Euclidean TSP, Motorcycle Graphs, and Other New Applications of Nearest-Neighbor Chains

02/19/2019

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by   Alon Efrat, et al.

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We show new applications of the nearest-neighbor chain algorithm, a technique that originated in agglomerative hierarchical clustering. We apply it to a diverse class of geometric problems: we construct the greedy multi-fragment tour for Euclidean TSP in O(n n) time in any fixed dimension and for Steiner TSP in planar graphs in O(n√(n) n) time; we compute motorcycle graphs (which are a central part in straight skeleton algorithms) in O(n^4/3+ε) time for any ε>0; we introduce a narcissistic variant of the k-attribute stable matching model, and solve it in O(n^2-4/(k(1+ε)+2)) time; we give a linear-time 2-approximation for a 1D geometric set cover problem with applications to radio station placement.