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

@article{Megiddo1982LineartimeAF,
title={Linear-time algorithms for linear programming in R3 and related problems},
author={Nimrod Megiddo},
journal={23rd Annual Symposium on Foundations of Computer Science (sfcs 1982)},
year={1982},
pages={329-338}
}
• N. Megiddo
• Published 3 November 1982
• Mathematics, Computer Science
• 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982)
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…
679 Citations

## Figures from this paper

Low-Dimensional Linear Programming with Violations
A simple algorithm in 2-d that runs in O((n + k/sup 2/) log n) expected time is given; this is faster than earlier algorithms by Everett, Robert, and van Kreveld (1993) and Matousek (1994) and is probably near-optimal for all k /spl Lt/ n/2.
Linear Time Algorithms for Testing Approximate Congruence in the Plane
The algorithm presented in this paper uses a generalization of the linear programming algorithm by Megiddo and solves the problem of finding a feasible solution for a general system of algebraic inequalities of bounded degree.
Fast and Optimal Parallel Multidimensional Search in PRAMs with Applications to Linear Programming and Related Problems
• Computer Science
SIAM J. Comput.
• 2000
We describe a deterministic parallel algorithm for linear programming in fixed dimension d that takes poly(log log n) time in the common concurrent read concurrent write (CRCW) PRAM model and does
Fixed-dimensional parallel linear programming via relative ε-approximations
We show that linear programming in IRd can be solved deterministically in O((loglogn)d) time using linear work in the PRAM model of computation, for any fixed constant d. Our method is developed for
Bounded-Independence Derandomization of Geometric Partitioning with Applications to Parallel Fixed-Dimensional Linear Programming
• Computer Science, Mathematics
Discret. Comput. Geom.
• 1997
The δ-relativeε-approximation method, developed for the CRCW variant of the PRAM parallel computation model, can be easily implemented to run in $O(\log n(\log\log n)^{d-1})$ time using linear work on an EREW PRAM.
Chapter 16 Fixed-Dimensional Parallel Linear Programming via Relative c-Approximations
• Computer Science, Mathematics
• 1999
We show that linear programming in IRd can be solved deterministically in O((loglogn)d) time using linear work in the PRAM model of computation, for any fixed constant d. Our method is developed for
Helly-type theorems and Generalized Linear Programming
• N. Amenta
• Mathematics
Discret. Comput. Geom.
• 1994
It is shown that there is a Helly-type theorem about the constraint set of every Generalized Linear-Programming problem, which leads to many applications, including linear expected time algorithms for finding line transversals and mini-max hyperplane fitting.

## References

SHOWING 1-10 OF 22 REFERENCES
The Complexity of Linear Programming
• Mathematics, Computer Science
Theor. Comput. Sci.
• 1980
A Lower Bound to Finding Convex Hulls
• A. Yao
• Mathematics, Computer Science
JACM
• 1981
It is shown that any algorithm in the quadratic decision-tree model must make cn log n tests for some input.
Closest-point problems
• Computer Science
16th Annual Symposium on Foundations of Computer Science (sfcs 1975)
• 1975
The purpose of this paper is to introduce a single geometric structure, called the Voronoi diagram, which can be constructed rapidly and contains all of the relevant proximity information in only linear space, and is used to obtain O(N log N) algorithms for most of the problems considered.
An Algorithmic Approach to Network Location Problems. II: The p-Medians
• Computer Science
• 1979
An algorithm is presented which finds a p-median of a tree (for $p > 1$) in time $O(n^2 \cdot p^2 )$.
Optimal location of a single service center of certain types
• Computer Science
• 1971
It is seen that for a class of problems, the geometric method is well suited and very efficient while the graph theoretic method, in general, will give only approximate solutions in spite of the increased efforts involved.
Mathematics
• Mathematics
Nature
• 1935
AbstractTHE calculus of matrices has had a curious history. It was first used by Hamilton in 1853 under the name of “Linear and Vector Functions”. Cay ley used the term matrix in 1854, and developed
Letter to the Editor - Some Aspects of a Minimax Location Problem
Some aspects of a minimax version of the Steiner-Weber location problem are studied and a geometric characterization of the minimax solution is discussed, and a closely related facility design problem is examined.
Applying parallel computation algorithms in the design of serial algorithms
• N. Megiddo
• Computer Science
22nd Annual Symposium on Foundations of Computer Science (sfcs 1981)
• 1981
It is pointed out that analyses of parallelism in computational problems have practical implications even when multi-processor machines are not available, and a unified framework for cases like this is presented.
A Minimax Location Problem on a Network