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

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 $\mathbb {R}^1$, 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 $\mathbb {R}^1$, the centroid problem in $\mathbb {R}^d$, and the 1-median problem in $\mathbb {R}^1$, we prove that the Round-robin scheduling algorithm is the best possible competitive algorithm. For the center of mass problem in $\mathbb {R}^d$, we provide an $O(\log {n})$-competitive algorithm. In addition, for the general case of the 1-center problem in $\mathbb {R}^d$ ($d \ge 2$), we argue that no algorithm can guarantee a bounded competitive ratio against the optimal algorithm.

This work was partially funded by NSERC Discovery Grants and the Institute for Computing, Information and Cognitive Systems (ICICS) at UBC.