Steiner Point Removal with distortion O( k), using the Noisy-Voronoi algorithm
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In the Steiner Point Removal (SPR) problem, we are given a weighted graph G=(V,E) and a set of terminals K⊂ 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(^5 k). Cheung [SODA2018] improved the analysis of the same algorithm, bounding the distortion by O(^2 k). We devise a novel and simpler algorithm (called the Noisy Voronoi algorithm) which incurs distortion O( k). This algorithm can be implemented in almost linear time (O(|E| |V|)).
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