design and analysis of algorithm

This file contains the exercises, hints, and solutions for Chapter 1 of the
book ”Introduction to the Design and Analysis of Algorithms,” 2nd edition, by
A. Levitin. The problems that might be challenging for at least some students
are marked by ; those that might be difficult for a majority of students are
marked by .
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. 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.
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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?
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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.
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.
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.
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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
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.
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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
algorithm makes exactly one division; it is the smallest number possible
for two positive numbers.
b. The answer is 5 divisions, which is made by Euclid’s algorithm in
computing gcd(5, 8). It is not too time consuming to get this answer by
examining the number of divisions made by the algorithm on all input
pairs 1 < m < n ≤ 10.
Note: A pertinent general result (see [KnuII], p. 360) is that for any
input pair m, n where 0 ≤ n < N, the number of divisions required by
Euclid’s algorithm to compute gcd(m, n) is at most logφ(3−φ)N) where
φ = (1 +
√
5)/2.
9. a. Here is a nonrecursive version:
Algorithm Euclid2(m, n)
//Computes gcd(m, n) by Euclid’s algorithm based on subtractions
//Input: Two nonnegative integers m and n not both equal to 0
//Output: The greatest common divisor of m and n
while n = 0 do
if m < n swap(m, n)
m ← m − n
return m
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b. It is not too diﬃcult to prove that the integers that can be written on
the board are the integers generated by the subtraction version of Euclid’s
algorithm and only them. Although the order in which they appear on
the board may vary, their total number always stays the same: It is equal
to m/ gcd(m, n), where m is the maximum of the initial numbers, which
includes two integers of the initial pair. Hence, the total number of
possible moves is m/ gcd(m, n)−2. Consequently, if m/ gcd(m, n) is odd,
one should choose to go ﬁrst; if it is even, one should choose to go second.
10. n/a
11. Since all the doors are initially closed, a door will be open after the last
pass if and only if it is toggled an odd number of times. Door i (1 ≤ i ≤ n)
is toggled on pass j (1 ≤ j ≤ n) if and only if j divides i. Hence, the total
number of times door i is toggled is equal to the number of its divisors.
Note that if j divides i, i.e. i = jk, then k divides i too. Hence all the
divisors of i can be paired (e.g., for i = 12, such pairs are 1 and 12, 2
and 6, 3 and 4) unless i is a perfect square (e.g., for i = 16, 4 does not
have another divisor to be matched with). This implies that i has an
odd number of divisors if and only if it is a perfect square, i.e., i = j2
.
Hence doors that are in the positions that are perfect squares and only
such doors will be open after the last pass. The total number of such
positions not exceeding n is equal to
√
n : these numbers are the squares
of the positive integers between 1 and
√
n inclusively.
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Exercises 1.2
1. Old World puzzle A peasant finds himself on a riverbank with a wolf,
a goat, and a head of cabbage. He needs to transport all three to the
other side of the river in his boat. However, the boat has room for only
the peasant himself and one other item (either the wolf, the goat, or the
cabbage). In his absence, the wolf would eat the goat, and the goat would
eat the cabbage. Solve this problem for the peasant or prove it has no
solution. (Note: The peasant is a vegetarian but does not like cabbage
and hence can eat neither the goat nor the cabbage to help him solve the
problem. And it goes without saying that the wolf is a protected species.)
2. New World puzzle There are four people who want to cross a bridge; they
all begin on the same side. You have 17 minutes to get them all across to
the other side. It is night, and they have one flashlight. A maximum of two
people can cross the bridge at one time. Any party that crosses, either one
or two people, must have the flashlight with them. The flashlight must be
walked back and forth; it cannot be thrown, for example. Person 1 takes
1 minute to cross the bridge, person 2 takes 2 minutes, person 3 takes 5
minutes, and person 4 takes 10 minutes. A pair must walk together at the
rate of the slower person’s pace. For example, if person 1 and person 4
walk across first, 10 minutes have elapsed when they get to the other side
of the bridge. If person 4 returns the flashlight, a total of 20 minutes have
passed and you have failed the mission. (Note: According to a rumor on
the Internet, interviewers at a well-known software company located near
Seattle have given this problem to interviewees.)
