J. Edmonds. Systems of distinct representatives and linear algebra. J. of Research and the National Bureau of Standards, 71B (1967), 241-245.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....from 0 up to W max , where W max = WN(G) G) and then interpolate to obtain p(x) The interpolation is the most costly step, and it involves solving a system of W max 1 equations with O(W max ) digit integer coefficients. This can be done in time polynomial in W max by the result of Edmonds [Ed], who proved that Gaussian elimination with pivoting is in P for exact rational arithmetic. A second approach is to obtain an upper bound for p k and then compute det B 0 A 0 for a single value of x that is large (or small) enough to ensure that all of the coefficients p k can be extracted ....

J. Edmonds. Systems of distinct representatives and linear algebra. J. of Research and the National Bureau of Standards, 71B (1967), 241-245.

....rows of (A ) may be linearly dependent. In this case, not all of the rows are needed. What is needed is a minimal, linearly independent set of equations de ning the ane hull of F . Such a set of rows can be found by Gaussian elimination, which was shown to run in polynomial time by Edmonds [5]. Thus, from now on we assume that the rows of (A ) are linearly independent. Since b is in the columnspace of A , this implies that also the rows of A are linearly independent. Next, we need the concept of Hermite normal form (see, e.g. 13, 15] De nition: A square ....

J. Edmonds (1967) Systems of distinct representatives and linear algebra. J. Res. Nat. Bur. Std. (B), 71, 233-240.

.... analysis of the average accuracy of Hadamard s bound can be found in [1] Once a bound is found, the determinant can be computed by a Gaussian elimination with the sizes of the intermediate integers controlled by exact division or more sophisticatedly by Bareiss s method [6] Another approach [25, 13] is to use matrix arithmetic modulo primes and Chinese remaindering (on this technique see [2, Theorem 8.9] or [8, Problem 4.2] The classical associated cost for the exact computation of the determinant, including a fast reduction of the matrix entries modulo the di erent primes, is [29, Chapter ....

J. Edmonds. Systems of distinct representatives and linear algebra. J. Res. Nat. Bur. Standards  B, 71(4):241245, 1967.

....However, to avoid the excessive growth of numbers, it is not necessary to reduce fractions by their greatest common divisor: one can perform pivots involving divisions for which the result is known to be integral. This form of integral pivoting is attributed to Camille Jordan [5, p. 69] see also [6, 2, 9, 7]. Edmonds [6] has shown that the numbers do not ex plode and hence, this algorithm runs in polynomial time. The same approach can be extended to Pfaffians [9] Thus, for evaluating determinants or Pfaffians of integers, this approach is superior to our division free algorithms because it takes ....

....excessive growth of numbers, it is not necessary to reduce fractions by their greatest common divisor: one can perform pivots involving divisions for which the result is known to be integral. This form of integral pivoting is attributed to Camille Jordan [5, p. 69] see also [6, 2, 9, 7] Edmonds [6] has shown that the numbers do not ex plode and hence, this algorithm runs in polynomial time. The same approach can be extended to Pfaffians [9] Thus, for evaluating determinants or Pfaffians of integers, this approach is superior to our division free algorithms because it takes only O(n 3) ....

Jack Edmonds, Systems of distinct representatives and linear algebra. J. Res. Nat. Bur. Standards, Sect. B 71 (1967), 241-245.

....its mathematical foundation in linear algebra and analytical geometry. Therefore, a central task is the implementation and integration of a generic linear algebra module. Since the dimension is now a parameter of the interface and since the solution of linear systems can be done in different ways [13, 4, 22], a linear algebra concept is part of the interface of the higher dimensional kernel models Cartesian d FT,LA and Homogeneous d RT,LA . The linear algebra concept provides a standard interface to matrix and vector types and the solution of linear systems of equations. 10 Conclusions Many of the ....

EDMONDS, J. Systems of distinct representatives and linear algebra. Journal of Research of the National Bureau of Standards 71(B) (1967), 241--245.

.... analysis of the average accuracy of Hadamard s bound can be found in [1] Once a bound is found, the determinant can be computed by a Gaussian elimination with the sizes of the intermediate integers controlled by exact division or more sophisticatedly by Bareiss s method [6] Another approach [25,13] is to use matrix arithmetic modulo primes and Chinese remaindering (on this technique see [2, Theorem 8.9] or [8, Problem 4.2] The classical associated cost for the exact computation of the determinant, including a fast reduction of the matrix entries modulo the di#erent primes, is [29, Chapter ....

J. Edmonds. Systems of distinct representatives and linear algebra. J. Res. Nat. Bur. Standards -- B, 71(4):241--245, 1967.

