A deterministic algorithm for sparse multivariate polynomial interpolation
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Fast parallel algorithms for sparse multivariate polynomial interpolation over finite fields
SIAM Journal on Computing
Learning read-once formulas over fields and extended bases
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Learning read-once formulas with queries
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Learning Arithmetic Read-Once Formulas
SIAM Journal on Computing
Interpolating Arithmetic Read-Once Formulas in Parallel
SIAM Journal on Computing
On interpolating arithmetic read-once formulas with exponentiation
Journal of Computer and System Sciences
Fast Probabilistic Algorithms for Verification of Polynomial Identities
Journal of the ACM (JACM)
Randomness efficient identity testing of multivariate polynomials
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Deterministic identity testing for multivariate polynomials
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
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Interpolation of depth-3 arithmetic circuits with two multiplication gates
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SIAM Journal on Computing
Hardness-randomness tradeoffs for bounded depth arithmetic circuits
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
CCC '08 Proceedings of the 2008 IEEE 23rd Annual Conference on Computational Complexity
Proving lower bounds via pseudo-random generators
FSTTCS '05 Proceedings of the 25th international conference on Foundations of Software Technology and Theoretical Computer Science
Improved Polynomial Identity Testing for Read-Once Formulas
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
Deterministic identity testing of depth-4 multilinear circuits with bounded top fan-in
Proceedings of the forty-second ACM symposium on Theory of computing
On the relation between polynomial identity testing and finding variable disjoint factors
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
Arithmetic Circuits: A survey of recent results and open questions
Foundations and Trends® in Theoretical Computer Science
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
FAW-AAIM'12 Proceedings of the 6th international Frontiers in Algorithmics, and Proceedings of the 8th international conference on Algorithmic Aspects in Information and Management
MFCS'12 Proceedings of the 37th international conference on Mathematical Foundations of Computer Science
Monomials, multilinearity and identity testing in simple read-restricted circuits
Theoretical Computer Science
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In this paper we study the problems of polynomial identity testing (PIT) and reconstruction of read-once formulas. The following are some deterministic algorithms that we obtain. An nO(k2) algorithm for checking whether given k ROFs sum to zero or not. An nO(d+k2) time algorithm for checking whether a black box holding the sum of k depth d ROFs computes the zero polynomial. In other words, we provide a hitting set of size nO(d+k2) for the sum of k depth d ROFs. This implies an nO(d) deterministic algorithm for the reconstruction of depth d ROFs. A hitting set of size exp(~O(√n+k2)) for the sum of k ROFs (without depth restrictions). This implies a sub-exponential time deterministic algorithm for black-box identity testing and reconstructing of ROFs. To the best of our knowledge our results give the first polynomial time (non black-box) and sub-exponential time (black-box) identity testing algorithms for the sum of (a constant number of) ROFs. In addition, we introduce and study the read-once testing problem (ROT for short): Given an arithmetic circuit computing a polynomial P(x), decide whether there is a ROF computing P(x). If there is such a formula then output it. Otherwise output "No". We show that most previous algorithms for polynomial identity testing can be strengthen to yield algorithms for the ROT problem. In particular we give ROT algorithms for: Depth-2 circuits (circuits computing sparse polynomials), Depth-3 circuits with bounded top fan-in (both in the black-box and non black-box settings, where the running time depends on the model), non-commutative formulas and sum of k ROFs. The running time of the ROT algorithm is essentially the same running time as the corresponding PIT algorithm for the class. The main tool in most of our results is a new connection between polynomial identity testing and reconstruction of read-once formulas. Namely, we show that in any model that is closed under partial derivatives (that is, a partial derivative of a polynomial computed by a circuit in the model, can also be computed by a circuit in the model) and that has an efficient deterministic polynomial identity testing algorithm, we can also answer the read-once testing problem.