Range Reporting for Moving Points on a Grid

TL;DR

This paper introduces a new data structure for efficient orthogonal range reporting on moving points on a grid, significantly improving query times compared to previous models.

Contribution

It presents a novel data structure that leverages grid constraints to reduce query time for moving points, surpassing existing kinetic model bounds.

Findings

Query time is $O(\sqrt{rac{\log U}{\log \log U}} + k)$, improving over the $\Omega(\log n)$ lower bound.

The data structure is efficient for points moving along linear trajectories on a $U imes U$ grid.

Methods may be of independent interest for related computational geometry problems.

Abstract

In this paper we describe a new data structure that supports orthogonal range reporting queries on a set of points that move along linear trajectories on a $U\times U$ grid. The assumption that points lie on a $U\times U$ grid enables us to significantly decrease the query time in comparison to the standard kinetic model. Our data structure answers queries in $O(\sqrt{\log U/\log \log U}+k)$ time, where $k$ denotes the number of points in the answer. The above improves over the $\Omega(\log n)$ lower bound that is valid in the infinite-precision kinetic model. The methods used in this paper could be also of independent interest.