New (alpha, beta) Spanners and Hopsets

Uri Ben-Levy, Merav Parter

Research output: Chapter in Book/Report/Conference proceedingConference contribution


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 (distG (u, v)) for every u, v ∈ V. A simple girth argument implies that any f(d)-spanner with O(n1+1/k) edges must satisfy that f(d)/d = Ω(⌈k/d⌉). A matching upper bound (even up to constants) for super-constant values of d is currently known only for d = Ω((log k)log k) as given by the well known (1 + ϵ, β) spanners of Elkin and Peleg, and its recent improvements by [Elkin-Neiman, SODA'17], and [Abboud-Bodwin-Pettie, SODA'18].We present new spanner constructions that achieve a nearly optimal stretch of O(⌈k/d⌉) for any distance value d ∈ [1, k1−o(1)] and d ≥ k1+o(1). We also show more optimized spanner constructions with nearly linear number of edges. Specifically for every ϵ ∈ (0, 1), we show the construction of (3 + ϵ, β) spanners for β = Oϵ(klog(3+8/ϵ)) with Õϵ(n) edges.In addition, we consider the related graph concept of hopsets introduced by [Cohen, J. ACM '00]. Informally, an hopset H is a weighted edge set that, when added to the graph G, allows one to get a path from each node u to a node v with at most β hops (i.e., edges) and length at most α · distG(u, v). We present a new family of (α, β) hopsets with Õ(k · n1+1/k) edges and α · β = O(k). Turning to nearly linear-size hopsets, we show a construction of (3 + ϵ, β) hopset with Õϵ(n) edges and hop-bound of β = Oϵ((log n)log(3+9/ϵ)), improving upon the state-of-the-art hop-bound of β = O(log log n)log log n.
Original languageEnglish
Title of host publicationProceedings of the Thirty-First Annual ACM-SIAM Symposium on Discrete Algorithms(SODA'20)
PublisherAssociation for Computing Machinery (ACM)
Number of pages20
StatePublished - 5 Jan 2020
Event31st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2020 - Hilton Salt Lake City Center, Salt Lake City, United States
Duration: 5 Jan 20208 Jan 2020
Conference number: 31st

Publication series

NameSymposium on Discrete Algorithms (SODA 2020)


Conference31st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2020
Abbreviated titleSODA
Country/TerritoryUnited States
CitySalt Lake City


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