trace steps for quick sort for the following 3 inputs 
answer.add(new int[]{12, 0, 67, -8, 54, 53, 95, 22});
answer.add(new int[]{-8, 0, 12, 22, 53, 54, 67, 95});
answer.add(new int[]{95, 67, 54, 53, 22, 12, 0, -8});

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.

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### 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?

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**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.

Explore detailed steps for Quick Sort algorithm with example arrays. Understand how Quick Sort partitions elements based on a chosen pivot and recursively sorts sub-arrays, leading to efficient sorting. Includes trace steps for arrays [12, 0, 67, -8, 54, 53, 95, 22], [-8, 0, 12, 22, 53, 54, 67, 95], and [95, 67, 54, 53, 22, 12, 0, -8].

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