Interpolation and Approximation of Sparse Multivariate Polynomials over $GF(2)$

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	Interpolation and Approximation of Sparse Multivariate Polynomials over $GF(2)$
	Ron M. Roth ; Gyora M. Benedek
刊名	SIAM Journal on Computing
	1991
卷号	20 期号:2 页码:291-314
合作状况	其它
英文摘要	A function $f:\{ 0,1\} ^n \to \{ 0,1\} $ is called $t$-sparse if the $n$-variable polynomial representation of $f$ over $GF(2)$ contains at most $t$ monomials. Such functions are uniquely determined by their values at the so-called critical set of all binary $n$-tuples of Hamming weight $ \geqq n - \lfloor \log _2 t \rfloor - 1$. An algorithm is presented for interpolating any $t$-sparse function $f$, given the values of $f$ at the critical set. The time complexity of the proposed algorithm is proportional to $n$, $t$, and the size of the critical set. Then, the more general problem of approximating 1-sparse functions is considered, in which case the approximating function may differ from $f$ at a fraction $\varepsilon $ of the space $\{ 0,1\} ^n $. It is shown that $O(({t / \varepsilon }) \cdot n)$ evaluation points are sufficient for the (deterministic) $\varepsilon $-approximation of any $t$-sparse function, and that an order $(t / \varepsilon )^{\alpha (t,\varepsilon )} \cdot \log n$ points are necessary for this purpose, where $\alpha (t,\varepsilon ) \geqq 0.694$ for a large range of $t$ and $\varepsilon $. Similar bounds hold for the $t$-term DNF case as well. Finally, a probabilistic polynomial-time algorithm is presented for the $\varepsilon $-approximation of any $t$-sparse function.
收录类别	其他
语种	英语
公开日期	2010-05-13
内容类型	期刊论文
源URL	[http://124.16.136.157/handle/311060/1302]
专题	软件研究所_软件所图书馆_期刊论文
推荐引用方式 GB/T 7714	Ron M. Roth,Gyora M. Benedek. Interpolation and Approximation of Sparse Multivariate Polynomials over $GF(2)$[J]. SIAM Journal on Computing,1991,20(2):291-314.
APA	Ron M. Roth,&Gyora M. Benedek.(1991).Interpolation and Approximation of Sparse Multivariate Polynomials over $GF(2)$.SIAM Journal on Computing,20(2),291-314.
MLA	Ron M. Roth,et al."Interpolation and Approximation of Sparse Multivariate Polynomials over $GF(2)$".SIAM Journal on Computing 20.2(1991):291-314.