3. Which of the following formulas can be considered an algorithm for com-
puting the area of a triangle whose side lengths are given positive numbers
a, b, and c?
a. S = p(p − a)(p − b)(p − c), where p = (a + b + c)/2
b. S = 1
2 bc sin A, where A is the angle between sides b and c
c. S = 1
2 aha, where ha is the height to base a
4. Write a pseudocode for an algorithm for finding real roots of equation
ax2
+ bx + c = 0 for arbitrary real coefficients a, b, and c. (You may
assume the availability of the square root function sqrt(x).)
5. Describe the standard algorithm for finding the binary representation of
a positive decimal integer
a. in English.
b. in a pseudocode.
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6. Describe the algorithm used by your favorite ATM machine in dispensing
cash. (You may give your description in either English or a pseudocode,
whichever you find more convenient.)
7. a. Can the problem of computing the number π be solved exactly?
b. How many instances does this problem have?
c. Look up an algorithm for this problem on the World Wide Web.
8. Give an example of a problem other than computing the greatest common
divisor for which you know more than one algorithm. Which of them is
simpler? Which is more efficient?
9. Consider the following algorithm for finding the distance between the two
closest elements in an array of numbers.
Algorithm MinDistance(A[0..n − 1])
//Input: Array A[0..n − 1] of numbers
//Output: Minimum distance between two of its elements
dmin ← ∞
for i ← 0 to n − 1 do
for j ← 0 to n − 1 do
if i = j and |A[i] − A[j]| < dmin
dmin ← |A[i] − A[j]|
return dmin
Make as many improvements as you can in this algorithmic solution to
the problem. (If you need to, you may change the algorithm altogether; if
not, improve the implementation given.)
10. One of the most influential books on problem solving, titled How To Solve
It [Pol57], was written by the Hungarian-American mathematician George
Polya (1887—1985). Polya summarized his ideas in a four-point summary.
Find this summary on the Web or, better yet, in his book, and compare
it with the plan outlined in Section 1.2. What do they have in common?
How are they different?
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1. The peasant would have to make several trips across the river, starting
with the only one possible.
2. Unlike the Old World puzzle of Problem 1, the first move solving this
puzzle is not obvious.
3. The principal issue here is a possible ambiguity.
4. Your algorithm should work correctly for all possible values of the coeffi-
cients, including zeros.
5. You almost certainly learned this algorithm in one of your introductory
programming courses. If this assumption is not true, you have a choice
between designing such an algorithm on your own or looking it up.
6. You may need to make a field trip to refresh your memory.
7. Question (a) is difficult though the answer to it–discovered in 1760s by
the German mathematician Johann Lambert –is well-known. By com-
parison, question (b) is incomparably simpler.
8. You probably know two or more different algorithms for sorting an array
of numbers.
9. You can: decrease the number of times the innermost loop is executed,
make that loop run faster (at least for some inputs), or, more significantly,
design a faster algorithm from scratch.
10. n/a
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1. Let P, w, g, and c stand for the peasant, wolf, goat, and cabbage head,
respectively. The following is one of the two principal sequences that
solve the problem:
Pwgc
P g
w c
g
Pw c
Pwg
c
w
P gc
Pw c
g
w c
P g
Pwgc
Note: This problem is revisited later in the book (see Section 6.6).
2. Let 1, 2, 5, 10 be labels representing the men of the problem, f represent
the flashlight’s location, and the number in the parenthesis be the total
amount of time elapsed. The following sequence of moves solves the
problem:
(0)
f,1,2,5,10
f,1,2
(2)
5,10
2
(3)
f,1,5,10
f,2,5,10
(13)
1
5,10
(15)
f,1,2
f,1,2,5,10
(17)
3. a. The formula can be considered an algorithm if we assume that we know
how to compute the square root of an arbitrary positive number.
b. The difficulty here lies in computing sin A. Since the formula says
nothing about how it has to be computed, it should not be considered an
algorithm. This is true even if we assume, as we did for the square root
function, that we know how to compute the sine of a given angle. (There
are several algorithms for doing this but only approximately, of course.)
The problem is that the formula says nothing about how to compute angle
A either.
c. The formula says nothing about how to compute ha.
