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Analysis of Algorithms

Question 1
What is time complexity of fun()? 
 int fun(int n) { int count = 0; for (int i = n; i > 0; i /= 2) for (int j = 0; j < i; j++) count += 1; return count; }
A
O(n^2)
B
O(nLogn)
C
O(n)
D
O(nLognLogn)

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Question 1 Explanation: 
For a input integer n, the innermost statement of fun() is executed following times. So time complexity T(n) can be written as T(n) = O(n + n/2 + n/4 + ... 1) = O(n) The value of count is also n + n/2 + n/4 + .. + 1
Question 2
What is the time complexity of fun()? 
 int fun(int n) { int count = 0; for (int i = 0; i < n; i++) for (int j = i; j > 0; j--) count = count + 1; return count; }
A
Theta (n)
B
Theta (n^2)
C
Theta (n*Logn)
D
Theta (nLognLogn)

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Question 2 Explanation: 
The time complexity can be calculated by counting number of times the expression "count = count + 1;" is executed. The expression is executed 0 + 1 + 2 + 3 + 4 + .... + (n-1) times. Time complexity = Theta(0 + 1 + 2 + 3 + .. + n-1) = Theta (n*(n-1)/2) = Theta(n^2)
Question 3
The recurrence relation capturing the optimal time of the Tower of Hanoi problem with n discs is. (GATE CS 2012)
A
T(n) = 2T(n – 2) + 2
B
T(n) = 2T(n – 1) + n
C
T(n) = 2T(n/2) + 1
D
T(n) = 2T(n – 1) + 1

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Question 3 Explanation: 
Following are the steps to follow to solve Tower of Hanoi problem recursively. 
Let the three pegs be A, B and C. The goal is to move n pegs from A to C. To move n discs from peg A to peg C: move n-1 discs from A to B. This leaves disc n alone on peg A move disc n from A to C move n?1 discs from B to C so they sit on disc n
The recurrence function T(n) for time complexity of the above recursive solution can be written as following. T(n) = 2T(n-1) + 1
Question 4
Let w(n) and A(n) denote respectively, the worst case and average case running time of an algorithm executed on an input of size n. which of the following is ALWAYS TRUE? (GATE CS 2012)
(A) A(n) = \Omega(W(n))
(B) A(n) = \Theta(W(n))
(C) A(n) = O(W(n))
(D) A(n) = o(W(n))
A
A
B
B
C
C
D
D

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Question 4 Explanation: 
The worst case time complexity is always greater than or same as the average case time complexity.
Question 5
Which of the following is not O(n^2)?
A
(15^10) * n + 12099
B
n^1.98
C
n^3 / (sqrt(n))
D
(2^20) * n

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Question 5 Explanation: 
The order of growth of option c is n^2.5 which is higher than n^2.
Question 6
Which of the given options provides the increasing order of asymptotic complexity of functions f1, f2, f3 and f4? 
 f1(n) = 2^n f2(n) = n^(3/2) f3(n) = nLogn f4(n) = n^(Logn)
A
f3, f2, f4, f1
B
f3, f2, f1, f4
C
f2, f3, f1, f4
D
f2, f3, f4, f1

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Question 7
Consider the following program fragment for reversing the digits in a given integer to obtain a new integer. Let n = D1D2…Dm 
 int n, rev; rev = 0; while (n > 0) { rev = rev*10 + n%10; n = n/10; }
The loop invariant condition at the end of the ith iteration is: (GATE CS 2004)
A
n = D1D2….Dm-i and rev = DmDm-1…Dm-i+1
B
n = Dm-i+1…Dm-1Dm and rev = Dm-1….D2D1
C
n != rev
D
n = D1D2….Dm and rev = DmDm-1…D2D1

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Question 8
What is the time complexity of the below function? 
 void fun(int n, int arr[]) { int i = 0, j = 0; for(; i < n; ++i) while(j < n && arr[i] < arr[j]) j++; }
A
O(n)
B
O(n^2)
C
O(nlogn)
D
O(n(logn)^2)

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Question 8 Explanation: 
In the first look, the time complexity seems to be O(n^2) due to two loops. But, please note that the variable j is not initialized for each value of variable i. So, the inner loop runs at most n times. Please observe the difference between the function given in question and the below function: 1
Question 9
In a competition, four different functions are observed. All the functions use a single for loop and within the for loop, same set of statements are executed. Consider the following for loops: 
 A) for(i = 0; i < n; i++) B) for(i = 0; i < n; i += 2) C) for(i = 1; i < n; i *= 2) D) for(i = n; i > -1; i /= 2)
If n is the size of input(positive), which function is most efficient(if the task to be performed is not an issue)?
A
A
B
B
C
C
D
D

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Question 9 Explanation: 
The time complexity of first for loop is O(n). The time complexity of second for loop is O(n/2), equivalent to O(n) in asymptotic analysis. The time complexity of third for loop is O(logn). The fourth for loop doesn't terminate.
Question 10
The following statement is valid. log(n!) = \theta(n log n).
A
True
B
False

