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38
On the fixed parameter complexity of graph enumeration problems definable in monadic secondorder logic
, 2001
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Fast parallel algorithms for sparse multivariate polynomial interpolation over finite fields
 SIAM J. COMPUT
, 1990
"... The authors consider the problem of reconstructing (i.e., interpolating) a tsparse multivariate polynomial given a black box which will produce the value of the polynomial for any value of the arguments. It is shown that, if the polynomial has coefficients in a finite field GF[q] and the black box ..."
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Cited by 52 (12 self)
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The authors consider the problem of reconstructing (i.e., interpolating) a tsparse multivariate polynomial given a black box which will produce the value of the polynomial for any value of the arguments. It is shown that, if the polynomial has coefficients in a finite field GF[q] and the black box can evaluate the polynomial in the field GF[qr2g,tnt+37], where n is the number of variables, then there is an algorithm to interpolate the polynomial in O(log (nt)) boolean parallel time and O(n2t log nt) processors. This algorithm yields the first efficient deterministic polynomial time algorithm (and moreover boolean NCalgorithm) for interpolating tsparse polynomials over finite fields and should be contrasted with the fact that efficient interpolation using a black box that only evaluates the polynomial at points in GF[q] is
Improved Sparse Multivariate Polynomial Interpolation Algorithms*
, 1988
"... . We consider the problem of interpolating sparse multivariate polynomials from their values. We discuss two algorithms for sparse interpolation, one due to BenOr and Tiwari (1988) and the other due to Zippel (1988). We present efficient algorithms for finding the rank of certain special Toeplitz s ..."
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Cited by 48 (11 self)
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. We consider the problem of interpolating sparse multivariate polynomials from their values. We discuss two algorithms for sparse interpolation, one due to BenOr and Tiwari (1988) and the other due to Zippel (1988). We present efficient algorithms for finding the rank of certain special Toeplitz systems arising in the BenOr and Tiwari algorithm and for solving transposed Vandermonde systems of equations, the use of which greatly improves the time complexities of the two interpolation algorithms. 1. Introduction We consider the problem of interpolating a multivariate polynomial over a field of characteristic zero from its values at several points. While techniques for interpolating dense polynomials have been known for a long time (e.g., Lagrangian interpolation formula for univariate polynomials), and probabilistic algorithms for interpolating sparse multivariate polynomials have existed since 1979 (Zippel 1979, 1988), until recently no algorithm was known to interpolate sparse mul...
Randomized Interpolation and Approximation of Sparse Polynomials
 SIAM Journal on Computing
, 1995
"... We present a randomized algorithm that interpolates a sparse polynomial in polynomial time in the bit complexity model. The algorithm can be also applied to approximate polynomials that can be approximated by sparse polynomials (the approximation is in the L_2 norm). ..."
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Cited by 39 (1 self)
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We present a randomized algorithm that interpolates a sparse polynomial in polynomial time in the bit complexity model. The algorithm can be also applied to approximate polynomials that can be approximated by sparse polynomials (the approximation is in the L_2 norm).
Polynomial Time Algorithms To Approximate Permanents And Mixed Discriminants Within A Simply Exponential Factor
 Random Structures & Algorithms
, 1999
"... We present real, complex, and quaternionic versions of a simple randomized polynomial time algorithm to approximate the permanent of a nonnegative matrix and, more generally, the mixed discriminant of positive semidefinite matrices. The algorithm provides an unbiased estimator, which, with high pro ..."
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Cited by 33 (3 self)
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We present real, complex, and quaternionic versions of a simple randomized polynomial time algorithm to approximate the permanent of a nonnegative matrix and, more generally, the mixed discriminant of positive semidefinite matrices. The algorithm provides an unbiased estimator, which, with high probability, approximates the true value within a factor of O(c n ), where n is the size of the matrix (matrices) and where c 0:28 for the real version, c 0:56 for the complex version and c 0:76 for the quaternionic version. We discuss possible extensions of our method as well as applications of mixed discriminants to problems of combinatorial counting.
Computational Complexity of Sparse Rational Interpolation
 SIAM J. Comput
, 1991
"... We analyze the computational complexity of sparse rational interpolation, and give the first genuine time (arithmetic complexity does not depend on the size of the coefficients) algorithm for this problem. 1 Max Planck Institute of Mathematics, 5300 Bonn 1, on leave from Steklov Institue of Math ..."
