A Near-linear Time ε-Approximation Algorithm for Geometric Bipartite Matching.

For point sets A, B ⊂ R^d, |A| = |B| = n, and for a parameter ε > 0, we present a Monte Carlo algorithm that computes, in O(npoly(logn, 1/ε)) time, an ε-approximate perfect matching of A and B under any L_p-norm with high probability; the previously best-known algorithm takes Ω(n^3/2) time. We approximate the L_p-norm using a distance function, d(·, ·) based on a randomly shifted quad-tree. The algorithm iteratively generates an approximate minimum-cost augmenting path under d(·, ·) in time proportional, within a polylogarithmic factor, to the length of the path. We show that the total length of the augmenting paths generated by the algorithm is O((n/ε) logn), implying that the running time of our algorithm is O(npoly(logn, 1/ε)). [ABSTRACT FROM AUTHOR]