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![Recursion: Quicksort Sub-
arrays
7 20 10 30 40 50 60 80 100
[0] [1] [2] [3] [4] [5] [6] [7] [8]
<= data[pivot] > data[pivot]](https://image.slidesharecdn.com/quicksort-150701093646-lva1-app6891/85/Quick-sort-2-320.jpg)
![Quicksort: Worst Case
Assume first element is chosen as pivot.
Assume we get array that is already in order:
2 4 10 12 13 50 57 63 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
too_big_index too_small_index](https://image.slidesharecdn.com/quicksort-150701093646-lva1-app6891/85/Quick-sort-11-320.jpg)
![1. While data[too_big_index] <= data[pivot]
++too_big_index
2. While data[too_small_index] > data[pivot]
--too_small_index
3. If too_big_index < too_small_index
swap data[too_big_index] and data[too_small_index]
4. While too_small_index > too_big_index, go to 1.
5. Swap data[too_small_index] and data[pivot_index]
2 4 10 12 13 50 57 63 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
too_big_index too_small_index](https://image.slidesharecdn.com/quicksort-150701093646-lva1-app6891/85/Quick-sort-12-320.jpg)
![1. While data[too_big_index] <= data[pivot]
++too_big_index
2. While data[too_small_index] > data[pivot]
--too_small_index
3. If too_big_index < too_small_index
swap data[too_big_index] and data[too_small_index]
4. While too_small_index > too_big_index, go to 1.
5. Swap data[too_small_index] and data[pivot_index]
2 4 10 12 13 50 57 63 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
too_big_index too_small_index](https://image.slidesharecdn.com/quicksort-150701093646-lva1-app6891/85/Quick-sort-13-320.jpg)
![1. While data[too_big_index] <= data[pivot]
++too_big_index
2. While data[too_small_index] > data[pivot]
--too_small_index
3. If too_big_index < too_small_index
swap data[too_big_index] and data[too_small_index]
4. While too_small_index > too_big_index, go to 1.
5. Swap data[too_small_index] and data[pivot_index]
2 4 10 12 13 50 57 63 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
too_big_index too_small_index](https://image.slidesharecdn.com/quicksort-150701093646-lva1-app6891/85/Quick-sort-14-320.jpg)
![1. While data[too_big_index] <= data[pivot]
++too_big_index
2. While data[too_small_index] > data[pivot]
--too_small_index
3. If too_big_index < too_small_index
swap data[too_big_index] and data[too_small_index]
4. While too_small_index > too_big_index, go to 1.
5. Swap data[too_small_index] and data[pivot_index]
2 4 10 12 13 50 57 63 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
too_big_index too_small_index](https://image.slidesharecdn.com/quicksort-150701093646-lva1-app6891/85/Quick-sort-15-320.jpg)
![1. While data[too_big_index] <= data[pivot]
++too_big_index
2. While data[too_small_index] > data[pivot]
--too_small_index
3. If too_big_index < too_small_index
swap data[too_big_index] and data[too_small_index]
4. While too_small_index > too_big_index, go to 1.
5. Swap data[too_small_index] and data[pivot_index]
2 4 10 12 13 50 57 63 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
too_big_index too_small_index](https://image.slidesharecdn.com/quicksort-150701093646-lva1-app6891/85/Quick-sort-16-320.jpg)
![1. While data[too_big_index] <= data[pivot]
++too_big_index
2. While data[too_small_index] > data[pivot]
--too_small_index
3. If too_big_index < too_small_index
swap data[too_big_index] and data[too_small_index]
4. While too_small_index > too_big_index, go to 1.
5. Swap data[too_small_index] and data[pivot_index]
2 4 10 12 13 50 57 63 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
too_big_index too_small_index](https://image.slidesharecdn.com/quicksort-150701093646-lva1-app6891/85/Quick-sort-17-320.jpg)
![1. While data[too_big_index] <= data[pivot]
++too_big_index
2. While data[too_small_index] > data[pivot]
--too_small_index
3. If too_big_index < too_small_index
swap data[too_big_index] and data[too_small_index]
4. While too_small_index > too_big_index, go to 1.
5. Swap data[too_small_index] and data[pivot_index]
2 4 10 12 13 50 57 63 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
> data[pivot]<= data[pivot]](https://image.slidesharecdn.com/quicksort-150701093646-lva1-app6891/85/Quick-sort-18-320.jpg)
![Improved Pivot Selection
Pick median value of three elements from data
array:
data[0], data[n/2], and data[n-1].
Use this median value as pivot.](https://image.slidesharecdn.com/quicksort-150701093646-lva1-app6891/85/Quick-sort-25-320.jpg)
![Improving Performance of
Quicksort
Improved selection of pivot.
