### Chapter 6 summary: Point Location

Given a map, graph, or other planar subdivision, how can a particular point be located quickly and efficiently? This is known as a planar point location problem, and it arises fairly frequently when dealing with geographic maps and informational systems. Once the coordinates of the sought point are known, the map must be divided into smaller sections that can then be searched. When the correct section has been found, the specific point can then be found within it. Methods of implementing such point-location systems have the potential to be prohibitively inefficient, particularly when dealing with large maps. The problem is to find a method of point location that uses a minimum of time and storage.

Glossary of terms

planar point location:
the process of locating a particular point in a plane, using its coordinates
planar subdivision:
a planar area divided into faces with edges and vertices (i.e., planar graph)
preprocessing:
in this instance, preparing a map for a point location query before one is made (by dividing into sections and storing the sections in a data structure)
refinement:
Map A is a refinement of Map B if every face of B lies completely within one face of A (I was not familiar with this term)

Keyword

• planar point location

Computational Geometry Bibliography Database Search

Results of a search on GeomBib (limited to ten items):
1. Udo Adamy and Raimund Seidel "Planar Point Location Close to the Information-Theoretic Lower Bound." Proc. 9th ACM-SIAM Sympos. Discrete Algorithms, 1998
2. M. J. Atallah and R. Cole and M. T. Goodrich "Cascading divide-and-conquer: A technique for designing parallel algorithms." Proc. 28th Annu. IEEE Sympos. Found. Comput. Sci., 1987
3. R. Batta and L. A. Leifer "On the accuracy of demand point solutions to the planar (Manhattan) metric p-median problem with and without barriers to travel." Comput. Oper. Res. volume 15 #3, 1998
4. S. W. Cheng and R. Janardan "New Results on Dynamic Planar Point Location." Technical Report, number TR 90-13, Dept. Comput. Sci., Univ. Minnesota, 1990
5. Y.-J. Chiang and F. P. Preparata and R. Tamassia "A Unified Approach to Dynamic Point Location, Ray Shooting, and Shortest Paths in Planar Maps." Proc. 4th ACM-SIAM Sympos. Discrete Algorithms, 1993
6. R. Cole and O. Zajicek "An optimal parallel algorithm for building a data structure for planar point location." J. Parallel Distrib. Comput., volume 8, 1990
7. Y.-J. Chiang and R. Tamassia "Dynamization of the trapezoid method for planar point location." Proc. 7th Annu. ACM Sympos. Comput. Geom., 1991
8. F. P. Preparata and R. Tamassia "Dynamic planar point location with optimal query time." Theoret. Comput. Sci., volume 74, 1990
9. Raimund Seidel "On the Exact Query Complexity of Planar Point Location." Abstracts 14th European Workshop Comput. Geom., 1998
10. F. P. Preparata "Planar point location revisited." Internat. J. Found. Comput. Sci., volume 1, #1, 1990

*This assignment took about 2.5 hours to complete.*