Bottleneck Partial-Matching Voronoi Diagrams and Applications

Given two finite point sets \(A\) and \(B\) in the plane, we study the minimization of the bottleneck distance between the smaller set \(B\) and an equally-sized subset of \(A\) under translations. We relate this problem to a Voronoi-type diagram and derive polynomial bounds for its complexity that are tight in the size of \(A\). We devise efficient algorithms for the construction of such a diagram and its lexicographic variant, which generalize to higher dimensions. We use the diagram to find the optimal bottleneck matching under translations, and to compute a connecting path of minimum bottleneck cost between two positions of \(B\).