Abstract
We propose data structures for answering a geodesic-distance query between two query points in a two-dimensional or three-dimensional dynamic environment, in which obstacles are deforming continuously. Each obstacle in the environment is modeled as the convex hull of a continuously deforming point cloud. The key to our approach is to avoid maintaining the convex hull of each point cloud explicitly but still able to retain sufficient geometric information to estimate geodesic distances in the free space.
Work on this paper is supported by NSF under grants CNS-05-40347, CFF-06-35000, and DEB-04-25465, by ARO grants W911NF-04-1-0278 and W911NF-07-1-0376, by an NIH grant 1P50-GM-08183-01, by a DOE grant OEG-P200A070505, and by a grant from the U.S.–Israel Binational Science Foundation. Part of the work was done while the last author was at Duke University.
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References
Agarwal, P.K., Aronov, B., O’Rourke, J., Schevon, C.: Star unfolding of a polytope with applications. SIAM J. Comput. 26, 1689–1713 (1997)
Agarwal, P.K., Guibas, L., Hershberger, J., Veach, E.: Maintaining the extent of a moving point set. Discrete Comput. Geom. 26, 353–374 (2001)
Agarwal, P.K., Sharathkumar, R., Yu, H.: Approximate Euclidean shortest paths amid convex obstacles. In: Proc. 20th ACM-SIAM Sympos. Discrete Algorithms (to appear)
Alexandron, G., Kaplan, H., Sharir, M.: Kinetic and dynamic data structures for convex hulls and upper envelopes. Comput. Geom. Theory Appl. 36, 144–158 (2007)
Arikati, S., Chen, D., Chew, L., Das, G., Smid, M., Zaroliagis, C.: Planar spanners and approximate shortest path queries among obstacles in the plane. In: Díaz, J. (ed.) ESA 1996. LNCS, vol. 1136, pp. 514–528. Springer, Heidelberg (1996)
Basch, J., Guibas, L.J., Hershberger, J.: Data structures for mobile data. J. Algorithms 31, 1–28 (1999)
Chen, D.: On the all-pairs Euclidean short path problem. In: Proc. 6th Annu. ACM-SIAM Sympos. Discrete Algorithms, pp. 292–301 (1995)
Chiang, Y.-J., Mitchell, J.S.B.: Two-point Euclidean shortest path queries in the plane. In: Proc. 10th Annu. ACM-SIAM Sympos. Discrete Algorithms, pp. 215–224 (1999)
Clarkson, K.: Approximation algorithms for shortest path motion planning. In: Proc. 19th Annu. ACM Sympos. Theory Comput., pp. 56–65 (1987)
Dobkin, D.P., Kirkpatrick, D.G.: Determining the separation of preprocessed polyhedra — a unified approach. In: Paterson, M. (ed.) ICALP 1990. LNCS, vol. 443, pp. 400–413. Springer, Heidelberg (1990)
Har-Peled, S.: Approximate shortest-path and geodesic diameter on convex polytopes in three dimensions. Discrete Comput. Geom. 21, 217–231 (1999)
Hershberger, J., Suri, S.: Practical methods for approximating shortest paths on a convex polytope in ℝ3. Comput. Geom. Theory Appl. 10, 31–46 (1998)
Hershberger, J., Suri, S.: An optimal algorithm for Euclidean shortest paths in the plane. SIAM J. Comput. 28, 2215–2256 (1999)
Ling, M., Manocha, D.: Collision and proximity queries. In: Goodman, J., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, 2nd edn., pp. 787–808. CRC Press, Boca Raton (2004)
Matoušek, J., Sharir, M., Welzl, E.: A subexponential bound for linear programming. Algorithmica 16, 498–516 (1996)
Overmars, M., van Leeuwen, J.: Maintenance of configurations in the plane. J. Comput. Syst. Sci. 23, 166–204 (1981)
Reif, J., Sharir, M.: Motion planning in the presence of moving obstacles. J. Assoc. Comput. Mach. 41, 764–790 (1994)
van den Berg, J.: Path Planning in Dynamic Environments. PhD thesis, Utrecht University (2007)
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Agarwal, P.K., Efrat, A., Sharathkumar, R., Yu, H. (2009). On Approximate Geodesic-Distance Queries amid Deforming Point Clouds. In: Chirikjian, G.S., Choset, H., Morales, M., Murphey, T. (eds) Algorithmic Foundation of Robotics VIII. Springer Tracts in Advanced Robotics, vol 57. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00312-7_22
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