On Counting Range Maxima Points in Plane

Abstract

We consider the problem of reporting and counting maximal points in a given orthogonal query range in two-dimensions. Our model of computation is the pointer machine model. Let P be a static set of n points in ℝ². A point is maximal in P if it is not dominated by any other point in P. We propose a linear space data structure that can support counting the number of maximal points inside a 3-sided orthogonal query rectangle unbounded on its right in O(logn) time. For counting the number of maximal points in a 4-sided orthogonal query rectangle, we propose an O(n logn) space data structure that can be constructed in O(n logn) time and queried upon in O(logn) time. This work proposes the first data structure for counting the number of maximal points in a query range. Das et al. proposed a data structure for the counting version in the word RAM model [WALCOM 2012].

For the corresponding reporting versions, we propose a linear size data structure for reporting maximal points inside a 3-sided query range in time O(logn + k), where k is the size of the output. We propose an O(n logn) size data structure for reporting the maximal points in a 4-sided orthogonal query range in time O(logn + k), where k is the size of the output. The methods we propose for reporting maximal points are simpler than previous methods and meet the best known bounds.