Dynamic data structures for fat objects and their applications

Abstract We present several efficient dynamic data structures for point-enclosure queries, involving convex fat objects in R 2 or R 3 . Our planar structures are actually fitted for a more general class of objects – (β,δ) -covered objects – which are not necessarily convex, see definition below. These structures are more efficient than alternative known structures, because they exploit the fatness of the objects. We then apply these structures to obtain efficient solutions to two problems: (i) finding a perfect containment matching between a set of points and a set of convex fat objects, and (ii) finding a piercing set for a collection of convex fat objects, whose size is optimal up to some constant factor.

[1]  Micha Sharir,et al.  Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications , 1995, SCG '95.

[2]  Jon Louis Bentley,et al.  Decomposable Searching Problems I: Static-to-Dynamic Transformation , 1980, J. Algorithms.

[3]  Mark de Berg,et al.  Linear Size Binary Space Partitions for Fat Objects , 1995, ESA.

[4]  Michael T. Goodrich,et al.  Almost optimal set covers in finite VC-dimension , 1995, Discret. Comput. Geom..

[5]  Mark H. Overmars,et al.  Range Searching and Point Location among Fat Objects , 1996, J. Algorithms.

[6]  Esther M. Arkin,et al.  Matching Points into Pairwise-Disjoint Noise Regions: Combinatorial Bounds and Algorithms , 1992, INFORMS J. Comput..

[7]  Mark H. Overmars,et al.  The Complexity of the Free Space for a Robot Moving Amidst Fat Obstacles , 1992, Comput. Geom..

[8]  Micha Sharir,et al.  Computing Depth Orders and Related Problems , 1994, SWAT.

[9]  Alon Itai,et al.  Improvements on bottleneck matching and related problems using geometry , 1996, SCG '96.

[10]  Mark H. Overmars,et al.  Point Location in Fat Subdivisions , 1992, Inf. Process. Lett..

[11]  Frank Nielsen,et al.  Dynamic Data Structures for Fat Objects and Their Applications , 1997, WADS.

[12]  Kenneth L. Clarkson,et al.  Algorithms for Polytope Covering and Approximation , 1993, WADS.

[13]  Jan van Leeuwen,et al.  Maintenance of Configurations in the Plane , 1981, J. Comput. Syst. Sci..

[14]  Jirí Matousek.  Range searching with efficient hierarchical cuttings , 1992, SCG '92.

[15]  Vasek Chvátal,et al.  A Greedy Heuristic for the Set-Covering Problem , 1979, Math. Oper. Res..

[16]  Micha Sharir,et al.  Davenport-Schinzel sequences and their geometric applications , 1995, Handbook of Computational Geometry.

[17]  Micha Sharir,et al.  Efficient Hidden Surface Removal for Objects with Small Union Size , 1992, Comput. Geom..

[18]  A. Frank van der Stappen,et al.  Motion planning amidst fat obstacles , 1993 .

[19]  Robert J. Fowler,et al.  Optimal Packing and Covering in the Plane are NP-Complete , 1981, Inf. Process. Lett..

[20]  Pankaj K. Agarwal,et al.  Ray shooting and other applications of spanning trees with low stabbing number , 1992, SCG '89.

[21]  Matthew J. Katz.  3-D Vertical Ray Shooting and 2-D Point Enclosure, Range Searching, and Arc Shooting Amidst Convex Fat Objects , 1997, Comput. Geom..

[22]  Jirí Matousek,et al.  Efficient partition trees , 1992, Discret. Comput. Geom..