Solution Manual For
Introduction to the Design and Analysis of Algorithms,‖ 2nd edition, by A. Levitin
Chapter 1-10
business principles.1.4. Preparing for Business ExamsPreparing for business exams involves mastering both conceptual understanding and practical application. Students are encouraged to stu
theories
Exercises 1.1
1. Do some research on al-Khorezmi (also al-Khwarizmi), the man from
whose name the word ―algorithm‖ is derived. In particular, you should
learn what the origins of the words ―algorithm‖ and ―algebra‖ have in
common.
2. Given that the official purpose of the U.S. patent system is the promo-
tion of the ―useful arts,‖ do you think algorithms are patentable in this
country? Should they be?
3. a. Write down driving directions for going from your school to your home
with the precision required by an algorithm.
b. Write down a recipe for cooking your favorite dish with the precision
required by an algorithm.
√
4. Design an algorithm for computing n for any positive integer n. Be-
sides assignment and comparison, your algorithm may only use the four
basic arithmetical operations.
5. a. Find gcd(31415, 14142) by applying Euclid‘s algorithm.
b. Estimate how many times faster it will be to find gcd(31415, 14142)
by Euclid‘s algorithm compared with the algorithm based on checking
consecutive integers from min{m, n} down to gcd(m, n).
6. D Prove the equality gcd(m, n) = gcd(n, m mod n) for every pair of positive
integers m and n.
7. What does Euclid‘s algorithm do for a pair of numbers in which the first
number is smaller than the second one? What is the largest number of
times this can happen during the algorithm‘s execution on such an input?
8. a. What is the smallest number of divisions made by Euclid‘s algorithm
among all inputs 1 ≤ m, n ≤ 10?
b. What is the largest number of divisions made by Euclid‘s algorithm
among all inputs 1 ≤ m, n ≤ 10?
9. a. Euclid‘s algorithm, as presented in Euclid‘s treatise, uses subtractions
rather than integer divisions. Write a pseudocode for this version of
Euclid‘s algorithm.
1
, business principles.1.4. Preparing for Business ExamsPreparing for business exams involves mastering both conceptual
understanding and practical application. Students are encouraged to study theories
b.► Euclid’s game (see [Bog]) starts with two unequal positive numbers
on the board. Two players move in turn. On each move, a player has
to write on the board a positive number equal to the difference of two
numbers already on the board; this number must be new, i.e., different
from all the numbers already on the board. The player who cannot move
loses the game. Should you choose to move first or second in this game?
10. The extended Euclid’s algorithm determines not only the greatest
common divisor d of two positive integers m and n but also integers (not
necessarily positive) x and y, such that mx + ny = d.
a. Look up a description of the extended Euclid‘s algorithm (see, e.g.,
[KnuI], p. 13) and implement it in the language of your choice.
b. Modify your program for finding integer solutions to the Diophan-
tine equation ax + by = c with any set of integer coefficients a, b, and
c.
11. Locker doors There are n lockers in a hallway numbered sequentially
from 1 to n. Initially, all the locker doors are closed. You make n passes
by the lockers, each time starting with locker #1. On the ith pass, i =
1, 2, ..., n, you toggle the door of every ith locker: if the door is closed,
you open it, if it is open, you close it. For example, after the first pass
every door is open; on the second pass you only toggle the even-numbered
lockers (#2, #4, ...) so that after the second pass the even doors are
closed and the odd ones are opened; the third time through you close the
door of locker #3 (opened from the first pass), open the door of locker
#6 (closed from the second pass), and so on. After the last pass, which
locker doors are open and which are closed? How many of them are open?
Hints to Exercises 1.1
1. It is probably faster to do this by searching the Web, but your library
should be able to help too.
2. One can find arguments supporting either view. There is a well established
principle pertinent to the matter though: scientific facts or mathematical
expressions of them are not patentable. (Why do you think it is the case?)
But should this preclude granting patents for all algorithms?
3. You may assume that you are writing your algorithms for a human rather
than a machine. Still, make sure that your descriptions do not contain ob-
vious ambiguities. Knuth [KnuI], p.6 provides an interesting comparison
between cooking recipes and algorithms.
4. There is a quite straightforward algorithm for this problem based on the
definition of [ n∫.
5. a. Just follow Euclid‘s algorithm as described in the text.
2
, b. Compare the number of divisions made by the two algorithms.
