Linear-time algorithms for linear programming in R3 and related problems

Abstract

Linear-time for Linear Programming in R2 and R3 are presented. The methods used are applicable for some other problems. For example, a linear-time algorithm is given for the classical problem of finding the smallest circle enclosing n given points in the plane. This disproves a conjecture by Shamos and Hoey that this problem requires Ω(n log n) time. An immediate consequence of the main result is that the problem of linear separability is solvable in linear-time. This corrects an error in Shamos and Hoey's paper, namely, that their O(n log n) algorithm for this problem in the plane was optimal. Also, a linear-time algorithm is given for the problem of finding the weighted center of a tree and algorithms for other common location-theoretic problems are indicated. The results apply also to the problem of convex quadratic programming in three-dimensions. The results have already been extended to higher dimensions and we know that linear programming can be solved in linear-time when the dimension is fixed. This will be reported elsewhere; a preliminary report is available from the author.