# Problem 32: The Value of a Reverse Pass

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Suggested by Andrew McGregor Kanpur 2009 https://sublinear.info/32

Multi-pass stream algorithms have been designed for a range of problems including longest increasing subsequences [LibenNowellVZ-06,GuhaM-08], graph matchings [McGregor-05], and various geometric problems [ChanC-07]. However, the existing literature almost exclusively considers the case when the multiple passes are in the same direction. One exception is recent work by Magniez et al. [MagniezMN-10] on the DYCK${}_2$ problem: given a length $n$ string in the alphabet “${\tt (}$,${\tt )}$,${\tt [}$,${\tt ]}$”, determine whether it is well-parenthesized, i.e., it can be generated by the grammar $S\rightarrow {\tt (}S{\tt)} ~|~ {\tt [}S{\tt]} ~|~ SS ~|~ \epsilon$? For this problem it can be shown that with one forward and one reverse pass over the input, the problem can be solved with $O(\log^2 n)$ space. On the other hand, any algorithm using $O(1)$ forward passes and no reverse passes, requires $\Omega(\sqrt{n})$ space [ChakrabartiCKM-10,JainN-10]. For what other natural problems is there such a large separation?