 
Summary: Parallel Linear Programming in Fixed Dimension
Almost Surely in Constant Time
Noga Alon and Nimrod Megiddo
IBM Almaden Research Center
San Jose, California 95120 and
School of Mathematical Sciences
Tel Aviv University, Israel
Revised January 1992
Abstract. For any xed dimension d, the linear programming problem with
n inequality constraints can be solved on a probabilistic CRCW PRAM with O(n)
processors almost surely in constant time. The algorithm always nds the correct
solution. With nd= log2 d processors, the probability that the algorithm will not
nish within O(d2 log2
d) time tends to zero exponentially with n.
1. Introduction
The linear programming problem in xed dimension is to maximize a linear function of
a xed number, d, of variables, subject to n linear inequality constraints, where n is not
xed. Megiddo 11] showed that for any d, this problem can be solved in O(n) time.
Clarkson 4] and Dyer 7] improved the constant of proportionality. Clarkson 5] later
developed lineartimeprobabilistic algorithms with even better complexity. The problem
