Approximate Range Counting Revisited

We study range-searching for colored objects, where one has to count (approximately) the number of colors present in a query range. The problems studied mostly involve orthogonal range-searching in two and three dimensions, and the dual setting of rectangle stabbing by points. We present optimal and near-optimal solutions for these problems. Most of the results are obtained via reductions to the approximate uncolored version, and improved data-structures for them. An additional contribution of this work is the introduction of nested shallow cuttings

Independent Range Sampling, Revisited

In the independent range sampling (IRS) problem, given an input set P of n points in R^d, the task is to build a data structure, such that given a range R and an integer t >= 1, it returns t points that are uniformly and independently drawn from P cap R. The samples must satisfy inter-query independence, that is, the samples returned by every query must be independent of the samples returned by all the previous queries. This problem was first tackled by Hu, Qiao and Tao in 2014, who proposed optimal structures for one-dimensional dynamic IRS problem in internal memory and one-dimensional static IRS problem in external memory. In this paper, we study two natural extensions of the independent range sampling problem. In the first extension, we consider the static IRS problem in two and three dimensions in internal memory. We obtain data structures with optimal space-query tradeoffs for 3D halfspace, 3D dominance, and 2D three-sided queries. The second extension considers weighted IRS problem. Each point is associated with a real-valued weight, and given a query range R, a sample is drawn independently such that each point in P cap R is selected with probability proportional to its weight. Walker\u27s alias method is a classic solution to this problem when no query range is specified. We obtain optimal data structure for one dimensional weighted range sampling problem, thereby extending the alias method to allow range queries

The Space Complexity of 2-Dimensional Approximate Range Counting

We study the problem of $2$-dimensional orthogonal range counting with additive error. Given a set $P$ of $n$ points drawn from an $n\times n$ grid and an error parameter \eps, the goal is to build a data structure, such that for any orthogonal range $R$, it can return the number of points in $P\cap R$ with additive error \eps n. A well-known solution for this problem is the {\em \eps-approximation}, which is a subset $A\subseteq P$ that can estimate the number of points in $P\cap R$ with the number of points in $A\cap R$. It is known that an \eps-approximation of size O(\frac{1}{\eps} \log^{2.5} \frac{1}{\eps}) exists for any $P$ with respect to orthogonal ranges, and the best lower bound is \Omega(\frac{1}{\eps} \log \frac{1}{\eps}). The \eps-approximation is a rather restricted data structure, as we are not allowed to store any information other than the coordinates of the points in $P$. In this paper, we explore what can be achieved without any restriction on the data structure. We first describe a simple data structure that uses O(\frac{1}{\eps}(\log^2\frac{1} {\eps} + \log n) ) bits and answers queries with error \eps n. We then prove a lower bound that any data structure that answers queries with error \eps n must use \Omega(\frac{1}{\eps}(\log^2\frac{1} {\eps} + \log n) ) bits. Our lower bound is information-theoretic: We show that there is a collection of $2^{\Omega(n\log n)}$ point sets with large {\em union combinatorial discrepancy}, and thus are hard to distinguish unless we use $\Omega(n\log n)$ bits.Comment: 19 pages, 5 figure