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.

We show the existence of ε-nets of size O`1 ε log log 1 ε´f or planar point sets and axis-parallel rectangular ranges. The same bound holds for points in the plane with "fat" triangular ranges, and for point sets in R 3 and axis-parallel boxes; these are the first known non-trivial bounds for these range spaces. Our technique also yields improved bounds on the size of ε-nets in the more general context considered by Clarkson and Varadarajan. For example, we show the existence of ε-nets of size O`1 ε log log log 1 ε´f or the dual range space of "fat" regions and planar point sets (where the regions are the ground objects and the ranges are subsets stabbed by points). Plugging our bounds into the technique of Brönnimann and Goodrich, we obtain improved approximation factors (computable in randomized polynomial time) for the hitting set or the set cover problems associated with the corresponding range spaces.

We study the question of finding a deepest point in an arrangement of regions, and provide a fast algorithm for this problem using random sampling, showing it sufficient to solve this problem when the deepest point is shallow. This implies, among other results, a fast algorithm for solving linear programming with violations approximately. We also use this technique to approximate the disk covering the largest number of red points, while avoiding all the blue points, given two such sets in the plane. Using similar techniques imply that approximate range counting queries have roughly the same time and space complexity as emptiness range queries.

We show that n arbitrary circles in the plane can be cut into O(n 3/2+ε ) arcs, for any ε > 0, such that any pair of arcs intersects at most once. This improves a recent result of Tamaki and Tokuyama [20]. We use this result to obtain improved upper bounds on the number of incidences between m points and n circles. An improved incidence bound is also obtained for graphs of polynomials of any constant maximum degree.

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We present an O((n + k) log(n + k))-time, O(n + k)-space algorithm for computing the furthest-site Voronoi diagram of k point sites with respect to the geodesic metric within a simple n-sided polygon.

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