Optimal Search in Planar Subdivisions

A planar subdivision is any partition of the plane into (possibly unbounded) polygonal regions. The subdivision search problem is the following: given a subdivision $S$ with $n$ line segments and a query point $p$, determine which region of $S$ contains $p$. We present a practical algorithm for subdivision search that achieves the same (optimal) worst case complexity bounds as the significantly more complex algorithm of Lipton and Tarjan, namely $O(\log n)$ search time with $O(n)$ storage. Our subdivision search structure can be constructed in linear time from the subdivision representation used in many applications.

[1]  Frank Harary,et al.  Graph Theory , 2016 .

[2]  Michael Ian Shamos,et al.  Geometric complexity , 1975, STOC.

[3]  Michael Ian Shamos,et al.  Geometric intersection problems , 1976, 17th Annual Symposium on Foundations of Computer Science (sfcs 1976).

[4]  Richard J. Lipton,et al.  Multidimensional Searching Problems , 1976, SIAM J. Comput..

[5]  D. T. Lee,et al.  Location of a point in a planar subdivision and its applications , 1976, STOC '76.

[6]  Robert E. Tarjan,et al.  Application of a Planar Separator Theorem , 1977, FOCS.

[7]  Robert E. Tarjan,et al.  Applications of a planar separator theorem , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[8]  R. Tarjan,et al.  A Separator Theorem for Planar Graphs , 1977 .

[9]  F. P. Preparata,et al.  Convex hulls of finite sets of points in two and three dimensions , 1977, CACM.

[10]  Warren Burton,et al.  Representation of many-sided polygons and polygonal lines for rapid processing , 1977, CACM.

[11]  Richard J. Lipton,et al.  A Batching Method for Coloring Planar Graphs , 1978, Information Processing Letters.

[12]  David E. Muller,et al.  Finding the Intersection of two Convex Polyhedra , 1978, Theor. Comput. Sci..

[13]  Kenneth B. Salomon.  An efficient point-in-polygon algorithm , 1978 .

[14]  Robert E. Tarjan,et al.  Triangulating a Simple Polygon , 1978, Inf. Process. Lett..

[15]  Hermann A. Maurer,et al.  A Note on Euclidean Near Neighbor Searching in the Plane , 1979, Inf. Process. Lett..

[16]  D. T. Lee,et al.  Location of Multiple Points in a Planar Subdivision , 1979, Inf. Process. Lett..

[17]  David G. Kirkpatrick,et al.  Efficient computation of continuous skeletons , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

[18]  Franco P. Preparata.  A Note on Locating a Set of Points in a Planar Subdivision , 1979, SIAM J. Comput..

[19]  D. T. Lee,et al.  Two-Dimensional Voronoi Diagrams in the Lp-Metric , 1980, J. ACM.

[20]  Franco P. Preparata,et al.  A New Approach to Planar Point Location , 1981, SIAM J. Comput..

[21]  D. T. Lee,et al.  Generalization of Voronoi Diagrams in the Plane , 1981, SIAM J. Comput..

[22]  David G. Kirkpatrick,et al.  Fast Detection of Polyhedral Intersections , 1982, ICALP.

[23]  I. G. Gowda,et al.  Dynamic Voronoi diagrams , 1983, IEEE Trans. Inf. Theory.