# Interpolating Arithmetic Read-Once Formulas in Parallel

@article{Bshouty1998InterpolatingAR, title={Interpolating Arithmetic Read-Once Formulas in Parallel}, author={Nader H. Bshouty and Richard Cleve}, journal={SIAM J. Comput.}, year={1998}, volume={27}, pages={401-413} }

A formula is read-once if each variable appears in it at most once. An arithmetic formula is one in which the operations are addition, subtraction, multiplication, and division (and constants are allowed). We present a randomized (Las Vegas) parallel algorithm for the exact interpolation of arithmetic read-once formulas over sufficiently large fields. More specifically, for $n$-variable read-once formulas and fields of size at least 3(n2+3n-2), our algorithm runs in $O(\log^2 n)$ parallel steps…

## 20 Citations

### Read-once polynomial identity testing

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A deterministic polynomial time canonization scheme for polynomials computed by read-once arithmetic formulas is proposed, and it is shown that when the arithmetic formula is allowed to read a variable twice, this problem is as hard as the graph isomorphism problem.

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The question of efficiently evaluating a polynomial given an oracle access to its power is studied and it is shown that a reconstruction algorithm for a circuit class C can be extended to handle f for f ∈ C, and an efficient deterministic algorithm for factoring sparse multiquadratic1 polynomials is found.

### Random Arithmetic Formulas can be Reconstructed Efficiently Full Version

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A randomized algorithm that given blackbox access to the polynomial f computed by an unknown/hidden arithmetic formula φ reconstructs, on the average, an equivalent or smaller formulaπ̂ in time polynomials in the size of its output φ̂.

### Characterizing Arithmetic Read-Once Formulae

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This work gives a simple characterization of an arithmetic Read-Once Formula in which the operations are { +, ×} and such that every input variable labels at most one leaf.

### Sums of read-once formulas: How many summands are necessary?

- Mathematics, Computer ScienceTheor. Comput. Sci.
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This work proves, for certain multilinear polynomials, a tight lower bound on the number of summands in such an expression.

### Complete Derandomization of Identity Testing and Reconstruction of Read-Once Formulas

- Mathematics, Computer ScienceElectron. Colloquium Comput. Complex.
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The first polynomial-time deterministic identity testing algorithm that operates in the black-box setting for ROFs, as well as some other related models, are obtained.

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