计算算法的时间复杂度

Question

So here is an algorithm that is supposed to return the polynomial value of P(x) of a given polynomial with any given x.

A[] is the coefficient array and P[] the power of x array.

(eg x^2 +2*x + 1 would have: A[] = {1,2,1} , P[]= {2,1,0})

Also, recPower() = O(logn)

int polynomial(int x, int A[], int P[], int l, int r)
{
    if (r - l == 1)
        return ( A[l] * recPower(x, P[l]) ) + ( A[r] * recPower (x, P[r]) );
    int m = (l + r) / 2;
    return polynomial(x, A, P, l, m) + polynomial(x, A, P, m, r);
}

How do I go about calculating this time complexity? I am perplexed due to the if statement. I have no idea what the recurrence relation will be.

Answer 1

Following observation might help: As soon as we have r = l + 1 , we spend O(logn) time and we are done.

My answer requires good understanding of Recursion Tree . So proceed wisely.

So our aim is to find : after how many iterations will we be able to tell that we have r = l + 1?

Lets find out:

Focusing on return polynomial(x, A, P, l, m) + polynomial(x, A, P, m, r);

Let us first consider left function polynomial(x, A, P, l, m) . Key thing to note is that l , remains constant , in all subsequent left function called recursively.

By left function I mean polynomial(x, A, P, l, m) and by right function I mean

polynomial(x, A, P, m, r) .

For left function polynomial(x, A, P, l, m) , We have:

First iteration

l = l and r = (l + r)/2

Second iteration

l = l and r = (l + (l + r)/2)/2

which means that

r = (2l + l + r)/2

Third iteration

l = l and r = (l + (l + (l + r)/2)/2)/2

which means that

r = (4l + 2l + l + r)/4

Fourth iteration

l = l and r = (l + (l + (l + (l + r)/2)/2)/2)/2

which means that

r = (8l + 4l + 2l + l + r)/8

This means in nth iteration we have:

r = (l(1 + 2 + 4 + 8 +......2^n-1) + r)/2^n 

and terminating condition is r = l + 1

Solving (l(1 + 2 + 4 + 8 +......2^n-1) + r)/2^n = l + 1 , we get

2^n = r - l

This means that n = log(r - l) . One might say that in all subsequent calls of left function we ignored the other call, that is right function call. The reason is this:

Since in the right function call we l = m , where m is already a reduced , as we take the mean, and r = r , which is even more averaged this asymptotically wont have any effect on time complexity.

So our recursion tree will have maximum depth = log(r - l) . Its true that not all levels will be fully populated, but for the sake of simplicity, we assume this in asymptotic analysis . So after reaching a depth of log(r - l) , we call function recPower , which takes O(logn) time. Total nodes ( assuming all levels above are full ) at depth log(r - l) is 2^(log(r - l) - 1) . For a single node , we take O(logn) time.

Therefore we have total time = O( logn*(2^(log(r - l) - 1)) ) .

Answer 2

This might help:

T(#terms) = 2T(#terms/2) + a
T(2) = 2logn + b

Where a and b are constants, and #terms refer to number of terms in polynomial. This recurrence relation can be solved using Master's Theorem or using the Recursion tree method.