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DOI: 10.4230/LIPIcs.FSTTCS.2009.2311
URN: urn:nbn:de:0030-drops-23111
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Braverman, Mark ; Cook, Stephen ; McKenzie, Pierre ; Santhanam, Rahul ; Wehr, Dustin

Fractional Pebbling and Thrifty Branching Programs

09005.BravemanMark.2311.pdf (0.1 MB)


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.

BibTeX - Entry

  author =	{Mark Braverman and Stephen Cook and Pierre McKenzie and Rahul Santhanam and Dustin Wehr},
  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 =	{Ravi Kannan and K. Narayan Kumar},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-23111},
  doi =		{10.4230/LIPIcs.FSTTCS.2009.2311},
  annote =	{Keywords: Branching programs, space complexity, tree evaluation, pebbling}

Keywords: Branching programs, space complexity, tree evaluation, pebbling
Collection: IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science
Issue Date: 2009
Date of publication: 14.12.2009

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