....However, to avoid the excessive growth of numbers, it is not necessary to reduce fractions by their greatest common divisor: one can perform pivots involving divisions for which the result is known to be integral. This form of integral pivoting is attributed to Camille Jordan [5, p. 69] see also [6, 2, 9, 7]. Edmonds [6] has shown that the numbers do not ex 74 G unter Rote A74:01 A74:02 A74:03 A74:04 A74:05 A74:06 A74:07 A74:08 A74:09 A74:10 A74:11 A74:12 A74:13 A74:14 A74:15 A74:16 A74:17 A74:18 A74:19 A74:20 A74:21 A74:22 A74:23 A74:24 A74:25 A74:26 A74:27 A74:28 A74:29 A74:30 A74:31 A74:32 ....

....excessive growth of numbers, it is not necessary to reduce fractions by their greatest common divisor: one can perform pivots involving divisions for which the result is known to be integral. This form of integral pivoting is attributed to Camille Jordan [5, p. 69] see also [6, 2, 9, 7] Edmonds [6] has shown that the numbers do not ex 74 G unter Rote A74:01 A74:02 A74:03 A74:04 A74:05 A74:06 A74:07 A74:08 A74:09 A74:10 A74:11 A74:12 A74:13 A74:14 A74:15 A74:16 A74:17 A74:18 A74:19 A74:20 A74:21 A74:22 A74:23 A74:24 A74:25 A74:26 A74:27 A74:28 A74:29 A74:30 A74:31 A74:32 A74:33 A74:34 ....

Jack Edmonds, Systems of distinct representatives and linear algebra. J. Res. Nat. Bur. Standards, Sect. B 71 (1967), 241-245.

....from 0 up to W max , where W max = WN(G) G) and then interpolate to obtain p(x) The interpolation is the most costly step, and it involves solving a system of W max 1 equations with O(W 2 max ) digit integer coefficients. This can be done in time polynomial in W max by the result of Edmonds [Ed], who proved that Gaussian elimination with pivoting is in P for exact rational arithmetic. A second approach is to obtain an upper bound for p k and then compute det B T 0 A 0 for a single value of x that is large (or small) enough to ensure that all of the coefficients p k can be extracted ....

J. Edmonds. Systems of distinct representatives and linear algebra. J. of Research and the National Bureau of Standards, 71B (1967), 241-245.

....0 up to W max , where W max = WN(G) G) and then interpolate to obtain p(x) The interpolation is the most costly step, and it involves solving a system of W max 1 equations with O(W 2 max ) digit integer coefficients. This can be done in time polynomial in W max by the result of Edmonds [Ed], who proved that Gaussian elimination with pivoting is in P for exact rational arithmetic. A second approach is to obtain an upper bound for p k and then compute det B T 0 A 0 for a single value of x that is large (or small) enough to ensure that all of the coefficients p k can be extracted ....

J. Edmonds. Systems of distinct representatives and linear algebra. J. of Research and the National Bureau of Standards, 71B (1967), 241-245.

....several bounds for the cardinality of subrange(A) We use the bound that for current purposes seems most useful. Let the BG rank of a f0; Sigma1g 6 matrix A be the rank of the matrix when the Sigma1s are replaced by reals that are algebraically independent over the rationals. According to Edmonds [1967], the BG rank can be computed as follows, where we assume that A has row index set X and column index set Y . Create a bipartite undirected graph G where X is the node set on one side and where Y is the node set on the other side. Connect node x 2 X with node y 2 Y by an undirected edge if the ....

.... Such a matching can be efficiently found by network flow techniques and thus is easily computed; for details, see, for example, Ahuja, Magnanti, and Orlin (1993) or Cook, Cunningham, Pulleyblank, and Schrijver (1998) A byproduct of that way of computing BG rank(A) is the following result due to Edmonds [1967]. 4.4) Theorem. Any f0; Sigma1g matrix A can be partitioned as (4.5) X 1 X 2 any entry Y 1 Y 2 A = 0 Partition of matrix A so that jX 2 j jY 1 j = BG rank(A) One may compute subrange(A) as follows. IB multiply an arbitrarily selected column of A once by 1 and a second time by ....

Edmonds, J. (1967), Systems of distinct representatives and linear algebra, Journal of Research of the National Bureau of Standards (B) 71B (1967) 241--245.

....at the Annual ACM SIAM Symposium on Discrete Algorithms (SODA99) Maryland, January 1999. 1 Introduction. 1.1 The problem and the background. The classical problem of computing det A, the determinant of an n Theta n matrix A, has long history (see e.g. Mu] Ma] Du] Fo] R52] [E67], B68] P88] BP] Recently, it turned out that some of the most fundamental problems of computational geometry (such as the computation of convex hulls and Voronoi diagrams) are reduced to the computation of det A or, more precisely, its sign, that is, testing whether det A = 0, det A 0, or ....