6. Prove that if d divides both m and n (i.e., m = sd and n = td for some
positive integers s and t), then it also divides both n and r = m mod n
and vice versa. Use the formula m = qn + r (0 ≤ r < n) and the fact that
if d divides two integers u and v, it also divides u + v and u — v. (Why?)
7. Perform one iteration of the algorithm for two arbitrarily chosen integers
m < n.
8. The answer to part (a) can be given immediately; the answer to part
(b) can be given by checking the algorithm‘s performance on all pairs
1 < m < n ≤ 10. business principles.1.4. Preparing for Business ExamsPreparing for business exams involves mastering both conceptual understandin
practical application. Students are encouraged to study theories
9. a. Use the equality
gcd(m, n) = gcd(m — n, n) for m ≥ n > 0.
b. The key is to figure out the total number of distinct integers that can be
written on the board, starting with an initial pair m, n where m > n 1.
You should exploit a connection of this question to the question of part
(a). Considering small examples, especially those with n = 1 and n = 2,
should help, too.
10. Of course, for some coefficients, the equation will have no solutions.
11. Tracing the algorithm by hand for, say, n = 10 and studying its outcome
should help answering both questions.
Solutions to Exercises 1.1
1. Al-Khwarizmi (9th century C.E.) was a great Arabic scholar, most fa-
mous for his algebra textbook. In fact, the word ―algebra‖ is derived
from the Arabic title of this book while the word ―algorithm‖ is derived
from a translation of Al-Khwarizmi‘s last name (see, e.g., [KnuI], pp. 1-2,
[Knu96], pp. 88-92, 114).
2. This legal issue has yet to be settled. The current legal state of affairs
distinguishes mathematical algorithms, which are not patentable, from
other algorithms, which may be patentable if implemented as computer
programs (e.g., [Cha00]).
3. n/a
4. A straightforward algorithm that does not rely on the availability of an
approximate value of n can check the squares of consecutive positive
integers until the first square exceeding n is encountered. The answer will
3
, be the number‘s immediate predecessor. Note: A much faster algorithm
for solving this problem can be obtained by using Newton‘s method (see
Sections 11.4 and 12.4).
5. a. gcd(31415, 14142) = gcd(14142, 3131) = gcd(3131, 1618) =
gcd(1618, 1513) = gcd(1513, 105) = gcd(1513, 105) = gcd(105, 43) =
gcd(43, 19) = gcd(19, 5) = gcd(5, 4) = gcd(4, 1) = gcd(1, 0) = 1.
b. To answer the question, we need to compare the number of divisions
the algorithms make on the input given. The number of divisions made
by Euclid‘s algorithm is 11 (see part a). The number of divisions made
by the consecutive integer checking algorithm on each of its 14142 itera-
tions is either 1 and 2; hence the total number of multiplications is be-
tween 1·14142 and 2·14142. Therefore, Euclid‘s algorithm will be between
1·14142/11 ≈ 1300 and 2·14142/11 ≈ 2600 times faster.
6. Let us first prove that if d divides two integers u and v, it also divides
both u + v and u v. By definition of division, there exist integers s and
t such that u = sd and v = td. Therefore
u ± v = sd ± td = (s ± t)d,
i.e., d divides both u + v and u — v.
business principles.1.4. Preparing for Business ExamsPreparing for business exams involves mastering both conceptual understanding and practical application. Students are encouraged to stu
theories
Also note that if d divides u, it also divides any integer multiple ku of
u. Indeed, since d divides u, u = sd. Hence
ku = k(sd) = (ks)d,
i.e., d divides ku.
Now we can prove the assertion in question. For any pair of positive
integers m and n, if d divides both m and n, it also divides both n and
r = m mod n = m qn. Similarly, if d divides both n and r = m mod n =
m qn, it also divides both m = r + qn and n. Thus, the two pairs
(m, n) and (n, r) have the same finite nonempty set of common divisors,
including the largest element in the set, i.e., gcd(m, n) = gcd(n, r).
7. For any input pair m, n such that 0 m < n, Euclid‘s algorithm simply
swaps the numbers on the first iteration:
gcd(m, n) = gcd(n, m)
because m mod n = m if m < n. Such a swap can happen only once since
gcd(m, n) = gcd(n, m mod n) implies that the first number of the new pair
(n) will be greater than its second number (m mod n) after every iteration
of the algorithm.
