Minkowski-Type Theorems and Least-Squares Partitioning

Franz Aurenhammer (joint work with Boris Aronov, Friedrich Hoffmann) Institut fü Grundlagen der Informationsverarbeitung Graz, Österreich Report B 92-09 April 92

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The power diagram of $n$ weighted sites in $d$-space partitions a given $m$-point set into clusters, one cluster for each region of the diagram. In this way, an assignment of points to sites is induced. We show the equivalence of such assignments to Euclidean least-squares assignments. As a corollary, there always exists a power diagram whose regions partition a given $d$-dimensional $m$-point set into clusters of prescribed sizes, no matter where the sites are taken. Another consequence is that least-squares assignments can be computed by finding suitable weights for the sites. In the plane, this takes roughly $O(n^2 m)$ time and optimal space $O(m)$ which improves on previous methods. We further show that least-squares assignments can be computed by solving a particular linear program in $n+1$ dimensions. This leads to a gradient method for iteratively improving the weights. Aside from the obvious application, least-squares assignments are shown to be useful in solving a certain transportation problem and in finding least-squares fittings when translation and scaling are allowed. Finally, we extend the concept of least-squares assignments to continious point sets, thereby obtaining results on power diagrams with prescribed region volumes that are related to Minkowski's Theorem for convex polytopes.