D. Y. GRIGORIEV, M. KARPINSKI, AND M. F. SINGER, Fast parallel algorithms for sparse multivariate polynomial interpolation over finite fields, SIAM J. Comput., 19 (1990), pp. 1059--1083.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the value of the function at a. In [3] Bshouty, Hancock and Hellerstein gave a polynomial time algorithm for interpolating arithmetic read once formula. Other arithmetic classes that can be interpolated in polynomial time are the classes of sparse polynomials and sparse rational functions. [1,2,4,5,7,8,10]. This work is a nontrivial generalization of the algorithm of Bshouty, Hancock and Hellerstein for polynomial time interpolating arithmetic read once formulas over the basis of addition, subtraction, multiplication and division. An arithmetic read once formula with exponentiation is a read once ....

D.Y. Grigoriev, M. Karpinski, and M. F. Singer. Fast parallel algorithms for sparse multivariate polynomial interpolation over finite fields. Technical report, Research Report No. 8523-C5, University of Bonn (1988.

.... As a particular example, Q(x; y) Pi 1il (y Gamma f i (x) would mean that the black box chooses to output one (or more) of l different functions f 1 (x) f l (x) at every input x (though it is not specified which ones) This is a generalization of the black box model used in [3] 15] [16], 28] and [29] where on input x, the black box outputs f(x) for f polynomial or rational function. While the target we set for our analysis is that we recover at least one function f i such that many of the y s satisfy y = f i (x) we can iterate this process, after stripping off points from f i ....

....black boxes. Previous Work and Our Results The setting where the black box represents a single polynomial or rational function, without noise, is the classic interpolation problem and is well studied. Efficient algorithms for sparse multivariate polynomial interpolation are given in [28] [16], 3] and [29] and for sparse rational functions in [15] The case where the black box represents a single function with some noise has also been studied previously. In [5] it is shown how to reconstruct a univariate polynomial from a ( 1 2 Gamma ffi ) noisy 1 polynomial black box and in [10] ....

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D. Grigoriev, M. Karpinski, M.F. Singer. Fast Parallel Algorithms for Sparse Multivariate Polynomial Interpolation over Finite Fields. SIAM J. Comput., Vol. 19, No. 6, pp 1059-1063, Dec. 1990.

....Results. The setting where the black box represents a single polynomial or rational function, without noise, is the classic interpolation problem and is well studied. Efficient algorithms for sparse multivariate polynomial interpolation are given by Zippel [40, 41] Grigoriev, Karpinski and Singer [21] and Borodin and Tiwari [3] and for sparse rational functions by Grigoriev, Karpinski and Singer [20] The case where the black box represents a single function with some noise has also been studied previously. Welch and Berlekamp [39, 5] see also [14] show how to reconstruct a univariate ....

....by determining all its coefficients) we allow the reconstruction algorithm to reconstruct the polynomial implicitly, i.e. by constructing a black box which computes the multivariate polynomial. If the polynomial turns out to be sparse then we can now use any sparse interpolation algorithm from [3, 20, 21, 40] to reconstruct an explicit representation of the polynomials in time polynomial in n, d and the number of non zero coefficients. On the other hand, by using the techniques of [25] we can also continue to manipulate the black boxes as they are for whatever purposes 1 . We now describe our ....

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D. Grigoriev, M. Karpinski, M.F. Singer. Fast Parallel Algorithms for Sparse Multivariate Polynomial Interpolation over Finite Fields. SIAM Journal on Computing, Vol. 19, No. 6, pp 1059-1063, Dec. 1990. 24

....for various classes of functions such as polynomials, rational functions, and algebraic functions (Grigoriev Karpinski Work supported by NSF grant CCR 9203062 y Work supported by NSF grant CCR 9123666. 1987, Clausen et al. 1988, Ben Or Tiwari 1988, KaltofenLakshman 1988, Borodin Tiwari 1990, Grigoriev KarpinskiSinger 1990,1991a,1991b, Mansour 1992) Sparse interpolation is well recognized as a very useful tool for controlling intermediate expression swell in computer algebra (Zippel 1990, Kaltofen Trager 1990) Sparse polynomials and rational functions can be evaluated quickly and that makes them attractive to ....

