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

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 $O(nlog n)$ time in any fixed dimension and for Steiner TSP in planar graphs in $O(nsqrt{n}log n)$ time; we compute motorcycle graphs (which are a central part in straight skeleton algorithms) in $O(n^{4/3+varepsilon})$ time for any $varepsilonu003e0$; we introduce a narcissistic variant of the $k$-attribute stable matching model, and solve it in $O(n^{2-4/(k(1+varepsilon)+2)})$ time; we give a linear-time $2$-approximation for a 1D geometric set cover problem with applications to radio station placement.