## Answers

1.

The 2 for loops in the code segment are not independent and so the product rule cannot be used to compute the time complexity.

Since both i loop and j loop are dependent upon each other, let's analyze how both the loops work using the following table :

Value of i | n | n/2 | n/4 | 1 | |||||

j loop runs for | log n times | log ( n / 2 ) times | log ( n / 4 ) times | log ( 1 ) times |

Let the running time of the algorithm be represented as T(n).

So, the time complexity is calculated as :

T(n) = Sum of the total number of times j loop runs

= log n + log ( n / 2 ) + log ( n / 4 ) + .................... log ( 1 )

= log ( n * n /2 * n / 4 .......... 1 ) + 0 [ logarithmic property ]

<= log ( n^n)

<= n log n

So,

T(n) = Θ(n log n)

Hence, the running time of the given algorithm is Θ(n log n).

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