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ESA'07 Proceedings of the 15th annual European conference on Algorithms
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An almost optimal unrestricted fast Johnson-Lindenstrauss transform
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Algebraic algorithms for linear matroid parity problems
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
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STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
IEEE Transactions on Information Theory
A Random Linear Network Coding Approach to Multicast
IEEE Transactions on Information Theory
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We consider the problem of computing the rank of an m × n matrix A over a field. We present a randomized algorithm to find a set of r = rank(A) linearly independent columns in Õ(|A| + rω) field operations, where |A| denotes the number of nonzero entries in A and ω r linearly independent columns is by Gaussian elimination, with deterministic running time O(mnrω-2). Our algorithm is faster when r m,n}, for instance when the matrix is rectangular. We also consider the problem of computing the rank of a matrix dynamically, supporting the operations of rank one updates and additions and deletions of rows and columns. We present an algorithm that updates the rank in Õ(mn) field operations. We show that these algorithms can be used to obtain faster algorithms for various problems in exact linear algebra, combinatorial optimization and dynamic data structure.