Abstract
A spanner of a graph is a subgraph that preserves lengths of shortest paths up to a multiplicative distortion. For every k, a spanner with size O(n^{1+1/k}) and stretch (2k+1) can be constructed by a simple centralized greedy algorithm, and this is tight assuming Erdős girth conjecture.
In this paper we study the problem of constructing spanners in a local manner, specifically in the Local Computation Model proposed by Rubinfeld et al. (ICS 2011).
We provide a randomized Local Computation Agorithm (LCA) for constructing (2r1)spanners with Õ(n^{1+1/r}) edges and probe complexity of Õ(n^{11/r}) for r ∈ {2,3}, where n denotes the number of vertices in the input graph. Up to polylogarithmic factors, in both cases, the stretch factor is optimal (for the respective number of edges). In addition, our probe complexity for r = 2, i.e., for constructing a 3spanner, is optimal up to polylogarithmic factors. Our result improves over the probe complexity of Parter et al. (ITCS 2019) that is Õ(n^{11/2r}) for r ∈ {2,3}. Both our algorithms and the algorithms of Parter et al. use a combination of neighborprobes and pairprobes in the abovementioned LCAs.
For general k ≥ 1, we provide an LCA for constructing O(k²)spanners with Õ(n^{1+1/k}) edges using O(n^{2/3}Δ²) neighborprobes, improving over the Õ(n^{2/3}Δ⁴) algorithm of Parter et al.
By developing a new randomized LCA for graph decomposition, we further improve the probe complexity of the latter task to be O(n^{2/3(1.5α)/k}Δ²), for any constant α > 0. This latter LCA may be of independent interest.
BibTeX  Entry
@InProceedings{arviv_et_al:LIPIcs.APPROX/RANDOM.2023.42,
author = {Arviv, Rubi and Chung, Lily and Levi, Reut and Pyne, Edward},
title = {{Improved Local Computation Algorithms for Constructing Spanners}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
pages = {42:142:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772969},
ISSN = {18688969},
year = {2023},
volume = {275},
editor = {Megow, Nicole and Smith, Adam},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/18867},
URN = {urn:nbn:de:0030drops188671},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2023.42},
annote = {Keywords: Local Computation Algorithms, Spanners}
}
Keywords: 

Local Computation Algorithms, Spanners 
Collection: 

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023) 
Issue Date: 

2023 
Date of publication: 

04.09.2023 