Translating a regular grid over a point set

Translating a regular grid over a point set

We consider the problem of translating a (finite or infinite) square grid G over a set S of n points in the plane in order to maximize some objective function. We say that a grid cell is k-occupied if it contains k or more points of S. The main set of problems we study have to do with translating an infinite grid so that the number of k-occupied cells is maximized or minimized. For these proble...

Translating a Star over a Point Set

We consider the problem of placing a star ~ R on a set S of n points in the plane in order to maximize a given objective function. A star ~ R is a set ofm rays {r1, . . . , rm} in R, emanating from a point p such that the angle between two consecutive rays is 2π/m. The cone defined by two consecutive rays with apex p is k-occupied if it contains at least k points of S. We design algorithms to c...

Detecting all regular polygons in a point set

In this paper, we analyze the time complexity of nding regular polygons in a set of n points. We combine two di erent approaches to nd regular polygons, depending on their number of edges. Our result depends on the parameter α, which has been used to bound the maximum number of isosceles triangles that can be formed by n points. This bound has been expressed as O(n ), and the current best value...

Rigid Resolution of a Finitely Generated Module Over a Regular Local Ring

Translating the Cantor set by a random

We determine the constructive dimension of points in random translates of the Cantor set. The Cantor set “cancels randomness” in the sense that some of its members, when added to Martin-Löf random reals, identify a point with lower constructive dimension than the random itself. In particular, we find the Hausdorff dimension of the set of points in a Cantor set translate with a given constructiv...