Optimal Expected-Time Algorithms for Closest Point Problems

Abstract : Geometric closest-point problems deal with the proximity relationships in k-dimensional point sets. Examples of closest-point problems include building minimum spanning trees, nearest neighbor searching, and triangulation construction. Shamos and Hoey (1975) have shown how the Voronoi diagram can be used to solve a number of planar closest-point problems in optimal worst-case time. In this paper we extend their work by giving optimal expected-time algorithms for solving a number of closest-point problems in k- space, including nearest neighbor searching, finding all nearest neighbors, and computing planar minimum spanning trees. In addition to establishing theoretical bounds, the algorithms in this paper can be implemented to solve practical problems very efficiently.

[1]  J. Hammersley.  A few seedlings of research , 1972 .

[2]  Richard O. Duda,et al.  Pattern classification and scene analysis , 1974, A Wiley-Interscience publication.

[3]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

[4]  Andrea Maggiolo-Schettini,et al.  Sequence-to-Sequence Recursiveness , 1975, Inf. Process. Lett..

[5]  Michael L. Fredman,et al.  On computing the length of longest increasing subsequences , 1975, Discret. Math..

[6]  Michael Ian Shamos,et al.  Closest-point problems , 1975, 16th Annual Symposium on Foundations of Computer Science (sfcs 1975).

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

[8]  Robert E. Tarjan,et al.  Finding Minimum Spanning Trees , 1976, SIAM J. Comput..

[9]  Gideon Yuval,et al.  Finding Nearest Neighbors , 1976, Inf. Process. Lett..

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

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

[12]  Jon Louis Bentley,et al.  An Algorithm for Finding Best Matches in Logarithmic Expected Time , 1977, TOMS.

[13]  Wlodzimierz Dobosiewicz,et al.  Sorting by Distributive Partitioning , 1978, Inf. Process. Lett..

[14]  Jon Louis Bentley,et al.  Fast Algorithms for Constructing Minimal Spanning Trees in Coordinate Spaces , 1978, IEEE Transactions on Computers.

[15]  B. Weide.  Statistical methods in algorithm design and analysis. , 1978 .

[16]  F. James Rohlf,et al.  A Probabilistic Minimum Spanning Tree Algorithm , 1978, Inf. Process. Lett..

[17]  Christos H. Papadimitriou.  Efficient Search for Rationals , 1979, Inf. Process. Lett..

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

[19]  John E. Hopcroft,et al.  A Note on Rabin's Nearest-Neighbor Algorithm , 1978, Inf. Process. Lett..

[20]  Jon Louis Bentley,et al.  Data Structures for Range Searching , 1979, CSUR.

[21]  Jon Louis Bentley,et al.  Multidimensional divide-and-conquer , 1980, CACM.

[22]  Andrew Chi-Chih Yao,et al.  On Constructing Minimum Spanning Trees in k-Dimensional Spaces and Related Problems , 1977, SIAM J. Comput..

[23]  H. Alt.  Nearest Neighbor Search in High Dimensions , .