TY - JOUR

T1 - Hopsets with constant hopbound, and applications to approximate shortest paths

AU - Elkin, Michael

AU - Neiman, Ofer

N1 - Funding Information:
∗Received by the editors January 23, 2018; accepted for publication (in revised form) June 6, 2019; published electronically August 29, 2019. A preliminary version of this paper [EN16a] appeared in FOCS’16. https://doi.org/10.1137/18M1166791 Funding: The first author was supported by ISF grant (724/15). The second author was supported in part by ISF grant (523/12) and BSF grant 2015813. †Computer Science, Ben-Gurion University, Beer-Sheva 84105, Israel (elkinm@cs.bgu.ac.il, neimano@cs.bgu.ac.il).
Publisher Copyright:
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - A (β, ϵ)-hopset for a weighted undirected n-vertex graph G = (V, E) is a set of edges, whose addition to the graph guarantees that every pair of vertices has a path between them that contains at most β edges, whose length is within 1 + ϵ of the shortest path. In her seminal paper, Cohen [J. ACM, 47 (2000), pp. 132-166] introduced the notion of hopsets in the context of parallel computation of approximate shortest paths, and since then it has found numerous applications in various settings, such as dynamic graph algorithms, distributed computing, and the streaming model. Cohen [J. ACM, 47 (2000), pp. 132-166] devised efficient algorithms for constructing hopsets with polylogarithmic in n number of hops. Her constructions remain the state of the art since the publication of her paper in the proceedings of STOC'94, i.e., for more than two decades. In this paper we exhibit the first construction of sparse hopsets with a constant number of hops. We also find efficient algorithms for hopsets in various computational settings, improving the best-known constructions. Generally, our hopsets strictly outperform the hopsets of [J. ACM, 47 (2000), pp. 132-166] in terms of both their parameters and the resources required to construct them. We demonstrate the applicability of our results for the fundamental problem of computing approximate shortest paths from s sources. Our results improve the running time for this problem in the parallel, distributed, and streaming models for a vast range of s.

AB - A (β, ϵ)-hopset for a weighted undirected n-vertex graph G = (V, E) is a set of edges, whose addition to the graph guarantees that every pair of vertices has a path between them that contains at most β edges, whose length is within 1 + ϵ of the shortest path. In her seminal paper, Cohen [J. ACM, 47 (2000), pp. 132-166] introduced the notion of hopsets in the context of parallel computation of approximate shortest paths, and since then it has found numerous applications in various settings, such as dynamic graph algorithms, distributed computing, and the streaming model. Cohen [J. ACM, 47 (2000), pp. 132-166] devised efficient algorithms for constructing hopsets with polylogarithmic in n number of hops. Her constructions remain the state of the art since the publication of her paper in the proceedings of STOC'94, i.e., for more than two decades. In this paper we exhibit the first construction of sparse hopsets with a constant number of hops. We also find efficient algorithms for hopsets in various computational settings, improving the best-known constructions. Generally, our hopsets strictly outperform the hopsets of [J. ACM, 47 (2000), pp. 132-166] in terms of both their parameters and the resources required to construct them. We demonstrate the applicability of our results for the fundamental problem of computing approximate shortest paths from s sources. Our results improve the running time for this problem in the parallel, distributed, and streaming models for a vast range of s.

KW - Graph algorithms

KW - Hopset

KW - Shortest path

UR - http://www.scopus.com/inward/record.url?scp=85072728837&partnerID=8YFLogxK

U2 - 10.1137/18M1166791

DO - 10.1137/18M1166791

M3 - Article

AN - SCOPUS:85072728837

VL - 48

SP - 1436

EP - 1480

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

SN - 0097-5397

IS - 4

ER -