Effiiently computing the smallest axis-parallel squares spanning all colors

Abstract

Abstract. For a set of colored points, a region is called color-spanning if it contains at least one point of each color. In this paper, we rst consider the problem of maintaining the smallest color-spanning interval for a set of n points with k colors on the real line, such that the insertion and deletion of an arbitrary point takes O(log 2 n) the worst-case time. Then, we exploit the data structure to show that there is O(n log 2 n) time algorithm to compute the smallest color-spanning square for a set of n points with k colors in the plane. This is a new way to improve O(nk log n) time algorithm presented by Abellanas et al. [1] when k = !(log n). We also consider the problem of computing the smallest color-spanning square in a special case in which we have, at most, two points from each color. We present O(n log n) time algorithm to solve the problem which improves the result presented by Arkin et al. [2] by a factor of log n.

Normal approximation in total variation for statistics in geometric probability

“…Maximal layers of points have been of considerable interest since [42, 29], and have a wide range of applications; see [11] for a brief review of their applications. One of the applications is the smallest colour-spanning interval [28], which is a linear function of the distances between maximal points and the edge. In this subsection, we demonstrate that Theorem 2.2 with marks can easily be applied to estimate the error of the normal approximation to the distribution of the sum of distances between different maximal layers if the points are from a Poisson point process.…”

Normal approximation in total variation for statistics in geometric probability

“…Maximal layers of points have been of considerable interest since [Rényi (1962), Kung (1975)] and have a wide range of applications, see [Chen, Hwang and Tsai (2003)] for a brief review of their applications. One of the applications is the smallest colorspanning interval [Khanteimouri et al (2017)] which is a linear function of the distances between maximal points and the edge. In this subsection, we demonstrate that Theorem 2.6 with marks can be easily applied to estimate the error of normal approximation to the distribution of the sum of distances between different maximal layers if the points are from a Poisson point process.…”