Ultra-Sparse Near-Additive Emulators

Elkin, Michael; Matar, Shaked

doi:10.48550/arxiv.2106.01036

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2021

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“…We focus on additive spanners and emulators, which only allow for additive error, but in general spanners and emulators can also have multiplicative error or mixed additive/multiplicative error (e.g. very recently [EM21]).…”

Section: Introductionmentioning

confidence: 99%

Better Lower Bounds for Shortcut Sets and Additive Spanners via an Improved Alternation Product

Williams¹,

Wein²,

Zixuan³

2021

Preprint

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We obtain improved lower bounds for additive spanners, additive emulators, and diameterreducing shortcut sets. Spanners and emulators are sparse graphs that approximately preserve the distances of a given graph. A shortcut set is a set of edges that when added to a directed graph, decreases its diameter. The previous best known lower bounds for these three structures are given by Huang and Pettie [HP18]. For O(n)-sized spanners, we improve the lower bound on the additive stretch from Ω(n 1/11 ) to Ω(n 2/21 ). For O(n)-sized emulators, we improve the lower bound on the additive stretch from Ω(n 1/18 ) to Ω(n 2/29 ). For O(m)-sized shortcut sets, we improve the lower bound on the graph diameter from Ω(n 1/11 ) to Ω(n 1/8 ).Our key technical contribution, which is the basis of all of our bounds, is an improvement of a graph product known as an alternation product.

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Section: Introductionmentioning

confidence: 99%