Near-Additive Spanners and Near-Exact Hopsets, A Unified View

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Abstract

Given an unweighted undirected graph G = (V, E), and a pair of parameters ✏ > 0, = 1,2,..., a subgraph G0 = (V,H), H ✓ E, of G is a (1 + ✏,)-spanner (aka, a near- additive spanner) of G if for every u, v 2 V , dG0 (u, v)  (1 + ✏)dG(u, v) + .
It was shown in [25] that for any n-vertex G as above, and any ✏ > 0 and  = 1, 2, . . ., there exists a (1 + ✏, )-spanner G0 with O✏,(n1+1/) edges, with log  !log 2 = EP = ✏ .
This bound remains state-of-the-art, and its dependence on ✏ (for the case of small ) was shown to be tight in [3].
Given a weighted undirected graph G = (V, E, !), and a pair of parameters ✏ > 0, =1,2,...,agraphG0 =(V,H,!0)isa(1+✏,)-hopset(aka,anear-exacthopset)ofGif
for every u, v 2 V, where d() 0 (u, v) stands for a -(hop)-bounded distance between u and v in the union graph dG(u,v)d() 0(u,v)(1+✏)dG(u,v), G[G 0 G[G
G[G. Itwasshownin[22]thatforanyn-vertexGand✏andasabove,thereexistsa (1 + ✏, )-hopset with O ̃(n1+1/) edges, with = EP.
Not only the two results of [25] and [22] are strikingly similar, but so are also their proof techniques. Moreover, Thorup-Zwick’s later construction of near-additive spanners [41] was also shown in [24, 29] to provide hopsets with analogous (to that of [41]) properties.
In this survey we explore this intriguing phenomenon, sketch the basic proof techniques used for these results, and highlight open questions.
Original languageEnglish
JournalEuropean association for theoretical computer science bulletin
Volume130
StatePublished - Feb 2020

Keywords

  • Computer Science - Data Structures and Algorithms

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