Let’s break down the problem into manageable steps and provide the necessary function to proceed with calculus:

Key Idea

The goal is to minimize the number of comparisons $C(n, k)$, where $k \in (0, 1)$ defines the fraction of the array length $n$ for pivot selection. Here's the reasoning:

1. In a divide-and-conquer algorithm like quicksort, the pivot divides the array into two subarrays:

   • Left subarray of size $pn = k \cdot n$
   • Right subarray of size $(1-k)n$

2. The total comparisons $C(n, k)$ in this process include:

   • Comparing the pivot with $n-1$ elements initially.
   • Recursively dividing and processing the two subarrays.

Total Comparisons

The recurrence relation for comparisons is: $C(n, k) = (n - 1) + C(k \cdot n, k) + C((1-k)n, k).$

Approximation for $C(n, k)$

Using continuous analysis and integration, the total comparisons $C(n, k)$ is approximated as: $C(n, k) \approx n \left[ 1 + k \ln(k) + (1-k) \ln(1-k) \right].$

Function for Optimization

To simplify: $f(k) = 1 + k \ln(k) + (1-k) \ln(1-k).$

Your goal is to minimize $f(k)$ on $k \in (0, 1)$.

Steps:

1. First Derivative: Differentiate $f(k)$ with respect to $k$: $f'(k) = \ln(k) + 1 - \ln(1-k) - 1.$

   Simplify: $f'(k) = \ln\left(\frac{k}{1-k}\right).$

2. Second Derivative: Differentiate $f'(k)$ to get $f''(k)$: $f''(k) = \frac{1}{k} + \frac{1}{1-k}.$

3. Find the Critical Point: Set $f'(k) = 0$: $\ln\left(\frac{k}{1-k}\right) = 0 \implies \frac{k}{1-k} = 1 \implies k = \frac{1}{2}.$

4. Second Derivative Test: Substitute $k = \frac{1}{2}$ into $f''(k)$: $f''(k) = \frac{1}{k} + \frac{1}{1-k} = 2 + 2 = 4 > 0.$

Result:

• The minimum value of $f(k)$ occurs at $k = \frac{1}{2}$, and $f''(k) > 0$ confirms it is a minimum.

Let me know if you need a deeper dive into the derivatives or verification of the result!

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Follow-up Questions:

1. What does minimizing $C(n, k)$ imply for practical algorithms like quicksort?
2. How would changing the definition of $f(k)$ affect the analysis?
3. What if $k$ could also be zero or one? How does this affect the solution?
4. Can you visualize $f(k)$ to confirm the result geometrically?
5. How does this relate to real-world performance tuning in sorting algorithms?

Tip: Always verify the second derivative to confirm the nature (min/max) of critical points.