Chapter 1
Preliminaries: Answers to Exercises
1. The required algorithm is shown below:
procedure M ULTIPLY(m, n)
p←0
while n > 0 do
p←p+m
n←n−1
6:
return p
1:
2:
3:
4:
5:
2. The answers are given below:
a. ⌊9.47⌋ = 9
b. ⌊−9.47⌋ = −10
c. ⌊75⌋ = 75
d. ⌊−227.9⌋ = −228
e. ⌊24.5 + 35.5⌋ = 60
f. ⌊24.5⌋ + ⌊35.5⌋ = 59
3. Compute each of the following:
a.
b.
c.
d.
e.
f.
⌈9.47⌉ = 10
⌈−9.47⌉ = −9
⌈75⌉ = 75
⌈−227.9⌉ = −227
⌈29.5 + 35.5⌉ = 65
⌈29.5⌉ + ⌈35.5⌉ = 66
4. If n is an odd integer, show that
n2
4
=
n−1
2
1
n+1
2
.
2
CHAPTER 1. PRELIMINARIES: ANSWERS TO EXERCISES
[Hint: Let n = 2k − 1]
1
n2
= k 2 − k + , which yields
With n = 2k − 1, we have n2 = 4k 2 − 4k + 1. Therefore,
4
4
2
n
n+1
n−1
2
= k − k = k(k − 1). But, n = 2k − 1 yields k =
and k − 1 =
, from
4
2
2
which we conclude that
2 n
n−1
n+1
=
.
4
2
2
5. If n is an odd integer, show that
n2
4
=
n2 + 3
.
4
[Hint: Let n = 2k − 1]
n2
1
With n = 2k − 1, we have we have n2 = 4k 2 − 4k + 1. Therefore,
= k 2 − k + , which
4
4
2 2
n
n
+
3
1
3
yields
.
= k2 − k + 1 = k2 − k +
+ =
4
4
4
4
6. Compute each of the following: [You may use a calculator]
a. 7! = 7 · 6 · 5 · 4 · 3 · 2 · 1 = 5040
b. 8! + 3! = 8 · 7 · 5 · 4 · 3 · 2 · 1 + 3 · 2 · 1 = 40320 + 6 = 40326
c. (4 + 3)! = 7! = 5040
7. Compute each of the following:
11!
= 11 · 10 · 9 · 8 · 7 · 6 · 5 · 4 · 3 = 6652800
3!
9!
9·8·7·6
b.
=
= 126
4!5!
4·3·2·1
11 · 10 · 9
990
11!
=
=
= 165
c.
(11 − 8)!8!
3·2·1
6
a.
8. Simplify each of the following expressions (your final answer should only contain positive
exponents):
x−1 y 4
y6
a. 5 −2 = 6
xy
x
20 −1/2
(x )
1
b.
= 14
4
x
x
9. Compute each of the following: [Do not use a calculator]
a. log2 128 = log2 = 27 = 7
b. log2 4096 = log2 212 = 12
10. Compute each of the following: [Do not use a calculator]
a. log3 243 = log3 35 = 5
3
b. log5 15625 = log5 56 = 6
c. log10 1000000 = log10 106 = 6
11. Compute each of the following: [Do not use a calculator]
a. ⌊log2 75⌋ Answer: 26 < 75 < 27 ⇒ ⌊log2 75⌋ = 6
b. ⌈log3 89⌉ Answer: 34 < 89 < 35 ⇒ ⌈log3 89⌉ = 5
c. ⌊log4 500⌋ Answer: 44 < 500 < 45 ⇒ ⌊log4 500⌋ = 4
12. Compute each of the following: [Do not use a calculator]
a. ⌈log10 65536⌉ Answer: 105 < 65536 < 106 ⇒ ⌈log10 65536⌉ = 6
b. ⌈log3 225⌉ Answer: 34 < 225 < 35 ⇒ ⌈log3 225⌉ = 5
c. ⌈log4 3500⌉ Answer: 45 < 3500 < 46 ⇒ ⌈log4 3500⌉ = 6
13. Simplify each of the following expressions. [Do not use a calculator]
2
a. 2log2 4 = 2log2 2 = 22 log2 2 = 22 =4 30
30
b. log2 30 − log2 5 − log2 3 = log2
= log2
= log2 2 = 1
5·3
15
14. Determine mathematically the final value of the variable count in each of the following
systems of loops. Assume that n has been initialized in a calling procedure appropriately.
Do not use any asymptotic notations. Express your answer in terms of n.
a.
count = 0
for i = 1 to n do
for j = n downto i do
count = count + 1
end for
end for
Answer: The desired answer is obtained by simplifying the following summation:
2
2
n X
n
n
n
n
n
X
X
X
X
X
n +n
n +n
2
+n =
1=
(n−i+1) =
n−
i+
1=n −
2
2
i=1 j=i
i=1
i=1
i=1
i=1
Another way to simplify:
n X
n
X
i=1 j=i
1=
n
X
(n − i + 1) =
i=1
which is obtained by letting k = n − i + 1.
count = 0
b.
for (i = 1; i < n; i = 2*i)
count = count + 1
for j = i+10 to i+17 do
count = count + 1
n
X
k=1
k=
n2 + n
2
4
CHAPTER 1. PRELIMINARIES: ANSWERS TO EXERCISES
The desired answer is obtained by first observing that the i-loop is executed for values of
i = 1, 2, 22 , · · · , 2k as long as 2k is smaller than n. Therefore, we obtain the desired answer
by simplifying the following summation:
X
i+17
X
1+
1
j=i+10
1≤2k <n
k=⌊log2 n⌋
!
=
X
(1 + 8) = 9
1≤2k <n
X
1≤2k <n
1=9
X
1 = 9(⌊log2 n⌋ + 1)
k=0
15. For each pair of functions given below, indicate which function, f (n), or g(n) grows asymptotically faster.
√
n+1
Answer: f (n) grows faster
n+2
b. f (n) = n1.59 , g(n) = n2 Answer: g(n) grows faster
c. f (n) = n3 log n + n2 , g(n) = n4 Answer: g(n) grows faster
a. f (n) =
n, g(n) =
16. Rewrite the following functions in the order of increasing growth rate from left to right.
0.01n2 log n, 2n3 , 0.0000001n!, 2000n log n, 10100 , 3n , 50n2 + n5 .
The rearranged list is shown below:
10100 , 2000n log n, 0.01n2 log n, 2n3 , 50n2 + n5 , 3n , 0.0000001n!.