Abstract
Considern independent identically distributed random vectors fromR d with common densityf, and letE (C) be the average complexity of an algorithm that finds the convex hull of these points. Most well-known algorithms satisfyE (C)=0(n) for certain classes of densities. In this note, we show thatE (C)=0(n) for algorithms that use a “throw-away” pre-processing step whenf is bounded away from 0 and ∞ on any nondegenerate rectangle ofR 2.
Zusammenfassung
Wir betrachtenn als unabhängig identisch verteilte Zufallsvektoren imR d mit der gemeinsamen Verteilungsdichtef. Die mittlere Konvexität eines Algorithmus zur Bestimmung der konvexen Hülle dieser Punkte seiE (C). Die meisten bekannten Algorithmen genügen für gewisse Klassen von Dichten der BedingungE (C)=0(n). In dieser Mitteilung zeigen wirE (C)=0(n) für Algorithmen, die im Vorlauf einen „Wegwerf-Schritt” benützen, wennf auf jedem nicht ausgearteten Rechteck desR 2 beschränkt ist und positiven Abstand von 0 besitzt.
This is a preview of subscription content, log in via an institution to check access.
References
Akl, S. G., Toussaint, G. T.: A fast convex hull algorithm. Information Processing Letters7, 219–222 (1978).
Bentley, J. L., Shamos, M. I.: Divide and conquer for linear expected time. Information Processing Letters7, 87–91 (1978).
Bentley, J. L., Kung, H. T., Schkolnick, M., Thompson, C. D.: On the average number of maxima in a set of vectors and applications. Journal of the ACM25, 536–543 (1978).
Bentley, J. L., Weide, B. W., Yao, A. C.: Optimal expected-time algorithms for closest-point problems. Allerton Conference, Urbana, Illinois, 1978.
Carnal, H.: Die konvexe Hülle vonn rotationssymmetrisch verteilten Punkten. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte. Gebiete15, 168–176 (1970).
Devroye, L., Klincsek, T.: Average time behavior of distributive sorting algorithms. Computing26, 1–7 (1980).
Devroye, L.: A note on finding convex hulls via maximal vectors. Information Processing Letters11, 53–56 (1980a).
Devroye, L.: How to reduce the average complexity of convex hull finding algorithms. Manuscript, McGill University, 1980 b.
Dvoretzky, A., Kiefer, J., Wolfowitz, J.: Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator. Annals of Mathematical Statistics27, 642–669 (1956).
Eddy, W. F.: A new convex hull algorithm for planar sets. ACM Transactions on Mathematical Software3, 398–403, 411–412 (1977).
Feller, W.: An Introduction to Probability Theory and Its Applications. Vol. 2, pp. 148. 1966.
Graham, R. L.: An efficient algorithm for determining the convex hull of a planar set. Information Processing Letters1, 132–133 (1972).
Jarvis, R. A.: On the identification of the convex hull of a finite set of points in the plane. Information Processing Letters2, 18–21 (1973).
Shamos, M. I.: Computational geometry. Ph. D. thesis, Yale University, May 1978.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Devroye, L., Toussaint, G.T. A note on linear expected time algorithms for finding convex hulls. Computing 26, 361–366 (1981). https://doi.org/10.1007/BF02237955
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02237955