Steiner Point Removal with Distortion $O(\log {k})$ using the Relaxed-Voronoi Algorithm

Abstract

In the Steiner point removal 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 [SIAM J. Comput., 44 (2015), pp. 975--995] devised a ball-growing algorithm with exponential distributions to show that the distortion is at most $O(\log^5 k)$. Cheung [Proceedings of the 29th Annual ACM/SIAM Symposium on Discrete Algorithms, 2018, pp. 1353--1360] improved the analysis of the same algorithm, bounding the distortion by $O(\log^2 k)$. We devise a novel and simpler algorithm (called the Relaxed-Voronoi algorithm) which incurs distortion $O(\log k)$. This algorithm can be implemented in almost linear time ($O(|E|\log |V|)$).