Recall that the algorithm of Section 3.2 uses two black-boxes that are impractical and have hidden To implement this algorithm for edge-generation, we must be able to compute a bidirectional mapping between an input point and the grid cell that contains the point. Using this, we will show the following lemma. Proof: We begin by proving (i) and (ii) and then use it to prove the lemma. No weights are assigned in the HK algorithm. The LR has a preprocessing step that computes the maximum cardinality matching for each piece.

Maximum cardinality bipartite matching is an important graph optimization problem with several applications.