General Computer Science 320201 GenCS I & II Lecture ... - Kwarc

n-ary Gates as Subgraphs
Idea: Identify (and abbreviate) frequently occurring subgraphs
Definition 402 AND(x1, . . . , xn) := 1 n
i=1 xi and OR(x1, . . . , xn) := 1 n
i=1 xi
Note: These can be realized as balanced binary trees Gn
Corollary 403 C(Gn) = n − 1 and dp(Gn) = ⌊log 2(n)⌋.
Notation 404
AND OR
c○: Michael Kohlhase 233
Using these building blocks, we can establish a worst-case result for the depth of a combinatory
circuit computing a given Boolean function.
Worst Case Depth Theorem for Combinational Circuits
Theorem 405 The worst case depth dp(G) of a combinational circuit G which realizes an
k × n-dimensional boolean function is bounded by dp(G) ≤ n + ⌈log 2(n)⌉ + 1.
Proof: The main trick behind this bound is that AND and OR are associative and that the
according gates can be arranged in a balanced binary tree.
P.1 Function f corresponding to the output oj of the circuit G can be transformed in DNF
P.2 each monomial consists of at most n literals
P.3 the possible negation of inputs for some literals can be done in depth 1
P.4 for each monomial the ANDs in the related circuit can be arranged in a balanced binary
tree of depth ⌈log 2(n)⌉
P.5 there are at most 2 n monomials which can be ORed together in a balanced binary tree
of depth ⌈log 2(2 n )⌉ = n.
c○: Michael Kohlhase 234
Of course, the depth result is related to the first worst-case complexity result for Boolean expressions
(Theorem 270); it uses the same idea: to use the disjunctive normal form of the Boolean
function. However, instead of using a Boolean expression, we become more concrete here and use
a combinational circuit.
An example of a DNF circuit
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