Trying to compare a recursive and an iterative algorithm

Question

I have two algorithms that solve this problem: Generate all sequences of bits within Hamming distance t. Now I want to compare them theoretically (I do have time measurements, i

Answer 1

At the most general level of time complexity, we have a "worst case" of t = n/2. Now, fix t and gradually increment n. Let's take a starting point of n=8, t=4

C(8 4) = 8*7*6*5*4*3*2*1 / (4*3*2*1 * 4*3*2*1)
    = 8*7*6*5 / 24
n <= n+1 ... n choose t is now

C(9 4) = ...
    = 9*8*7*6 / 24
    = 9/5 of the previous value.

Now, the progression is a little easier to watch.

C( 8 4) = 8*7*6*5 / 24
C( 9 4) =  9/5 * C( 8 4)
C(10 4) = 10/6 * C( 9 4)
C(11 4) = 11/7 * C(10 4)
...
C( n 4) = n/(n-4) * C(n-1 4)

Now, as lemmas for the student:

• Find the base complexity, n! / ( (n-1)! ^ 2)
• Find the combinatorial complexity of product (n / (n-c)) for constant c