Steiner Point Removal - Distant Terminals Don't (Really) Bother

Given a weighted graph G = (V,E,w) with a set of k terminals T ⊂ V , the Steiner Point Removal problem seeks for a minor of the graph with vertex set T , such that the distance between every pair of terminals is preserved within a small multiplicative distortion. Kamma, Krauthgamer and Nguyen (SODA 2014, SICOMP 2015) used a ball-growing algorithm to show that the distortion is at most O(log k) for general graphs. In this paper, we improve the distortion bound to O(log k). The improvement is achieved based on a known algorithm that constructs terminal-distance exact-preservation minor with O(k4) (which is independent of |V |) vertices, and also two tail bounds on the sum of independent exponential random variables, which allow us to show that it is unlikely for a non-terminal being contracted to a distant terminal.