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.
- Sanjeev Arora and Boaz Barak. 2009. Computational Complexity—A Modern Approach. Cambridge University Press. I--XXIV, 1--579. Google Scholar
Digital Library
- David A. Mix Barrington and Pierre McKenzie. 1991. Oracle branching programs and Logspace versus P. Inf. Comput. 95, 1 (1991), 96--115. DOI:http://dx.doi.org/10.1016/0890-5401(91)90017-V Google Scholar
Digital Library
- Stephen A. Cook. 1974. An observation on time-storage trade off. J. Comput. Syst. Sci. 9, 3 (1974), 308--316. Google Scholar
Digital Library
- Stephen A. Cook, Pierre McKenzie, Dustin Wehr, Mark Braverman, and Rahul Santhanam. 2012. Pebbles and branching programs for tree evaluation. TOCT 3, 2 (2012), 4. Google Scholar
Digital Library
- Jeff Edmonds, Chung Keung Poon, and Dimitris Achlioptas. 1999. Tight lower bounds for st-connectivity on the NNJAG model. SIAM J. Comput. 28 (1999), 2257--2284. Google Scholar
Digital Library
- Anna Gál, Michal Koucký, and Pierre McKenzie. 2008. Incremental branching programs. Theory of Computing Systems 43, 2 (2008), 159--184. Google Scholar
Cross Ref
- Stasys Jukna. 2012. Boolean Function Complexity—Advances and Frontiers. Springer. Google Scholar
Digital Library
- Stasys Jukna and Stanislav Zák. 2001. On uncertainty versus size in branching programs. Electronic Colloquium on Computational Complexity (ECCC) 8, 39 (2001).Google Scholar
- Alexander A. Razborov. 1991. Lower bounds for deterministic and nondeterministic branching programs. In FCT, Lecture Notes in Computer Science, Vol. 259, Lothar Budach (Ed.). Springer, 47--60. Google Scholar
Digital Library
- Ivan Hal Sudborough. 1978. On the tape complexity of deterministic context-free languages. J. ACM 25, 3 (1978), 405--414. Google Scholar
Digital Library
- F. Vanderzwet. 2013. Fractional pebbling game lower bounds. ArXiv e-prints (May 2013).Google Scholar
- Heribert Vollmer. 1999. Introduction to Circuit Complexity—A Uniform Approach. Springer. I--XI, 1--270. Google Scholar
Digital Library
- Ingo Wegener. 1987. On the complexity of branching programs and decision trees for clique functions. In TAPSOFT, Vol. 1, Lecture Notes in Computer Science, Vol. 249, Hartmut Ehrig, Robert A. Kowalski, Giorgio Levi, and Ugo Montanari (Eds.). Springer, 1--12. Google Scholar
Digital Library
- Dustin Wehr. 2011. Lower bound for deterministic semantic-incremental branching programs solving GEN. CoRR abs/1101.2705 (2011).Google Scholar
- Stanislav Zák. 1984. An exponential lower bound for one-time-only branching programs. In MFCS, Lecture Notes in Computer Science, Vol. 176, Michal Chytil and Václav Koubek (Eds.). Springer, 562--566. Google Scholar
Digital Library
Index Terms
Pebbling, Entropy, and Branching Program Size Lower Bounds
Recommendations
Pebbles and Branching Programs for Tree Evaluation
We introduce the tree evaluation problem, show that it is in LogDCFL (and hence in P), and study its branching program complexity in the hope of eventually proving a superlogarithmic space lower bound. The input to the problem is a rooted, balanced d-...
Parity graph-driven read-once branching programs and an exponential lower bound for integer multiplication
Branching programs are a well-established computation model for Boolean functions, especially read-once branching programs have been studied intensively. Exponential lower bounds for read-once branching programs are known for a long time. On the other ...
On uncertainty versus size in branching programs
We propose an information-theoretic approach to proving lower bounds on the size of branching programs. The argument is based on Kraft type inequalities for the average amount of uncertainty about (or entropy of) a given input during the various stages ...






Comments