On the ultimate convex hull algorithm in practice

Kirkpatrick and Seidel [13,14] recently proposed an algorithm for computing the convex hull of n points in the plane that runs in O(n log h) worst case time, where h denotes the number of points on the convex hull of the set. Here a modification of their algorithm is proposed that is believed to run in O(n) expected time for many reasonable distributions of points. The above O(n log h) algorithmsare experimentally compared to the O(n log n) 'throw-away' algorithms of Akl, Devroye and Toussaint [2, 8, 20]. The results suggest that although the O(n Log h) algorithms may be the 'ultimate' ones in theory, they are of little practical value from the point of view of running time.

[1]  Ronald L. Graham,et al.  An Efficient Algorithm for Determining the Convex Hull of a Finite Planar Set , 1972, Inf. Process. Lett..

[2]  Ray A. Jarvis,et al.  On the Identification of the Convex Hull of a Finite Set of Points in the Plane , 1973, Inf. Process. Lett..

[3]  Godfried T. Toussaint,et al.  Time- and storage-efficient implementation of an optimal planar convex hull algorithm , 1983, Image Vis. Comput..

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

[5]  Andrew Chi-Chih Yao,et al.  A Lower Bound to Finding Convex Hulls , 1981, JACM.

[6]  David G. Kirkpatrick,et al.  The Ultimate Planar Convex Hull Algorithm? , 1986, SIAM J. Comput..

[7]  Manuel Blum,et al.  Time Bounds for Selection , 1973, J. Comput. Syst. Sci..

[8]  Godfried T. Toussaint,et al.  Computational Geometric Problems in Pattern Recognition , 1982 .

[9]  Peter van Emde Boas,et al.  On the Omega(n log n) Lower Bound for Convex Hull and Maximal Vector Determination , 1980, Inf. Process. Lett..

[10]  Selim G. Akl,et al.  A Fast Convex Hull Algorithm , 1978, Inf. Process. Lett..

[11]  Michael Ian Shamos,et al.  Divide and Conquer for Linear Expected Time , 1978, Inf. Process. Lett..

[12]  Josef Kittler,et al.  Pattern Recognition Theory and Applications , 1987, NATO ASI Series.

[13]  William F. Eddy,et al.  A New Convex Hull Algorithm for Planar Sets , 1977, TOMS.

[14]  Davis Avis,et al.  On the complexity of finding the convex hull of a set of points , 1982, Discret. Appl. Math..

[15]  Selim G. Akl,et al.  EFFICIENT CONVEX HULL ALGORITHMS FOR PATTERN RECOGNITION APPLICATIONS. , 1979 .

[16]  William F. Eddy,et al.  Algorithm 523: CONVEX, A New Convex Hull Algorithm for Planar Sets [Z] , 1977, TOMS.

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

[18]  Franco P. Preparata,et al.  An optimal real-time algorithm for planar convex hulls , 1979, CACM.