Covering a set of line segments with a few squares

TL;DR

This paper investigates three geometric covering problems involving line segments and trajectories with small axis-parallel squares, focusing on decision, data structure, and optimization aspects in the plane.

Contribution

It introduces new problems inspired by trajectory analysis and develops solutions for covering line segments and subtrajectories with a limited number of small squares.

Findings

Decidability of covering line segments with four squares.

Data structure for efficient subtrajectory coverability queries.

Algorithm for longest subtrajectory covered by two squares.

Abstract

We study three covering problems in the plane. Our original motivation for these problems come from trajectory analysis. The first is to decide whether a given set of line segments can be covered by up to four unit-sized, axis-parallel squares. The second is to build a data structure on a trajectory to efficiently answer whether any query subtrajectory is coverable by up to three unit-sized axis-parallel squares. The third problem is to compute a longest subtrajectory of a given trajectory that can be covered by up to two unit-sized axis-parallel squares.