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Pebbling, Entropy, and Branching Program Size Lower Bounds

Published:11 May 2015Publication History
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Abstract

We contribute to the program of proving lower bounds on the size of branching programs solving the Tree Evaluation Problem introduced by Cook et al. [2012]. Proving a superpolynomial lower bound for the size of nondeterministic thrifty branching programs would be an important step toward separating NL from P using the tree evaluation problem. First, we show that Read-Once Nondeterministic Thrifty BPs are equivalent to whole black-white pebbling algorithms, thus showing a tight lower bound (ignoring polynomial factors) for this model. We then introduce a weaker restriction of nondeterministic thrifty branching programs called Bitwise Independence. The best known [Cook et al. 2012] nondeterministic thrifty branching programs (of size O(kh/2 + 1)) for the tree evaluation problem are Bitwise Independent. As our main result, we show that any Bitwise Independent Nondeterministic Thrifty Branching Program solving BT2(h, k) must have at least (k2)h/2 states. Prior to this work, lower bounds were known for nondeterministic thrifty branching programs only for fixed heights h = 2, 3, 4 [Cook et al. 2012]. We prove our results by associating a fractional black-white pebbling strategy with any bitwise independent nondeterministic 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 Zák [2001] originally in the context of proving lower bounds for read-once branching programs. We also show that the previous lower bounds known [Cook et al. 2012] for deterministic branching programs for the 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 the Tree Evaluation Problem when the instances are restricted to have the same group operation in all internal nodes.

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      • Published in

        cover image ACM Transactions on Computation Theory
        ACM Transactions on Computation Theory  Volume 7, Issue 2
        May 2015
        101 pages
        ISSN:1942-3454
        EISSN:1942-3462
        DOI:10.1145/2775140
        Issue’s Table of Contents

        Copyright © 2015 ACM

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        Association for Computing Machinery

        New York, NY, United States

        Publication History

        • Published: 11 May 2015
        • Accepted: 1 August 2014
        • Revised: 1 July 2014
        • Received: 1 June 2013
        Published in toct Volume 7, Issue 2

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