Grigoriev, D., Karpinski, M., and Singer, M. (1990), "Fast parallel algorithms for sparse multivariate polynomial interpolation over finite fields," SIAM J.Comp, Vol. 19, pp. 1059--1063.

....the simulation of one of the tn equivalence queries returned YES although the hypothesis is not equivalent to the target function. This happens with probability at most . An algorithm which learns multivariate polynomials using only membership queries is called an interpolation algorithm (e.g. [10, 30, 59, 25, 51, 21, 32]; for more background and references see [60] In [10] it is shown how to interpolate polynomials over infinite fields using only 2t membership queries. Algorithms for interpolating polynomials over finite fields are given in [21, 32] provided that the fields are big enough (in [32] the field ....

D. Y. Grigoriev, M. Karpinski, and M. F. Singer. Fast parallel algorithms for sparse multivariate polynomial interpolation over finite fields. SIAM Journal on Computing, 19(6):1059--1063, 1990.

.... A well known application of Pad# approximation is to computing the minimum span for a linear recurrence, also known as the Berlekamp Massey problem and having important applications to algebraic coding theory, sparse polynomial interpolation, and parallel matrix computations [Berlekamp, 1968, Grigoryev et al. 1990, Kaltofen et al. 1990, Zippel, 1993, Bini and Pan, 1994] Given a natural s and 2s numbers v 0 ; v 2s Gamma1 , compute the minimum natural n s and n numbers t 0 ; t n Gamma1 such that v i = t n Gamma1 v i Gamma1 Delta Delta Delta t 0 v i Gamman ; for i = n; n 1; ....

....the number of nonzero terms of the input polynomial, and O ( Delta) indicates that some polylogarithmic factors may have been omitted. There exists a deterministic version of this algorithm with higher, but still polynomial, complexity. For further information, see [Kaltofen and Lakshman, 1988, Grigoryev et al. 1990, Zippel, 1993] Another, historically the rst, approach is due to [Ben Or and Tiwari, 1988] The algorithm of [Ben Or and Tiwari, 1988] does not need any degree bounds, but uses a bound on the actual number of terms t, on which both the algorithm and its estimates complexity, O (ndt) depend. ....

Grigoryev, D.Y., Karpinski, M., and Singer, M.F. 1990. Fast parallel algorithms for sparse multivariate polynomial interpolation over ønite øelds. SIAM J. Computing, 19(6):10591063.

....large numbers that can occur in the process. The worst situation is with the small finite fields different than GF 2. Here, it has been shown that the solution of t sparse interpolation exists only in the extension fields that are large enough to allow the computation required by the algorithm [6]. Much less is known about finding a solution when the degree is not known in advance, as in our problem. One known algorithm [15] is probabilistic; it is not guaranteed to complete successfully. The algorithm relies on using univariate interpolations to fit the one dimensional projections of the ....

M. Karpinski Grigoriev, D. Y. and M. F. Singer. Fast parallel algorithms for sparse multivariate polynomial interpolation over finite fields. SIAM Journal of Computing, 19(6):1059--1063, December 1990.

....probability, a counterexample to our hypothesis (if such a counterexample exists) We then proceed as before (i.e. modify the counterexample to the domain Sigma n etc. An algorithm which learns multivariate polynomials using only membership queries is called an interpolation algorithm (e.g. [10, 28, 47, 24, 43, 30]; for more background and references see [48] In [10] it is shown how to interpolate polynomials over infinite fields using only 2t membership queries. In [47] it is shown how to interpolate polynomials over finite fields with Omega Gamma t 2 k tkn 2 ) elements. If the number of elements ....

D. Y. Grigoriev, M. Karpinski, and M. F. Singer. Fast parallel algorithms for sparse multivariate polynomial interpolation over finite fields. SIAM J. Comput., 19(6):1059-- 1063, 1990.

