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

Electronic Colloquium on Computational Complexity (ECCC) 01/2000; 7. DOI: 10.1007/3-540-44693-1_49
Source: DBLP


Ordered binary decision diagrams (OBDDs) nowadays belong 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 reasonable small OBDDs even for the multiplication of relatively short integers, since for only a 64-bit multiplication millions of nodes are required. Further, a first non-trivial upper bound of 7/3 ċ 24n/3 for the OBDD size of MULn-1,n is provided.

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    • ". The best known upper and lower bounds for the exponent of OBDD size for MUL k,n . The lower bound is by Woelfel [15]. The upper bound consists of three lines corresponding to three intervals in Theorem 9. "
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    ABSTRACT: We show that the middle bit of the multiplication of two n-bit integers can be computed by an ordered binary decision diagram (OBDD) of size less than 2.8·26n/5. This improves the previously known upper bound of by Woelfel (New Bounds on the OBDD-size of integer multiplication via Universal Hashing, J. Comput. System Sci. 71(4) (2005) 520–534). The experimental results suggest that our exponent of 6n/5 is optimal or at least very close to optimal. A general upper bound of O(23n/2) on the OBDD size of each output bit of the multiplication is also presented.
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    • "There are well-known exponential lower bounds on the OBDD-size of the middle bit MULT n,n−1 (see [22]). The π-OBDD size of the highest bit MULT n,2n−1 for any nontrivial variable order π ∈ Σ 2n has been open so far [21, Problem 4.12]. "
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    ABSTRACT: Ordered Binary Decision Diagrams (OBDDs) are a data structure for Boolean functions which is successfully applied in many areas like Integer Programming, Model Checking, and Relational Alge- bra. Nevertheless, many basic graph problems like Connectivity, Reacha- bility, Single-Source Shortest-Paths, and Flow Maximization are known to be PSPACE-hard if their input graphs are represented by OBDDs. This holds even for input OBDDs of constant width. We extend these results by concrete exponential lower bounds on the space complexity of OBDD-based algorithms for the Reachability Problem, the Single-Source Shortest-Paths Problem, and the Maximum Flow Problem. This involves the rst exponential lower bound on the OBDD size for the highest bit of Integer Multiplication w. r. t. the natural interleaved variable order.
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    • "Universal hashing is introduced in the next section. We remark, that following the conference version [17] of this paper, quite some progress has been made in proving lower bounds for the BDD-size of MUL n−1,n . E.g., in [2] a lower bound of Ω(2 n/4 ) was proven for read-once branching programs (improving the earlier 2 "
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    ABSTRACT: Bryant [On the complexity of VLSI implementations and graph representations of boolean functions with applications to integer multiplication, IEEE Trans. Comput. 40 (1991) 205–213] has shown that any OBDD for the function MULn-1,n, i.e. the middle bit of the n-bit multiplication, requires at least 2n/8 nodes. In this paper a stronger lower bound of essentially 2n/2/61 is proven by a new technique, using a universal family of hash functions. As a consequence, one cannot hope anymore to verify e.g. 128-bit multiplication circuits using OBDD-techniques because the representation of the middle bit of such a multiplier requires more than 3·1017OBDD-nodes. Further, 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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