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 works as well as 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：本文研究了在欧几里得空间中，对以有界速度运动的物体计算其1-中心、质心和1-中位数的问题。我们每单位时间只能获取常数个物体（通常为一个）的精确位置，但其他每个物体的潜在位置集合（称为物体的不确定区域）仅受速度限制而增长。因此，物体的中心可能位于多个可能位置，即中心的可能位置集合，称为中心的不确定区域。针对每个中心问题，我们设计了查询策略以最小化中心不确定区域的大小，并将其性能与一种已知物体轨迹但仍需查询以减少不确定性的最优查询策略进行比较。对于R^1中1-中心问题的静态情形，我们展示了一种算法，该算法每单位时间查询四个物体，其性能与每单位时间查询一次的最优算法相当。对于R^1中1-中心问题的一般情形、R^d中的质心问题以及R^1中的1-中位数问题，我们证明了轮询调度算法是最优的竞争算法。对于R^d中的质心问题，我们提供了一种O(log n)竞争的算法。此外，对于R^d（d≥2）中1-中心问题的一般情形，我们论证了没有任何算法能保证与最优算法相比具有有界竞争比。