Question

I’m reading my algorithms text book, and I’m reading about recurrence relations and finding the algorithms big O complexity. I run across this line

"In the case of the merge-sort algorithm, we get the recurrence equation:

    t(n) = b                  if n < 2

         = 2t(n/2) +bn        if n >= 2

for b > 0

my response was “how the heck did we know that?!?!”

So i’m wondering if there is a systematic approach, or just a logical way of getting these recurrence relations from the algorithms

can some one explain where the b and the two 2’s come from?

Answer 1

The merge sort algorithm can be summarized as:

mergesort (array A) {
   mergesort (first half of A);
   mergesort (second half of A);
   merge the two halves and return;
}

There is an O(N) algorithm to merging two arrays of length N, i.e. its complexity is bN for some constant b > 0.

Assuming the complexity of mergesort is T(N). Since half of N is N/2, we see that:

mergesort (array A) {                    T(N)    =
   mergesort (first half of A);          T(N/2)  +
   mergesort (second half of A);         T(N/2)  +
   merge the two halves and return;       bN
}

which explains the N ≥ 2 case.

The N < 2 case (the base case where you stop the recursion) is trivial.