comes with a constant factor loss in the dimension, and this constant factor slack appears in the exponents of the size and lightness bounds.…”

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confidence: 89%

“…Existential near-optimality In PODC'16, Filtser and Solomon [25] studied the optimality of the greedy spanner in doubling metrics, which is wider than the family of low-dimensional Euclidean spaces. They showed that the greedy spanner is existentially near-optimal with respect to both the size and the lightness.…”

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confidence: 99%

“…Our lightness analysis of the greedy algorithm builds on exciting developments on light spanners from recent years, which started from the works of Gottlieb [29] and Chechik and Wulff-Nilsen [15] on nongreedy spanners. Using the result of [25], the framework of [15] was refined in the works of Borradaile, Le and Wulff-Nilsen [7,8]. As mentioned, it was shown in [8] that the greedy spanner in metrics of doubling dimension d has lightness −O (d) .…”

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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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Truly Optimal Euclidean Spanners

Solomon

2019

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

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“…Hence the greedy algorithm produces multiplicative spanners with optimal tradeoff between stretch and number of edges. (See also [39].) However, the greedy algorithm is problematic from algorithmic perspective.…”

Section: Setting Definitionsmentioning

confidence: 99%

Efficient Algorithms for Constructing Very Sparse Spanners and Emulators

Elkin

Neiman

2017

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

155

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Miller et al. [43] devised a distributed 1 algorithm in the CONGEST model, that given a parameter k = 1, 2, . . ., constructs an O(k)-spanner of an input unweighted nvertex graph with O(n 1+1/k ) expected edges in O(k) rounds of communication. In this paper we improve the result of [43], by showing a k-round distributed algorithm in the same model, that constructs a (2k − 1)-spanner with O(n 1+1/k / ) edges, with probability 1 − , for any > 0. Moreover, when k = ω(log n), our algorithm produces (still in k rounds) ultra-sparse spanners, i.e., spanners of size n(1 + o(1)), with probability 1 − o(1). To our knowledge, this is the first distributed algorithm in the CONGEST or in the PRAM models that constructs spanners or skeletons (i.e., connected spanning subgraphs) that sparse. Our algorithm can also be implemented in linear time in the standard centralized model, and for large k, it provides spanners that are sparser than any other spanner given by a known (near-)linear time algorithm.We also devise improved bounds (and algorithms realizing these bounds) for (1 + , β)-spanners and emulators. In particular, we show that for any unweighted n-vertex graph and any > 0, there exists a (1 + , ( log log n ) log log n )-emulator with O(n) edges. All previous constructions of (1 + , β)-spanners and emulators employ a superlinear number of edges, for all choices of parameters.Finally, we provide some applications of our results to approximate shortest paths' computation in unweighted graphs. * Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel. Email: elkinm@cs.bgu.ac.il. 1 They actually showed a PRAM algorithm. The distributed algorithm with these properties is implicit in [43].

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“…However, since Filtser and Solomon have shown that greedy spanners are (nearly) optimal in their lightness [10], our result implies that the greedy spanner for H-minor-free graphs is also light.…”

Section: Introductionmentioning

confidence: 66%

Minor-Free Graphs Have Light Spanners

Borradaile

Wulff-Nilsen

2017

2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)

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We show that every H-minor-free graph has a light (1+ )-spanner, resolving an open problem of Grigni and Sissokho [13] and proving a conjecture of Grigni and Hung [12]. Our lightness bound iswhere σ H = |V (H)| log |V (H)| is the sparsity coefficient of H-minor-free graphs. That is, it has a practical dependency on the size of the minor H. Our result also implies that the polynomial time approximation scheme (PTAS) for the Travelling Salesperson Problem (TSP) in H-minor-free graphs by Demaine, Hajiaghayi and Kawarabayashi [7] is an efficient PTAS whose running time is 2where O H ignores dependencies on the size of H. Our techniques significantly deviate from existing lines of research on spanners for H-minor-free graphs, but build upon the work of Chechik and Wulff-Nilsen for spanners of general graphs [6].

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The Greedy Spanner is Existentially Optimal

Cited by 32 publications

References 62 publications

Truly Optimal Euclidean Spanners

Truly Optimal Euclidean Spanners

Efficient Algorithms for Constructing Very Sparse Spanners and Emulators

Minor-Free Graphs Have Light Spanners

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