4. Algorithm Quadratic(a, b, c)
//The algorithm finds real roots of equation ax2
+ bx + c = 0
//Input: Real coefficients a, b, c
//Output: The real roots of the equation or a message about their absence
if a = 0
D ← b ∗ b − 4 ∗ a ∗ c
if D > 0
temp ← 2 ∗ a
x1 ← (−b + sqrt(D))/temp
x2 ← (−b − sqrt(D))/temp
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return x1, x2
else if D = 0 return −b/(2 ∗ a)
else return ‘no real roots’
else //a = 0
if b = 0 return −c/b
else //a = b = 0
if c = 0 return ‘all real numbers’
else return ‘no real roots’
Note: See a more realistic algorithm for this problem in Section 11.4.
5. a. Divide the given number n by 2: the remainder rn (0 or 1) will be
the next (from right to left) digit of the binary representation in question.
Replace n by the quotient of the last division and repeat this operation
until n becomes 0.
b. Algorithm Binary(n)
//The algorithm implements the standard method for ﬁnding
//the binary expansion of a positive decimal integer
//Input: A positive decimal integer n
//Output: The list bk bk−1...b1 b0 of n’s binary digits
k ← 0
while n = 0
bk ← n mod 2
n ← n/2
k ← k + 1
6. n/a
7. a. π, as an irrational number, can be computed only approximately.
b. It is natural to consider, as an instance of this problem, computing
π’s value with a given level of accuracy, say, with n correct decimal digits.
With this interpretation, the problem has inﬁnitely many instances.
8. n/a
9. The following improved version considers the same pair of elements only
once and avoids recomputing the same expression in the innermost loop:
Algorithm MinDistance2(A[0..n − 1])
//Input: An array A[0..n − 1] of numbers
//Output: The minimum distance d between two of its elements
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dmin ← ∞
for j ← i + 1 to n − 1 do
temp ← |A[i] − A[j]|
if temp < dmin
dmin ← temp
return dmin
A faster algorithm is based on the idea of presorting (see Section 6.1).
10. Polya’s general four-point approach is:
1. Understand the problem
2. Devise a plan
3. Implement the plan
4. Look back/check
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Exercises 1.3
1. Consider the algorithm for the sorting problem that sorts an array by
counting, for each of its elements, the number of smaller elements and
then uses this information to put the element in its appropriate position
in the sorted array:
Algorithm ComparisonCountingSort(A[0..n − 1], S[0..n − 1])
//Sorts an array by comparison counting
//Input: Array A[0..n − 1] of orderable values
//Output: Array S[0..n − 1] of A’s elements sorted in nondecreasing order
Count[i] ← 0
for j ← i + 1 to n − 1 do
if A[i] < A[j]
Count[j] ← Count[j] + 1
else Count[i] ← Count[i] + 1
S[Count[i]] ← A[i]
a. Apply this algorithm to sorting the list 60, 35, 81, 98, 14, 47.
b. Is this algorithm stable?
c. Is it in place?
2. Name the algorithms for the searching problem that you already know.
Give a good succinct description of each algorithm in English. (If you
know no such algorithms, use this opportunity to design one.)
3. Design a simple algorithm for the string-matching problem.
4. Königsberg bridges The Königsberg bridge puzzle is universally accepted
as the problem that gave birth to graph theory. It was solved by the great
Swiss-born mathematician Leonhard Euler (1707—1783). The problem
asked whether one could, in a single stroll, cross all seven bridges of the
city of Königsberg exactly once and return to a starting point. Following
is a sketch of the river with its two islands and seven bridges:
a. State the problem as a graph problem.
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b. Does this problem have a solution? If you believe it does, draw such
a stroll; if you believe it does not, explain why and indicate the small-
est number of new bridges that would be required to make such a stroll
possible.
5. Icosian Game A century after Euler’s discovery (see Problem 4), an-
other famous puzzle–this one invented by the renown Irish mathemati-
cian Sir William Hamilton (1805-1865)–was presented to the world under
the name of the Icosian Game. The game was played on a circular wooden
board on which the following graph was carved:
Find a Hamiltonian circuit–a path that visits all the graph’s vertices
exactly once before returning to the starting vertex–for this graph.
