One line and n points

Authors:
Bernd Gärtner;Falk Tschirschnitz;Emo Welzl;József Solymosi;Pavel Valtr
Affiliations:
Institut für Theoretische Informatik, ETH Zürich, ETI-I Zentrum, CH-8092 Zürich, Switzerland;Institut für Theoretische Informatik, ETH Zürich, ETI-I Zentrum, CH-8092 Zürich, Switzerland;Institut für Theoretische Informatik, ETH Zürich, ETI-I Zentrum, CH-8092 Zürich, Switzerland;Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, Canada V6T 1Z2;Department of Applied Mathematics, Charles University, Prague Malostranské nám. 25, 118 00 Praha 1, Czech Republic
Venue:
Random Structures & Algorithms
Year:
2003

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Abstract

We analyze a randomized pivoting process involving one line and n points in the plane. The process models the behavior of the RANDOM-EDGE simplex algorithm on simple polytopes with n facets in dimension n - 2. We obtain a tight O(log2 n) bound for the expected number of pivot steps. This is the first nontrivial bound for RANDOM-EDGE, which goes beyond bounds for specific polytopes. The process itself can be interpreted as a simple algorithm for certain 2-variable linear programming problems, and we prove a tight Θ(n) bound for its expected runtime.