An Optimal Algorithm for Roundness Determination on Convex Polygons

In tolerancing, the Out-Of-Roundness factor determines the relative circularity of planar shapes. The measurement of concern in this work is the Minimum Radial Separation, as recommended by the American National Standards Institute (ANSI). Here presented is a further clarification of the complexity of a previously presented algorithm of Van-Ban Le and D. T. Lee to determine the Minimum Radial Separation of simple polygons, which is found to be Θ(n2). Secondly, an optimal O(n) time algorithm to compute the Minimum Radial Separation of convex polygons is presented, which represents not only a factor n improvement over the previously best known algorithm, but also a factor of log n improvement over Le and Lee's conjectured complexity for the problem.

[1]  榎原 博之,et al.  A Practical Algorithm for Computing the Roundness , 1992 .

[2]  George S. Lueker,et al.  A data structure for orthogonal range queries , 1978, 19th Annual Symposium on Foundations of Computer Science (sfcs 1978).

[3]  Bernard Chazelle,et al.  Lower bounds for orthogonal range searching: I. The reporting case , 1990, JACM.

[4]  Arne Andersson,et al.  Improved Behaviour of Tries by Adaptive Branching , 1993, Inf. Process. Lett..

[5]  Robert Giegerich,et al.  Suffix Trees in the Functional Programming Paradigm , 1994, ESOP.

[6]  Peter Bro Miltersen,et al.  On data structures and asymmetric communication complexity , 1994, STOC '95.

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

[8]  Sivan Toledo,et al.  Applications of parametric searching in geometric optimization , 1992, SODA '92.

[9]  Gaston H. Gonnet,et al.  Efficient Text Searching of Regular Expressions , 1989, WADS.

[10]  Hermann A. Maurer,et al.  Efficient worst-case data structures for range searching , 1978, Acta Informatica.

[11]  Xerox Polo,et al.  A Space-Economical Suffix Tree Construction Algorithm , 1976 .

[12]  Esko Ukkonen,et al.  On{line Construction of Suux Trees 1 , 1995 .

[13]  F. Frances Yao,et al.  Computational Geometry , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[14]  Leonidas J. Guibas,et al.  A linear-time algorithm for computing the voronoi diagram of a convex polygon , 1989, Discret. Comput. Geom..

[15]  Alberto Apostolico,et al.  The Myriad Virtues of Subword Trees , 1985 .

[16]  Michael L. Fredman,et al.  Surpassing the Information Theoretic Bound with Fusion Trees , 1993, J. Comput. Syst. Sci..

[17]  Bernard Chazelle,et al.  A Functional Approach to Data Structures and Its Use in Multidimensional Searching , 1988, SIAM J. Comput..

[18]  Friedhelm Meyer auf der Heide,et al.  Dynamic perfect hashing: upper and lower bounds , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[19]  Jon Louis Bentley,et al.  Transforming static data structures to dynamic structures , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

[20]  Eugene W. Myers,et al.  Suffix arrays: a new method for on-line string searches , 1993, SODA '90.

[21]  Peter Weiner,et al.  Linear Pattern Matching Algorithms , 1973, SWAT.

[22]  Edward M. McCreight,et al.  Priority Search Trees , 1985, SIAM J. Comput..

[23]  Robert E. Tarjan,et al.  Fast Algorithms for Finding Nearest Common Ancestors , 1984, SIAM J. Comput..

[24]  Bernard Chazelle,et al.  Lower bounds for orthogonal range searching: part II. The arithmetic model , 1990, JACM.

[25]  D. T. Lee,et al.  Out-of-Roundness Problem Revisited , 1991, IEEE Trans. Pattern Anal. Mach. Intell..

[26]  D. T. Lee,et al.  Medial Axis Transformation of a Planar Shape , 1982, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[27]  Leonidas J. Guibas,et al.  Optimal Point Location in a Monotone Subdivision , 1986, SIAM J. Comput..

[28]  Gaston H. Gonnet,et al.  Handbook Of Algorithms And Data Structures , 1984 .

[29]  Michael L. Fredman,et al.  BLASTING through the information theoretic barrier with FUSION TREES , 1990, STOC '90.

[30]  Arne Andersson,et al.  Faster Searching in Tries and Quadtrees - An Analysis of Level Compression , 1994, ESA.

[31]  Bernard Chazelle.  Slimming down search structures: A functional approach to algorithm design , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

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

[33]  Peter Bro Miltersen.  Lower bounds for union-split-find related problems on random access machines , 1994, STOC '94.

[34]  Jon Louis Bentley,et al.  Decomposable Searching Problems , 1979, Inf. Process. Lett..