Recursive algorithms

September 14, 2013

A recursive algorithm calls itself with a subset of the initial data until its base case is reached (e.g. data cannot be split further).

The canonical example is the factorial operation: x! = x (x-1) (x-2) \cdots 2 .
A straightforward Python function that implements this idea follows:

def fact(x):
    if x <= 1:
        return 1
        return x*fact(x-1)

In practice, recursion actually can speed up the execution of divide-and-conquer (DNQ) algorithms (i.e. in which the initial problem is sequentially split in more manageable sub-problems), as discussed in the next section.

Analysis: the Master Theorem
An asymptotic estimate of the running time T of a recursive DNQ algorithm with deterministically equal size subproblems is given by the following formula: \displaystyle T(n) \leq a T(\frac{n}{b}) + O(n^d), where n is the data size, a \geq 1 and b >  1 are the number of recursive calls per level (i.e. the branching factor of the recursion tree) and the associated reduction in input size, and O(n^d) describes the computational effort associated with preprocessing (i.e. all that is done before the recursions begin).
One can then identify three cases (for simplicity, n is assumed to be a power of b):

  • If a = b^d then T(n) = O(n^d \log n)
  • If a < b^d then T(n) = O(n^d)
  • If a > b^d then T(n) = O(n^{\log_b a})

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