Fast and Efficient Linear Programming and Linear Least-squares Computationst

We present a new parallel algorithm for computing a least-squares solution to a sparse overdetermined system of linear equations Ax = b such that the m x n matrix A is sparse and the graph, G = (V, E), of the matrix has an s(m +n)-separator family, i.e. either IV[ 0, where ~ is an m × n matrix. Hereafter it is assumed that m/> n. The recent algorithm by Karmarkar gives the best-known upper estimate [O (m35L) arithmetic operations, where L is the input size] for the cost of the solution of this problem in the worst case. We prove an asymptotic improvement of that result in the case where the graph of the associated matrix H has an s (m + n)-separator family; then our algorithm can be implemented using O (mL log m log'-s (m + n)) parallel arithmetic steps, s3 (m + n) processors and a total of O (mLs 3 (m + n) log m log 2 s (m + n)) arithmetic operations. In many cases of practical importance this is a considerable improvement on the known estimates: for example, s(m +n)= ~+~ if G is planar [as occurs in many operations research applications: for instance, in the problem of computing the maximum multicommodity flow with a bounded number of commodities in a network having an s (m + n)-separator family], so that the processor bound is only 8 x/8 (m + n) I-5 and the total number of arithmetic steps is O …