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Algorithms for Dynamic Computational Geometry with Applications

Laurence Boxer

TL;DR

The paper studies dynamic computational geometry for moving point objects whose coordinates are polynomials in time within Atallah's framework. It develops sequential and coarse grained parallel algorithms for three problems, namely too close, too far, and 3 aligned, leveraging lower envelopes, piecewise representations, and Manhattan distance analysis. The main results establish worst case optimal runtimes: Θ(n) for the sequential tasks Too Close and Too Far (and Θ(n/p) on CGM), and Θ(n^2) for the sequential solution to the 3 aligned problem, with the 3 aligned approach relying on piecewise decomposition and switchpoint bounds. These contributions offer provably efficient, scalable methods for safety and communication related questions in dynamic geometric settings, and provide a foundation for further parallelization on CGM architectures.

Abstract

Most of the literature of computational geometry concerns geometric properties of sets of static points. M.J. Atallah introduced dynamic computational geometry, concerned with both momentary and long-term geometric properties of sets of moving point-objects. This area of research seems to have been dormant recently. The current paper examines new problems in dynamic computational geometry.

Algorithms for Dynamic Computational Geometry with Applications

TL;DR

The paper studies dynamic computational geometry for moving point objects whose coordinates are polynomials in time within Atallah's framework. It develops sequential and coarse grained parallel algorithms for three problems, namely too close, too far, and 3 aligned, leveraging lower envelopes, piecewise representations, and Manhattan distance analysis. The main results establish worst case optimal runtimes: Θ(n) for the sequential tasks Too Close and Too Far (and Θ(n/p) on CGM), and Θ(n^2) for the sequential solution to the 3 aligned problem, with the 3 aligned approach relying on piecewise decomposition and switchpoint bounds. These contributions offer provably efficient, scalable methods for safety and communication related questions in dynamic geometric settings, and provide a foundation for further parallelization on CGM architectures.

Abstract

Most of the literature of computational geometry concerns geometric properties of sets of static points. M.J. Atallah introduced dynamic computational geometry, concerned with both momentary and long-term geometric properties of sets of moving point-objects. This area of research seems to have been dormant recently. The current paper examines new problems in dynamic computational geometry.

Paper Structure

This paper contains 16 sections, 16 theorems, 24 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Motion assumptions
  4. On more general motion
  5. Piecewise defined functions
  6. Coarse grained multicomputer
  7. "Too close" problem
  8. Sequential solution
  9. CGM solution
  10. "Too far" problem
  11. "3 aligned" problem
  12. Preliminaries
  13. Solving the 3 aligned problem
  14. Sequential solution
  15. Further remarks
  16. ...and 1 more sections

Key Result

Lemma 2.4

A continuous function $f: [0,M] \to {\mathbb R}$ has $s$ switchpoints based on $\{f_i\}_{i=1}^s$ if and only if $f$ has $s+1$ pieces based on $\{f_i\}_{i=1}^s$ on $[0,M]$.

Figures (1)

  • Figure 1: Assume we use the Euclidean or the Manhattan metric. Left: For a triangle as shown, with non-collinear vertices, $C < A + B$. Right: For collinear points as shown, $C = A + B$.

Theorems & Definitions (35)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Theorem 3.2
  • proof
  • ...and 25 more