On Density, Threshold and Emptiness Queries for Intervals in the Streaming Model.

In this paper, we study the maximum density, threshold and emptiness queries for intervals in the streaming model. input is a stream S n points in the real line R and a floating closed interval W of width alpha. specific problems we consider in this paper are as follows.- Maximum density: find a placement of W in R containing themaximum number of points of S.- Threshold query: find a placement of W in R, if it exists,that contains at least Delta elements of S. - Emptiness query: find, if possible, a placement of W within the extent of S so that the interior of W does not contain any element of S.The stream S, being huge, does not fit into main memory and can be read sequentially at most a constant number of times, usually once. The problems studied here in the geometric setting have relations to frequency estimation and heavy hitter identification in a stream of data. We provide lower bounds and results on trade-off between extra space and quality of solution. We also discuss generalizations for the higher dimensional variants for a few cases.