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Cited by 17 (5 self)
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We analyze the computational complexity of sparse rational interpolation, and give the first genuine time (arithmetic complexity does not depend on the size of the coefficients) algorithm for this problem. 1 Max Planck Institute of Mathematics, 5300 Bonn 1, on leave from Steklov Institue of Mathematics, Sov. Acad. of Sciences, Leningrad 191011 2 Dept. of Computer Science, University of Bonn, 5300 Bonn 1, and the International Computer Science Institute, Berkeley, California. Supported in part by Leibniz Center for Research in Computer Science, by the DFG, Grant KA 673/41 and by the SERC Grant GRE 68297 3 Dept. of Mathematics, North Carolina State University, Raleigh, Nc 276958205. Supported in part by NSF Grant DMS8803109. Introduction In this paper we present an algorithm which, given a black box to evaluate a t sparse (a quotient of two tsparse polynomials) nvariable rational function f with integer coefficients, can find the coefficients and ppowers (for some s...
Sparse Polynomial Interpolation in Nonstandard Bases
"... In this paper, we consider the problem of interpolating univariate polynomials over a field of characteristic zero that are sparse in (a) the Pochhammer basis or, (b) the Chebyshev basis. The polynomials are assumed to be given by black boxes, i.e., one can obtain the value of a polynomial at any po ..."
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Cited by 17 (3 self)
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In this paper, we consider the problem of interpolating univariate polynomials over a field of characteristic zero that are sparse in (a) the Pochhammer basis or, (b) the Chebyshev basis. The polynomials are assumed to be given by black boxes, i.e., one can obtain the value of a polynomial at any point by querying its black box. We describe efficient new algorithms for these problems. Our algorithms may be regarded as generalizations of BenOr and Tiwari's (1988) algorithm (based on the BCH decoding algorithm) for interpolating polynomials that are sparse in the standard basis. The arithmetic complexity of the algorithms is O(t 2 + t log d) which is also the complexity of the univariate version of the BenOr and Tiwari algorithm. That algorithm and those presented here also share the requirement of 2t evaluation points. Key words: polynomial interpolation, sparsity, Chebyshev polynomial, Pochhamer basis, BCH codes. AMS(MOS) subject classifications: 68Q40, 12Y05, 41A05. Abbreviated...
Fast Multivariate Power Series Multiplication in Characteristic Zero
 SADIO Electronic Journal on Informatics and Operations Research
, 2001
"... Let k be a eld of characteristic zero. We present a fast algorithm for multiplying multivariate power series over k truncated in total degree. Up to logarithmic factors, its complexity is optimal, i.e. linear in the number of coecients of the series. 1. ..."
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Cited by 16 (6 self)
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Let k be a eld of characteristic zero. We present a fast algorithm for multiplying multivariate power series over k truncated in total degree. Up to logarithmic factors, its complexity is optimal, i.e. linear in the number of coecients of the series. 1.
A ZeroTest and an Interpolation Algorithm for the Shifted Sparse Polynomials
"... Recall that a polynomial f 2 F [X1 ; : : : ; Xn ] is t sparse, if f = P ff I X I contains at most t terms. In [BT 88], [GKS 90] (see also [GK 87] and [Ka 89]) the problem of interpolation of tsparse polynomial given by a blackbox for its evaluation has been solved. In this paper we shall ass ..."
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Cited by 14 (4 self)
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Recall that a polynomial f 2 F [X1 ; : : : ; Xn ] is t sparse, if f = P ff I X I contains at most t terms. In [BT 88], [GKS 90] (see also [GK 87] and [Ka 89]) the problem of interpolation of tsparse polynomial given by a blackbox for its evaluation has been solved. In this paper we shall assume that F is a field of characteristic zero. One can consider a t sparse polynomial as a polynomial represented by a straightline program or an arithmetic circuit of the depth 2 where on the first level there are multiplications with unbounded fanin and on the second level there is an addition with fanin t. In the present paper we consider a generalization of the notion of sparsity, namely we say that a polynomial g(X1 ; : : : ; Xn) 2 F [X1 ; : : : ; Xn ] is shifted tsparse if for a suitable nonsingular n \Theta n matrix A and a vector B the polynomial g(A(X1 ; : : : ; Xn) T + B) is tsparse. One could consider g as being represented by a straightline program of the depth 3 w...