Abstract
Consider a collection of entities moving with bounded speed, but otherwise unpredictably, in some low-dimensional space. Two such entities encroach upon one another at a fixed time if their separation is less than some specified threshold. Encroachment, of concern in many settings such as collision avoidance, may be unavoidable. However, the associated difficulties are compounded if there is uncertainty about the precise location of entities, giving rise to potential encroachment and, more generally, potential congestion within the full collection.
We adopt a model in which entities can be queried for their current location (at some cost) and the uncertainty region associated with an entity grows in proportion to the time since that entity was last queried. The goal is to maintain low potential congestion, measured in terms of the (dynamic) intersection graph of uncertainty regions, using the lowest possible query cost. Previous work, in the same uncertainty model, described query schemes that minimize several measures of congestion potential for point entities, using location queries of some fixed frequency. These schemes were shown to be O(1)-competitive, 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 design a scheme that is competitive in terms of its query granularity (minimum spacing between queries), over all sufficiently large time intervals, while guaranteeing a fixed bound on collision potential (defined as the maximum degree of the intersection graph of uncertainty regions), for entities with positive extent. Our complementary optimization objective necessitates surprisingly different algorithms and analyses from that in previous work. Nevertheless, we also show that the competitive factor of our scheme is best possible, up to a constant factor, in the worst case.
This work was funded in part by Discovery Grants from the Natural Sciences and Engineering Research Council of Canada.
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Evans, W., Kirkpatrick, D. (2023). A Frequency-Competitive Query Strategy for Maintaining Low Collision Potential Among Moving Entities. In: Byrka, J., Wiese, A. (eds) Approximation and Online Algorithms . WAOA 2023. Lecture Notes in Computer Science, vol 14297. Springer, Cham. https://doi.org/10.1007/978-3-031-49815-2_2
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