Better upper bounds on the QOBDD size of integer multiplication
Discrete Applied Mathematics
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It is well known that the hardest bit of integer multiplication is the middle bit, i.e. MULn-1,n. This paper contains several new results on its complexity. First, the size s of randomized read-k branching programs, or, equivalently, its 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. Applying Nechiporukýs technique, lower bounds of \Omega (n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} /\log n) and \Omega (n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}}), respectively, are obtained. Moreover, by bounding the number of subfunctions of MUL_n-1,n, it is proven that Nechiporukýs technique cannot provide larger lower bounds than 0(n^{{7 \mathord{\left/ {\vphantom {7 4}} \right. \kern-\nulldelimiterspace} 4}} /\log n)and 0(n^{{7 \mathord{\left/ {\vphantom {7 4}} \right. \kern-\nulldelimiterspace} 4}} ), respectively.