....freely (black box model) and or that the degree of the polynomial is known in advance, neither of which is true in our case. Several algorithms have been devised for this problem, including a parallel algorithm which runs in O(log 3 (nt) time with O(t 6 n 2 log 2 (nt) processors [2], where t is the number of terms and n is the number of variables in the polynomial. Little is known about finding a solution when the degree is not known in advance and the points cannot be selected freely, as in our problem. There exists a Monte Carlo probabilistic algorithm [13] which uses ....

M. Karpinski D. Y. Grigoriev and M. F. Singer. Fast parallel algorithms for sparse multivariate polynomial interpolation over finite fields. SIAM Journal of Computing, 19(6):1059--1063, December 1990.

....distinct vectors j = j1 ; jn) j 0 = j 0 1 ; j 0 n ) 2 Z n such that j1 Delta Delta Delta jn d; j 0 1 Delta Delta Delta j 0 n d and fij = fij 0 . Observe that each (n Theta n) minor of D is non zero being a Vandermonde determinant modulo p (cf. GKS 90] It follows that for each pair of distinct vectors j; j 0 as above, there exist at most n Gamma 1 rows fi such that fij = fij 0 . The obtained contradiction proves the lemma. Fix the row fi of D satisfying Lemma 1. Consider the Gamma n d n Delta Theta n matrix E whose (i; k) entry ....

Grigoriev, D., Karpinski, M., Singer, M., Fast parallel algorithms for sparse multivariate polynomial interpolation over finite fields, SIAM J. Comput., 19, 1990, 1059--1063.

....f = p=q is a minimal (t 1 ; t 2 ) sparse representation if f is minimally (t 1 ; t 2 ) sparse and p is t 1 sparse and q is t 2 sparse. We will need a zero test for (t 1 ; t 2 ) sparse rational functions. This is similar 4 to the well known zero test for t sparse polynomials (c.f. 1] 9] [11]) We assume that we are given a black box for an n variable rational function f with integer coefficients in which we can put points with rational coefficients. The output of the black box is either the value of the function at this point or some special sign, e.g. 1 , if the denominator of the ....

....of at most (t 1 t 2 ) 0( t 1 t 2 )n) hyperplanes determined by linear forms with integer coefficients. We now proceed to describe an algorithm to find p powers of the exponents of a minimally (t 1 ; t 2 ) sparse normalized rational function f . For any c 0 using the construction from ([11] or [12] Lemma) one can explicitly produce, for suitable c 1 0, c 2 0, N = t 1 t 2 ) c 1 (t 1 t 2 )n vectors (i) i) 1 ; i) n ) 1 i N where the integers 1 (i) j (t 1 t 2 ) c 2 (t 1 t 2 )n such that for any family of (t 1 t 2 ) c(t 1 t 2 )n ....

Grigoriev, D.Yu., Karpinski, M., Singer, M., Fast Parallel Algorithms for Sparse Multivariate Polynomial Interpolation over Finite Fields, SIAM J. Comp., 19, No. 6, (1990), pp. 1059--1063.

....University Park, PA 16802 2 Dept. of Computer Science, University of Bonn, 5300 Bonn 1, and International Computer Science Institute, Berkeley, California Abstract. 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 t sparse polynomial given by a black box 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 ....

....: Xn) T B) depends on d n 4 where d is the degree of g, we could first interpolate g within time d O(n) and suppose that g is given explicitly. It would be interesting to get rid of d in the complexity bounds as it is usually done in the interpolation of sparse polynomials ( BT 88] GKS 90] Ka 89] The main technical tool we rely on is the criterium of t sparsity based on Wronskian ( GKS 91] GKS 92] the latter criterium has a parametrical nature (so we can select t sparse polynomials from a given parametrical family of polynomials) unlike the approach in [BT 88] using ....

Grigoriev, D., Karpinski, M. & Singer, M., Fast parallel algorithms for sparse multivariate polynimial interpolation over finite fields, SIAM J. Comput. 19, N 6, 1990, pp. 1059-1063.

