Minimizing Co-location Potential of Moving Entities

We study the problem of maintaining knowledge of the locations of $n$ entities that are moving, each with some, possibly different, upper bound on their speed. We assume a setting where we can query the current location of any one entity, but this query takes a unit of time, during which we cannot query any other entities. In this model, we can never know the exact locations of all entities at any one time. Instead, we wish to minimize uncertainty concerning the locations of all entities at some target time that is t units in the future. We measure uncertainty by the ply of the potential locations: the maximum over all points $x$ of the number of entities that could potentially be at $x$. Since the ply could be large for every query strategy, we analyze the performance of our query strategy in a competitive framework: we consider the worst-case ratio of the ply achieved by our strategy to the intrinsic ply (the smallest ply achievable by any strategy, even one that knows in advance the full trajectories o...