New Bounds on the OBDD-Size of Integer Multiplication via Universal Hashing

Electronic Colloquium on Computational Complexity - ECCC 12/2000; 7(46):563-574. DOI:10.1007/3-540-44693-1_49

ABSTRACT Ordered binary decision diagrams (OBDDs) nowadays be- long to the most common representation types for Boolean functions.
Although they allow important operations such as satisfiability test and equality test to be performed efficiently, their
limitation lies in the fact that they may require exponential size for important functions. Bryant [8] has shown that any OBDD-representation of the function MULn-1;n, which computes the middle bit of the product of two n-bit numbers, requires at least 2n/8 nodes. In this paper a stronger bound of 2n/2/61 is proven by a new technique, using a recently found universal family of hash functions [23]. As a result, one cannot hope anymore to find reason- able small OBDDs even for the multiplication of relatively short integers,
since for only a 64-bit multiplication millions of nodes are required. Fur- ther, a first non-trivial upper bound of 7/3.
24n/3 for the OBDD size of MULn-1;n is provided.

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Philipp Woelfel