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 [13] used a ball-growing algorithm to show that the distortion is at most O(log5k) for general graphs. In this paper, we improve the distortion bound to O(log2k). 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 sums of independent exponential random variables, which allow us to show that it is unlikely for a non-terminal being contracted to a distant terminal.