6. Consider the following problem: Design an algorithm to determine the
best route for a subway passenger to take from one designated station to
another in a well-developed subway system similar to those in such cities
as Washington, D.C., and London, UK.
a. The problem’s statement is somewhat vague, which is typical of real-
life problems. In particular, what reasonable criterion can be used for
deﬁning the “best” route?
b. How would you model this problem by a graph?
7. a. Rephrase the traveling salesman problem in combinatorial object terms.
b. Rephrase the graph-coloring problem in combinatorial object terms.
8. Consider the following map:
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a
b
c
d
e
f
a. Explain how we can use the graph-coloring problem to color the map
so that no two neighboring regions are colored the same.
b. Use your answer to part (a) to color the map with the smallest number
of colors.
9. Design an algorithm for the following problem: Given a set of n points
in the Cartesian plane, determine whether all of them lie on the same
circumference.
10. Write a program that reads as its inputs the (x, y) coordinates of the
endpoints of two line segments P1Q1 and P2Q2 and determines whether
the segments have a common point.
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1. Trace the algorithm on the input given. Use the definitions of stability
and being in place that were introduced in the section.
2. If you do not recall any searching algorithms, you should design a simple
searching algorithm (without succumbing to the temptation to find one in
the latter chapters of the book).
3. This algorithm is introduced later in the book but you should have no
trouble to design it on your own.
4. If you have not encountered this problem in your previous courses, you
may look up the answers on the Web or in a discrete structures textbook.
The answers are, in fact, surprisingly simple.
5. No efficient algorithm for solving this problem for an arbitrary graph is
known. This particular graph does have Hamiltonian circuits which are
not difficult to find. (You need to find just one of them.)
6. a. Put yourself (mentally) in a passenger’s place and ask yourself what
criterion for the “best” route you would use. Then think of people that
may have different needs.
b. The representation of the problem by a graph is straightforward. Give
some thoughts though to stations where trains can be changed.
7. a. What are tours in the traveling salesman problem?
b. It would be natural to consider vertices colored the same color as
elements of the same subset.
8. Create a graph whose vertices represent the map’s regions. You will have
to decide on the edges on your own.
9. Assume that the circumference in question exists and find its center first.
Also, do not forget to give a special answer for n ≤ 2.
10. Be careful not to miss some special cases of the problem.
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1. a. Sorting 60, 35, 81, 98, 14, 47 by comparison counting will work as
follows:
Array A[0..5] 60 35 81 98 14 47
Initially Count[] 0 0 0 0 0 0
After pass i = 0 Count[] 3 0 1 1 0 0
After pass i = 1 Count[] 1 2 2 0 1
After pass i = 2 Count[] 4 3 0 1
After pass i = 3 Count[] 5 0 1
After pass i = 4 Count[] 0 2
Final state Count[] 3 1 4 5 0 2
Array S[0..5] 14 35 47 60 81 98
b. The algorithm is not stable. Consider, as a counterexample, the
result of its application to 1 , 1 .
c. The algorithm is not in place because it uses two extra arrays of size
n: Count and S.
2. Answers may vary but most students should be familiar with sequential
search, binary search, binary tree search and, possibly, hashing from their
introductory programming courses.
3. Align the pattern with the beginning of the text. Compare the corre-
sponding characters of the pattern and the text left-to right until either
all the pattern characters are matched (then stop–the search is success-
ful) or the algorithm runs out of the text’s characters (then stop–the
search is unsuccessful) or a mismatching pair of characters is encountered.
In the latter case, shift the pattern one position to the right and resume
the comparisons.
4. a. If we represent each of the river’s banks and each of the two islands by
vertices and the bridges by edges, we will get the following graph:
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b
a
c
d
b
a
c
d
(This is, in fact, a multigraph, not a graph, because it has more than one
edge between the same pair of vertices. But this doesn’t matter for the is-
sue at hand.) The question is whether there exists a path (i.e., a sequence
of adjacent vertices) in this multigraph that traverses all the edges exactly
once and returns to a starting vertex. Such paths are called Eulerian cir-
cuits; if a path traverses all the edges exactly once but does not return to
its starting vertex, it is called an Eulerian path.
b. Euler proved that an Eulerian circuit exists in a connected (multi)graph
if and only if all its vertices have even degrees, where the degree of a ver-
tex is deﬁned as the number of edges for which it is an endpoint. Also,
an Eulerian path exists in a connected (multi)graph if and only if it has
exactly two vertices of odd degrees; such a path must start at one of those
two vertices and end at the other. Hence, for the multigraph of the puz-
zle, there exists neither an Eulerian circuit nor an Eulerian path because
all its four vertices have odd degrees.
