We consider the problem of using location queries to monitor the congestion potential among a collection of entities moving, with bounded speed but otherwise unpredictably, in $d$-dimensional Euclidean space. Uncertainty in entity locations due to potential motion between queries gives rise to a space of possible entity configurations at each moment in time, with possibly very different congestion properties. We define different measures of what we call the congestion potential of such spaces, in terms of the (dynamic) intersection graph of the uncertainty regions associated with entities, to describe the congestion that might actually occur. Previous work [SoCG'13, EuroCG'14, SICOMP'16, SODA'19], in the same uncertainty model, addressed the problem of minimizing congestion potential using location queries of some bounded frequency. It was shown that it is possible to design a query scheme that is $O(1)$-competitive, in terms of worst-case congestion potential, with other, even clairvoyant query schemes (that know the trajectories of all entities), subject to the same bound on query frequency. In this paper we address the dual problem: how to guarantee a fixed bound on congestion potential while minimizing the query frequency, measured in terms of total number of queries or the minimum spacing between queries (granularity), over any fixed time interval. This complementary objective necessitates quite different algorithms and analyses. Nevertheless, our results parallel those of the earlier papers, specifically tight competitive bounds on required query frequency, with a few surprising differences.

翻译：我们考虑在d维欧氏空间中，利用位置查询来监测一组以有界速度（但其他运动不可预测）移动的实体间的拥塞潜力问题。由于查询间隔内潜在运动导致的实体位置不确定性，每个时刻都会产生一个可能的实体配置空间，其中可能包含截然不同的拥塞特性。我们基于实体相关不确定区域的（动态）交图，定义了此类空间的“拥塞潜力”的不同度量，以描述可能实际发生的拥塞。先前工作[SoCG'13, EuroCG'14, SICOMP'16, SODA'19]在同一不确定性模型下，研究了利用有界频率的位置查询来最小化拥塞潜力的问题。研究表明，在相同的查询频率约束下，可以设计一种查询方案，在最坏情况拥塞潜力方面，与其它（甚至能预知所有实体轨迹的）先验查询方案相比，达到O(1)的竞争比。本文关注其对偶问题：如何在固定时间区间内，通过最小化查询频率（以总查询次数或最小查询间隔（粒度）衡量）来保证拥塞潜力的固定界。这一互补目标需要截然不同的算法与分析。然而，我们的结果与早期论文相呼应，特别是在所需查询频率的严格竞争界方面，但存在几处令人惊讶的差异。