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- W256395497 abstract "It is well known that the hardest bit of integer multiplication is the middle bit, i.e., 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 $$k = {mathcal{O}}(hbox{log}, n)$$ (implying time $${mathcal{O}}(n, hbox{log}, n))$$ , space $${mathcal{O}}(hbox{log}, n)$$ and error probability n ?c for arbitrarily chosen constants c is presented.Second, the size of general branching programs and formulas is investigated. Applying Nechiporuk's technique, lower bounds of $$Omega (n^{3/2}/ hbox{log}, n)$$ and ? (n 3/2), respectively, are obtained. Moreover, by bounding the number of subfunctions of MUL n?1,n , it is proven that Nechiporuk's technique cannot provide larger lower bounds than $${mathcal{O}}(n^{5/3}/ hbox{log}, n)$$ and $${mathcal{O}}(n^{5/3})$$ , respectively." @default.
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- W256395497 date "2004-01-01" @default.
- W256395497 modified "2023-09-23" @default.
- W256395497 title "New Results on the Complexity of the Middle Bit of Multiplication" @default.
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