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On the Complexity of Integer Multiplication in Branching Programs with Multiple Tests and in ReadOnce Branching Programs with Limited Nondeterminism (Extended Abstract)
"... Branching Programs (BPs) are a wellestablished computation and representation model for Boolean functions. Although exponential lower bounds for restricted BPs such as ReadOnce Branching Programs (BP1s) have been known for a long time, the proof of lower bounds for important selected functions is ..."
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Branching Programs (BPs) are a wellestablished computation and representation model for Boolean functions. Although exponential lower bounds for restricted BPs such as ReadOnce Branching Programs (BP1s) have been known for a long time, the proof of lower bounds for important selected functions is sometimes difficult. Especially the complexity of fundamental functions such as integer multiplication in different BP models is interesting. In [4], the first strongly exponential lower bound of � � 2n/4 � has been proven for the complexity of integer multiplication in the deterministic BP1 model. Here, we consider two wellstudied BP models which generalize BP1s by allowing a limited amount of nondeterminism and multiple variable tests, respectively. More precisely, we prove a
A Lower Bound Technique for nondeterministic graphdriven readonce branching programs and its applications
 Proc. of MFCS 2002
, 2002
"... Abstract. We present a new lower bound technique for a restricted Branching Program model, namely for nondeterministic graphdriven readonce Branching Programs (g.d.BP1s). The technique is derived by drawing a connection between ωnondeterministic g.d.BP1s and ωnondeterministic communication comp ..."
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Abstract. We present a new lower bound technique for a restricted Branching Program model, namely for nondeterministic graphdriven readonce Branching Programs (g.d.BP1s). The technique is derived by drawing a connection between ωnondeterministic g.d.BP1s and ωnondeterministic communication complexity (for the nondeterministic acceptance modes ω ∈ {∨, ∧, ⊕}). We apply the technique in order to prove an exponential lower bound for integer multiplication for ωnondeterministic wellstructured g.d.BP1s. (For ω = ⊕ an exponential lower bound was already obtained in [5] by using a different technique.) Further, we use the lower bound technique to prove for an explicitly defined fnction which can be represented by polynomial size ωnondeterministic BP1s that it has exponential complexity in the ωnondeterministic wellstructured g.d.BP1 model for ω ∈ {∨, ⊕}. This answers an open question from Brosenne, Homeister, and Waack [7], whether the nondeterministic BP1 model is in fact more powerful than the wellstructured graphdriven variant. 1
General Terms Theory
"... We prove exponential size lower bounds for nondeterministic and randomized readk BPs as well as a timespace tradeoff lower bound for unrestricted, deterministic multiway BPs computing the middle bit of integer multiplication. The lower bound for randomized readk BPs is superpolynomial as long as ..."
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We prove exponential size lower bounds for nondeterministic and randomized readk BPs as well as a timespace tradeoff lower bound for unrestricted, deterministic multiway BPs computing the middle bit of integer multiplication. The lower bound for randomized readk BPs is superpolynomial as long as the error probability is superpolynomially small. For polynomially small error, we have a polynomial upper bound on the size of approximating readonce BPs for this function. The lower bounds follow from a more general result for the graphs of universal hash classes that is applicable to the graphs of arithmetic functions such as integer multiplication, convolution, and finite field multiplication.