Ultra-Sparse Near-Additive Emulators

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(n1+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 n1+1/κ edges suffice.