  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.
