A near-linear time ε-approximation algorithm for geometric bipartite matching

For point sets A,B ⊂ Rd, |A|=|B|=n, and for a parameter ε > 0, we present an algorithm that computes, in O(n poly(log n, 1/ε)) time, an ε-approximate perfect matching of A and B with high probability; the previously best known algorithm takes Ω(n3/2) time. We approximate the Lp-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 to the length of the path. We show that the total length of the augmenting paths generated by the algorithm is O((n/ε)log n), implying that the running time of our algorithm is O(n poly(log n,1/ε)).