A137202 Number of nodes in the BDD for the hidden weighted bit function h_n under the best possible ordering of variables.

3, 3, 5, 9, 16, 23, 33, 46, 63, 82, 109, 139, 178, 224, 282, 348, 434, 531, 653, 796, 973, 1176, 1433, 1725, 2090

Offset: 1

Author

Don Knuth, Apr 23 2008

Keywords

• nonn
• more

Comments

In this problem we don't consider "complement bits" to shorten the BDD.

The best method presently known to find a(n) takes something like 2.5^n steps.

Examples

			For example, when n=8 the smallest BDD is obtained when one tests first x8 (1 node), then x7 (2 nodes), then x1 (4), then x6 (6), then x2 (9), then x5 (12), then x4 (8), then x3 (2). The total number of nodes is 46, including the two sink nodes.
		

References

• D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4.

Links

• Beate Bollig, Martin Löbbing, Martin Sauerhoff and Ingo Werner, On the complexity of the hidden weighted bit function for various BDD models, Theoretical Informatics and Applications, 33 (1999), 103-115, Theorem 4.4.
• Randal E. Bryant, On the complexity of VLSI implementations and graph representations of Boolean functions with application to integer multiplication, IEEE Transactions on Computers C-40 (1991), 205-213.

Crossrefs

Cf. A136445.