New Results on the Most Significant Bit of Integer Multiplication

Integer multiplication as one of the basic arithmetic functions has been in the focus of several complexity theoretical investigations and ordered binary decision diagrams (OBDDs) are one of the most common dynamic data structures for Boolean functions. Analyzing the limits of symbolic graph algorithms for the reachability problem Sawitzki (2006) has presented the first exponential lower bound on the pi-OBDD size for the most significant bit of integer multiplication according to one predefined variable order pi. Since the choice of the variable order is a main issue to obtain OBDDs of small size the investigation is continued. As a result a new upper bound method and the first non-trivial tipper bound oil the size of OBDDs according to an arbitrary variable order is presented. Furthermore, Sawitzki's lower bound is improved.