2 Search Results for "Wehr, Dustin"


Document
Pebbling, Entropy and Branching Program Size Lower Bounds

Authors: Balagopal Komarath and Jayalal M. N. Sarma

Published in: LIPIcs, Volume 20, 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)


Abstract
We contribute to the program of proving lower bounds on the size of branching programs solving the Tree Evaluation Problem introduced in (Stephen A. Cook, Pierre McKenzie, Dustin Wehr, Mark Braverman, and Rahul Santhanam, 2012). Proving an exponential lower bound for the size of the non-deterministic thrifty branching programs would separate NL from P under the thrifty hypothesis. In this context, we consider a restriction of non-deterministic thrifty branching programs called bitwise-independence. We show that any bitwise-independent non-deterministic thrifty branching program solving BT_2(h,k) must have at least 1/2 k^{h/2} states. Prior to this work, lower bounds were known for general branching programs only for fixed heights h=2,3,4 (Stephen A. Cook, Pierre McKenzie, Dustin Wehr, Mark Braverman, and Rahul Santhanam, 2012). Our lower bounds are also tight (up to a factor of k), since the known (Stephen A. Cook, Pierre McKenzie, Dustin Wehr, Mark Braverman, and Rahul Santhanam, 2012) non-deterministic thrifty branching programs for this problem of size O(k^{h/2+1}) are bitwise-independent. We prove our results by associating a fractional pebbling strategy with any bitwise-independent non-deterministic thrifty branching program solving the Tree Evaluation Problem. Such a connection was not known previously even for fixed heights. Our main technique is the entropy method introduced by Jukna and Zak (S. Jukna and S. Žák, 2003) originally in the context of proving lower bounds for read-once branching programs. We also show that the previous lower bounds known (Stephen A. Cook, Pierre McKenzie, Dustin Wehr, Mark Braverman, and Rahul Santhanam, 2012) for deterministic branching programs for Tree Evaluation Problem can be obtained using this approach. Using this method, we also show tight lower bounds for any k-way deterministic branching program solving Tree Evaluation Problem when the instances are restricted to have the same group operation in all internal nodes.

Cite as

Balagopal Komarath and Jayalal M. N. Sarma. Pebbling, Entropy and Branching Program Size Lower Bounds. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 622-633, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)


Copy BibTex To Clipboard

@InProceedings{komarath_et_al:LIPIcs.STACS.2013.622,
  author =	{Komarath, Balagopal and Sarma, Jayalal M. N.},
  title =	{{Pebbling, Entropy and Branching Program Size Lower Bounds}},
  booktitle =	{30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)},
  pages =	{622--633},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-50-7},
  ISSN =	{1868-8969},
  year =	{2013},
  volume =	{20},
  editor =	{Portier, Natacha and Wilke, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2013.622},
  URN =		{urn:nbn:de:0030-drops-39709},
  doi =		{10.4230/LIPIcs.STACS.2013.622},
  annote =	{Keywords: Pebbling, Entropy Method, Branching Programs, Size Lower Bounds.}
}
Document
Fractional Pebbling and Thrifty Branching Programs

Authors: Mark Braverman, Stephen Cook, Pierre McKenzie, Rahul Santhanam, and Dustin Wehr

Published in: LIPIcs, Volume 4, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2009)


Abstract
We study the branching program complexity of the {\em tree evaluation problem}, introduced in \cite{BrCoMcSaWe09} as a candidate for separating \nl\ from\logcfl. The input to the problem is a rooted, balanced $d$-ary tree of height$h$, whose internal nodes are labelled with $d$-ary functions on$[k]=\{1,\ldots,k\}$, and whose leaves are labelled with elements of $[k]$.Each node obtains a value in $[k]$ equal to its $d$-ary function applied to the values of its $d$ children. The output is the value of the root. Deterministic $k$-way branching programs as related to black pebbling algorithms have been studied in \cite{BrCoMcSaWe09}. Here we introduce the notion of {\em fractional pebbling} of graphs to study non-deterministicbranching program size. We prove that this yields non-deterministic branching programs with $\Theta(k^{h/2+1})$ states solving the Boolean problem ``determine whether the root has value 1'' for binary trees - this isasymptotically better than the branching program size corresponding toblack-white pebbling. We prove upper and lower bounds on the fractionalpebbling number of $d$-ary trees, as well as a general result relating thefractional pebbling number of a graph to the black-white pebbling number. We introduce a simple semantic restriction called {\em thrifty} on $k$-way branching programs solving tree evaluation problems and show that the branchingprogram size bound of $\Theta(k^h)$ is tight (up to a constant factor) for all $h\ge 2$ for deterministic thrifty programs. We show that thenon-deterministic branching programs that correspond to fractional pebbling are thrifty as well, and that the bound of $\Theta(k^{h/2+1})$ is tight for non-deterministic thrifty programs for $h=2,3,4$. We hypothesise that thrifty branching programs are optimal among $k$-way branching programs solving the tree evaluation problem - proving this for deterministic programs would separate \lspace\ from \logcfl\, and proving it for non-deterministic programs would separate \nl\ from \logcfl.

Cite as

Mark Braverman, Stephen Cook, Pierre McKenzie, Rahul Santhanam, and Dustin Wehr. Fractional Pebbling and Thrifty Branching Programs. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 4, pp. 109-120, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)


Copy BibTex To Clipboard

@InProceedings{braverman_et_al:LIPIcs.FSTTCS.2009.2311,
  author =	{Braverman, Mark and Cook, Stephen and McKenzie, Pierre and Santhanam, Rahul and Wehr, Dustin},
  title =	{{Fractional Pebbling and Thrifty Branching Programs}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science},
  pages =	{109--120},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-13-2},
  ISSN =	{1868-8969},
  year =	{2009},
  volume =	{4},
  editor =	{Kannan, Ravi and Narayan Kumar, K.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2009.2311},
  URN =		{urn:nbn:de:0030-drops-23111},
  doi =		{10.4230/LIPIcs.FSTTCS.2009.2311},
  annote =	{Keywords: Branching programs, space complexity, tree evaluation, pebbling}
}
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