Need help with recurrence relation and postcondition of a function

Question

I just wanted to make sure I'm on the right track regarding this.

Here's the function that I'm dealing with:

import math

def Mystery2(A, f, l):
    if f == l:
        return A[f]
    m = math.floor((f+l)/2)
    x = Mystery2(A, f, m)
    c = 0
    i = f
    while i <= l:
        if A[i] == x:
            c = c + 1
        i = i + 1
    if c > (l-f+1)/2:
        return x
    return Mystery2(A, m+1, l)

The precondition is given as: A is an array of integers, and $f$ and $l$ are integers $0 \leq f \leq l < len(A)$.

From what I can see by running the function, the postcondition appears to be that it outputs the element within f and l indices that occurs more than $ \frac{l-f+1}{2}$ times or A[l].

I'm not sure how to find a closed form for the recurrence relation representing the worst case runtime.

So far I figured that with $T(n)$ being the worst runtime of the function and $n = l-f$, $T(n)$ will be constant if $n = 0$ and $T(\left \lfloor \frac{n}{2} \rfloor \right) + T(\left \lceil \frac{n}{2} \right \rceil - 1) + n*c_1 + c$, if $n > 0$. I'm not sure how to convert this into a closed form.

Is it okay to simplify this into $T(\left \lfloor \frac{n}{2} \rfloor \right) + T(\left \lceil \frac{n}{2} \right \rceil) + n*c_1 + c$ for the purpose of comparing it to the master theorem?

Thanks in advance for any help

Answer 1

Without having checked if $T(n)$ is correct, you can find the closed form by:

$$T(n) = T(\lfloor\dfrac{n}{2}\rfloor) + T(\lceil\dfrac{n}{2}\rceil - 1) + n*c_1 + c$$

If n is odd, you can rewrite $T(n)$ as:

$$T(n) = T(\lfloor\dfrac{n}{2}\rfloor) + T(\lfloor\dfrac{n}{2}\rfloor) + n*c_1 + c = $$

$$T(n) = 2T(\lfloor\dfrac{n}{2}\rfloor) + n*c_1 + c $$

If you substitute $T(\lfloor\dfrac{n}{2}\rfloor)$ you get

$$T(n) = 2( 2T(\lfloor\dfrac{n}{4}\rfloor) + \dfrac{n}{2}*c_1 + c ) + n*c_1 + c =$$

$$T(n) = 4T(\lfloor\dfrac{n}{4}\rfloor) + 2n*c_1 + 3c $$

If you substitute $T(\lfloor\dfrac{n}{4}\rfloor)$ you get

$$ T(n) = 4 (2T(\lfloor\dfrac{n}{8}\rfloor) + \dfrac{n}{4}*c_1 + c) + 2n*c_1 + 3c = $$

$$ T(n) = 8T(\lfloor\dfrac{n}{8}\rfloor) + 3n*c_1 + 7c$$

Now you can observe the pattern and see that after substituting $\dfrac{n}{2^{k-1}}$

$$ T(n) = 2^kT(\lfloor\dfrac{n}{2^k}\rfloor) + k*n*c_1 + (2^0 + 2^1 +...+ 2^k)c$$

Let $n = 2^k$

$$ T(n) = nT(1) + n*log_2n*c_1 + (2^0 + 2^1 +...+ n)c = $$

$$ T(n) = n + n*log_2n*c_1 + (2^n - 1)c = $$

$$ T(n) = n + n*log_2n*c_1 + 2^n*c - c = $$

So you have $O(2^n)$

Now if n is even, you have

$$T(n) = T(\dfrac{n}{2}) + T(\dfrac{n}{2} - 1) + n*c_1 + c $$

And you follow the same method as before (i.e., start by substituting n/2 and find the pattern after a few iterations).