Minkowski-Type Theorems and Least-Squares Clustering

Abstract

Dissecting Euclidean d-space with the power diagram of n weighted point sites partitions a given m-point set into clusters, one cluster for each region of the diagram. In this manner, an assignment of points to sites is induced. We show the equivalence of such assignments to constrained 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 placed. Another consequence is that constrained 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 a constrained least-squares assignment can be computed by solving a specially structured linear program in n + 1 dimensions. This leads to an algorithm for iteratively improving the weights, based on the gradient-descent method. Besides having the obvious optimization property, least-squares assignments are shown to be useful in solving a certain transportation problem, and in finding a least-squares fitting of two point sets where translation and scaling are allowed. Finally, we extend the concept of a constrained least-squares assignment to continuous distributions of points, thereby obtaining existence results for power diagrams with prescribed region volumes. These results are related to Minkowski's theorem for convex polytopes. The aforementioned iterative method for approximating the desired power diagram applies to continuous distributions as well.

The Continuous Assignment Problem and Its Application to Preemptive and Non-Preemptive Scheduling with Irregular Cost Functions

“…Thus, CAP shows that there exists a polyhedron whose facets have prescribed orientation V i / V i and d-volume V i within the prism T × R always exists when the absolute value of the last coordinate of i v i is equal to the d-volume vol(T ). Thus, this result -which was previously stated in [3] can be seen as a projective variant of Minkowski's theorem. 4 Application to earliness-tardiness scheduling…”

“…A closely related problem is the capacity constrained least square assignment problem in which the space has to be partitioned into a given number of regions of prescribed volumes. As shown by Aurenhammer et al [3], these theorems are related to a theorem by Minkowski (see e.g. [10]) about the existence of a convex polytope subject to some constraints on its facets.…”

“…For example (see Figure 3), with T = [0, 8] (d = 1), we can define f 0 (t) = f 1 (t) = t and f 2 (t) = f 3 (t) = 6 − t and p i = 2 for any i ∈ J = {0, 1, 2, 3}. The vector u u u = (u 1 , u 2 , u 3 ) T = 0 corresponds to a degenerated dual solution with T u u u ({0, 1}) = (0, 3) and T u u u ({1, 2}) = (3,8). We have…”

“…Minkowski's theorem can be stated as follows (this the formulation given by Aurenhammer et al [3]). Let V be a collection of n + 1 non-zero non-parallel vector that span R d+1 and sum up to zero.…”

Measuring the Compactness of Political Districting Plans

“…This is a complex combinatorial problem for which existing algorithms are inadequate. We solve this problem by showing that optimal districting plans are akin to so-called power diagrams 17 and then modifying an algorithm presented in Aurenhammer, Hoffmann, and Aronov (1998) to create a power diagram. The key ingredient in the algorithm is the centroid, or geometric center, of existing districts, 18 a point that is provided in census data from the GeoLytics database.…”