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| Interpolation and Approximation of Sparse Multivariate Polynomials over $GF(2)$ | |
| Ron M. Roth; Gyora M. Benedek | |
| 1991 | |
| Source | SIAM Journal on Computing
 |
| Volume | 20Issue:2Pages:291-314 |
| English Abstract | 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. |
| Indexed Type | 其他 |
| Cooperation Status | 其它 |
| Language | 英语 |
| Content Type | 期刊论文 |
| URI | http://ir.iscas.ac.cn/handle/311060/1302 |
| Collection | 中国科学院软件研究所 |
| Recommended Citation 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. |
| Files in This Item: | ||||||
| File Name/Size | DocType | Version | Access | License | ||
| bj01130931.pdf（1396KB） | 开放获取 | License | Application Full Text | |||
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