New Results on the Complexity of the Middle Bit of Multiplication

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, their space (log s) is investigated. A randomized algorithm for MULn−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 Nechiporuk’s technique, lower bounds of Ω n3/2/ log n and Ω n3/2 , respectively, are obtained. Moreover, by bounding the number of subfunctions of MULn−1,n, it is proven that Nechiporuk’s technique cannot provide larger lower bounds than O(n5/3/ log n) and O(n5/3), respectively. ∗ Supported in part by DFG grant WO 1232/1-1 

• Type: Conference
• Year: 2005
• Where: COCO
• Authors: Ingo Wegener, Philipp Woelfel
• Keywords: Branching Programs, COCO 2005, Lower Bounds, Read-k Branching Programs