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

significant digit, we follow the process of
• adding the two digits with carry from the previous column
• recording the sum bit as the result, and
• passing the carry bit on to the next column
until one of the numbers ends.
The Carry Chain Adder
The inductively designed circuit of the carry chain adder.
n = 1: the CCA 1 consists of a full adder
n > 1: the CCA n consists of an (n − 1)-bit carry chain adder CCA n−1 and a full adder
that sums up the carry of CCA n−1 and the last two bits of a and b
Definition 419 An n-bit carry chain adder CCA n is inductively defined as
(f 1 CCA (a0, b0, c)) = (FA 1 (a0, b0, c))
(f n CCA (〈an−1, . . . , a0〉, 〈bn−1, . . . , b0〉, c ′ )) = 〈c, sn−1, . . . , s0〉 with
〈c, sn−1〉 = (FA n−1 (an−1, bn−1, cn−1))
〈cn−1, . . . , cs〉0 = (f n−1
CCA (〈an−2, . . . , a0〉, 〈bn−2, . . . , b0〉, c ′ ))
Lemma 420 (Cost) C(CCA n ) ∈ O(n)
Proof Sketch: C(CCA n ) = C(CCA n−1 ) + C(FA 1 ) = C(CCA n−1 ) + 5 = 5n
Lemma 421 (Depth) dp(CCA n ) ∈ O(n)
Proof Sketch: dp(CCA n ) ≤ dp(CCA n−1 ) + dp(FA 1 ) ≤ dp(CCA n−1 ) + 3 ≤ 3n
The carry chain adder is simple, but cost and depth are high. (depth is critical (speed))
Question: Can we do better?
Problem: the carry ripples up the chain (upper parts wait for carries from lower part)
c○: Michael Kohlhase 246
A consequence of using the carry chain adder is that if we go from a 32-bit architecture to a 64-bit
architecture, the speed of additions in the chips would not increase, but decrease (by 50%). Of
course, we can carry out 64-bit additions now, a task that would have needed a special routine
at the software level (these typically involve at least 4 32-bit additions so there is a speedup for
such additions), but most addition problems in practice involve small (under 32-bit) numbers, so
we will have an overall performance loss (not what we really want for all that cost).
If we want to do better in terms of depth of an n-bit adder, we have to break the dependency
on the carry, let us look at a decimal addition example to get the idea. Consider the following
snapshot of an carry chain addition
135