An Improved Random Shift Algorithm for Spanners and Low Diameter Decompositions

TL;DR

This paper introduces a deterministic clustering algorithm that constructs efficient graph spanners and low diameter decompositions with improved bounds on running time and stretch, surpassing previous probabilistic methods.

Contribution

The authors present a clustering algorithm that guarantees deterministic bounds on spanner stretch and decomposition diameter, improving upon prior probabilistic approaches.

Findings

Constructed a spanner with stretch 2k-1 and size O(n^{1+1/k}) in k rounds.

Achieved low diameter decompositions with diameter O(log n / β) and controlled inter-cluster edge probability.

Improved bounds are independent of random choices, unlike previous high-probability guarantees.

Abstract

Spanners have been shown to be a powerful tool in graph algorithms. Many spanner constructions use a certain type of clustering at their core, where each cluster has small diameter and there are relatively few spanner edges between clusters. In this paper, we provide a clustering algorithm that, given $k\geq 2$, can be used to compute a spanner of stretch $2k-1$ and expected size $O(n^{1+1/k})$ in $k$ rounds in the CONGEST model. This improves upon the state of the art (by Elkin, and Neiman [TALG'19]) by making the bounds on both running time and stretch independent of the random choices of the algorithm, whereas they only hold with high probability in previous results. Spanners are used in certain synchronizers, thus our improvement directly carries over to such synchronizers. Furthermore, for keeping the \emph{total} number of inter-cluster edges small in low diameter decompositions, our clustering algorithm provides the following guarantees. Given $\beta\in (0,1]$, we compute a low diameter decomposition with diameter bound $O\left(\frac{\log n}{\beta}\right)$ such that each edge $e\in E$ is an inter-cluster edge with probability at most $\beta\cdot w(e)$ in $O\left(\frac{\log n}{\beta}\right)$ rounds in the CONGEST model. Again, this improves upon the state of the art (by Miller, Peng, and Xu [SPAA'13]) by making the bounds on both running time and diameter independent of the random choices of the algorithm, whereas they only hold with high probability in previous results.