Maximizing Range Sum in Trajectory Data

Maximizing Range Sum (MaxRS) query is a basic operation in computational geometry and database communities. Given a set of weighted objects in 2-dimensional space and a rectangle, MaxRS query aims to find an optimal position of the rectangle to maximize the total weight of covered objects (i.e., Range Sum). All the existing literature for MaxRS query commonly assumes that every object is associated with a unique point. In real applications, however, every object (e.g., GPS-enabled moving vehicle) is related to a trajectory including a sequence of points, which goes beyond this restrictive assumption. How to tackle the problem of MaxRS query in trajectory data (MaxRST) is important and challenging. In this paper, we propose the definition of MaxRST query where a trajectory is covered by a rectangle if at least one of points in the trajectory is enclosed by the rectangle. We propose a novel method to solve MaxRST query by converting it to rectilinear polygon intersection problem. Then, an interval-tree-based partitioning technique is developed to efficiently settle rectilinear polygon intersection problem. To further shorten the response time, we present ( $\epsilon, \delta$ ) -approximate MaxRST query, which returns an approximate answer having the relative error $\epsilon$ to the optimal covered weight with probability at least $\delta$ . Furthermore, two complementary sampling-based ( $\epsilon, \delta$ ) -approximate MaxRST algorithms are proposed. One performs random sampling with replacements on rectilinear polygons and the sample size is irrelevant to the number of trajectories. The other employs grid shifting technique to reduce sample size yet requires an extra cost for grid construction. The theoretical analysis and experimental results show that our proposed algorithms have high performance in terms of efficiency and accuracy.