Minimizing Co-location Potential of Moving Entities

Abstract

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 of all entities). We describe an efficient strategy that, knowing only an upper bound on the speed of individual entities, is $O(k)$-competitive, provided the lead time t is at least 2n and the number of different entity speed classes (groups of entities whose speed bounds differ by at most a factor of two) is at most $k$. (This contrasts with the fact that, even given the full trajectories, the problem of computing the intrinsic ply is NP-hard.) If t is small, though at least $n$, and the entities move in any constant dimension $d$, our strategy is $O(k(\frac{\widetilde{T}}{n})^{d-\frac{d}{d+1}})$-competitive, where $\widetilde{T}$ is the median of the lengths of time since the $n$ entity locations were last known precisely. Matching lower bounds demonstrate that our strategy, in all cases, is optimally competitive, up to constant factors.