# New (α,β) Spanners and Hopsets

An f(d)-spanner of an unweighted n-vertex graph G=(V,E) is a subgraph H satisfying that dist_H(u, v) is at most f(dist_G(u, v)) for every u,v ∈ V. We present new spanner constructions that achieve a nearly optimal stretch of O( k /d ) for any distance value d ∈ [1,k^1-o(1)], and d ≥ k^1+o(1). We show the following: 1. There exists an f(d)-spanner H ⊆ G with f(d)≤ 7k for any d ∈ [1,√(k)/2] with expected size O_k(n^1+1/k). This in particular gives (α,β) spanners with α=O(√(k)) and β=O(k). 2. For any ϵ∈ (0,1/2], there exists an (α,β)-spanner with α=O(k^ϵ), β=O_ϵ(k) and of expected size O_k(n^1+1/k). This implies a stretch of O( k/d ) for any d ∈ [√(k)/2, k^1-ϵ], and for every d≥ k^1+ϵ. In particular, it provides a constant stretch already for vertex pairs at distance k^1+o(1) (improving upon d=( k)^ k that was known before). Up to the o(1) factor in the exponent, and the constant factor in the stretch, this is the best possible by the girth argument. 3. For any ϵ∈ (0,1), there is a (3+ϵ, β)-spanner with β=O_ϵ(k^(3+8/ϵ)). We also consider the related graph concept of hopsets introduced by [Cohen, J. ACM '00]. We present a new family of (α,β) hopsets with O(k · n^1+1/k) edges and α·β=O(k). Most notably, we show a construction of (3+ϵ,β) hopset with O_ϵ(n) edges and hop-bound of β=O_ϵ(( n)^(3+9/ϵ)), improving upon the state-of-the-art hop-bound of β=O( n)^ n by [Elkin-Neiman, '17] and [Huang-Pettie, '17]

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