语言识别问题的流算法
Streaming algorithms for language recognition problems
摘要：
我们研究了流模型中以下问题的复杂性。\DLIN的成员测试我们证明了\DLIN中的每种语言都可以被具有逆多项式单侧误差的随机单通道$O(\log n)$空间算法和确定性p通道$O(n/p)$空间算法识别。我们证明了这些算法是最优的。成员资格测试\LL $(k)$对于语言生成的\LL $(k)$文法与一个界的$r$的数量在任何阶段的非终端在最左边的推导，我们表明，成员资格可以测试一个随机的一遍$O(r\log n)$空间算法与逆多项式（在$n$）单侧错误。\DCFL的成员资格测试我们证明了，像上面描述的\DLIN \和$\LL(k)$（它们是\DCFL的子类）一样有效的随机算法不可能存在于所有的\DCFL中：在\VPL \（\DCFL的子类）中有一种语言，对于这种语言，任何具有$ε< 1/2$误差边界的随机p-遍算法必须使用$Ω(n/p)$空间。度序列问题我们研究的问题确定，给定一个序列$d_1, d_2,..., d_n$和一个图$G$，是否度序列的$G$正是$d_1, d_2,..., d_n$。我们给出了一个具有逆多项式单边错误概率的随机化单遍$O(\log n)$空间算法。我们表明，我们的算法是最优的。我们的随机化算法是基于Magniez等人最近的工作。\cite{MMN09}，我们的下界是通过考虑相关的通信复杂性问题。
英文摘要：
We study the complexity of the following problems in the streaming model. Membership testing for \DLIN We show that every language in \DLIN\ can be recognised by a randomized one-pass $O(\log n)$ space algorithm with inverse polynomial one-sided error, and by a deterministic p-pass $O(n/p)$ space algorithm. We show that these algorithms are optimal. Membership testing for \LL$(k)$ For languages generated by \LL$(k)$ grammars with a bound of $r$ on the number of nonterminals at any stage in the left-most derivation, we show that membership can be tested by a randomized one-pass $O(r\log n)$ space algorithm with inverse polynomial (in $n$) one-sided error. Membership testing for \DCFL We show that randomized algorithms as efficient as the ones described above for \DLIN\ and $\LL(k)$ (which are subclasses of \DCFL) cannot exist for all of \DCFL: there is a language in \VPL\ (a subclass of \DCFL) for which any randomized p-pass algorithm with error bounded by $ε< 1/2$ must use $Ω(n/p)$ space. Degree sequence problem We study the problem of determining, given a sequence $d_1, d_2,..., d_n$ and a graph $G$, whether the degree sequence of $G$ is precisely $d_1, d_2,..., d_n$. We give a randomized one-pass $O(\log n)$ space algorithm with inverse polynomial one-sided error probability. We show that our algorithms are optimal. Our randomized algorithms are based on the recent work of Magniez et al. \cite{MMN09}; our lower bounds are obtained by considering related communication complexity problems.
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