New Results on the Most Significant Bit of Integer Multiplication

Authors:
Beate Bollig;Jochen Klump
Affiliations:
LS2 Informatik, TU Dortmund, Dortmund, Germany 44221;LS2 Informatik, TU Dortmund, Dortmund, Germany 44221
Venue:
ISAAC '08 Proceedings of the 19th International Symposium on Algorithms and Computation
Year:
2008

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Abstract

Integer multiplication as one of the basic arithmetic functionshas been in the focus of several complexity theoreticalinvestigations and ordered binary decision diagrams (OBDDs) are oneof the most common dynamic data structures for Boolean functions.Analyzing the limits of symbolic graph algorithms for thereachability problem Sawitzki (2006) has presented the firstexponential lower bound on the π-OBDD size for the mostsignificant bit of integer multiplication according to onepredefined variable order π. Since the choice of thevariable order is a main issue to obtain OBDDs of small size theinvestigation is continued. As a result a new upper bound methodand the first non-trivial upper bound on the size of OBDDsaccording to an arbitrary variable order is presented. Furthermore,Sawitzki's lower bound is improved.