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Exercises of The design and Analysis of Algorithms
Chapter 1
1. For each of the following pairs of functions, indicate whether the first function of each of the
following pairs has a lower, same, or higher order of growth (to within a constant multiple)
than the second function.
a. n ( n + 1) and 2000 n
3
b. log 2
n and ln n
c. 2
n −
and 2
n
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2. Use the informal definitions of O, , and to determine whether the following assertions are
true or false.
a. n ( n + 1) / 2 = O( n
3
) b. n ( n + 1) / 2 = O( n
2
c. n ( n + 1) / 2 = ( n
3
) d. n ( n + 1) / 2 = ( n )
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3. Prove the following assertions by using the definitions of the notations involved, or
disprove them by giving a specific counterexample.
a. If t ( n ) = O( g ( n )) , then g ( n ) = ( t ( n )).
b. ( αg ( n )) = ( g ( n )) , where α > 0_._
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4. Consider the following algorithm.
ALGORITHM Mystery ( n )
//Input: A nonnegative integer n
S ←
for i ←1 to n do
S ← S + i ∗ i
return S
a. What does this algorithm compute?
b. What is its basic operation?
c. Compute the complexity of this algorithm.
d. Suggest an improvement, or a better algorithm altogether, and indicate its complexity. If
you cannot do it, try to prove that, in fact, it cannot be done.
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5. Consider the following recursive algorithm for computing the sum of the first n cubes:
6. Consider the following recursive algorithm
ALGORITHM Q ( n )
//Input: A positive integer n
if n = 1 return 1
else return Q ( n − 1) + 2 ∗ n − 1
a. Set up a recurrence relation for the number of multiplications made by this algorithm
and solve it.
b. Compute the complexity of the algorithm.
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7. a. Design a recursive algorithm for computing 2
n
for any nonnegative integer
n that is based on the formula 2
n
n −
n −
b. Compute the complexity of this algorithm.
c. Is it a good algorithm for solving this problem?
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8. Consider the following recursive algorithm.
ALGORITHM Riddle ( A [0 ..n − 1])
//Input: An array A [0 ..n − 1] of real numbers
if n = 1 return A [0]
else temp ← Riddle(A [0 ..n − 2] )
if temp ≤ A [ n − 1] return temp
else return A [ n − 1]
a. What does this algorithm compute?
b. Compute the complexity of the algorithm.
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Chapter 2
1. Write a brute-force algorithm to compute a
n
then compute the time complexity of it.
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2. a. Design a brute-force algorithm for computing the value of a polynomial
p ( x ) = a n
xn + a n −
xn −1 +... + a 1
x + a 0
at a given point x 0
and determine its complexity.
b. If the algorithm you designed is in ( n
2
) , design a linear algorithm for this problem.
c. Is it possible to design an algorithm with a better-than-linear efficiency for this problem?
5. The closest-pair problem can be posed in the k -dimensional space, in which
the Euclidean distance between two points p’ ( x’ 1
,... , x’ k
) and p ( x” 1
,... , x” k
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Chapter 3
1. a. Write pseudocode for a divide-and-conquer algorithm for finding the position of the
largest element in an array of n numbers.
b. Compute the complexity of the algorithm made by you.
c. How does this algorithm compare with the brute-force algorithm for this problem
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2. a. Write pseudocode for a divide-and-conquer algorithm for finding values of the
smallest elements in an array of n numbers.
b. Compute the complexity of the algorithm made by you.
c. How does this algorithm compare with the brute-force algorithm for this problem?
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3. a. Write pseudocode for a divide-and-conquer algorithm for the exponentiation problem
of computing a
n
where n is a positive integer.
b. Compute the complexity of the algorithm made by you.
c. How does this algorithm compare with the brute-force algorithm for this problem?
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Chapter 4
1. Consider the problem of finding the distance between the two closest numbers in an array of n
numbers. (The distance between two numbers x and y is computed as | x − y |).
a. Design a presorting-based algorithm for solving this problem and determine its complexity.
b. Compare the efficiency of this algorithm with that of the brute-force algorithm
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Let A = { a
1
,... , a
n
} and B = { b
1
,... , b
m
} be two sets of numbers. Consider the problem
of finding their intersection, i.e., the set C of all the numbers that are in both A and B.
a. Design a brute-force algorithm for solving this problem and determine its complexity.
b. Design a presorting-based algorithm for solving this problem and determine its complexity.
