Section 8.2
227
also involves the multiplication of a number with perhaps
n
+ 2 digits by 2, requires at most
3n
+
[3(n
+ 2) 
2]
=
6n
+ 4 operations. The next step requires at most
6n
+ 7 operations, and
so on. Assuming
k
steps, the number of multiplications and divisions by 2 is at most
(6n
 2) +
(6n
+
1)
+
(6n
+ 4) +
(6n
+ 7) +
... +
(6n
+
3k
 5)
=
k(6n)
+ [2 +
1
+ 4 +
... +
(3k
 5)]
=
k(6n)
+
~[2(
2)
+
(k
 1)3]
=
6kn
+
~(3k
 7)
=
~k2

~k
+
6kn
~ ~k2
+
6kn,
using the fact that the sum of the first
k
terms of the arithmetic sequence
a, a
+
d, a
+
2d,
...
is
~[2a
+
(k

l)d]see
(7) on p. 234. By part (a),
k
<
1 + nlog21O, so the number of
multiplications and divisions by 2 is at most
(c) We must add at most
k
integers, each with at most
n
+
k
digits. Using Problem 16, we see that
this requires at most
k[2(n
+
k)
+
k]
~
(1
+ nlog21O)((2n + 3 + 3nlog2 1O)
=
0(n
2
)
basic operations.
(d) We use the result of Exercise 9. Since the number of operations required by each of the two stages
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This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.
 Summer '10
 any
 Graph Theory, Multiplication

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