Intersection queries for curved objects (extended abstract)

The following class of query problems is studied: Given a set of n arcs (disks, circles, circular arcs, Jordan arcs) in the plane, preprocess it into a data structure, such that for a query line (or segment) 1, one can quickly (i) report all arcs intersecting /?, or (ii) count the number of arcs intersecting L We also stud y the ray shooting problem for disjoint Jordan arcs and for circular arcs. Most of the data structures presented here use linear or near to linear space and have query time near to 0(/E+K) or 0(n2/3+K), where K is the size of the output. The query time of some of our algorithms can be improved by allowing more space.

[1]  Micha Sharir,et al.  Sharp upper and lower bounds on the length of general Davenport-Schinzel sequences , 1989, J. Comb. Theory A.

[2]  Micha Sharir,et al.  Storing line segments in partition trees , 1990, BIT.

[3]  Leonidas J. Guibas,et al.  Ray Shooting in Polygons Using Geodesic Triangulations , 1991, ICALP.

[4]  David P. Dobkin,et al.  Space Searching for Intersecting Objects , 1987, J. Algorithms.

[5]  Pankaj K. Agarwal,et al.  A deterministic algorithm for partitioning arrangements of lines and its application , 1989, SCG '89.

[6]  David Haussler,et al.  ɛ-nets and simplex range queries , 1987, Discret. Comput. Geom..

[7]  Mark H. Overmars,et al.  Intersection Queries in Sets of Disks , 1990, SWAT.

[8]  Bernard Chazelle,et al.  Quasi-optimal upper bounds for simplex range searching and new zone theorems , 1990, SCG '90.

[9]  Robert E. Tarjan,et al.  Making data structures persistent , 1986, STOC '86.

[10]  Bernard Chazelle,et al.  Quasi-optimal range searching in spaces of finite VC-dimension , 1989, Discret. Comput. Geom..

[11]  David P. Dobkin,et al.  Partitioning Space for Range Queries , 1989, SIAM J. Comput..

[12]  Bernard Chazelle,et al.  Filtering search: A new approach to query-answering , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[13]  Herbert Edelsbrunner,et al.  Algorithms in Combinatorial Geometry , 1987, EATCS Monographs in Theoretical Computer Science.

[14]  Mark H. Overmars,et al.  Intersection Queries in Curved Objects , 1993, J. Algorithms.

[15]  Dan E. Willard,et al.  Polygon Retrieval , 1982, SIAM journal on computing (Print).

[16]  Alok Aggarwal,et al.  Solving query-retrieval problems by compacting Voronoi diagrams , 1990, STOC '90.

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

[18]  Jirí Matousek,et al.  Good splitters for counting points in triangles , 1989, SCG '89.

[19]  Herbert Edelsbrunner,et al.  Halfplanar Range Search in Linear Space and O(n^(0.695)) Query Time , 1986, Inf. Process. Lett..