If we are to be satisﬁed with an Eulerian path, two of the multigraph’s
vertices must be made even. This can be accomplished by adding one new
bridge connecting the same places as the existing bridges. For example,
a new bridge between the two islands would make possible, among others,
the walk a − b − c − a − b − d − c − b − d
b
a
c
d
b
a
c
d
If we want a walk that returns to its starting point, all the vertices in the
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corresponding multigraph must be even. Since a new bridge/edge changes
the parity of two vertices, at least two new bridges/edges will be needed.
For example, here is one such “enhancement”:
b
a
c
d
b
a
c
d
This would make possible a − b − c − a − b − d − c − b − d − a, among
several other such walks.
5. A Hamiltonian circuit is marked on the graph below:
6. a. At least three “reasonable”criteria come to mind: the fastest trip, a
trip with the smallest number of train stops, and a trip that requires the
smallest number of train changes. Note that the ﬁrst criterion requires
the information about the expected traveling time between stations and
the time needed for train changes whereas the other two criteria do not
require such information.
b. A natural approach is to mimic subway plans by representing sta-
tions by vertices of a graph, with two vertices connected by an edge if
there is a train line between the corresponding stations. If the time spent
on changing a train is to be taken into account (e.g., because the station
in question is on more than one line), the station should be represented
by more then one vertex.
19

7. a. Find a permutation of n given cities for which the sum of the distances
between consecutive cities in the permutation plus the distance between
its last and ﬁrst city is as small as possible.
b. Partition all the graph’s vertices into the smallest number of disjoint
subsets so that there is no edge connecting vertices from the same subset.
8. a. Create a graph whose vertices represent the map’s regions and the
edges connect two vertices if and only if the corresponding regions have a
common border (and therefore cannot be colored the same color). Here
is the graph for the map given:
b
c
b
a d
c
e f
Solving the graph coloring problem for this graph yields the map’s color-
ing with the smallest number of colors possible.
b. Without loss of generality, we can assign colors 1 and 2 to vertices
c and a, respectively. This forces the following color assignment to the
remaining vertices: 3 to b, 2 to d, 3 to f, 4 to e. Thus, the smallest number
of colors needed for this map is four.
Note: It’s a well-known fact that any map can be colored in four colors
or less. This problem–known as the Four-Color Problem–has remained
unresolved for more than a century until 1976 when it was ﬁnally solved by
the American mathematicians K. Appel and W. Haken by a combination
of mathematical arguments and extensive computer use.
9. If n = 2, the answer is always “yes”; so, we may assume that n ≥ 3.
Select three points P1, P2, and P3 from the set given. Write an equation
of the perpendicular bisector l1 of the line segment with the endpoints at
P1 and P2, which is the locus of points equidistant from P1 and P2. Write
an equation of the perpendicular bisector l2 of the line segment with the
endpoints at P2 and P3, which is the locus of points equidistant from P2
and P3. Find the coordinates (x, y) of the intersection point P of the lines
l1 and l2 by solving the system of two equations in two unknowns x and
y. (If the system has no solutions, return “no”: such a circumference
does not exist.) Compute the distances (or much better yet the distance
squares!) from P to each of the points Pi, i = 3, 4, ..., n and check whether
all of them are the same: if they are, return “yes,” otherwise, return “no”.
20

Exercises 1.4
1. Describe how one can implement each of the following operations on an
array so that the time it takes does not depend on the array’s size n.
a. Delete the ith element of an array (1 ≤ i ≤ n).
b. Delete the ith element of a sorted array (the remaining array has
to stay sorted, of course).
2. If you have to solve the searching problem for a list of n numbers, how
can you take advantage of the fact that the list is known to be sorted?
Give separate answers for
a. lists represented as arrays.
b. lists represented as linked lists.
