Prerequiste: Analysis of Algorithms

**1. What is the time, space complexity of following code:**

`int` `a = 0, b = 0; ` `for` `(i = 0; i < N; i++) { ` ` ` `a = a + ` `rand` `(); ` `} ` `for` `(j = 0; j < M; j++) { ` ` ` `b = b + ` `rand` `(); ` `} ` |

**Options:**

- O(N * M) time, O(1) space
- O(N + M) time, O(N + M) space
- O(N + M) time, O(1) space
- O(N * M) time, O(N + M) space

Output:

3. O(N + M) time, O(1) space

**Explanation:** The first loop is O(N) and the second loop is O(M). Since we don’t know which is bigger, we say this is O(N + M). This can also be written as O(max(N, M)).

Since there is no additional space being utilized, the space complexity is constant / O(1)

**2. What is the time complexity of following code:**

`int` `a = 0; ` `for` `(i = 0; i < N; i++) { ` ` ` `for` `(j = N; j > i; j--) { ` ` ` `a = a + i + j; ` ` ` `} ` `} ` |

**Options:**

- O(N)
- O(N*log(N))
- O(N * Sqrt(N))
- O(N*N)

Output:

4. O(N*N)

**Explanation:**

The above code runs total no of times

= N + (N – 1) + (N – 2) + … 1 + 0

= N * (N + 1) / 2

= 1/2 * N^2 + 1/2 * N

O(N^2) times.

**3. What is the time complexity of following code:**

`int` `i, j, k = 0; ` `for` `(i = n / 2; i <= n; i++) { ` ` ` `for` `(j = 2; j <= n; j = j * 2) { ` ` ` `k = k + n / 2; ` ` ` `} ` `} ` |

**Options:**

- O(n)
- O(nLogn)
- O(n^2)
- O(n^2Logn)

Output:

2. O(nLogn)

**Explanation:**If you notice, j keeps doubling till it is less than or equal to n. Number of times, we can double a number till it is less than n would be log(n).

Let’s take the examples here.

for n = 16, j = 2, 4, 8, 16

for n = 32, j = 2, 4, 8, 16, 32

So, j would run for O(log n) steps.

i runs for n/2 steps.

So, total steps = O(n/ 2 * log (n)) = **O(n*logn)**

**4. What does it mean when we say that an algorithm X is asymptotically more efficient than Y?**

**Options:**

- X will always be a better choice for small inputs
- X will always be a better choice for large inputs
- Y will always be a better choice for small inputs
- X will always be a better choice for all inputs

2. X will always be a better choice for large inputs

**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.

##### 5. What is the time complexity of following code:

`int` `a = 0, i = N; ` `while` `(i > 0) { ` ` ` `a += i; ` ` ` `i /= 2; ` `} ` |

**Options:**

- O(N)
- O(Sqrt(N))
- O(N / 2)
- O(log N)

Output:

4. O(log N)

**Explanation:** We have to find the smallest x such that N / 2^x N

x = log(N)

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