Steiner Point Removal with distortion $O(\log k)$, using the Noisy-Voronoi algorithm

TL;DR

This paper introduces a new, simpler algorithm for the Steiner Point Removal problem that achieves a significantly improved distortion bound of O(log k) while maintaining near-linear runtime, advancing the state of the art.

Contribution

The authors present the Noisy Voronoi algorithm, which simplifies previous methods and improves distortion bounds for Steiner Point Removal to O(log k).

Findings

Achieves O(log k) distortion bound.

Runs in almost linear time, O(|E| log |V|).

Simplifies previous algorithms with better theoretical guarantees.

Abstract

In the Steiner Point Removal (SPR) problem, we are given a weighted graph $G=(V,E)$ and a set of terminals $K\subset V$ of size $k$. The objective is to find a minor $M$ of $G$ with only the terminals as its vertex set, such that distances between the terminals will be preserved up to a small multiplicative distortion. Kamma, Krauthgamer and Nguyen [SICOMP2015] devised a ball-growing algorithm with exponential distributions to show that the distortion is at most $O(\log^5 k)$. Cheung [SODA2018] improved the analysis of the same algorithm, bounding the distortion by $O(\log^2 k)$. We devise a novel and simpler algorithm (called the Noisy Voronoi algorithm) which incurs distortion $O(\log k)$. This algorithm can be implemented in almost linear time ($O(|E|\log |V|)$).