Near-Additive Spanners and Near-Exact Hopsets, A Unified View
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Given an unweighted undirected graph G = (V,E), and a pair of parameters ϵ > 0, β = 1,2,..., a subgraph G' =(V,H), H ⊆ E, of G is a (1+ϵ,β)-spanner (aka, a near-additive spanner) of G if for every u,v ∈ V, d_G'(u,v) < (1+ϵ)d_G(u,v) + β . It was shown in <cit.> that for any n-vertex G as above, and any ϵ > 0 and κ = 1,2,..., there exists a (1+ϵ,β)-spanner G' with O_ϵ,κ(n^1+1/κ) edges, with β = β_EP = (logκϵ)^logκ - 2 . This bound remains state-of-the-art, and its dependence on ϵ (for the case of small κ) was shown to be tight in <cit.>. Given a weighted undirected graph G = (V,E,ω), and a pair of parameters ϵ > 0, β = 1,2,..., a graph G'= (V,H,ω') is a (1+ϵ,β)-hopset (aka, a near-exact hopset) of G if for every u,v ∈ V, d_G(u,v) < d_G∪ G'^(β)(u,v) < (1+ϵ)d_G(u,v) , where d_G∪ G'^(β)(u,v) stands for a β-(hop)-bounded distance between u and v in the union graph G ∪ G'. It was shown in <cit.> that for any n-vertex G and ϵ and κ as above, there exists a (1+ϵ,β)-hopset with Õ(n^1+1/κ) edges, with β = β_EP. Not only the two results of <cit.> and <cit.> are strikingly similar, but so are also their proof techniques. Moreover, Thorup-Zwick's later construction of near-additive spanners <cit.> was also shown in <cit.> to provide hopsets with analogous (to that of <cit.>) properties. In this survey we explore this intriguing phenomenon, sketch the basic proof techniques used for these results, and highlight open questions.
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