3. a. Show the stack after each operation of the following sequence that
starts with the empty stack:
push(a), push(b), pop, push(c), push(d), pop
b. Show the queue after each operation of the following sequence that
starts with the empty queue:
enqueue(a), enqueue(b), dequeue, enqueue(c), enqueue(d), dequeue
4. a. Let A be the adjacency matrix of an undirected graph. Explain what
property of the matrix indicates that
i. the graph is complete.
ii. the graph has a loop, i.e., an edge connecting a vertex to itself.
iii. the graph has an isolated vertex, i.e., a vertex with no edges incident
to it.
b. Answer the same questions for the adjacency list representation.
5. Give a detailed description of an algorithm for transforming a free tree
into a tree rooted at a given vertex of the free tree.
6. Prove the inequalities that bracket the height of a binary tree with n
vertices:
log2 n ≤ h ≤ n − 1.
7. Indicate how the ADT priority queue can be implemented as
a. an (unsorted) array.
21

b. a sorted array.
c. a binary search tree.
8. How would you implement a dictionary of a reasonably small size n if
you knew that all its elements are distinct (e.g., names of 50 states of the
United States)? Specify an implementation of each dictionary operation.
9. For each of the following applications, indicate the most appropriate data
structure:
a. answering telephone calls in the order of their known priorities.
b. sending backlog orders to customers in the order they have been re-
ceived.
c. implementing a calculator for computing simple arithmetical expres-
sions.
10. Anagram checking Design an algorithm for checking whether two given
words are anagrams, i.e., whether one word can be obtained by permut-
ing the letters of the other. (For example, the words tea and eat are
anagrams.)
22

1. a. Take advantage of the fact that the array is not sorted.
b. We used this trick in implementing one of the algorithms in Section
1.1.
2. a. For a sorted array, there is a spectacularly efficient algorithm you al-
most certainly have heard about.
b. Unsuccessful searches can be made faster.
3. a. Push(x) puts x on the top of the stack, pop deletes the item from the
top of the stack.
b. Enqueue(x) adds x to the rear of the queue, dequeue deletes the item
from the front of the queue.
4. Just use the definitions of the graph properties in question and data struc-
tures involved.
5. There are two well-known algorithms that can solve this problem. The
first uses a stack, the second uses a queue. Although these algorithms
are discussed later in the book, do not miss this chance to discover them
by yourself!
6. The inequality h ≤ n−1 follows immediately from the height’s definition.
The lower bound inequality follows from inequality 2h+1
− 1 ≥ n, which
can be proved by considering the largest number of vertices a binary tree
of height h can have.
7. You need to indicate how each of the three operations of the priority queue
will be implemented.
8. Because of insertions and deletions, using an array of the dictionary’s
elements (sorted or unsorted) is not the best implementation possible.
9. You need to know about the postfix notation in order to answer one of
these questions. (If you are not familiar with it, find the information on
the Internet.)
10. There are several algorithms for this problem. Keep in mind that the
words may contain multiple occurrences of the same letter.
23

1. a. Replace the ith element with the last element and decrease the array
size by 1.
b. Replace the ith element with a special symbol that cannot be a value
of the array’s element (e.g., 0 for an array of positive numbers) to mark
the ith position as empty. (This method is sometimes called the “lazy
deletion”.)
2. a. Use binary search (see Section 4.3 if you are not familiar with this
algorithm).
b. When searching in a sorted linked list, stop as soon as an element
greater than or equal to the search key is encountered.
3. a.
d
push(a) push(b) b pop push(c) c push(d) c pop c
a a a a a a
b.
enqueue(a) enqueue(b) dequeue enqueue(c) enqueue(d) dequeue
a ab b bc bcd cd
4. a. For the adjacency matrix representation:
i. A graph is complete if and only if all the elements of its adjacency
matrix except those on the main diagonal are equal to 1, i.e., A[i, j] = 1
for every 1 ≤ i, j ≤ n, i = j.
ii. A graph has a loop if and only if its adjacency matrix has an ele-
ment equal to 1 on its main diagonal, i.e., A[i, i] = 1 for some 1 ≤ i ≤ n.
iii. An (undirected, without loops) graph has an isolated vertex if and
only if its adjacency matrix has an all-zero row.
b. For the adjacency list representation:
i. A graph is complete if and only if each of its linked lists contains
all the other vertices of the graph.
ii. A graph has a loop if and only if one of its adjacency lists contains the
24

vertex defining the list.
iii. An (undirected, without loops) graph has an isolated vertex if and
only if one of its adjacency lists is empty.
