Difference between revisions of "Open Problems:32"

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(Created page with "{{DISPLAYTITLE:Problem 32: The Value of a Reverse Pass}} {{Infobox |label1 = Proposed by |data1 = Andrew McGregor |label2 = Source |data2 = [[Workshops:Kanpur_2009|Kanpur 2009...")
 
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|title=The Value of a Reverse Pass
|label1 = Proposed by
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|source=kanpur09
|data1 = Andrew McGregor
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|who=Andrew McGregor
|label2 = Source
 
|data2 = [[Workshops:Kanpur_2009|Kanpur 2009]]
 
|label3 = Short link
 
|data3 = http://sublinear.info/32
 
 
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Multi-pass stream algorithms have been designed for a range of problems including longest increasing subsequences {{cite|LibenNowellVZ-06|GuhaM-08}}, graph matchings {{cite|McGregor-05}}, and various geometric problems {{cite|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. {{cite|MagniezMN-10}} on  the '''DYCK'''${}_2$ problem: given a length $n$ string in the alphabet
 
Multi-pass stream algorithms have been designed for a range of problems including longest increasing subsequences {{cite|LibenNowellVZ-06|GuhaM-08}}, graph matchings {{cite|McGregor-05}}, and various geometric problems {{cite|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. {{cite|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 {{cite|ChakrabartiCKM-10|JainN-10}}. For what other natural problems is there such a large separation?
 
“${\tt (}$,${\tt )}$,${\tt [}$,${\tt ]}$”, 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 {{cite|ChakrabartiCKM-10|JainN-10}}. For what other natural problems is there such a large separation?

Revision as of 05:25, 16 November 2012

Suggested by Andrew McGregor
Source Kanpur 2009
Short link 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?