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Volume 8, numb :I 4 INFORMATION PROCESSING LETTERS 30 April 1979 COMMENTS DN CONVEX HULL OF A FINITE SET OF POINTS IN TWO DIMENSIONS* Alain FOURNIER Programs Sn Mat~~ematical Sciences, The University of Texas, Dallas, TX 75080, U.S.A. Received 27 November 1978; revised version received 8 January 1979 Convex hull, computational geometry A paper by A. Bykat [2], presents an algorithm to determine the convex hull of a finite set of points in the plane. In the analysis of the complexity of the algorithm the author claimed a worst case running time of O(n log n) (the notations of [2] will be used throughout). Unfortunately the case analysed (a regular n sided polygon) is not the worst case. Indeed, it is one of the most favorable, since at each step each of the two subproblems is about half the size of the previous problem. It is easy to construct an example where the run- ning time of the given algorithm is O(n*), by making at each step the point H the neighbour of R (or of L) in the subset LR. Then at each step one of the sub- problems will be of size p - 1 if the previcirs one is of size p, for a running time of O(n*) (see Fig. 1 for illustration). The complexity of the expected case for this algo- rithm is of course dependent upon the assumed dis- tribution of the points in the plane. Relevant referen- Fig. 1. Worst case. ces and rest&s, and a convex hull algorithm with an O(n) expected case can be found in [ 11. References [I] J.L. Bentley and MI. Shamos, Divide and conquer for linear expected time, Information Processing Lett. 7 (1978) 87-91. [2] A. Bykat, Convex hull of a finite set of points in two dimensions, Information Processing Lett. 7 (1978) 296-298. * Supported in part by the National Science Foundation under Grant MCS-77-03905. 173

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Page 1: Comments on convex hull of a finite set of points in two dimensions

Volume 8, numb :I 4 INFORMATION PROCESSING LETTERS 30 April 1979

COMMENTS DN CONVEX HULL OF A FINITE SET OF POINTS IN TWO DIMENSIONS *

Alain FOURNIER Programs Sn Mat~~ematical Sciences, The University of Texas, Dallas, TX 75080, U.S.A.

Received 27 November 1978; revised version received 8 January 1979

Convex hull, computational geometry

A paper by A. Bykat [2], presents an algorithm to determine the convex hull of a finite set of points in the plane. In the analysis of the complexity of the algorithm the author claimed a worst case running time of O(n log n) (the notations of [2] will be used throughout).

Unfortunately the case analysed (a regular n sided polygon) is not the worst case. Indeed, it is one of the most favorable, since at each step each of the two subproblems is about half the size of the previous problem.

It is easy to construct an example where the run- ning time of the given algorithm is O(n*), by making at each step the point H the neighbour of R (or of L) in the subset LR. Then at each step one of the sub- problems will be of size p - 1 if the previcirs one is of size p, for a running time of O(n*) (see Fig. 1 for illustration).

The complexity of the expected case for this algo- rithm is of course dependent upon the assumed dis- tribution of the points in the plane. Relevant referen-

Fig. 1. Worst case.

ces and rest&s, and a convex hull algorithm with an O(n) expected case can be found in [ 11.

References

[I] J.L. Bentley and MI. Shamos, Divide and conquer for linear expected time, Information Processing Lett. 7 (1978) 87-91.

[2] A. Bykat, Convex hull of a finite set of points in two dimensions, Information Processing Lett. 7 (1978) 296-298.

* Supported in part by the National Science Foundation under Grant MCS-77-03905.

173

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