Ultra-Sparse Near-Additive Emulators
Pages 235 - 246
Abstract
Near-additive (aka (1+ε,β)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 = łeft(\fracłog κ ε \right)^łog κ . At their sparsest, these emulators employ c.n edges, for some constant c≥ 2. We tighten this bound, and show that in fact precisely n^1+1/κ edges suffice.
Supplementary Material
We prove that ultra-sparse near-additive emulators exist and describe efficient centralized as well as distributed CONGEST algorithms that construct such emulators. Given an unweighted, undirected graph G=(V,E), a graph H=(V,E?) is a near-additive emulator for G if for every pair of vertices u,v in V it holds that: d_G(u,v) \leq d_H(u,v) \leq (1+\epsilon)d_G(u,v) +\beta. We show that for every n-vertex graph G, there exists near-additive emulator that is ultra-sparse, i.e., with n+o(n) edges. We also show how one can construct such emulators efficiently in the centralized and in the distributed CONGEST model.
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Published: 23 July 2021
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