Trajectory Minimum Touching Ball

TL;DR

This paper develops algorithms to find the smallest sphere intersecting all given trajectories, with efficient solutions for simple cases and approximate methods for complex configurations, advancing geometric intersection problems.

Contribution

It introduces algorithms for minimum intersecting sphere problems on trajectories, including LP-type reduction for simple cases and near-linear approximation algorithms for complex cases.

Findings

Linear time solution for single-segment trajectories

Unbounded LP-type complexity for multi-segment trajectories

Near-linear time approximation algorithm

Abstract

We present algorithms to find the minimum radius sphere that intersects every trajectory in a set of $n$ trajectories composed of at most $k$ line segments each. When $k=1$, we can reduce the problem to the LP-type framework to achieve a linear time complexity. For $k \geq 4$ we provide a trajectory configuration with unbounded LP-type complexity, but also present an almost $O\left((nk)^2\log n\right)$ algorithm through the farthest line segment Voronoi diagrams. If we tolerate a relative approximation, we can reduce to time near-linear in $n$.