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3. Consider the problem of finding the smallest and largest elements in an array of n numbers.
a. Design a presorting-based algorithm for solving this problem and determine its complexity.
b. Compare the efficiency of the three algorithms: (i) the brute-force algorithm, (ii)
this presorting-based algorithm, and (iii) the divide-and-conquer algorithm.
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4. You have an array of n real numbers and another integer s. Find out whether the array
contains two elements whose sum is s. (For example, for the array 5, 9, 1, 3 and s = 6 , the answer
is yes, but for the same array and s = 7 , the answer is no.) Design an algorithm for this problem
with a better than quadratic time efficiency.
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5. Consider the following brute-force algorithm for evaluating a polynomial.
ALGORITHM BruteForcePolynomialEvaluation(P[0..n], x)
p←0.
for i ←n downto 0 do
power ←
for j ←1 to i do
power ←power ∗
x p←p + P[i] ∗ power
and indicate its complexity.
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Chapter 5
1. Binomial coefficient: Design an efficient algorithm for computing the binomial coefficient
C ( n, k ) that uses no multiplications. Compute the complexity of algorithms designed.
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2. Design a dynamic programming algorithm to find the minimum number of coins
of
denominations d 1
<d 2
<.. .<d m
with d 1
=1 so that the sum of their values is n.
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3. a. Find a recurrence relation and the initial conditions for the number of ways to climb n stairs
if the person climbing the stairs can take one stair or two stairs at a time.
a. Design a brute force algorithm for computing the number of ways to climb n stairs
established by the recurrence relation in question a.
b. Design a dynamic programming algorithm for computing the number of ways to climb n stairs
established by the recurrence relation in question a.
c. Design a transform-and conquer algorithm for computing the number of ways to climb n stairs
established by the recurrence relation in question a.
d.
Design a algorithm with the complexity O(log 2
n ) for computing the number of ways to climb
n stairs established by the recurrence relation in question a.
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4. Present executing of the dynamic programming algorithm that pick up the maximum
amount of coin with the coin row of denominations 7, 1, 3, 8, 5, 3.
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Chapter 6
1. There are n people who need to be assigned to execute n jobs, one person per job. (That is,
each person is assigned to exactly one job and each job is assigned to exactly one person). The
cost that would accrue if the i th person is assigned to the j th job is a known quantity C [ i, j ] for
each pair i, j = 1 , 2 ,... , n. The problem is to find an assignment with the minimum total cost.
Design a greedy algorithm for the assignment problem. Execute the greedy algorithm with the
table entries representing the assignment costs C [ i, j ] as follows:
Job 1 Job 2 Job 3 Job 4
Person 1 9 2 7 8
Person 2 6 4 3 7
Person 3 5 8 1 8
Person 4 7 6 9 4
Design a a greedy algorithm to find the minimum number of coins of denominations d 1
<d 2
. .<d m
with d 1
=1 so that the sum of their values is n.
3. Present executing of the algorithm that colours the graph below
Chapter 7
1.
Design a backtracking algorithm for finding a subset of a given set A = { a 1
,... , a n
} of n
positive integers whose sum is equal to a given positive integer d. For example, for A = {1 , 2 , 5 ,
6 , 8} and d = 9 , there are two solutions: {1 , 2 , 6} and {1 , 8}. Of course, some instances of this
problem may have no solutions.
2.
Design a backtracking algorithm for generating all permutations of {1 , 2 , …, n }. Compute
the complexity of the algorithm.
3.
Design a backtracking algorithm for generating all bit strings of length n that do not have two
consecutive 0s. Compute the complexity of the algorithm.
4.
Design a backtracking algorithm for generating all bit strings of length n that do not have two
consecutive 1s. Compute the complexity of the algorithm.
5.
Present executing of the branch-and-bound algorithm to find the shortest path of the
traveler with the cost matrix as follows.