5. The first algorithm works as follows. Mark a vertex to serve as the root
of the tree, make it the root of the tree to be constructed, and initialize
a stack with this vertex. Repeat the following operation until the stack
becomes empty: If there is an unmarked vertex adjacent to the vertex on
the top to the stack, mark the former vertex, attach it as a child of the
top’s vertex in the tree, and push it onto the stack; otherwise, pop the
vertex off the top of the stack.
The second algorithm works as follows. Mark a vertex to serve as the
root of the tree, make it the root of the tree to be constructed, and ini-
tialize a queue with this vertex. Repeat the following operations until
the queue becomes empty: If there are unmarked vertices adjacent to the
vertex at the front of the queue, mark all of them, attach them as children
to the front vertex in the tree, and add them to the queue; then dequeue
the queue.
6. Since the height is defined as the length of the longest simple path from
the tree’s root to its leaf, such a pass will include no more than n vertices,
which is the total number of vertices in the tree. Hence, h ≤ n − 1.
The binary tree of height h with the largest number of vertices is the full
tree that has all its h + 1 levels filled with the largest number of vertices
possible. The total number of vertices in such a tree is h
l=0 2l
= 2h+1
−1.
Hence, for any binary tree with n vertices and height h
2h+1
− 1 ≥ n.
This implies that
2h+1
≥ n + 1
or, after taking binary logarithms of both hand sides and taking into
account that h + 1 is an integer,
h + 1 ≥ log2(n + 1) .
Since log2(n + 1) = log2 n + 1 (see Appendix A), we finally obtain
h + 1 ≥ log2 n + 1 or h ≥ log2 n .
7. a. Insertion can be implemented by adding the new item after the ar-
ray’s last element. Finding the largest element requires a standard scan
25

through the array to find its largest element. Deleting the largest ele-
ment A[i] can be implemented by exchanging it with the last element and
decreasing the array’s size by 1.
b. We will assume that the array A[0..n − 1] representing the priority
queue is sorted in ascending order. Inserting a new item of value v can be
done by scanning the sorted array, say, left to right until an element A[j]
≥ v or the end of the array is reached. (A faster algorithm for finding
a place for inserting a new element is binary search discussed in Section
4.3.) In the former case, the new item is inserted before A[j] by first mov-
ing A[n − 1], ..., A[j] one position to the right; in the latter case, the new
item is simply appended after the last element of the array. Finding the
largest element is done by simply returning the value of the last element
of the sorted array. Deletion of the largest element is done by decreasing
the array’s size by one.
c. Insertion of a new element is done by using the standard algorithm
for inserting a new element in a binary search tree: recursively, the new
key is inserted in the left or right subtree depending on whether it is
smaller or larger than the root’s key. Finding the largest element will
require finding the rightmost element in the binary tree by starting at
the root and following the chain of the right children until a vertex with
no right subtree is reached. The key of that vertex will be the largest
element in question. Deleting it can be done by making the right pointer
of its parent to point to the left child of the vertex being deleted;. if the
rightmost vertex has no left child, this pointer is made “null”. Finally, if
the rightmost vertex has no parent, i.e., if it happens to be the root of the
tree, its left child becomes the new root; if there is no left child, the tree
becomes empty.
8. Use a bit vector, i.e., an array on n bits in which the ith bit is 1 if
the ith element of the underlying set is currently in the dictionary and
0 otherwise. The search, insertion, and deletion operations will require
checking or changing a single bit of this array.
9. Use: (a) a priority queue; (b) a queue; (c) a stack (and reverse Polish
notation–a clever way of representing arithmetical expressions without
parentheses, which is usually studied in a data structures course).
10. The most straightforward solution is to search for each successive letter
of the first word in the second one. If the search is successful, delete the
first occurrence of the letter in the second word, stop otherwise.
Another solution is to sort the letters of each word and then compare
26

them in a simple parallel scan.
We can also generate and compare “letter vectors” of the given words:
Vw[i] = the number of occurrences of the alphabet’s ith letter in the word
w. Such a vector can be generated by initializing all its components to
0 and then scanning the word and incrementing appropriate letter counts
in the vector.
27

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