Streaming algorithms for language recognition problems

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 $\epsilon < 1/2$ must use $\Omega(n/p)$ 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

Streaming algorithms for language recognition problems

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 recognized by a randomized one-pass O(log n) space algorithm with an 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 an 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 an error bounded by epsilon < 1/2 must use Omega(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 an 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. [1]; our lower bounds are obtained by considering related communication complexity problems. (c) 2013 Elsevier B.V. All rights reserved