STEINER POINT REMOVAL with DISTORTION /bfitO (log /bfitk ) USING the RELAXED-VORONOI ALGORITHM

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(log⁵ 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(log² 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 | )).