8. a. For any input pair m n 1, in which m is a multiple of n, Euclid‘s
4
Introduction to the Design and Analysis of Algorithms,‖ 2nd edition, by A. Levitin
Chapter 1-10
business principles.1.4. Preparing for Business ExamsPreparing for business exams involves mastering both conceptual understanding and practical application. Students are encouraged to stu
theories
Exercises 1.1
1. Do some research on al-Khorezmi (also al-Khwarizmi), the man from
whose name the word ―algorithm‖ is derived. In particular, you should
learn what the origins of the words ―algorithm‖ and ―algebra‖ have in
common.
2. Given that the official purpose of the U.S. patent system is the promo-
tion of the ―useful arts,‖ do you think algorithms are patentable in this
country? Should they be?
3. a. Write down driving directions for going from your school to your home
with the precision required by an algorithm.
b. Write down a recipe for cooking your favorite dish with the precision
required by an algorithm.
√
4. Design an algorithm for computing n for any positive integer n. Be-
sides assignment and comparison, your algorithm may only use the four
basic arithmetical operations.
5. a. Find gcd(31415, 14142) by applying Euclid‘s algorithm.
b. Estimate how many times faster it will be to find gcd(31415, 14142)
by Euclid‘s algorithm compared with the algorithm based on checking
consecutive integers from min{m, n} down to gcd(m, n).
6. D Prove the equality gcd(m, n) = gcd(n, m mod n) for every pair of positive
integers m and n.
7. What does Euclid‘s algorithm do for a pair of numbers in which the first
number is smaller than the second one? What is the largest number of
times this can happen during the algorithm‘s execution on such an input?
8. a. What is the smallest number of divisions made by Euclid‘s algorithm
among all inputs 1 ≤ m, n ≤ 10?
b. What is the largest number of divisions made by Euclid‘s algorithm
among all inputs 1 ≤ m, n ≤ 10?
9. a. Euclid‘s algorithm, as presented in Euclid‘s treatise, uses subtractions
rather than integer divisions. Write a pseudocode for this version of
Euclid‘s algorithm.
1
, business principles.1.4. Preparing for Business ExamsPreparing for business exams involves mastering both conceptual
understanding and practical application. Students are encouraged to study theories
b.► Euclid’s game (see [Bog]) starts with two unequal positive numbers
on the board. Two players move in turn. On each move, a player has
to write on the board a positive number equal to the difference of two
numbers already on the board; this number must be new, i.e., different
from all the numbers already on the board. The player who cannot move
loses the game. Should you choose to move first or second in this game?
10. The extended Euclid’s algorithm determines not only the greatest
common divisor d of two positive integers m and n but also integers (not
necessarily positive) x and y, such that mx + ny = d.
a. Look up a description of the extended Euclid‘s algorithm (see, e.g.,
[KnuI], p. 13) and implement it in the language of your choice.
b. Modify your program for finding integer solutions to the Diophan-
tine equation ax + by = c with any set of integer coefficients a, b, and
c.
11. Locker doors There are n lockers in a hallway numbered sequentially
from 1 to n. Initially, all the locker doors are closed. You make n passes
by the lockers, each time starting with locker #1. On the ith pass, i =
1, 2, ..., n, you toggle the door of every ith locker: if the door is closed,
you open it, if it is open, you close it. For example, after the first pass
every door is open; on the second pass you only toggle the even-numbered
lockers (#2, #4, ...) so that after the second pass the even doors are
closed and the odd ones are opened; the third time through you close the
door of locker #3 (opened from the first pass), open the door of locker
#6 (closed from the second pass), and so on. After the last pass, which
locker doors are open and which are closed? How many of them are open?
Hints to Exercises 1.1
1. It is probably faster to do this by searching the Web, but your library
should be able to help too.
2. One can find arguments supporting either view. There is a well established
principle pertinent to the matter though: scientific facts or mathematical
expressions of them are not patentable. (Why do you think it is the case?)
But should this preclude granting patents for all algorithms?
3. You may assume that you are writing your algorithms for a human rather
than a machine. Still, make sure that your descriptions do not contain ob-
vious ambiguities. Knuth [KnuI], p.6 provides an interesting comparison
between cooking recipes and algorithms.
4. There is a quite straightforward algorithm for this problem based on the
definition of [ n∫.
5. a. Just follow Euclid‘s algorithm as described in the text.
2
, b. Compare the number of divisions made by the two algorithms.
