The document discusses three methods to solve recurrences in algorithm analysis: the substitution method, recursion tree method, and master method. It details the substitution method, which involves guessing the form of a solution and proving it by induction. Additionally, there is a midterm exam section with multiple choice and true/false questions evaluating concepts in algorithm analysis and complexity.

1. CS341
Algorithms Analysis and
Design
Section 7

2. Solving Recurrences
Three methods are used in “solving” recurrences
•The Substitution Method
•The Recursion Tree Method
•The Master Method
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3. Substitution Method
Substitution method has two steps
• Guess the form of the solution
• Use induction to prove that the solution is correct
The substitution method can be used to establish an
upper bound on difficult recurrences.
Its use is based on the strength of the guess
• applied in cases when it’s easy to guess the form of
answer
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4. Substitution Method (1)
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5. Substitution Method (1)
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6. Substitution Method (2)
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7. Substitution Method (2)
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8. Substitution Method (2)
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9. Midterm Exam
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10. Question One: Multiple Choice
• 1- Algorithm analysis is determining the amount of ______ resources necessary to
execute it.
a) one b) two c) three d) four
• 2- Algorithm ______ means how does the input/output relation match algorithm
requirement?
a) clarity b) optimality c) simplicity d) correctness
• 3- ______ essential approaches are used to analyze the algorithm.
a) Five b) Four c) Three d) Two
• 4- The ______ order of growth rate means that when N doubles, runtime increases
fourfold.
a) 2N b) N² c) N2 d) 2N
• 5- The insertion and removal in the ______ has an O(1) in the best, average, and
worst cases.
a) tree b) stack c) graph d) queue
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11. Question Two: True or False.
• 1- Many problems are in a complexity class for which no
practical algorithms are known.
• 2- The semantics of an algorithm is its actual
representation (body).
• 3- The efficiency of algorithm determines its complexity.
• 4- Analyzing the average case of an algorithm is the most
difficult in practice.
• 5- An algorithm is a procedure that maps one or more
input(s) to one correct output.
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12. Question Two: True or False.
• 6- The approach to analyzing algorithms must be
independent of four influences.
• 7- Being “O(n3)” is a running time, or a function which
gives the running time of algorithm.
• 8- f(n) ∈ O(g(n)) ≡ g(n) ∈ Ω(f(n))
• 9- Asymptotic growth rate is not always useful for analysis
on fixed-size inputs.
• 10- The bubble, insertion, merge and selection algorithms
are incremental comparison sort.
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13. Question Three: Essay Question
1- Consider the following algorithm:
• n = read input from user
• pos = 0
• neg = 0
• i = 0
• while i < n
• number = read input from user
• if (number > 0)
• pos = pos + 1
• else if (number < 0)
• neg = neg + 1
• i = i + 1
• write pos to user
• write neg to user
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14. Question Three: Essay Question
1- Consider the following algorithm:
• n = read input from user 1
• pos = 0 1
• neg = 0 1
• i = 0 1
• while i < n 1
• number = read input from user 1
• if (number > 0) 1
• pos = pos + 1
• else if (number < 0) OR
• neg = neg + 1 1
• i = i + 1 1
• write pos to user 1
• write neg to user 1
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n

15. Question Three: Essay Question
• A. What is the computation time, T(n), in terms of the input size n?
• 5n+6
• B. What is the tight upper bound?
• O(n)
• C. Give an input sequence that satisfies the best case.
• All positives (number>0)
• D. Give an input sequence that satisfies the average case.
• Positives, Zero and Negatives
• E. Give an input sequence that satisfies the worst case.
• All negatives (number<0)
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