Linear-Time Algorithms for Linear Programming in R³ and Related Problems

Linear-time algorithms for linear programming in R² and R³ are presented. The methods used are applicable for other graphic and geometric problems as well as quadratic programming. 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 Proc. 16th IEEE Symposium on Foundations of Computer Science, 1975 that this problem requires (n 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 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 version is available from the author.