TY - JOUR

T1 - Linear-Size hopsets with small hopbound, and constant-hopbound hopsets in RNC

AU - Elkin, Michael

AU - Neiman, Ofer

N1 - Funding Information:
A preliminary version of this paper was published in SPAA’19 []. M. Elkin: This research was supported by the ISF grant No. (724/15). O. Neiman: Supported in part by ISF grant No. (1817/17) and by BSF grant No. 2015813.
Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2022/10/1

Y1 - 2022/10/1

N2 - Hopsets are a fundamental graph-theoretic and graph-algorithmic construct, and they are widely used for distance-related problems in a variety of computational settings. Currently existing constructions of hopsets produce hopsets either with Ω (nlog n) edges, or with a hopbound nΩ (1). In this paper we devise a construction of linear-size hopsets with hopbound (ignoring the dependence on ϵ) (log log n) loglogn+O(1). This improves the previous hopbound for linear-size hopsets almost exponentially. We also devise efficient implementations of our construction in PRAM and distributed settings. The only existing PRAM algorithm [19] for computing hopsets with a constant (i.e., independent of n) hopbound requires nΩ (1) time. We devise a PRAM algorithm with polylogarithmic running time for computing hopsets with a constant hopbound, i.e., our running time is exponentially better than the previous one. Moreover, these hopsets are also significantly sparser than their counterparts from [19]. We apply these hopsets to achieve the following online variant of shortest paths in the PRAM model: preprocess a given weighted graph within polylogarithmic time, and then given any query vertex v, report all approximate shortest paths from v in constant time. All previous constructions of hopsets require either polylogarithmic time per query or polynomial preprocessing time.

AB - Hopsets are a fundamental graph-theoretic and graph-algorithmic construct, and they are widely used for distance-related problems in a variety of computational settings. Currently existing constructions of hopsets produce hopsets either with Ω (nlog n) edges, or with a hopbound nΩ (1). In this paper we devise a construction of linear-size hopsets with hopbound (ignoring the dependence on ϵ) (log log n) loglogn+O(1). This improves the previous hopbound for linear-size hopsets almost exponentially. We also devise efficient implementations of our construction in PRAM and distributed settings. The only existing PRAM algorithm [19] for computing hopsets with a constant (i.e., independent of n) hopbound requires nΩ (1) time. We devise a PRAM algorithm with polylogarithmic running time for computing hopsets with a constant hopbound, i.e., our running time is exponentially better than the previous one. Moreover, these hopsets are also significantly sparser than their counterparts from [19]. We apply these hopsets to achieve the following online variant of shortest paths in the PRAM model: preprocess a given weighted graph within polylogarithmic time, and then given any query vertex v, report all approximate shortest paths from v in constant time. All previous constructions of hopsets require either polylogarithmic time per query or polynomial preprocessing time.

UR - http://www.scopus.com/inward/record.url?scp=85133166422&partnerID=8YFLogxK

U2 - 10.1007/s00446-022-00431-z

DO - 10.1007/s00446-022-00431-z

M3 - Article

AN - SCOPUS:85133166422

SN - 0178-2770

VL - 35

SP - 419

EP - 437

JO - Distributed Computing

JF - Distributed Computing

IS - 5

ER -