Categorical Range Reporting with Frequencies

In this paper, we consider a variant of the color range reporting problem called color reporting with frequencies. Our goal is to pre-process a set of colored points into a data structure, so that given a query range Q, we can report all colors that appear in Q, along with their respective frequencies. In other words, for each reported color, we also output the number of times it occurs in Q. We describe an external-memory data structure that uses O(N(1+log^2D/log N)) words and answers one-dimensional queries in O(1 +K/B) I/Os, where N is the total number of points in the data structure, D is the total number of colors in the data structure, K is the number of reported colors, and B is the block size. Next we turn to an approximate version of this problem: report all colors sigma that appear in the query range; for every reported color, we provide a constant-factor approximation on its frequency. We consider color reporting with approximate frequencies in two dimensions. Our data structure uses O(N) space and answers two-dimensional queries in O(log_B N +log^*B + K/B) I/Os in the special case when the query range is bounded on two sides. As a corollary, we can also answer one-dimensional approximate queries within the same time and space bounds

Stabbing Colors in One Dimension

Given n horizontal segments, each associated with a color from [σ], the Categorical Segment Stabbing problem is to find the distinct K colors stabbed by a vertical line. When the end-points of the segments are distinct and lie in [1, 2n], we present an (2 + ϵ)n log σ + O(n)-bit index with O(K/ϵ) query time, where ϵ(0, 1].When the end-points are arbitrary real numbers, a standard reduction to the above scenario improves the existing bounds of Janardan and Lopez. We also present results for few other variations: • reporting the top-k colors that are stabbed, where each color has a fixed priority. • handling these scenarios when the given segments form a tree range