Efficiently Computing The Smallest Axis-Parallel Squares Spanning All Colors

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 first 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(nlog(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 = omega(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. (C) 2017 Sharif University of Technology. All rights reserved.