Math Problem Statement
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});
Solution
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]
- Partition around
-
Step 2: Recursively sort sub-arrays:
- Left side:
[12, 0, -8]
- Right side:
[54, 53, 95, 67]
- Left side:
-
Left side (pivot = -8):
- Partition around
-8
→[-8, 0, 12]
- Recursively sort
[0, 12]
→[0, 12]
(already sorted)
- Partition around
-
Right side (pivot = 67):
- Partition around
67
→[54, 53, 22, 67, 95]
- Recursively sort
[54, 53]
→[53, 54]
- Recursively sort
[95]
→[95]
(already sorted)
- Partition around
-
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
- Partition around
-
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
- Partition around
-
Step 3: Continue with smaller pivots and realize array is already sorted:
- Pivot =
54
, then53
, then22
, then12
, then0
, then-8
- No changes needed.
- Pivot =
-
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
- Partition around
-
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
- Partition around
-
Step 3: Choose pivot =
0
- Partition around
0
→[-8, 0, 54, 53, 22, 12, 67, 95]
- Recursively sort
[54, 53, 22, 12, 67]
→ continue
- Partition around
-
Step 4: Choose pivot =
67
- Partition around
67
→[-8, 0, 12, 22, 53, 54, 67, 95]
- Recursively sort
[12, 22, 53, 54]
→ continue
- Partition around
-
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:
- How does the choice of pivot affect the efficiency of Quick Sort?
- Can Quick Sort perform worse than other sorting algorithms? When?
- What is the average time complexity of Quick Sort?
- How can Quick Sort be optimized to avoid the worst-case scenario?
- 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.
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Math Problem Analysis
Mathematical Concepts
Sorting Algorithms
Quick Sort
Formulas
-
Theorems
-
Suitable Grade Level
Not Applicable
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