Recurrences
Question 1 |
(a) Use the patterns given to prove that
(You are not permitted to employ induction)
(b) Use the result obtained in (a) to prove that
Theory Explanation. |
Question 2 |
For parameters a and b, both of which are ω(1), T(n) = T(n1/a)+1, and T(b)=1.
Then T(n) is
θ(loga logb n) | |
θ(logb loga n)
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θ(log2 log2 n)
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θ(logab n)
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Question 2 Explanation:
T(n) = T(n1/a+1, T(b) = 1
T(n) = [T(n1/a2)+1] + 1
= [T(n1/a3)+1] + 2
= [T(n1/a3)] + 3
= [T(n1/ak)] + b
= logb n = ak
= log logb n = k log a
= k= loga logb n
T(n)=1+loga logb n
T(n)=O(loga logb n)
T(n) = [T(n1/a2)+1] + 1
= [T(n1/a3)+1] + 2
= [T(n1/a3)] + 3
= [T(n1/ak)] + b
= logb n = ak
= log logb n = k log a
= k= loga logb n
T(n)=1+loga logb n
T(n)=O(loga logb n)
Question 3 |
For constants a ≥ 1 and b > 1, consider the following recurrence defined on the non-negative integers:
Which one of the following options is correct about the recurrence T(n)?
Which one of the following options is correct about the recurrence T(n)?
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Question 3 Explanation:
Question 4 |
The Lucas sequence Ln is defined by the recurrence relation:
Ln = Ln-1 + Ln-2 , for n >= 3,
with L1 = 1 and L2 = 3.
Which one of the options given is TRUE?
Ln = Ln-1 + Ln-2 , for n >= 3,
with L1 = 1 and L2 = 3.
Which one of the options given is TRUE?
A | |
B | |
C | |
D |
Question 5 |
Let f and g be functions of natural numbers given by f(n)=n and g(n)=n2
Which of the following statements is/are TRUE?
Which of the following statements is/are TRUE?
f ∈ O(g) | |
f ∈ Ω(g) | |
f ∈ o(g) | |
f ∈ Θ(g) |
Question 6 |
Consider functions Function 1 and Function 2 expressed in pseudocode as follows:
Let f1(n) and f2(n) denote the number of times the statement “x = x + 1” is executed in Function 1 and Function 2, respectively. Which of the following statements is/are TRUE?
Let f1(n) and f2(n) denote the number of times the statement “x = x + 1” is executed in Function 1 and Function 2, respectively. Which of the following statements is/are TRUE?
f1(n) ε Θ(f2(n)) | |
f1(n) ε o(f2(n)) | |
f1(n) ε ω(f2(n)) | |
f1(n) ε O(n) |
Question 7 |
The solution of the recurrence relation T(m) = T(3m/4) + 1 is :
θ(lg m) | |
θ(m) | |
θ(mlg m) | |
θ(lglg m)
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Question 7 Explanation:
Using Masters Method:
a = 1, b = 4/3, k = 0, p = 0
Here, a = bk
So, T(m) = nlog a/ log b logp+1 n
T(m) = θ(log m)
a = 1, b = 4/3, k = 0, p = 0
Here, a = bk
So, T(m) = nlog a/ log b logp+1 n
T(m) = θ(log m)
Question 8 |
The recurrence relation capturing the optimal execution time of the towers of Hanoi problem with n discs is:
T(n)=2T(n-2)+2 | |
T(n)=2T(n/2)+1 | |
T(n)=2T(n-2)+n | |
T(n)=2T(n-1)+1 |
Question 8 Explanation:
The recurrence equation for given recurrence function is
T(n) = 2T(n – 1) + 1
= 2 [2T(n – 2) + 1] + 1
= 2 2 T(n – 2) + 3
⋮
= 2 k T( n – k) + (2 k – 1)
n – k = 1
= 2 n-1 T(1) + (2 n-1 – 1)
= 2 n-1 + 2 n-1 – 1
= 2 n – 1
≌ O(2 n )
T(n) = 2T(n – 1) + 1
= 2 [2T(n – 2) + 1] + 1
= 2 2 T(n – 2) + 3
⋮
= 2 k T( n – k) + (2 k – 1)
n – k = 1
= 2 n-1 T(1) + (2 n-1 – 1)
= 2 n-1 + 2 n-1 – 1
= 2 n – 1
≌ O(2 n )
There are 8 questions to complete.