### New (α,β) Spanners and Hopsets [article]

Uri Ben-Levy, Merav Parter
2020 arXiv   pre-print
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
more » ... ny 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=(log k)^log 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) and integer k≥ 1, there is a (3+ϵ, β)-spanner with β=O_ϵ(k^log(3+8/ϵ)) and O_k,ϵ(n^1+1/k) edges. 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_k,ϵ(n^1+1/k) edges and hop-bound of β=O_ϵ(k^log(3+9/ϵ)), improving upon the state-of-the-art hop-bound of β=O(log k /ϵ)^log k by [Elkin-Neiman, '17] and [Huang-Pettie, '17].