Realistic Input Models for Geometric Algorithms

The traditional worst-case analysis often fails to predict the actual behavior of the running time of geometric algorithms in practical situations. One reason is that worst-case scenarios are often very contrived and do not occur in practice. To avoid this, models are needed that describe the properties that realistic inputs have, so that the analysis can take these properties into account.We try to bring some structure to this emerging research direction. In particular, we present the following results: • We show the relations between various models that have been proposed in the literature. • For several of these models, we give algorithms to compute the model parameter(s) for a given (planar) scene; these algorithms can be used to verify whether a model is appropriate for typical scenes in some application area. • As a case study, we give some experimental results on the appropriateness of some of the models for one particular type of scene often encountered in geographic information systems, namely certain triangulated irregular networks.

[1]  Joseph S. B. Mitchell,et al.  Query-Sensitive Ray Shooting , 1997, Int. J. Comput. Geom. Appl..

[2]  Mark de Berg,et al.  Computational geometry: algorithms and applications , 1997 .

[3]  Micha Sharir,et al.  Computing Depth Orders for Fat Objects and Related Problems , 1995, Comput. Geom..

[4]  T. M. Murali,et al.  Binary space partitions for fat rectangles , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[5]  Matthew J. Katz,et al.  On the union of κ-curved objects , 1998, SCG '98.

[6]  Frank Nielsen,et al.  Dynamic data structures for fat objects and their applications , 1997, Comput. Geom..

[7]  Mark de Berg,et al.  Linear Size Binary Space Partitions for Uncluttered Scenes , 2000, Algorithmica.

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

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

[10]  Micha Sharir,et al.  Fat Triangles Determine Linearly Many Holes , 1994, SIAM J. Comput..

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

[12]  Matthew J. Kaltz 3-D vertical ray shooting and 2-D point enclosure, range searching, and arc shooting amidst convex fat objects , 1997 .

[13]  Mark de Berg,et al.  Motion Planning in Environments with Low Obstacle Density , 1998, Discret. Comput. Geom..

[14]  Otfried Cheong,et al.  Range Searching in Low-Density Environments , 1996, Inf. Process. Lett..

[15]  J. Schwartz,et al.  On the “piano movers'” problem I. The case of a two‐dimensional rigid polygonal body moving amidst polygonal barriers , 1983 .

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

[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]  Marc J. van Kreveld On fat partitioning, fat covering and the union size of polygons , 1998, Comput. Geom..

[20]  Micha Sharir,et al.  On the complexity of the union of fat objects in the plane , 1997, SCG '97.

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

[22]  David Eppstein,et al.  Provably Good Mesh Generation , 1994, J. Comput. Syst. Sci..