Bipartite Matching in the Plane I'm currently working on a problem that I came across:


  You are given a set $B$ of $n$ points in the plane, and a set $R$ of $n$ points in the plane. Each point is given by its coordinates. I have to Suggest an $O(n^{3})$ time algorithm for determining if there is a perfect matching between them, with the constraint that each point in $R$ is matched to a point in $B$ point whose distance is at most 1 unit away. 


I have tried solving using the minimum distance using bipartite algorithm but could not arrive at a solution. Can anyone provide me with any ideas?
This problem has been solved by Alon Efrat, Alon Itai and Matthew J. Katz in their paper Geometry Helps in Bottleneck Matching and Related Problems. The running time of the algorithm is $O(n^{1.5} \log n)$.