6. Prove that if d divides both m and n (i.e., m = sd and n = td for some
positive integers s and t), then it also divides both n and r = m mod n
and vice versa. Use the formula m = qn + r (0 ≤ r < n) and the fact that
if d divides two integers u and v, it also divides u + v and u — v. (Why?)
7. Perform one iteration of the algorithm for two arbitrarily chosen integers
m < n.
8. The answer to part (a) can be given immediately; the answer to part
(b) can be given by checking the algorithm‘s performance on all pairs
1 < m < n ≤ 10. business principles.1.4. Preparing for Business ExamsPreparing for business exams involves mastering both conceptual understandin
practical application. Students are encouraged to study theories
9. a. Use the equality
gcd(m, n) = gcd(m — n, n) for m ≥ n > 0.
b. The key is to figure out the total number of distinct integers that can be
written on the board, starting with an initial pair m, n where m > n 1.
You should exploit a connection of this question to the question of part
(a). Considering small examples, especially those with n = 1 and n = 2,
should help, too.
10. Of course, for some coefficients, the equation will have no solutions.
11. Tracing the algorithm by hand for, say, n = 10 and studying its outcome
should help answering both questions.
Solutions to Exercises 1.1
1. Al-Khwarizmi (9th century C.E.) was a great Arabic scholar, most fa-
mous for his algebra textbook. In fact, the word ―algebra‖ is derived
from the Arabic title of this book while the word ―algorithm‖ is derived
from a translation of Al-Khwarizmi‘s last name (see, e.g., [KnuI], pp. 1-2,
[Knu96], pp. 88-92, 114).
2. This legal issue has yet to be settled. The current legal state of affairs
distinguishes mathematical algorithms, which are not patentable, from
other algorithms, which may be patentable if implemented as computer
programs (e.g., [Cha00]).
3. n/a
4. A straightforward algorithm that does not rely on the availability of an
approximate value of n can check the squares of consecutive positive
integers until the first square exceeding n is encountered. The answer will
3
, be the number‘s immediate predecessor. Note: A much faster algorithm
for solving this problem can be obtained by using Newton‘s method (see
Sections 11.4 and 12.4).
5. a. gcd(31415, 14142) = gcd(14142, 3131) = gcd(3131, 1618) =
gcd(1618, 1513) = gcd(1513, 105) = gcd(1513, 105) = gcd(105, 43) =
gcd(43, 19) = gcd(19, 5) = gcd(5, 4) = gcd(4, 1) = gcd(1, 0) = 1.
b. To answer the question, we need to compare the number of divisions
the algorithms make on the input given. The number of divisions made
by Euclid‘s algorithm is 11 (see part a). The number of divisions made
by the consecutive integer checking algorithm on each of its 14142 itera-
tions is either 1 and 2; hence the total number of multiplications is be-
tween 1·14142 and 2·14142. Therefore, Euclid‘s algorithm will be between
1·14142/11 ≈ 1300 and 2·14142/11 ≈ 2600 times faster.
6. Let us first prove that if d divides two integers u and v, it also divides
both u + v and u v. By definition of division, there exist integers s and
t such that u = sd and v = td. Therefore
u ± v = sd ± td = (s ± t)d,
i.e., d divides both u + v and u — v.
business principles.1.4. Preparing for Business ExamsPreparing for business exams involves mastering both conceptual understanding and practical application. Students are encouraged to stu
theories
Also note that if d divides u, it also divides any integer multiple ku of
u. Indeed, since d divides u, u = sd. Hence
ku = k(sd) = (ks)d,
i.e., d divides ku.
Now we can prove the assertion in question. For any pair of positive
integers m and n, if d divides both m and n, it also divides both n and
r = m mod n = m qn. Similarly, if d divides both n and r = m mod n =
m qn, it also divides both m = r + qn and n. Thus, the two pairs
(m, n) and (n, r) have the same finite nonempty set of common divisors,
including the largest element in the set, i.e., gcd(m, n) = gcd(n, r).
7. For any input pair m, n such that 0 m < n, Euclid‘s algorithm simply
swaps the numbers on the first iteration:
gcd(m, n) = gcd(n, m)
because m mod n = m if m < n. Such a swap can happen only once since
gcd(m, n) = gcd(n, m mod n) implies that the first number of the new pair
(n) will be greater than its second number (m mod n) after every iteration
of the algorithm.
8. a. For any input pair m n 1, in which m is a multiple of n, Euclid‘s
4
