A data structure for dynamic trees
Journal of Computer and System Sciences
Algorithms for proportional matrices in reals and integers
Mathematical Programming: Series A and B
Polynomial Methods for Separable Convex Optimization in Unimodular Linear Spaces with Applications
SIAM Journal on Computing
Combinatorial optimization
Minimum ratio cover of matrix columns by extreme rays of its induced cone
ISCO'12 Proceedings of the Second international conference on Combinatorial Optimization
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A given nonnegative n x n matrix A = (aij) is to be scaled, by multiplying its rows and columns by unknown positive multipliers λi and μj, such that the resulting matrix (aijλiμj) has specified row and column sums ri and sj. We give an algorithm that achieves the desired row and column sums with a maximum absolute error ε in O(n4(log n + log h/ε)) steps, where h is the overall total of the result matrix. Our algorithm is a scaling algorithm. It solves a sequence of more and more refined discretizations. The discretizations are minimum-cost network flow problems with convex piecewise linear costs. These discretizations are interesting in their own right because they arise in proportional elections.