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Minimizing the Size of the Uncertainty Regions for Centers of Moving Entities

William Evans, Seyed Ali Tabatabaee

TL;DR

The paper investigates maintaining small uncertainty regions for centers of moving objects with bounded speed by designing competitive query strategies. It shows that in $\mathbb{R}^1$ the static and moving 1-center problems can be handled with a $1$-competitive or optimal Round-robin-based approach, while in higher dimensions no algorithm can guarantee a bounded competitive ratio. For the centroid problem, Round-robin is $1$-competitive and the center of mass with heterogeneous speeds admits an $O(\log n)$-competitive algorithm. The work leverages scheduling techniques such as pinwheel scheduling and Round-robin to structure queries, providing rigorous bounds on the growth of center uncertainty and identifying fundamental limits in multi-dimensional settings. These results advance understanding of real-time query strategies for moving data and have implications for dynamic facility location and tracking systems.

Abstract

In this paper, we study the problems of computing the 1-center, centroid, and 1-median of objects moving with bounded speed in Euclidean space. We can acquire the exact location of only a constant number of objects (usually one) per unit time, but for every other object, its set of potential locations, called the object's uncertainty region, grows subject only to the speed limit. As a result, the center of the objects may be at several possible locations, called the center's uncertainty region. For each of these center problems, we design query strategies to minimize the size of the center's uncertainty region and compare its performance to an optimal query strategy that knows the trajectories of the objects, but must still query to reduce their uncertainty. For the static case of the 1-center problem in R^1, we show an algorithm that queries four objects per unit time and is 1-competitive against the optimal algorithm with one query per unit time. For the general case of the 1-center problem in R^1, the centroid problem in R^d, and the 1-median problem in R^1, we prove that the Round-robin scheduling algorithm is the best possible competitive algorithm. For the center of mass problem in R^d, we provide an O(log n)-competitive algorithm. In addition, for the general case of the 1-center problem in R^d (d >= 2), we argue that no algorithm can guarantee a bounded competitive ratio against the optimal algorithm.

Minimizing the Size of the Uncertainty Regions for Centers of Moving Entities

TL;DR

The paper investigates maintaining small uncertainty regions for centers of moving objects with bounded speed by designing competitive query strategies. It shows that in the static and moving 1-center problems can be handled with a -competitive or optimal Round-robin-based approach, while in higher dimensions no algorithm can guarantee a bounded competitive ratio. For the centroid problem, Round-robin is -competitive and the center of mass with heterogeneous speeds admits an -competitive algorithm. The work leverages scheduling techniques such as pinwheel scheduling and Round-robin to structure queries, providing rigorous bounds on the growth of center uncertainty and identifying fundamental limits in multi-dimensional settings. These results advance understanding of real-time query strategies for moving data and have implications for dynamic facility location and tracking systems.

Abstract

In this paper, we study the problems of computing the 1-center, centroid, and 1-median of objects moving with bounded speed in Euclidean space. We can acquire the exact location of only a constant number of objects (usually one) per unit time, but for every other object, its set of potential locations, called the object's uncertainty region, grows subject only to the speed limit. As a result, the center of the objects may be at several possible locations, called the center's uncertainty region. For each of these center problems, we design query strategies to minimize the size of the center's uncertainty region and compare its performance to an optimal query strategy that knows the trajectories of the objects, but must still query to reduce their uncertainty. For the static case of the 1-center problem in R^1, we show an algorithm that queries four objects per unit time and is 1-competitive against the optimal algorithm with one query per unit time. For the general case of the 1-center problem in R^1, the centroid problem in R^d, and the 1-median problem in R^1, we prove that the Round-robin scheduling algorithm is the best possible competitive algorithm. For the center of mass problem in R^d, we provide an O(log n)-competitive algorithm. In addition, for the general case of the 1-center problem in R^d (d >= 2), we argue that no algorithm can guarantee a bounded competitive ratio against the optimal algorithm.
Paper Structure (21 sections, 16 theorems, 6 equations, 3 figures)

This paper contains 21 sections, 16 theorems, 6 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Model and Definitions
  3. Contribution and Organization
  4. Background
  5. The 1-Center Problem
  6. The Static Case in R1̂
  7. The General Case in R1̂
  8. The General Case in Rd̂
  9. The Centroid Problem
  10. The 1-Median Problem
  11. Conclusion
  12. Proof of Lemma \ref{['lem:sched']}
  13. Proof of Proposition \ref{['prop:1c1dOmega']}
  14. Proof of Proposition \ref{['prop:1cdiffspeed']}
  15. Proof of Proposition \ref{['prop:1csizeunb']}
  16. ...and 6 more sections

Key Result

lemma thmcounterlemma

Given a multiset of positive integers $A$ where $d(A) \leq 2$, it is possible to partition $A$ into three subsets, which are themselves multisets, such that each subset has a pinwheel schedule.

Figures (3)

  • Figure 1: The uncertainty regions for three weighted objects $A$, $B$, and $C$, along with the potential locations (uncertainty region) of their center of mass.
  • Figure 2: The uncertainty regions of the objects and the strip-like region that contains the $1$-center of all objects other than $B$ for the example and the algorithm provided in the proof of Theorem \ref{['thm:1crd']} at the time (a) before the $(n-3)$-th query and (b) after the $(5n-15)$-th query.
  • Figure 3: Illustration for the example provided in the proof of Proposition \ref{['prop:1csizeunb']}.

Theorems & Definitions (32)

  • lemma thmcounterlemma
  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem
  • proof
  • proposition thmcounterproposition
  • proposition thmcounterproposition
  • proposition thmcounterproposition
  • theorem thmcountertheorem
  • proof
  • ...and 22 more