For sub-arrays of size 3 or less, apply brute
force search:
Sub-array of size 1: trivial
Sub-array of size 2:
if(data[first] > data[second]) swap them
Sub-array of size 3: left as an exercise.](https://image.slidesharecdn.com/quicksort-150701093646-lva1-app6891/85/Quick-sort-26-320.jpg)
![Recursion: Quicksort Sub-
arrays
7 20 10 30 40 50 60 80 100
[0] [1] [2] [3] [4] [5] [6] [7] [8]
<= data[pivot] > data[pivot]](https://image.slidesharecdn.com/quicksort-150701093646-lva1-app6891/75/Quick-sort-2-2048.jpg)
![Quicksort: Worst Case
Assume first element is chosen as pivot.
Assume we get array that is already in order:
2 4 10 12 13 50 57 63 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
too_big_index too_small_index](https://image.slidesharecdn.com/quicksort-150701093646-lva1-app6891/75/Quick-sort-11-2048.jpg)
![1. While data[too_big_index] <= data[pivot]
++too_big_index
2. While data[too_small_index] > data[pivot]
--too_small_index
3. If too_big_index < too_small_index
swap data[too_big_index] and data[too_small_index]
4. While too_small_index > too_big_index, go to 1.
5. Swap data[too_small_index] and data[pivot_index]
2 4 10 12 13 50 57 63 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
too_big_index too_small_index](https://image.slidesharecdn.com/quicksort-150701093646-lva1-app6891/75/Quick-sort-12-2048.jpg)
![1. While data[too_big_index] <= data[pivot]
++too_big_index
2. While data[too_small_index] > data[pivot]
--too_small_index
3. If too_big_index < too_small_index
swap data[too_big_index] and data[too_small_index]
4. While too_small_index > too_big_index, go to 1.
5. Swap data[too_small_index] and data[pivot_index]
2 4 10 12 13 50 57 63 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
too_big_index too_small_index](https://image.slidesharecdn.com/quicksort-150701093646-lva1-app6891/75/Quick-sort-13-2048.jpg)
![1. While data[too_big_index] <= data[pivot]
++too_big_index
2. While data[too_small_index] > data[pivot]
--too_small_index
3. If too_big_index < too_small_index
swap data[too_big_index] and data[too_small_index]
4. While too_small_index > too_big_index, go to 1.
5. Swap data[too_small_index] and data[pivot_index]
2 4 10 12 13 50 57 63 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
too_big_index too_small_index](https://image.slidesharecdn.com/quicksort-150701093646-lva1-app6891/75/Quick-sort-14-2048.jpg)
![1. While data[too_big_index] <= data[pivot]
++too_big_index
2. While data[too_small_index] > data[pivot]
--too_small_index
3. If too_big_index < too_small_index
swap data[too_big_index] and data[too_small_index]
4. While too_small_index > too_big_index, go to 1.
5. Swap data[too_small_index] and data[pivot_index]
2 4 10 12 13 50 57 63 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
too_big_index too_small_index](https://image.slidesharecdn.com/quicksort-150701093646-lva1-app6891/75/Quick-sort-15-2048.jpg)
![1. While data[too_big_index] <= data[pivot]
++too_big_index
2. While data[too_small_index] > data[pivot]
--too_small_index
3. If too_big_index < too_small_index
swap data[too_big_index] and data[too_small_index]
4. While too_small_index > too_big_index, go to 1.
5. Swap data[too_small_index] and data[pivot_index]
2 4 10 12 13 50 57 63 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
too_big_index too_small_index](https://image.slidesharecdn.com/quicksort-150701093646-lva1-app6891/75/Quick-sort-16-2048.jpg)
![1. While data[too_big_index] <= data[pivot]
++too_big_index
2. While data[too_small_index] > data[pivot]
--too_small_index
3. If too_big_index < too_small_index
swap data[too_big_index] and data[too_small_index]
4. While too_small_index > too_big_index, go to 1.
5. Swap data[too_small_index] and data[pivot_index]
2 4 10 12 13 50 57 63 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
too_big_index too_small_index](https://image.slidesharecdn.com/quicksort-150701093646-lva1-app6891/75/Quick-sort-17-2048.jpg)
![1. While data[too_big_index] <= data[pivot]
++too_big_index
2. While data[too_small_index] > data[pivot]
--too_small_index
3. If too_big_index < too_small_index
swap data[too_big_index] and data[too_small_index]
4. While too_small_index > too_big_index, go to 1.
5. Swap data[too_small_index] and data[pivot_index]
2 4 10 12 13 50 57 63 100pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
> data[pivot]<= data[pivot]](https://image.slidesharecdn.com/quicksort-150701093646-lva1-app6891/75/Quick-sort-18-2048.jpg)
![Improved Pivot Selection
Pick median value of three elements from data
array:
data[0], data[n/2], and data[n-1].
Use this median value as pivot.](https://image.slidesharecdn.com/quicksort-150701093646-lva1-app6891/75/Quick-sort-25-2048.jpg)
![Improving Performance of
Quicksort
Improved selection of pivot.