J. Edmonds, Systems of Distinct Representatives and Linear Algebra, J. Res. Nat. Bur. Standards, Sect. B, 71, 4, 241--245, 1967.

....V is a linear subspace of Mat n (K) Obviously, the conjugacy problem is equivalent to finding a nonsingular matrix in the subspace V . Thus, the conjugacy problem can be considered as a special case of finding nonsingular matrices in linear subspaces of Mat n (K) which was formulated by Edmonds [6]. Note that if our ground field K is sufficiently large then this problem admits an efficient randomized solution: If there exists a nonsingular matrix in the linear subspace V of Mat n (K) then a random matrix from V (i.e. a random linear combination of a basis) is nonsingular with high ....

Edmonds, J. System of distinct representatives and linear algebra. Journal of Research of the National Bureau of Standards 718, 4 (1967), 241--245.

....Proof. We apply Algorithm LIP to the LP. Each LLS step and each small step is carried out in exact rational arithmetic, but after the step, we truncate y to O(L) bits, where L is the total number of bits in A; b; c. The step involves solving linear equations, which can be done in polynomial time [4, 26]. It can be proved that this truncation preserves approximate centering; see, for example, Chapter 3 of [26] The running time is O(n 3:5 c(A) interior point steps, and from (63) and the lemma we see that c(A) O(LA log m) 10 Application to flow problems We can apply Algorithm LIP to flow ....

J. Edmonds. Systems of distinct representatives and linear algebra. J. Res. Nat. Bur. Standards, 71B:241--245, 1967.

....A the Tutte matrix of G 0 , even though it is not quite unique. Given subsets I; J of V 0 , both of size k, we want to determine whether the submatrix A[I; J ] is nonsingular, that is, whether its determinant is nonzero (as a polynomial) or more generally, to determine its rank. Edmonds [8] seems to have been the first to emphasize such algorithmic questions. For example, he proposed the problem of finding a polynomial time algorithm to compute the rank of a matrix whose entries are multivariate polynomials with integral coefficients. This problem is in NP, because it is not ....

J. Edmonds, Systems of distinct representatives and linear algebra, J. Res. Nat. Bur. Standards Sect. B 71:241--245, 1967.

....Proof. We apply Algorithm LIP to the LP. Each LLS step and each small step is carried out in exact rational arithmetic, but after the step, we truncate y to O(L) bits, where L is the total number of bits in A; b; c. The step involves solving linear equations, which can be done in polynomial time [6, 35]. It can be proved that this truncation preserves approximate centering; see, for example, Chapter 3 of [35] The running time is O(n 3:5 c(A) interior point steps, and from (67) and the lemma we see that c(A) O(LA log n) 12 Application to flow problems We can apply Algorithm LIP to flow ....

J. Edmonds. Systems of distinct representatives and linear algebra. J. Res. Nat. Bur. Standards, 71B:241--245, 1967.

....column operations as needed to get the matrix into a form in which its rank can be determined by inspection. There are many ways to accomplish this, and we describe one such way. Suppose V 2 Z M ThetaN and rank(V) r. Then by doing integer preserving Gaussian elimination with full pivoting [Edm67] we can establish the following factorization: HVP = R 11 R 12 0 0 (5) where H is an M Theta M invertible matrix representing the row operations, P is an N Theta N unimodular matrix representing the column operations (which are permutations) R 11 is an r Theta r upper triangular ....

....full pivoting requires O(rMN r 3 ) arithmetic operations. Note that r is at most min(M;N ) A subtle point is that this complexity measure ignores the sizes of matrix entries during the elimination, but it is a well known result that the sizes of numbers involved grow polynomially at the worst[Edm67]. By exploiting the fact that the matrices arising in alignment are sparse and by reordering them so that symbolic constants are used late in the elimination, it may be possible to speed up this algorithm if necessary. Our framework is robust enough that we can add additional constraints to ....

Jack Edmonds. Systems of distinct representatives and linear algebra. Journal of research of national bureau of standards (Sect. B), 71(4):241--245, 1967.

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J. Edmonds, Systems of Distinct Representatives and Linear Algebra, J. Res. Nat. Bur. Standarts, 71B, 1967, pp. 241-245.

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J. Edmonds, Systems of Distinct Representatives and Linear Algebra, J. Res. Nat. Bur. Stand., 71B(4), 241--245, 1967.

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J. Edmonds (1967b): Systems of distinct representatives and linear algebra, J. Res. Nat.

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