William Evans
ABSTRACT
We study the problem of maintaining the locations of a collection of n entities that are moving with some fixed upper bound on their speed. We assume a setting where we may query the current location of entities, but handling this query takes a certain 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 maintain a representation of the potential locations of all entities. We measure the quality of this representation by its ply: the maximum over all points p of the number of entities that could potentially be at p.
Since the ply could be large for every query strategy, we analyse the performance of our algorithms in a competitive framework: we consider the worst case ratio of the ply achieved by our algorithms to the intrinsic ply (the smallest ply achievable by any algorithm, even one that knows in advance the full trajectories of all entities). We show that, if our goal is to mimimise the ply at some number τ of time units in the future, an O(1)-competitive algorithm exists, provided τ is sufficiently large. If τ is small and the n entities move in any constant dimension d, our algorithm is O((l/τ + 1)d-d/(d+1))-competitive, where l is the median of the lengths of time since the $n$ entity locations were last known precisely.
We also provide matching lower bounds, and we show that computing the intrinsic ply exactly is NP-hard, even when the trajectories are known in advance.
- P. K. Agarwal, J. Erickson, and L. J. Guibas. Kinetic BSPs for intersecting segments and disjoint triangles. In Proc. 9th ACM-SIAM Symp. Discrete Algorithms, pages 107--116, 1998. Google Scholar
Digital Library
- J. Almeida and R. Araujo. Tracking multiple moving objects in a dynamic environment for autonomous navigation. In Advanced Motion Control, 2008. AMC'08. 10th IEEE International Workshop on, pages 21--26, 2008.Google ScholarCross Ref
- J. Basch, L. J. Guibas, C. Silverstein, and L. Zhang. A practical evaluation of kinetic data structures. In Proc. 13th ACM Symp. Comput. Geom., pages 388--390, 1997. Google ScholarDigital Library
- R. Bruce, M. Hoffmann, D. Krizanc, and R. Raman. Efficient update strategies for geometric computing with uncertainty. Theory of Computing Systems, 38(4):411--423, 2005. Google ScholarDigital Library
- K. Buchin, M. Löffler, P. Morin, and W. Mulzer. Delaunay triangulation of imprecise points simplified and extended. Algorithmica, 61(3):674--693, 2011. Google ScholarDigital Library
- M. Cho, D. M. Mount, and E. Park. Maintaining nets and net trees under incremental motion. In Proc. 20th International Symposium on Algorithms and Computation, ISAAC '09, pages 1134--1143. Springer, 2009. Google ScholarDigital Library
- M. de Berg, M. Roeloffzen, and B. Speckmann. Kinetic convex hulls and Delaunay triangulations in the black-box model. In Proc. 27th ACM Symp. on Comput. Geom., pages 244--253, 2011. Google ScholarDigital Library
- S. B. Eisenman. People-centric mobile sensing networks. PhD thesis, Columbia University, New York, NY, USA, 2008. Google ScholarDigital Library
- D. Eppstein, M. T. Goodrich, and M. Löffler. Tracking moving objects with few handovers. In Proc. 12th Algorithms and Data Structures Symposium, pages 362--373, 2011. Google ScholarDigital Library
- P. G. Franciosa, C. Gaibisso, G. Gambosi, and M. Talamo. A convex hull algorithm for points with approximately known positions. International Journal of Computational Geometry and Applications, 4(2):153--163, 1994.Google ScholarCross Ref
- J. Gao, L. Guibas, and A. Nguyen. Deformable spanners and their applications. Computational Geometry: Theory and Applications, 35:2--19, 2006. Google ScholarDigital Library
- L. Guibas, J. Hershberger, S. Suri, and L. Zhang. Kinetic connectivity for unit disks. In Proc. 16th ACM Symp. Comput. Geom., pages 331--340, 2000. Google ScholarDigital Library
- L. J. Guibas. Kinetic data structures -- a state of the art report. In P. K. Agarwal, L. E. Kavraki, and M. Mason, editors, Proc. Workshop Algorithmic Found. Robot., pages 191--209. A. K. Peters, Wellesley, MA, 1998. Google ScholarDigital Library
- M. Hoffmann, T. Erlebach, D. Krizanc, M. Mihalák, and R. Raman. Computing minimum spanning trees with uncertainty. In STACS, pages 277--288, 2008.Google Scholar
- S. H. Kahan. Real-Time Processing of Moving Data. PhD thesis, University of Washington, 1991. Google ScholarDigital Library
- M. Löffler and M. van Kreveld. Largest and smallest convex hulls for imprecise points. Algorithmica, 56(2):235--269, 2010. Google ScholarDigital Library
- G. Miller, S. Teng, W. Thurston, and S. Vavasis. Separators for sphere-packings and nearest neighbor graphs. Journal of the ACM, 44(1):1--29, 1992. Google ScholarDigital Library
- D. M. Mount, N. S. Netanyahu, C. D. Piatko, R. Silverman, and A. Y. Wu. A computational framework for incremental motion. In Proc. 20th ACM Symp. on Comput. Geom., pages 200--209, 2004. Google ScholarDigital Library
- M. Schneider. Moving Objects in Databases and GIS: State-of-the-Art and Open Problems. Research Trends in Geographic Information Science, pages 169--187, 2009.Google ScholarCross Ref
- K. Sreenath, F. L. Lewis, and D. O. Popa. Simultaneous adaptive localization of a wireless sensor network. SIGMOBILE Mob. Comput. Commun. Rev., 11(2):14--28, 2007. Google ScholarDigital Library
- K.-C. R. Tseng and D. G. Kirkpatrick. Input-thrifty extrema testing. In ISAAC, pages 554--563, 2011. Google ScholarDigital Library
- J. van den Berg and M. Overmars. Planning time-minimal safe paths amidst unpredictably moving obstacles. International Journal of Robotics Research, 27(11--12):1274--1294, 2008.Google Scholar
- K. Yi and Q. Zhang. Multi-dimensional online tracking. In Proc. 20th ACM-SIAM Symposium on Discrete Algorithms, pages 1098--1107. SIAM, 2009. Google ScholarDigital Library
- C. Zhu, L. Shu, T. Hara, L. Wang, and S. Nishio. Research issues on mobile sensor networks. In Communications and Networking in China (CHINACOM), 2010 5th International ICST Conference on, pages 1--6. IEEE, 2010.Google ScholarCross Ref
Index Terms
- Competitive query strategies for minimising the ply of the potential locations of moving points
Recommendations
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 Frequency-Competitive Query Strategy for Maintaining Low Collision Potential Among Moving Entities
Approximation and Online AlgorithmsAbstractConsider 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. ...
Minimizing Query Frequency to Bound Congestion Potential for Moving Entities at a Fixed Target Time
Fundamentals of Computation TheoryAbstractConsider 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 ...
Comments