Near-Optimal Light Spanners

Chechik, Shiri; Wulff-Nilsen, Christian

doi:10.1137/1.9781611974331.ch63

Cited by 21 publications

(78 citation statements)

References 15 publications

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“…Is the lightness analysis of [ENS14] for the greedy spanner optimal, or can one refine it to derive a stronger bound? In particular, is the spanner of [CW16] lighter than the greedy spanner?…”

Section: Introductionmentioning

confidence: 99%

The Greedy Spanner is Existentially Optimal

Filtser

Solomon

2016

Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing

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The greedy spanner is arguably the simplest and most well-studied spanner construction. Experimental results demonstrate that it is at least as good as any other spanner construction, in terms of both the size and weight parameters. However, a rigorous proof for this statement has remained elusive.In this work we fill in the theoretical gap via a surprisingly simple observation: The greedy spanner is existentially optimal (or existentially near-optimal) for several important graph families. Focusing on the weight parameter, the state-of-the-art spanner constructions for both general graphs (due to Chechik and Wulff-Nilsen [SODA'16]) and doubling metrics (due to Gottlieb [FOCS'15]) are complex. Plugging our observation on these results, we conclude that the greedy spanner achieves near-optimal weight guarantees for both general graphs and doubling metrics, thus resolving two longstanding conjectures in the area.Further, we observe that approximate-greedy algorithms are existentially near-optimal as well. Consequently, we provide an O(n log n)-time construction of (1 + )-spanners for doubling metrics with constant lightness and degree. Our construction improves Gottlieb's construction, whose runtime is O(n log 2 n) and whose number of edges and degree are unbounded, and remarkably, it matches the state-of-the-art Euclidean result (due to Gudmundsson et al. [SICOMP'02]) in all the involved parameters (up to dependencies on and the dimension).

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Section: Introductionmentioning

confidence: 99%

The Greedy Spanner is Existentially Optimal

Filtser

Solomon

2016

Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing

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“…At a high level, our proof shares several ideas with the work of Chechik and Wulff-Nilsen [6] who prove that (general) graphs have (2k − 1) · (1 + )-spanners with lightness O (n 1/k ), removing a factor of k/ log k from the previous best-known bound and matching Erdős's girth conjecture [9] up to a 1 + factor. Our work differs from Chechik and Wulff-Nilsen in two major aspects.…”

Section: Techniquesmentioning

confidence: 60%

“…where 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].…”

Section: Introductionmentioning

confidence: 99%

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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“…A possible approach would be to further improve the bound in Lemma 7.1 toÕ(log k/ρ), combined with an improved light spanner construction. In a very recent work (published in these proceedings), [14] construct a spanner with (2t − 1) · (1 + ) distortion and O (n 1/t ) lightness. Choosing t = log n and = O(1), they obtain a spanner with distortion O(log n) and constant lightness.…”

Section: Lower Bound On the Trade-off Betweenmentioning

confidence: 99%

On Notions of Distortion and an Almost Minimum Spanning Tree with Constant Average Distortion

Bartal

Filtser

Neiman

2015

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

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Minimum Spanning Trees of weighted graphs are fundamental objects in numerous applications. In particular in distributed networks, the minimum spanning tree of the network is often used to route messages between network nodes. Unfortunately, while being most efficient in the total cost of connecting all nodes, minimum spanning trees fail miserably in the desired property of approximately preserving distances between pairs. While known lower bounds exclude the possibility of the worst case distortion of a tree being small, it was shown in [4] that there exists a spanning tree with constant average distortion. Yet, the weight of such a tree may be significantly larger than that of the MST. In this paper, we show that any weighted undirected graph admits a spanning tree whose weight is at most (1 + ρ) times that of the MST, providing constant average distortion O(1/ρ 2 ). 1 The constant average distortion bound is implied by a stronger property of scaling distortion, i.e., improved distortion for smaller fractions of the pairs. The result is achieved by first showing the existence of a low weight spanner with small prioritized distortion, a property allowing to prioritize the nodes whose associated distortions will be improved. We show that prioritized distortion is essentially equivalent to coarse scaling distortion via a general transformation, which has further implications and may be of independent interest. In particular, we obtain an embedding for arbitrary metrics into Eu-

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Near-Optimal Light Spanners

Abstract: A spanner H of a weighted undirected graph G is a "sparse" subgraph that approximately preserves

Cited by 21 publications

References 15 publications

The Greedy Spanner is Existentially Optimal

The Greedy Spanner is Existentially Optimal

Minor-Free Graphs Have Light Spanners

On Notions of Distortion and an Almost Minimum Spanning Tree with Constant Average Distortion

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