Explain master theorem and solve the recurrence T(n)=9T(n/3)+n with master method.

To solve the recurrence T(n) = 9T(n/3) + n using the master theorem, we need to determine the complexity of the recurrence based on its form.

The master theorem is a tool used to solve recurrences of the form T(n) = aT(n/b) + f(n), where a ≥ 1, b > 1, and f(n) is an asymptotically positive function.

In our case, we have a = 9, b = 3, and f(n) = n.

To apply the master theorem, we compare f(n) with n^log_b(a). If f(n) is smaller, the complexity is dominated by the term T(n/b), and if f(n) is larger, the complexity is dominated by f(n).

In our case, n^log_b(a) = n^log_3(9) = n^2. Since f(n) = n is smaller than n^2, we can conclude that the complexity of the recurrence is dominated by the term T(n/b).

Now, we need to determine the value of f(n)/n^log_b(a). In our case, f(n)/n^log_b(a) = n/n^2 = 1/n.

According to the master theorem, if f(n)/n^log_b(a) is a constant, the complexity is O(n^log_b(a) * log n). In our case, since 1/n is a constant, the complexity of the recurrence T(n) = 9T(n/3) + n is O(n^2 * log n).

Therefore, the solution to the recurrence T(n) = 9T(n/3) + n using the master theorem is O(n^2 * log n).