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Trajectory Minimum Touching Ball

Jeff M. Phillips, Jens Kristian Refsgaard Schou

TL;DR

This work studies the trajectory minimum touching ball (TMTB) problem: given $n$ trajectories each with at most $k$ segments, find the smallest ball intersecting all trajectories. It shows that for $k=1$ the problem is LP-type and solvable in linear time, while for larger $k$ the LP-type approach can be unbounded, yet an almost $O\left((nk)^2\log n\right)$ exact algorithm via farthest trajectory Voronoi diagrams achieves a practical solution, with a near-linear-time option via a relative approximation. The paper then develops approximation schemes based on sausage neighborhoods and ghost trajectories that extend to high dimensions, providing $(3,\rho)$- and $(1+\varepsilon)$-approximation guarantees with explicit runtimes that scale with $n$, $k$, and dimension $d$. Overall, the results delineate the limits of LP-type formulations for multi-segment trajectories and deliver scalable exact and approximate algorithms for trajectory data analysis in the plane and beyond.

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 $n$.

Trajectory Minimum Touching Ball

TL;DR

This work studies the trajectory minimum touching ball (TMTB) problem: given trajectories each with at most segments, find the smallest ball intersecting all trajectories. It shows that for the problem is LP-type and solvable in linear time, while for larger the LP-type approach can be unbounded, yet an almost exact algorithm via farthest trajectory Voronoi diagrams achieves a practical solution, with a near-linear-time option via a relative approximation. The paper then develops approximation schemes based on sausage neighborhoods and ghost trajectories that extend to high dimensions, providing - and -approximation guarantees with explicit runtimes that scale with , , and dimension . Overall, the results delineate the limits of LP-type formulations for multi-segment trajectories and deliver scalable exact and approximate algorithms for trajectory data analysis in the plane and beyond.

Abstract

We present algorithms to find the minimum radius sphere that intersects every trajectory in a set of trajectories composed of at most line segments each. When , we can reduce the problem to the LP-type framework to achieve a linear time complexity. For we provide a trajectory configuration with unbounded LP-type complexity, but also present an almost algorithm through the farthest line segment Voronoi diagrams. If we tolerate a relative approximation, we can reduce to time near-linear in .
Paper Structure (12 sections, 7 theorems, 6 equations, 9 figures, 2 algorithms)

This paper contains 12 sections, 7 theorems, 6 equations, 9 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The minimum touching ball.
  4. Reduction to LP-Type Problems
  5. Segment TMTB reduction to LP-type
  6. General TMTB does not have Bounded Combinatorial Dimension
  7. Furthest Trajectory Voronoi diagrams
  8. Approximate Trajectory Minimum Touching Ball
  9. Constant Factor Approximate TMTB
  10. $(1+\varepsilon)$-Approximate TMTB
  11. High-dimensional TMTB
  12. Figures for Lemma \ref{['lemma:traj_lp_monster']} and Theorem \ref{['thm:FTVD=MTB']}

Key Result

Lemma 1

Line segment MTB in $\mathbb{R}^d$ is an LP-type problem with combinatorial dimension $d+1$.

Figures (9)

  • Figure 1: The trajectory configuration of Lemma \ref{['lemma:traj_lp_monster']} that has unbounded LP-type dimension, each trajectory has a different color and the TMTB is visualized in black. The trajectories are supposed to be on top of each other on the horizontal segment, but for clarity of presentation, they are raised.
  • Figure 2: Farthest Trajectory Voronoi Diagram of the lower-bound construction presented in Lemma \ref{['lemma:traj_lp_monster']}, note that the TMTB is on a curved edge of the diagram - in line with Theorem \ref{['thm:FTVD=MTB']}
  • Figure 3: A trajectory sausage $S_{T,\tau}$ covered by ghost trajectories $\mathcal{G}_{T,\tau,\varepsilon}$.
  • Figure 4: The construction of Lemma \ref{['lemma:traj_lp_monster']}, without $T_1$. Left, a simple visualization, and right, with the Farthest Trajectory Voronoi Diagram.
  • Figure 5: The construction of Lemma \ref{['lemma:traj_lp_monster']}, without $T_2$. Left, a simple visualization, and right, with the FTVD.
  • ...and 4 more figures

Theorems & Definitions (7)

  • Lemma 1
  • Lemma 2
  • Theorem 3
  • Lemma 4
  • Corollary 5
  • Lemma 6
  • Theorem 7