A deterministic algorithm for sparse multivariate polynomial interpolation
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Learning read-once formulas over fields and extended bases
COLT '91 Proceedings of the fourth annual workshop on Computational learning theory
Learning Arithmetic Read-Once Formulas
SIAM Journal on Computing
Reducing randomness via irrational numbers
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Checking polynomial identities over any field: towards a derandomization?
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of 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
Probabilistic algorithms for sparse polynomials
EUROSAM '79 Proceedings of the International Symposiumon on Symbolic and Algebraic Computation
Primality and identity testing via Chinese remaindering
Journal of the ACM (JACM)
Multilinear- NC" " Multilinear- NC"
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Extractors with weak random seeds
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Deterministic polynomial identity testing in non-commutative models
Computational Complexity
Derandomizing polynomial identity tests means proving circuit lower bounds
Computational Complexity
Polynomial Identity Testing for Depth 3 Circuits
Computational Complexity
Locally Decodable Codes with Two Queries and Polynomial Identity Testing for Depth 3 Circuits
SIAM Journal on Computing
Read-once polynomial identity testing
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Hardness-randomness tradeoffs for bounded depth arithmetic circuits
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Lower Bounds and Separations for Constant Depth Multilinear Circuits
CCC '08 Proceedings of the 2008 IEEE 23rd Annual Conference on Computational Complexity
CCC '08 Proceedings of the 2008 IEEE 23rd Annual Conference on Computational Complexity
Diagonal Circuit Identity Testing and Lower Bounds
ICALP '08 Proceedings of the 35th international colloquium on Automata, Languages and Programming, Part I
Arithmetic Circuits: A Chasm at Depth Four
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
A Lower Bound for the Size of Syntactically Multilinear Arithmetic Circuits
SIAM Journal on Computing
Interpolation of Depth-3 Arithmetic Circuits with Two Multiplication Gates
SIAM Journal on Computing
The monomial ideal membership problem and polynomial identity testing
ISAAC'07 Proceedings of the 18th international conference on Algorithms and computation
Proving lower bounds via pseudo-random generators
FSTTCS '05 Proceedings of the 25th international conference on Foundations of Software Technology and Theoretical Computer Science
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
Sans constraints? feature diagrams vs. feature models
SPLC'10 Proceedings of the 14th international conference on Software product lines: going beyond
Arithmetic Circuits: A survey of recent results and open questions
Foundations and Trends® in Theoretical Computer Science
Black-box identity testing of depth-4 multilinear circuits
Proceedings of the forty-third annual ACM symposium on Theory of computing
Blackbox identity testing for bounded top fanin depth-3 circuits: the field doesn't matter
Proceedings of the forty-third annual ACM symposium on Theory of computing
Recent results on polynomial identity testing
CSR'11 Proceedings of the 6th international conference on Computer science: theory and applications
An Almost Optimal Rank Bound for Depth-3 Identities
SIAM Journal on Computing
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Reconstruction of depth-4 multilinear circuits with top fan-in 2
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Testers and their applications
Proceedings of the 5th conference on Innovations in theoretical computer science
Hi-index | 0.00 |
An arithmetic read-once formula (ROF for short) is a formula (a circuit whose underlying graph is a tree) in which the operations are { + ,×} and such that every input variable labels at most one leaf. In this paper we study the problems of giving deterministic identity testing and reconstruction algorithms for ROFs. Our main result is an $n^{{\mathcal{O}}(k + \log n)}$ time deterministic algorithm for checking whether a black box holding the sum of k n -variate ROFs computes the zero polynomial. In other words, we provide a hitting set of size $n^{{\mathcal{O}}(k + \log n)}$ for the sum of k ROFs. This result greatly improves [27] where an $n^{{\mathcal{O}}(k^2 + \sqrt n)}$ algorithm was given for the problem. Using our new results we obtain a deterministic reconstruction algorithms for read-once formulas that runs in time $n^{{\mathcal{O}}(\log n)}$. In fact, our results also hold for the more general model of preprocessed read-once formulas that we define in this paper. In this model we are allowed to replace each variable x i with a polynomial T i (x i ). Our techniques are very close to the techniques in [27]. The main difference is that we obtain several tighter versions of the tools first used there. In particular we obtain a better version of the hardness of representation approach which was first used in [27]. This technique can be thought of as a very explicit way of transforming (mild) hardness of a very structured polynomial to an identity testing algorithm.