Range Aggregate Maximal Points in the Plane

Abstract

In this work, we study the problem of reporting and counting maximal points in a query rectangle for a set of n integer points that lie on an n×n grid. A point is said to be maximal inside a query rectangle if it is not dominated by any other point inside the query rectangle. Our model of computation is unit-cost RAM model with word size of O(logn) bits. For the reporting version of the problem, we present a data structure of size $O(n\frac{\log n}{\log\log n})$ words and support querying in $O(\frac{\log n}{\log\log n}+k)$ time where k is the size of the output. For the counting version, we present a data structure of size $O(n\frac{\log^{2} n}{\log\log n})$ words which supports querying in $O(\frac{\log^{\frac{3}{2}}n} {\log\log n})$. Both the data structures are static in nature. The reporting version of the problem has been studied in [1] and [5]. To the best of our knowledge, this is the first sub-logarithmic result for the reporting version and the first work for the counting version of the problem.