Quine-McCluskey Minimization Algorithm

Section 4.4

What is it?

• Karnaugh maps are nice, but only works for up to 6 variables because it is “manual.”

• Quine-McCluskey algorithm is similar, but is automated with a computer.

• Like Karnaugh maps, Q-M relies on “Theorem T10,” that is– AB + AB’ = A.

Quine-McCluskey Algorithm• Computerized• Theoretically no limit to number of input variables• Effective, always finds all the prime implicants.• However, the complexity (worst case) is exponential

in the number of variables!• But that may not be really “bad,” because the number

of minterms that you’ll have to input is also exponential!

• Q-M Algorithm, as is, is pretty “plain” (but still interesting enough)– No good strategy for always running it in a reasonable

amount of time.

Representation of Product Terms

• Must represent 3 possibilities for each variable– Uncomplemented (perhaps use 1)– Complemented (perhaps use 0)– Doesn’t appear (perhaps use X)

Representation of Product Terms(cont’d)

Representation of Product Terms(cont’d)

Quine-McCluskey Algorithm has two halves

• Generate all prime implicants– A prime implicant is a product term where

there’s no smaller product term that covers it– That is, a prime implicant corresponds to a

biggest possible circle in a K-map where you circle groups of 1’s

• Search for a set of prime implicants that give you a minimum-cost cover for all the minterms

The Algorithm – Intuitive explanation of the Tabular Method

• For hand-computed version• Start with truth table (i.e. all minterms)• Group all minterms by the number of 1’s they contain in the

input columns• Combine product terms from adjacent groups • Two product terms can be combined if they differ by exactly 1

bit (one minterm has a true version of the variable, the other a complemented version of same)

• The combined product term has a 0 in that position• We’ll do an example on the board• The computer version doesn’t group minterms into groups like

this – maybe that should be updated

The Algorithm – Intuitive explanation (cont’d)

• After combining a pair of minterms or product terms in general, create new groups

• Each member of a new group must have the same number of 1’s and -’s.– The dashes represent missing variables

• Also, the -’s (missing variables) must be in the same positions.

The Algorithm – Intuitive explanation (cont’d)

• Combine as much as possible.

• Finally, when no more combination is possible, select prime implicants (like a biggest “circle” in a K-map) to “cover” all ones in the function.

Cube manipulations:

(a) Determining whether 2 cubes are combinable by T10.

(b) Combining cubes using T10.

Functions for comparing and combining cubes