Linear-time algorithms for linear programming in R3 and related problems
- 1 November 1982
- conference paper
- Published by Institute of Electrical and Electronics Engineers (IEEE)
- Vol. xx (02725428) , 329-338
- https://doi.org/10.1109/sfcs.1982.24
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.Keywords
This publication has 19 references indexed in Scilit:
- An efficient algorith for determining the convex hull of a finite planar setPublished by Elsevier ,2002
- Impact of small process geometries on microarchitectures in systems on a chipProceedings of the IEEE, 2001
- An O(nlogn) randomizing algorithm for the weighted euclidean 1-center problemJournal of Algorithms, 1986
- A Lower Bound to Finding Convex HullsJournal of the ACM, 1981
- Single Facility $l_p $-Distance Minimax LocationSIAM Journal on Algebraic Discrete Methods, 1980
- The complexity of linear programmingTheoretical Computer Science, 1980
- An Algorithmic Approach to Network Location Problems. II: Thep-MediansSIAM Journal on Applied Mathematics, 1979
- Convex hulls of finite sets of points in two and three dimensionsCommunications of the ACM, 1977
- Geometrical Solutions for Some Minimax Location ProblemsTransportation Science, 1972
- Optimal location of a single service center of certain typesNaval Research Logistics Quarterly, 1971