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Question 10 Explanation: 
Order of growth of [Tex]\log n![/Tex] and [Tex]n\log n[/Tex] is same for large values of [Tex]n[/Tex], i.e., [Tex]\theta (\log n!) = \theta (n\log n)[/Tex]. So time complexity of fun() is [Tex] \theta (n\log n)[/Tex]. The expression [Tex]\theta (\log n!) = \theta (n\log n)[/Tex] can be easily derived from following Stirling's approximation (or Stirling's formula). [Tex] \log n! = n\log n - n +O(\log(n))\ [/Tex]
Question 11
What does it mean when we say that an algorithm X is asymptotically more efficient than Y?
A
X will be a better choice for all inputs
B
X will be a better choice for all inputs except small inputs
C
X will be a better choice for all inputs except large inputs
D
Y will be a better choice for small inputs

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Question 11 Explanation: 
In asymptotic analysis we consider growth of algorithm in terms of input size. An algorithm X is said to be asymptotically better than Y if X takes smaller time than y for all input sizes n larger than a value n0 where n0 > 0.
Question 12
What is the time complexity of Floyd–Warshall algorithm to calculate all pair shortest path in a graph with n vertices?
A
O(n^2logn)
B
Theta(n^2logn)
C
Theta(n^4)
D
Theta(n^3)

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Question 12 Explanation: 
Floyd–Warshall algorithm uses three nested loops to calculate all pair shortest path. So, time complexity is Thete(n^3). Read here for more details.
Question 13
Consider the following functions: 
 f(n) = 2^n g(n) = n! h(n) = n^logn
Which of the following statements about the asymptotic behavior of f(n), g(n), and h(n) is true?
(A) f(n) = O(g(n)); g(n) = O(h(n))
(B) f(n) = \Omega(g(n)); g(n) = O(h(n))
(C) g(n) = O(f(n)); h(n) = O(f(n))
(D) h(n) = O(f(n)); g(n) = \Omega(f(n))
A
A
B
B
C
C
D
D

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Question 13 Explanation: 
According to order of growth: h(n) < f(n) < g(n) (g(n) is asymptotically greater than f(n) and f(n) is asymptotically greater than h(n) ) We can easily see above order by taking logs of the given 3 functions
 lognlogn < n < log(n!) (logs of the given f(n), g(n) and h(n)).
Note that log(n!) = [Tex]\theta[/Tex](nlogn)
Question 14
In the following C function, let n >= m. 
 int gcd(n,m) { if (n%m ==0) return m; n = n%m; return gcd(m, n); }
How many recursive calls are made by this function?
(A) \theta(logn)
(B) \Omega(n)
(C) \theta(loglogn)
(D) \theta(sqrt(n))
A
A
B
B
C
C
D
D

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Question 14 Explanation: 
Above code is implementation of the Euclidean algorithm for finding Greatest Common Divisor (GCD). Please see http://mathworld.wolfram.com/EuclideanAlgorithm.html for time complexity.
Question 15
Consider the following functions formula Which of the following is true? (GATE CS 2000)
(a) h(n) is 0(f(n))
(b) h(n) is 0(g(n))
(c) g(n) is not 0(f(n))
(d) f(n) is 0(g(n))
A
a
B
b
C
c
D
d

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Question 15 Explanation: 
g(n) = 2^[Tex](\sqrt{n} \log{n} )[/Tex] = n^[Tex](\sqrt{n})[/Tex] f(n) and g(n) are of same asymptotic order and following statements are true. f(n) = O(g(n)) g(n) = O(f(n)). (a) and (b) are false because n! is of asymptotically higher order than n^[Tex](\sqrt{n})[/Tex].
Question 16
Consider the following three claims I (n + k)^m = \theta(n^m), where k and m are constants II 2^(n + 1) = 0(2^n) III 2^(2n + 1) = 0(2^n) Which of these claims are correct? (GATE CS 2003)
A
I and II
B
I and III
C
II and III
D
I, II and III

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Question 16 Explanation: 
 (I) (n+m)^k = n^k + c1*n^(k-1) + ... k^m = [Tex]\theta[/Tex](n^k) (II) 2^(n+1) = 2*2^n = O(2^n)
Question 17
Let s be a sorted array of n integers. Let t(n) denote the time taken for the most efficient algorithm to determined if there are two elements with sum less than 1000 in s. which of the following statements is true? (GATE CS 2000)
a) t (n) is 0 (1)
b) n < t (n) < n {log_2 n}
c) n log 2 n < t (n) < {n \choose 2}
d) t (n) = {n \choose 2}
A
a
B
b
C
c
D
d

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Question 17 Explanation: 
Let array be sorted in ascending order, if sum of first two elements is less than 1000 then there are  two elements with sum less than 1000 otherwise not. For array sorted in descending order we need to check last two elements. For an array data structure, number of operations are fixed in both the cases and not dependent on n, complexity is O(1)
Question 18
Consider the following function 
 int unknown(int n) { int i, j, k = 0; for (i = n/2; i <= n; i++) for (j = 2; j <= n; j = j * 2) k = k + n/2; return k; }
What is the returned value of the above function? (GATE CS 2013)
 (A) \Theta(n^2) (B) \Theta(n^2Logn) (C) \Theta(n^3) (D) \Theta(n^3Logn)
A
A
B
B
C
C
D
D

