{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article6624","name":"Fractional Pebbling and Thrifty Branching Programs","abstract":"We study the branching program complexity of the {\\em tree evaluation problem},\r\nintroduced 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.\r\n\r\nDeterministic $k$-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\r\nprograms 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.\r\n\r\nWe 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\r\n$h\\ge 2$ for deterministic thrifty programs. We show that thenon-deterministic branching programs that correspond to fractional pebbling are\r\nthrifty as well, and that the bound of $\\Theta(k^{h\/2+1})$ is tight for\r\nnon-deterministic thrifty programs for $h=2,3,4$. We hypothesise that thrifty\r\nbranching programs are optimal among $k$-way branching programs solving the\r\ntree evaluation problem - proving this for deterministic programs would\r\nseparate \\lspace\\ from \\logcfl\\, and proving it for non-deterministic programs\r\nwould separate \\nl\\ from \\logcfl.","keywords":["Branching programs","space complexity","tree evaluation","pebbling"],"author":[{"@type":"Person","name":"Braverman, Mark","givenName":"Mark","familyName":"Braverman"},{"@type":"Person","name":"Cook, Stephen","givenName":"Stephen","familyName":"Cook"},{"@type":"Person","name":"McKenzie, Pierre","givenName":"Pierre","familyName":"McKenzie"},{"@type":"Person","name":"Santhanam, Rahul","givenName":"Rahul","familyName":"Santhanam"},{"@type":"Person","name":"Wehr, Dustin","givenName":"Dustin","familyName":"Wehr"}],"position":10,"pageStart":109,"pageEnd":120,"dateCreated":"2009-12-14","datePublished":"2009-12-14","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Braverman, Mark","givenName":"Mark","familyName":"Braverman"},{"@type":"Person","name":"Cook, Stephen","givenName":"Stephen","familyName":"Cook"},{"@type":"Person","name":"McKenzie, Pierre","givenName":"Pierre","familyName":"McKenzie"},{"@type":"Person","name":"Santhanam, Rahul","givenName":"Rahul","familyName":"Santhanam"},{"@type":"Person","name":"Wehr, Dustin","givenName":"Dustin","familyName":"Wehr"}],"copyrightYear":"2009","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.FSTTCS.2009.2311","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6207","volumeNumber":4,"name":"IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science","dateCreated":"2009-12-07","datePublished":"2009-12-07","editor":[{"@type":"Person","name":"Kannan, Ravi","givenName":"Ravi","familyName":"Kannan"},{"@type":"Person","name":"Narayan Kumar, K.","givenName":"K.","familyName":"Narayan Kumar"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article6624","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6207"}}}