大O符号和递归

Question

I am having some trouble working out the Big O notation for these 2 recursive functions:

int calc (int n) 
{
  if (n <= 0)
    return 0 ;
  else if (n > 10) 
    return n ;
  else
    return calc (5 + calc(5n));
}

In the case above I think the Big O notation might be O(n^2) because of the nested iterations in the data set?

boolean method (int k ,int [] arr, int i, int j)
{
    if (i > j)
       return false;
    if (arr [(i+j)/2] == k)
       return true;
    if (arr [(i+j)/2] < k)
       return method (k, arr, i, ( (i+j)/2) - 1) ;
    else
       return method (k, arr, ((i+j)/2)+1, j) ;
}

Here I think the big O notation might be O(log N) because the input data set is halved with each iteration?

I am, however, very new to Big O notation and any help or explanations would be much appreciated!

Answer 1

For calc :

This function will never be called more than 5 times during recursion. It's easy to see from a brief analysis and substituting a few values for n . Thus it's O(1) . Hint: the function will get called more times the smaller n is (above some threshold).

Maybe a bit of a bold statement, but I believe any function (assuming n is the input / input size) with if (n > max) return const; has to be O(1) (just let the "constant" be the maximum time taken for n <= max ).

For method :

Yes, it's O(log n) .

The function is actually binary search, which is a good thing to know.

Answer 2

Dukeling is correct. The first is O(1) and the second is O(log N)

But given the title of this question I think it is important to remember that recursion is not special. Any recursive function can be re-written as a loop there is no difference when looking at them versus a standard loop.