递归 function 的时间复杂度

Question

I'm having some trouble determining the time complexity of this recursive function in terms of Big-O notation.

double expRecursive(double x, int n) {
    if (n <= 4) {
        return expIterativ(x, n);
    }

    return expRecursive(x, n/2) *
           expRecursive(x, (n + 1)/2);
}

(The time complexity of the method expIterative is O(n))

Is the recurrence relation of expRecursive() T(n) = 2T(n/2) + n?

If this is the case I suppose T(n) = O(nlogn) after applying the master theorem?

My biggest problem is coming up with the correct recurrence relation...

Answer 1

To get the recurrence relation, it helps to first annotate the code:

double expRecursive(double x, int n) {
    if (n <= 4) {
        return expIterativ(x, n);       // O(1) since n is bounded
    }

    return expRecursive(x, n/2)         // T(n/2)
           *                            // O(1)
           expRecursive(x, (n + 1)/2);  // T(n/2)
}

So the recurrence relation is

T(n) = O(1),              for 0 <= n <= 4
T(n) = 2 * T(n/2) + O(1), for n > 4

which has the solution T(n) ∈ Ө(n) .