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Question 18 Explanation: 
The outer loop runs n/2 or [Tex]\Theta(n)[/Tex] times. The inner loop runs [Tex]\Theta(Logn)[/Tex] times (Note that j is divide by 2 in every iteration). So the statement "k = k + n/2;" runs [Tex]\Theta(nLogn)[/Tex] times. The statement increases value of k by n/2. So the value of k becomes n/2* [Tex]\Theta(nLogn)[/Tex] which is [Tex]\Theta(n^2Logn)[/Tex]
Question 19
Consider the following two functions. What are time complexities of the functions? 
 int fun1(int n) { if (n <= 1) return n; return 2*fun1(n-1); } 
 int fun2(int n) { if (n <= 1) return n; return fun2(n-1) + fun2(n-1); }
A
O(2^n) for both fun1() and fun2()
B
O(n) for fun1() and O(2^n) for fun2()
C
O(2^n) for fun1() and O(n) for fun2()
D
O(n) for both fun1() and fun2()

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Question 19 Explanation: 
Time complexity of fun1() can be written as T(n) = T(n-1) + C which is O(n) Time complexity of fun2() can be written as T(n) = 2T(n-1) + C which is O(2^n)
Question 20
Consider the following segment of C-code: 
 int j, n; j = 1; while (j <= n) j = j*2;
The number of comparisons made in the execution of the loop for any n > 0 is: Base of Log is 2 in all options.
A
CEIL(logn) + 1
B
n
C
CEIL(logn)
D
FLOOR(logn) + 1

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Question 20 Explanation: 
We can see it by taking few examples like n = 1, n = 3, etc.
Question 21
Consider the following C-program fragment in which i, j and n are integer variables.
 for (i = n, j = 0; i >0; i /= 2, j += i);
Let val(j) denote the value stored in the variable j after termination of the for loop. Which one of the following is true? (A) val(j) = \theta(logn) (B) vaI(j) = \theta(sqrt(n)) (C) val(j) = \theta(n) (D) val(j) = \theta(nlogn)
A
A
B
B
C
C
D
D

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Question 21 Explanation: 
See question 1 of http://www.geeksforgeeks.org/c-language-set-6/
Question 22
The minimum number of comparisons required to find the minimum and the maximum of 100 numbers is _________________.
A
147.1 to 148.1
B
145.1 to 146.1
C
140 to 146
D
140 to 148

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Question 23
Consider the following pseudo code. What is the total number of multiplications to be performed? 
 D = 2 for i = 1 to n do for j = i to n do for k = j + 1 to n do D = D * 3
A
Half of the product of the 3 consecutive integers.
B
One-third of the product of the 3 consecutive integers.
C
One-sixth of the product of the 3 consecutive integers.
D
None of the above.

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Question 23 Explanation: 
See question 2 of http://www.geeksforgeeks.org/data-structures-algorithms-set-33/
Question 24
Consider the following C-function: 
 double foo (int n) { int i; double sum; if (n = = 0) return 1.0; else { sum = 0.0; for (i = 0; i < n; i++) sum += foo (i); return sum; } }
The space complexity of the above function is:
A
O(1)
B
O(n)
C
O(n!)
D
O(nn)

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Question 24 Explanation: 
Note that the function foo() is recursive. Space complexity is O(n) as there can be at most O(n) active functions (function call frames) at a time.
Question 25
Consider the following C-function: 
 double foo (int n) { int i; double sum; if (n = = 0) return 1.0; else { sum = 0.0; for (i = 0; i < n; i++) sum += foo (i); return sum; } }
Suppose we modify the above function foo() and store the values of foo (i), 0 < = i < n, as and when they are computed. With this modification, the time complexity for function foo() is significantly reduced. The space complexity of the modified function would be:
A
O(1)
B
O(n)
C
O(n!)
D
O(nn)

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Question 25 Explanation: 
Space complexity now is also O(n). We would need an array of size O(n). The space required for recursive calls would be O(1) as the values would be taken from stored array rather than making function calls again and again.
Question 26
Two matrices M1 and M2 are to be stored in arrays A and B respectively. Each array can be stored either in row-major or column-major order in contiguous memory locations. The time complexity of an algorithm to compute M1 × M2 will be
A
best if A is in row-major, and B is in column- major order
B
best if both are in row-major order
C
best if both are in column-major order
D
independent of the storage scheme

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Question 26 Explanation: 
This is a trick question. Note that the questions asks about time complexity, not time taken by the program. for time complexity, it doesn't matter how we store array elements as it always constant or O(1) time to do random access in arrays, the constants may differ for different schemes, but not the time complexity.
There are 26 questions to complete.


    

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