A Lower Bound for Integer Multiplication on Randomized Read-Once Branching Programs (original) (raw)

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A lower bound for integer multiplication on randomized ordered read-once branching programs

Information and Computation, 2003

We prove an exponential lower bound 2 (n= logn) on the size of any randomized ordered read-once branching program computing integer multiplication. Our proof depends on proving a new lower bound on Yao's randomized one-way communication complexity of certain boolean functions. It generalizes to some other common models of randomized branching programs. In contrast, we prove that testing integer multiplication, contrary even to nondeterministic situation, can be computed by randomized ordered read-once branching program in polynomial size. It is also known that computing the latter problem with deterministic read-once branching programs is as hard as factoring integers.

A Lower Bound for Integer Multiplication with Read-Once Branching Programs

SIAM Journal on Computing, 1998

We prove that read-once branching programs computing integer multiplication require size 2 Ω(√ n). This is the first nontrivial lower bound for multiplication on branching programs that are not oblivious. By the appropriate problem reductions, we obtain the same lower bound for other arithmetic functions.

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Electronic Colloquium on Computational Complexity, 1998

We de ne the notion of a randomized branching program in the natural way similar to the de nition of a randomized circuit. We exhibit an explicit boolean function f n : f0;1g n ! f0;1g 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;

A lower bound for read-once-only branching programs

Journal of Computer and System Sciences, 1987

We give a C" lower bound for read-once-only branching programs computing an explicit Boolean function. For n = (;), the function computes the parity of the number of triangles in a graph on v vertices. This improves previous exp(c &) lower bounds for other graph functions by Wegener and Zak. The result implies a linear lower bound for the space complexity of this Boolean function on "eraser machines," i.e., machines that erase each input bit immediately after having read it.

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We de ne the notion of a randomized branching program in the natural way similar to the de nition of a randomized circuit. We exhibit an explicit boolean function f n : f0;1g n ! f0;1g 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;

New results on the complexity of the middle bit of multiplication

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Abstract. It is well known that the hardest bit of integer multiplication is the middle bit, ie, MUL n��� 1, n. This paper contains several new results on its complexity. First, the size s of randomized read-k branching programs, or, equivalently, their space (log s) is investigated. A randomized algorithm for MUL n��� 1, n with k= O (log\, n)(implying time O (n\, log\, n)), space O (log\, n) and error probability n��� c for arbitrarily chosen constants c is presented. Second, the size of general branching programs and formulas is investigated.

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The seminar \Structure and Complexity" was the third Dagstuhl Seminar devoted to the structural aspects of Computational Complexity Theory. It was attented by 40 scientists who in 27 talks presented new results in this eld. The following topics were among the main subjects covered by the talks: Kolmogorov complexity, isomorphism theory, resource-bounded measures, relativizations, randomness, leaf language characterizations, circuit theory, logical characterizations of complexity classes, interactive proof systems, oneway functions, and computational models.