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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    • "Progress in the analysis of MUL n−1,n has been achieved by a new approach using universal hashing. Woelfel [20] has improved Bryant's lower bound to Ω(2 n/2 ) and Bollig and Woelfel [5] have presented a lower bound of Ω(2 n/4 ) for read-once branching programs. Exponential lower bounds have also been proved for more general read-once branching program models that allow limited nondeterminism and for models where some but not all variables may be tested multiple times (see e.g. "
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    ABSTRACT: Integer multiplication as one of the basic arithmetic functions has been in the focus of several complexity theoretical investigations. Ordered binary decision diagrams (OBDDs) are the most common dynamic data structure for boolean functions. Among the many areas of application are verification, model checking, computer-aided design, relational algebra, and symbolic graph algorithms. In this paper it is shown that the OBDD complexity of the most significant bit of integer multiplication is exponential answering an open question posed by Wegener (2000).
    Theoretical Computer Science 04/2011; 412(18):1686-1695. DOI:10.1016/j.tcs.2010.12.043 · 0.66 Impact Factor
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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.
    Discrete Applied Mathematics 05/2007; 155(10-155):1224-1232. DOI:10.1016/j.dam.2006.11.010 · 0.80 Impact Factor
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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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