Provide the solution for the following recurrence: T(n) = 2T(n/2) + logn

Recurrence Relation:

$$T(n) = 2T(n/2) + \log n$$

Solution:

To solve this recurrence relation, we can use the following steps:

1. Guess the form of the solution:

Based on the recurrence relation, we can guess that the solution is of the form:

$$T(n) = a \log n + b$$

where a and b are constants.

1. Verify the guess:

To verify the guess, we need to substitute it into the recurrence relation and see if it holds.

Substituting the guess into the recurrence relation, we get:

$$a \log(\frac{n}{2}) + b = 2(a \log(\frac{n}{2}) + b) + \log n$$

Simplifying the equation, we get:

$$(a-2a) \log(\frac{n}{2}) + (b-2b) = \log n$$

$$-a \log(\frac{n}{2}) - b = \log n$$

$$a \log n + b = \log n$$

$$a = 1$$

$$b = 0$$

Therefore, the guess is correct.

1. Final Solution:

The final solution to the recurrence relation is:

$$T(n) = \log n$$

1. Complexity Analysis:

The complexity of the solution is O(log n). This is because the recurrence relation has a constant number of subproblems, and each subproblem is solved in O(1) time. Therefore, the total time complexity is O(log n).