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

Abstract

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 ( n poly(log n , 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)log n ), implying that the running time of our algorithm is O ( n poly(log n , 1/ε)).