.... much interest in the design of efficient algorithms for computing sparse representations for various classes of functions such as polynomials, rational functions, and algebraic functions (Grigoriev Karpinski 1987, Clausen et al. 1988, Ben Or Tiwari 1988, Kaltofen Lakshman 1988, Borodin Tiwari 1990, Grigoriev Karpinski Singer 1990,1991,1992, 1993, 1994, Mansour 1992, Lakshman Saunders 1993, 1994) The problem of finding sparsifying invertible linear tranformations for polynomials in F [x 1 ; x 2 ; x n ] was first addressed in a recent paper by Grigoriev and Karpinski (Grigoriev Karpinski 1992) where they provide an ....

Grigoriev, D., Karpinski, M., and Singer, M. (1990), "Fast parallel algorithms for sparse multivariate polynomial interpolation over finite fields," SIAM J.Comp, Vol. 19, pp. 1059--1063.

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D. Grigoriev, M. Karpinski and M. Singer, `Fast parallel algorithm for sparse multivariate polynomials over finite fields', SIAM J. Comput., 19 (1990), 1059-- 1063.

....Computer Science Institute, Berkeley, California, E Mail: marek cs.uni bonn.de. Research supported in part by the DFG Grant KA 673 4 1, by the ESPRIT BR Grants 7097 and ECU030, and by the Volkswagen Stiftung. Introduction The models considered so far require exact computations, see [3], 4] 5] but in practice exact computations of values of sparse polynomials are very difficult. Indeed, we cannot even compute values of sparse polynomials in small integer points such as 2,3, Since the lengths of the values will be exponential in the size of input in the general case. We ....

Grigoriev D.Y., Karpinski M., Singer M.F., Fast Parallel Algorithms for Sparse Multivariate Polynomial Interpolation over Finite Fields. SIAM Journal of Comput. 19, #6 (1990) pp. 1059--1063.

....technical contribution of this paper. Please note that the condition whether f = 0 is satisfiable can be checked deterministically for arbitrary polynomial f 2 GF [q] x 1 ; x n ] within the bounds stated above because of the following (for a problem of a black box interpolation of f , see [GKS 90] Proposition 1. Let f 2 GF [q] x 1 ; Delta Delta Delta ; x n ] and c 2 GF [q] the equation f = c is satisfiable if and only if g = f Gamma c) q Gamma1 Gamma 1 has at least one nonconstant term. Proof. f = c is satisfiable iff (f Gamma c) q Gamma1 = 0 is satisfiable iff the ....

Grigoriev, D., Karpinski, M., Singer, M., Fast Parallel Algorithms for Sparse Multivariate Polynomial Interpolation over Finite Fields, SIAM Journal on Computing 19 (1990), pp. 1059--1063.

....f = p=q is a minimal (t 1 ; t 2 ) sparse representation if f is minimally (t 1 ; t 2 ) sparse and p is t 1 sparse and q is t 2 sparse. We will need a zero test for (t 1 ; t 2 ) sparse rational functions. This is similar to the well known zero test for t sparse polynomials (c.f. 1] 9] [11]) We assume that we are given a black box for an n variable rational function f with integer coefficients in which we can put points with rational coefficients. The output of the black box is either the value of the function at this point or some special sign, e.g. 1 , if the denominator of the ....

....of at most (t 1 t 2 ) 0( t 1 t 2 )n) hyperplanes determined by linear forms with integer coefficients. We now proceed to describe an algorithm to find p powers of the exponents of a minimally (t 1 ; t 2 ) sparse normalized rational function f . For any c 0 using the construction from ([11] or [12] Lemma) one can explicitly produce, for suitable c 1 0, c 2 0, N = t 1 t 2 ) c 1 (t 1 t 2 )n vectors (i) i) 1 ; i) n ) 1 i N where the integers 1 (i) j (t 1 t 2 ) c 2 (t 1 t 2 )n such that for any family of (t 1 t 2 ) c(t 1 t 2 )n ....

