7. Acknowledgments 8. References 6. Concluding Remarks

Optimal algorithms for computing the minimum distance between two finite planar sets, " Proc. A fast algorithm for the planar convex hull problem, " internal manuscript, [25] B. K. Bhattacharya and G. T. Toussaint. " A time-and-storage efficient implementation of an optimal planar convex hull algorithm, " Divide and conquer for linear expected time, " Inform. A linear algorithm for finding the convex hull of a simple polygon, " Inform. Applications of a two-dimensional hidden-line algorithm to other geometric problems, " Tech. [17] J. L. Bentley, et al., " On the average number of maxima in a set of vectors and applications , " [18] L. Devroye, " A note on finding convex hulls via maximal vectors, " Inform. [20] R. Graham, " An efficient algorithm for determining the convex hull of a planar set, " Inform .It should be noted that if metrics other than the euclidean are used, then the O(n log n) complexity can be reduced for the set-of-points problem. In [13] it is shown that with the L 1 and L ∞ metrics, the maximum distance between two arbitrary sets of points can be computed in O(n) worst-case time in two dimensions. Furthermore, in d dimensions the L ∞ maximum distance can be computed in O(nd) worst-case time. Finally, we note that for the euclidean metric we can obtain O(n) expected running time for any fixed dimensions d with a slight modification of the algorithm MAXDIST-1. Lemma 2.1 generalizes to higher dimensions. Furthermore, the 2 d sets of maximal vectors [17] of S i are a superset of the vertices of the convex hull of S i. Thus in the modified versions of MAXDIST-1 we first find the maximal vector sets S mi of S i , i = 1, 2 and then we use BRUTE-FORCE to compute d max (S m1 , S m2). Bentley et al. [17] have shown that the maximal vectors can be found in O(n) expected time whenever the underlying density f can be written as a d-fold product of densities: f(x 1 ,..., x d) =. These are very general conditions which are satisfied for example for the normal density. Furthermore, Devroye [18] has shown that, under the above condition, if an O(n p) (p ≥ 1) algorithm is used on the resulting set of maximal vectors the algorithm has average complexity O(n). Since computing the maximum distance by BRUTE-FORCE …