A Hierarchy of Lower Bounds for Sublinear Additive Spanners

Abboud, Amir; Bodwin, Greg; Pettie, Seth

doi:10.1137/16m1105815

Cited by 49 publications

(92 citation statements)

References 33 publications

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“…, and this estimate stays the state-of-the-art. Based on [1], Abboud et al [2] showed a lower bound of β = Ω…”

Section: Introduction 1background and Our Resultsmentioning

confidence: 99%

“…Near-additive emulators and spanners were a subject of intensive research in the last two decades [2,12,13,15,17,19,20,22,27,37,38,40,45]. They found numerous applications for computing almost shortest paths and distance oracles in various computational settings [4,5,12,17,21,22].…”

Section: Introduction 1background and Our Resultsmentioning

confidence: 99%

“…A significant research effort was put into decreasing the sparsity level of near-additive emulators and spanners [2,17,37,38,40]. Pettie [38] showed that one can efficiently construct near-additive spanners of size O(n • (log log n) ϕ ), where ϕ ≈ 1.44 (with κ = log n and β = log log n ϵ ϕlog log n…”

Section: Introduction 1background and Our Resultsmentioning

confidence: 99%

“…For κ = log n this becomes O(n • log log n). Subsequent improvements in the sparsity level of near-additive emulators and spanners [2,17,27,37,38,40] optimized the degree sequence deд 0 , deд 1 , . .…”

Section: Technical Overviewmentioning

confidence: 99%

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Ultra-Sparse Near-Additive Emulators

Elkin

Matar

2021

Proceedings of the 2021 ACM Symposium on Principles of Distributed Computing

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Near-additive (aka (1 + ϵ, β)-) emulators and spanners are a fundamental graph-algorithmic construct, with numerous applications for computing approximate shortest paths and related problems in distributed, streaming and dynamic settings.Known constructions of near-additive emulators enable one to trade between their sparsity (i.e., number of edges) and the additive stretch β. Specifically, for any pair of parameters ϵ > 0, κ = 1, 2, . . . , one can have a (1 + ϵ, β)-emulator with O(n 1+ 1 κ ) edges, with β = log κ ϵ log κ

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“…, and this estimate stays the state-of-the-art. Based on [1], Abboud et al [2] showed a lower bound of β = Ω…”

Section: Introduction 1background and Our Resultsmentioning

confidence: 99%

Section: Introduction 1background and Our Resultsmentioning

confidence: 99%

Section: Introduction 1background and Our Resultsmentioning

confidence: 99%

Section: Technical Overviewmentioning

confidence: 99%

See 2 more Smart Citations

Ultra-Sparse Near-Additive Emulators

Elkin

Matar

2021

Proceedings of the 2021 ACM Symposium on Principles of Distributed Computing

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“…These lower bounds begin from the construction of graphs in which numerous pairs of vertices have shortest paths that are unique, edge-disjoint, and relatively long. Such graphs were independently discovered by Alon [4], Hesse [16], and Coppersmith and Elkin [12]; see also [1,2]. Given such a "base graph," derived graphs can be obtained through a variety of graph products such as the Table 1 Upper and Lower bounds on shortcutting sets.…”

Section: Introductionmentioning

confidence: 99%

Lower Bounds on Sparse Spanners, Emulators, and Diameter-Reducing Shortcuts

Huang¹,

Pettie²

2021

SIAM J. Discrete Math.

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We prove better lower bounds on additive spanners and emulators, which are lossy compression schemes for undirected graphs, as well as lower bounds on shortcut sets, which reduce the diameter of directed graphs. We show that any O(n)-size shortcut set cannot bring the diameter below Ω(n 1/6 ), and that any O(m)-size shortcut set cannot bring it below Ω(n 1/11 ). These improve Hesse's [16] lower bound of Ω(n 1/17 ). By combining these constructions with Abboud and Bodwin's [1] edge-splitting technique, we get additive stretch lower bounds of +Ω(n 1/13 ) for O(n)-size spanners and +Ω(n 1/18 ) for O(n)-size emulators. These improve Abboud and Bodwin's +Ω(n 1/22 ) lower bounds.

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Fast approximate shortest paths in the congested clique

et al. 2020

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We design fast deterministic algorithms for distance computation in the Congested Clique model. Our key contributions include:A $$(2+\epsilon )$$(2+ϵ)-approximation for all-pairs shortest paths in $$O(\log ^2{n} / \epsilon )$$O(log2n/ϵ) rounds on unweighted undirected graphs. With a small additional additive factor, this also applies for weighted graphs. This is the first sub-polynomial constant-factor approximation for APSP in this model.A $$(1+\epsilon )$$(1+ϵ)-approximation for multi-source shortest paths from $$O(\sqrt{n})$$O(n) sources in $$O(\log ^2{n} / \epsilon )$$O(log2n/ϵ) rounds on weighted undirected graphs. This is the first sub-polynomial algorithm obtaining this approximation for a set of sources of polynomial size. Our main techniques are new distance tools that are obtained via improved algorithms for sparse matrix multiplication, which we leverage to construct efficient hopsets and shortest paths. Furthermore, our techniques extend to additional distance problems for which we improve upon the state-of-the-art, including diameter approximation, and an exact single-source shortest paths algorithm for weighted undirected graphs in $$\tilde{O}(n^{1/6})$$O~(n1/6) rounds.

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A Hierarchy of Lower Bounds for Sublinear Additive Spanners

Cited by 49 publications

References 33 publications

Ultra-Sparse Near-Additive Emulators

Ultra-Sparse Near-Additive Emulators

Lower Bounds on Sparse Spanners, Emulators, and Diameter-Reducing Shortcuts

Fast approximate shortest paths in the congested clique

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