Grigoriev, D.Yu., Karpinski, M., Singer, M., Fast Parallel Algorithms for Sparse Multivariate Polynomial Interpolation over Finite Fields, SIAM J. Comp., 19, No. 6, (1990), pp. 1059--1063.

.... It gives a polynomial time approximation algorithm even in case of growing q (the previous one needs q to be fixed as contains log q in the exponent) Then we consider a related question about the zero testing of t sparse multivariate polynomials over F q in the black box model of [2] [5], 16] For the case on a non prime field we obtain several improvements of previously known results. It is hoped they can be applied to the more general problem of polynomial interpolation. Finally we show that in some cases, the image size of a univariate t sparse polynomials can be estimated ....

....f , using (m log d log q log(1=ffi) O(1) arithmetical operations over F q and having the probability of the correct answer at least 1 Gamma ffi. Note that several deterministic algorithms are known for this problem but for q growing all of them are exponential with respect to q (see [2] [5], 16] All these algorithms are based on evaluations of the polynomial in several points over some extension F q l computed from a primitive root of this field. So, in order to get an effective algorithm we should find a primitive root firstly. All known (probabilistic and deterministic) ....

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Grigoriev, D., Karpinski, M., and Singer, M., Fast Parallel Algorithm for Sparse Multivariate Polynomials over Finite Fields, in SIAM J. Comput. 19, 1990, pp. 1059-1063.

....is to design an efficient algorithm testing if f is identical to zero and using as little of calls of B as possible. In a number of papers this question was considered for polynomials, rational functions and algebraic functions belonging various families of functions over various algebraic domains [1, 2, 3, 4, 5, 6, 7, 8, 14, 16, 18], some additional references can be found in Section 4.4 of [15] and in Chapter 12 of [17] In this paper we consider similar questions for multivariate polynomials over p adic fields. As usual Q p denotes the p adic completion of the field of rationals, and C p the p adic completion of its ....

....0:25p s jr i Gamma r j j n. Hence, obtain ord p (g r i Gamma g r j ) log p 4n; 1 i j t: Finally we derive ord p Delta 0:5t(t Gamma 1) log p 4n k which contradicts the inequality (4) For m 2 we use the reduction to the univariate case which for the first time was used in [6]. Let l be the smallest prime number exceeding mt(t Gamma 1) Obviously l 2mt(t Gamma 1) Integers 0 c uv l Gamma 1 we define from the congruences c uv j 1 u v (mod l) u; v = 1; l Gamma 1) 2: The matrix C = c ij ) l Gamma1 i;j=1 is a Cauchy matrix which has the property ....

D. Grigoriev, M. Karpinski and M. Singer, `Fast parallel algorithm for sparse multivariate polynomials over finite fields', SIAM J. Comput., 19(1990), 1059--1063.

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D. Y. GRIGORIEV, M. KARPINSKI, AND M. F. SINGER, Fast parallel algorithms for sparse multivariate polynomial interpolation over finite fields, SIAM J. Comput., 19 (1990), pp. 1059--1083.

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D.Y. Grigoriev, M. Karpinski, and M.F. Singer. Fast parallel algorithms for sparse multivariate polynomial interpolation over finite fields. SIAM Journal on Computing, 19, 1990.

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D. Grigoriev, M. Karpinski and M. Singer, `Fast parallel algorithm for sparse multivariate polynomials over finite fields', SIAM J. Comput., 19 (1990), 1059--1063.

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D. Y. Grigoriev, M. Karpinski, and M. F. Singer. Fast parallel algorithms for sparse multivariate polynomial interpolation over nite elds. SIAM Journal on Computing, 19(6):1059-1063, 1990.

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D.Y. Grigoriev, M. Karpinski, and M.F. Singer. Fast parallel algorithms for sparse multivariate polynomial interpolation over finite fields. Technical report, Research Report No. 8523-C5, University of Bonn (1988), 1988. To appear in SIAM J. Comp.

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D. GRIGORIEV, M. KARPINSKI AND M. SINGER, Fast parallel algorithms for sparse multivariate polynomial interpolation over finite fields, SIAM J. Comp. 19, 1990, pp. 1059--1063.

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