Article

# On the Power of Randomized Branching Programs

Electronic Colloquium on Computational Complexity (ECCC) 01/1995; 2.
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ABSTRACT

We define the notion of a randomized branching program in the natural way similar to the definition of a randomized circuit. We exhibit an explicit function fn for which we prove that: 1) f n can be computed by polynomial size randomized read-once ordered branching program with a small one-sided error; 2) fn cannot be computed in polynomial size by deterministic readonce branching programs; 3) fn cannot be computed in polynomial size by deterministic read- k-times ordered branching program for k = o(n= log n) (the required deterministic size is exp GammaOmega Gamma n k DeltaDelta ). 1 Preliminaries Different models of branching programs introduced in [13, 15], have been studied extensively in the last decade (see for example [19]). A survey of known lower bounds for different models of branching programs can be found in [17]. Developments in the field of digital design and verification have led to the introduction of restricted forms of branching programs. In parti...

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##### Article: A Lower Bound for Randomized Read-k-Times Branching Programs
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ABSTRACT: Randomized branching programs are a probabilistic model of computation defined in analogy to the well-known probabilistic Turing machines. In this paper, we contribute to the complexity theory of randomized read-k-times branching programs. We first consider the case read-k-times = 1 and present a function which has nondeterministic read-once branching programs of polynomial size, but for which every randomized read-once branching program with two-sided error at most 27/128 is exponentially large. The same function also exhibits an exponential gap between the randomized read-once branching program sizes for different constant worst-case errors, which shows that there is no “probability amplification” technique for read-once branching programs which allows to decrease the error to an arbitrarily small constant by iterating probabilistic computations. Our second result is a lower bound for randomized read-k-times branching programs with two-sided error, where k > 1 is allowed. The bound is exponential for k < clog n, c an appropriate constant. Randomized read-k-times branching programs are thus one of the most general types of branching programs for which an exponential lower bound result could be established.
Preview · Article · Jul 1997
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##### Chapter: On the Size of Randomized OBDDs and Read-Once Branching Programs for k -Stable Functions
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ABSTRACT: In this paper, a simple technique which unifies the known approaches for proving lower bound results on the size of deterministic, nondeterministic, and randomized OBDDs and kOBDDs is described. This technique is applied to establish a generic lower bound on the size of randomized OBDDs with bounded error for the so-called “k-stable” functions which have been studied in the literature on read-once branching programs and OBDDs for a long time. It follows by our result that several standard functions are not contained in the analog of the class BPP for OBDDs. It is well-known that k-stable functions are hard for deterministic read-once branching programs. Nevertheless, there is no generic lower bound on the size of randomized read-once branching programs for these functions as for OBDDs. This is proven by presenting a randomized read-once branching program of polynomial size, even with zero error, for a certain k-stable function. As a consequence, we obtain that P ≠⊂ ZPP ∩ NP ∩ coNP for the analogs of these classes defined in terms of the size of read-once branching programs.
Preview · Chapter · Dec 1998
• ##### Conference Paper: Tradeoffs between Nondeterminism and Complexity for Communication Protocols and Branching Programs
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ABSTRACT: In this paper, lower bound and tradeoff results relating the computational power of determinism, nondeterminism, and randomness for communication protocols and branching programs are presented. The main results can be divided into the following three groups. (i) One of the few major open problems concerning nondeterministic communication complexity is to prove an asymptotically exact tradeoff between complexity and the number of available advice bits. This problem is solved here for the case of one-way communication. (ii) Multipartition protocols are introduced as a new type of communication protocols using a restricted form of non-obliviousness. In order to be able to study methods for proving lower bounds on multilective and/or non-oblivious computation, these protocols are allowed to either deterministically or nondeterministically choose between different partitions of the input. Here, the first results showing the potential increase of the computational power by non-obliviousness as well as boundaries on this power are derived. (iii) The above results (and others) are applied to obtain several new exponential lower bounds for different types of oblivious branching programs, which also yields new insights into the power of nondeterminism and randomness for the considered models. The proofs rely on a general technique described here which allows to prove explicit lower bounds on the size of oblivious branching programs in an easy and transparent way.
No preview · Conference Paper · Feb 2000