To trace the steps for the Quick Sort algorithm, let's break down how it would work on each of the given inputs.

Quick Sort Overview

Quick Sort is a divide-and-conquer algorithm that works by selecting a "pivot" element from the array and partitioning the other elements into two sub-arrays, according to whether they are less than or greater than the pivot. The sub-arrays are then sorted recursively.

Steps for Quick Sort

For simplicity, let's assume that we always choose the last element as the pivot.

---

1. Input: {12, 0, 67, -8, 54, 53, 95, 22}

Initial Array: [12, 0, 67, -8, 54, 53, 95, 22]

• Step 1: Choose pivot = 22

  • Partition around 22 → [12, 0, -8, 22, 54, 53, 95, 67]

• Step 2: Recursively sort sub-arrays:

  • Left side: [12, 0, -8]
  • Right side: [54, 53, 95, 67]

• Left side (pivot = -8):

  • Partition around -8 → [-8, 0, 12]
  • Recursively sort [0, 12] → [0, 12] (already sorted)

• Right side (pivot = 67):

  • Partition around 67 → [54, 53, 22, 67, 95]
  • Recursively sort [54, 53] → [53, 54]
  • Recursively sort [95] → [95] (already sorted)

• Final Sorted Array: [-8, 0, 12, 22, 53, 54, 67, 95]

---

2. Input: {-8, 0, 12, 22, 53, 54, 67, 95}

Initial Array: [-8, 0, 12, 22, 53, 54, 67, 95]

• Step 1: Choose pivot = 95

  • Partition around 95 → [-8, 0, 12, 22, 53, 54, 67, 95] (no change needed)
  • Recursively sort [-8, 0, 12, 22, 53, 54, 67] → continue

• Step 2: Choose pivot = 67

  • Partition around 67 → [-8, 0, 12, 22, 53, 54, 67] (no change needed)
  • Recursively sort [-8, 0, 12, 22, 53, 54] → continue

• Step 3: Continue with smaller pivots and realize array is already sorted:

  • Pivot = 54, then 53, then 22, then 12, then 0, then -8
  • No changes needed.

• Final Sorted Array: [-8, 0, 12, 22, 53, 54, 67, 95]

---

3. Input: {95, 67, 54, 53, 22, 12, 0, -8}

Initial Array: [95, 67, 54, 53, 22, 12, 0, -8]

• Step 1: Choose pivot = -8

  • Partition around -8 → [-8, 67, 54, 53, 22, 12, 0, 95]
  • Recursively sort [67, 54, 53, 22, 12, 0, 95] → continue

• Step 2: Choose pivot = 95

  • Partition around 95 → [-8, 67, 54, 53, 22, 12, 0, 95] (no change needed)
  • Recursively sort [67, 54, 53, 22, 12, 0] → continue

• Step 3: Choose pivot = 0

  • Partition around 0 → [-8, 0, 54, 53, 22, 12, 67, 95]
  • Recursively sort [54, 53, 22, 12, 67] → continue

• Step 4: Choose pivot = 67

  • Partition around 67 → [-8, 0, 12, 22, 53, 54, 67, 95]
  • Recursively sort [12, 22, 53, 54] → continue

• Final Sorted Array: [-8, 0, 12, 22, 53, 54, 67, 95]

---

In each case, the final sorted array is [-8, 0, 12, 22, 53, 54, 67, 95], but the steps and partitions differ based on the initial input array.

Do you have any questions or need further details?

---

Relative Questions:

1. How does the choice of pivot affect the efficiency of Quick Sort?
2. Can Quick Sort perform worse than other sorting algorithms? When?
3. What is the average time complexity of Quick Sort?
4. How can Quick Sort be optimized to avoid the worst-case scenario?
5. How does Quick Sort compare with Merge Sort?

Tip: Always consider using a random pivot or median-of-three to improve Quick Sort's performance on already sorted or nearly sorted arrays.