Power of Optimal Algorithm Design

I just ran across a great illustration of how a small design detail can mean the difference between ok and great performance. The referenced slides show various versions of a method to calculate the result of raising one integer to another (e.g. 353=42,78535^3 = 42,785). The first version in the slides is given as a recursive function, but an equivalent and intuitive way of doing this is by using a loop like so (Python).

def power(x, n):
  if n == 0: return 1
  if n == 1: return x
  while n > 1:
    y = y * x
    n = n - 1
  return y

It is easy to see that there will be n1n - 1 multiplications performed. If one is not concerned too much with performance, then we are done.

But can we do better?

One approach to take when looking to solve a problem involving counting or iterating is the divide and conquer strategy, where one attempts to reduce a problem into smaller steps in hopes of saving some computation. For instance, the case for n=4n = 4 is the same as n=2n = 2 but with x=x×xx = x \times x. The case for an odd nn is less compelling as we will see, but the reduction in the number of multiplications is there as well too.

Here is the second version, this time with recursion (a function invoking itself):

def power(x, n):
  if n == 0: return 1
  if n == 1: return x
  if n % 2 == 0:
    return power(x * x, n / 2)
  return power(x * x, n / 2) * x

We can easily see that the recursion will quickly stop since n is being halved each time power calls itself. For n=256n = 256, we actually perform a multiplication only 8 times, a much better result than n1=255n - 1 = 255 times in the first program above. In “Big-Oh” notation, the original code runs in O(n) while the recursive one executes in O(lg n).

Now I personally have never had to handwrite a power function; I just rely on whatever math library is available. However, I have written plenty of loops in my time, and my guess is that there is at least one time where I could have improved performance by thinking a bit more about what I was calculating.