6 Concluding Remarks

if (pp<min) { dis-; } else { swap(dis,min); for (inc=min+1; inc<=dis;) { if (prec(min,dis)) dis-; else if (prec(dis,min)) swap(dis-,min); else swap(dis,inc++); } min++; } } Figure 1: The minima algorithm, in C heuristically there is no reason to keep track of the hulls of two diierent sets. Perhaps a simpler but still provably good algorithm can found, particularly for d 3. Appendix The triviality of the algorithm has the fortunate consequence that implementation is also trivial. The C code fragment of Figure 6 is part of an implementation of the algorithm; the elements of the set are in an array, and pairs of elements at two locations are compared with the prec function, and swapped by the swap function. The minima are kept at the beginning of the array, and the non-minimal, discarded elements are kept at the end. Note that in the usual case, where an element is discarded, no swapping is done. References 1] D. Avis and K. Fukuda. A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra. Fast linear expected-time algorithms for computing maxima and convex hulls. 3] B. Chazelle and J. Matou sek. Derandomizing an output-sensitive convex hull algorithm in three and so is in one of the connict lists for these facets. By examining these lists, such a p 2 S can be found and added to E. A witness vector, that proves that a point p is outside conv E, can be found in O(logA) time, assuming appropriate preprocessing of E in A O(1) time. For d = 3, a planar point location data structure or Dobkin and Kirkpatrick's hierarchical decomposition can be used for this purpose. In higher dimensions, the best algorithm for this is due to Matou sek and Schwarzkopff7]. To bound the running time of this algorithm, note that a point is tested with respect to containment in conv E at most once, and as just mentioned, this requires O(logA) time. To nd d facets of conv R whose normals contain a given vector requires time at most O(r bd=2c) = O(A d), and this is done A times. The connict lists of these d facets contain O(n=r) logr = O(n=jEj) sites, with probability 1 ? 1=A (1) , using standard random sampling arguments, and so the total time to search these lists is no more than O(n) P 1j<A 1=j = O(n logA). The maintenance of …