Abstract
Consider a collection of entities moving continuously 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 consider 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, at specified (possibly all) times, using the lowest possible query cost. Previous work [SoCG’13, EuroCG’14, SICOMP’16, SODA’19], in the same uncertainty model, addressed the problem of minimizing the congestion potential of point entities using location queries of some bounded frequency. It was shown that it is possible to design query schemes that are O(1)-competitive, in terms of worst-case congestion potential, with other, even clairvoyant query schemes (that exploit knowledge of the trajectories of all entities), subject to the same bound on query frequency.
In this paper we initiate the treatment of a more general problem with the dual optimization objective: minimizing the query frequency, measured as the reciprocal of the minimum time between queries (granularity), while guaranteeing a fixed bound on congestion potential of entities with positive extent at one specified target time. This complementary objective necessitates quite different schemes and analyses. Nevertheless, our results parallel those of the earlier papers, specifically tight competitive bounds on required query frequency.
This work was funded in part by Discovery Grants from the Natural Sciences and Engineering Research Council of Canada.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Basch, J., Guibas, L.J., Hershberger, J.: Data structures for mobile data. J. Algorithms 31(1), 1–28 (1999)
de Berg, M., Roeloffzen, M., Speckmann, B.: Kinetic compressed quadtrees in the black-box model with applications to collision detection for low-density scenes. In: Epstein, L., Ferragina, P. (eds.) ESA 2012. LNCS, vol. 7501, pp. 383–394. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-33090-2_34
de Berg, M., Roeloffzen, M., Speckmann, B.: Kinetic convex hulls, Delaunay triangulations and connectivity structures in the black-box model. J. Comput. Geom. 3(1), 222–249 (2012)
de Berg, M., Roeloffzen, M., Speckmann, B.: Kinetic 2-centers in the black-box model. In: Symposium on Computational Geometry, pp. 145–154 (2013)
Busto, D., Evans, W., Kirkpatrick, D.: Minimizing interference potential among moving entities. In: Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 2400–2418 (2019)
Erlebach, T., Hoffmann, M.: Query-competitive algorithms for computing with uncertainty. Bull. Eur. Assoc. Theor. Comput. Sci. 2(116) (2015)
Evans, W., Kirkpatrick, D.: Frequency-competitive query strategies to maintain low congestion potential among moving entities (2023). arXiv:2205.09243
Evans, W., Kirkpatrick, D., Löffler, M., Staals, F.: Competitive query strategies for minimising the ply of the potential locations of moving points. In: Symposium on Computational Geometry, pp. 155–164 (2013)
Evans, W., Kirkpatrick, D., Löffler, M., Staals, F.: Query strategies for minimizing the ply of the potential locations of entities moving with different speeds. In: Abstr. 30th European Workshop on Computational Geometry (EuroCG) (2014)
Evans, W., Kirkpatrick, D., Löffler, M., Staals, F.: Minimizing co-location potential of moving entities. SIAM J. Comput. 45(5), 1870–1893 (2016)
Guibas, L.J.: Kinetic data structures: a state of the art report. In: Proceedings of the Third Workshop on the Algorithmic Foundations of Robotics on Robotics: The Algorithmic Perspective, pp. 191–209. WAFR ’98, A. K. Peters Ltd, USA (1998)
Guibas, L.J., Roeloffzen, M.: Modeling motion. In: Toth, C.D., O’Rourke, J., Goodman, J.E. (eds.) Handbook of Discrete and Computational Geometry, chap. 53, pp. 1401–1420. CRC Press (2017)
Kahan, S.: A model for data in motion. In: Twenty-third Annual ACM Symposium on Theory of Computing, pp. 265–277. STOC ’91 (1991)
Kahan, S.: Real-Time Processing of Moving Data. Ph.D. thesis, University of Washington (1991)
Löffler, M., Snoeyink, J.: Delaunay triangulation of imprecise points in linear time after preprocessing. Comput. Geom.: Theory Appl. 43(3), 234–242 (2010)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Evans, W., Kirkpatrick, D. (2023). Minimizing Query Frequency to Bound Congestion Potential for Moving Entities at a Fixed Target Time. In: Fernau, H., Jansen, K. (eds) Fundamentals of Computation Theory. FCT 2023. Lecture Notes in Computer Science, vol 14292. Springer, Cham. https://doi.org/10.1007/978-3-031-43587-4_12
Download citation
DOI: https://doi.org/10.1007/978-3-031-43587-4_12
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-43586-7
Online ISBN: 978-3-031-43587-4
eBook Packages: Computer ScienceComputer Science (R0)