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.

翻译：本文研究在欧几里得空间中，以有界速度运动的物体的1-中心、质心和1-中位数的计算问题。我们每单位时间只能获取恒定数量的物体（通常为一个）的精确位置，但对于其他每个物体，其潜在位置集合（称为物体的不确定性区域）仅在速度限制条件下增长。因此，物体的中心可能位于多个可能的位置，称为中心的不确定性区域。针对每个中心问题，我们设计查询策略以最小化中心不确定性区域的大小，并将其性能与了解物体轨迹但仍需通过查询来减少其不确定性的最优查询策略进行比较。对于R^1中静态情况的1-中心问题，我们提出一种每单位时间查询四个物体且与每单位时间查询一次的最优算法相比具有1-竞争性的算法。对于R^1中一般情况的1-中心问题、R^d中的质心问题以及R^1中的1-中位数问题，我们证明轮询调度算法是最优的竞争性算法。对于R^d中的重心问题，我们提供一种O(log n)-竞争性算法。此外，对于R^d（d ≥ 2）中一般情况的1-中心问题，我们认为没有任何算法能保证相对于最优算法有界竞争比。