For sub-arrays of size 3 or less, apply brute
force search:
Sub-array of size 1: trivial
Sub-array of size 2:
if(data[first] > data[second]) swap them
Sub-array of size 3: left as an exercise.](https://image.slidesharecdn.com/quicksort-150701093646-lva1-app6891/75/Quick-sort-26-2048.jpg)
More Related Content
Quick sort
- 2. Recursion: Quicksort Sub- arrays 720 10 30 40 50 60 80 100 [0] [1] [2] [3] [4] [5] [6] [7] [8] <= data[pivot] > data[pivot]
- 3. Quicksort Analysis Assumethat keys are random, uniformly distributed. What is best case running time?
- 4. Quicksort Analysis Assumethat keys are random, uniformly distributed. What is best case running time? Recursion: 1. Partition splits array in two sub-arrays of size n/2 2. Quicksort each sub-array
- 5. Quicksort Analysis Assumethat keys are random, uniformly distributed. What is best case running time? Recursion: 1. Partition splits array in two sub-arrays of size n/2 2. Quicksort each sub-array Depth of recursion tree?
- 6. Quicksort Analysis Assumethat keys are random, uniformly distributed. What is best case running time? Recursion: 1. Partition splits array in two sub-arrays of size n/2 2. Quicksort each sub-array Depth of recursion tree? O(log2n)
- 7. Quicksort Analysis Assumethat keys are random, uniformly distributed. What is best case running time? Recursion: 1. Partition splits array in two sub-arrays of size n/2 2. Quicksort each sub-array Depth of recursion tree? O(log2n) Number of accesses in partition?
- 8. Quicksort Analysis Assumethat keys are random, uniformly distributed. What is best case running time? Recursion: 1. Partition splits array in two sub-arrays of size n/2 2. Quicksort each sub-array Depth of recursion tree? O(log2n) Number of accesses in partition? O(n)
- 9. Quicksort Analysis Assumethat keys are random, uniformly distributed. Best case running time: O(n log2n)
- 10. Quicksort Analysis Assumethat keys are random, uniformly distributed. Best case running time: O(n log2n) Worst case running time?
- 11. Quicksort: Worst Case Assume first element is chosen as pivot. Assume we get array that is already in order: 2 4 10 12 13 50 57 63 100pivot_index = 0 [0] [1] [2] [3] [4] [5] [6] [7] [8] too_big_index too_small_index
- 12. 1. While data[too_big_index]<= data[pivot] ++too_big_index 2. While data[too_small_index] > data[pivot] --too_small_index 3. If too_big_index < too_small_index swap data[too_big_index] and data[too_small_index] 4. While too_small_index > too_big_index, go to 1. 5. Swap data[too_small_index] and data[pivot_index] 2 4 10 12 13 50 57 63 100pivot_index = 0 [0] [1] [2] [3] [4] [5] [6] [7] [8] too_big_index too_small_index
- 13. 1. While data[too_big_index]<= data[pivot] ++too_big_index 2. While data[too_small_index] > data[pivot] --too_small_index 3. If too_big_index < too_small_index swap data[too_big_index] and data[too_small_index] 4. While too_small_index > too_big_index, go to 1. 5. Swap data[too_small_index] and data[pivot_index] 2 4 10 12 13 50 57 63 100pivot_index = 0 [0] [1] [2] [3] [4] [5] [6] [7] [8] too_big_index too_small_index
- 14. 1. While data[too_big_index]<= data[pivot] ++too_big_index 2. While data[too_small_index] > data[pivot] --too_small_index 3. If too_big_index < too_small_index swap data[too_big_index] and data[too_small_index] 4. While too_small_index > too_big_index, go to 1. 5. Swap data[too_small_index] and data[pivot_index] 2 4 10 12 13 50 57 63 100pivot_index = 0 [0] [1] [2] [3] [4] [5] [6] [7] [8] too_big_index too_small_index
- 15. 1. While data[too_big_index]<= data[pivot] ++too_big_index 2. While data[too_small_index] > data[pivot] --too_small_index 3. If too_big_index < too_small_index swap data[too_big_index] and data[too_small_index] 4. While too_small_index > too_big_index, go to 1. 5. Swap data[too_small_index] and data[pivot_index] 2 4 10 12 13 50 57 63 100pivot_index = 0 [0] [1] [2] [3] [4] [5] [6] [7] [8] too_big_index too_small_index
- 16. 1. While data[too_big_index]<= data[pivot] ++too_big_index 2. While data[too_small_index] > data[pivot] --too_small_index 3. If too_big_index < too_small_index swap data[too_big_index] and data[too_small_index] 4. While too_small_index > too_big_index, go to 1. 5. Swap data[too_small_index] and data[pivot_index] 2 4 10 12 13 50 57 63 100pivot_index = 0 [0] [1] [2] [3] [4] [5] [6] [7] [8] too_big_index too_small_index
- 17. 1. While data[too_big_index]<= data[pivot] ++too_big_index 2. While data[too_small_index] > data[pivot] --too_small_index 3. If too_big_index < too_small_index swap data[too_big_index] and data[too_small_index] 4. While too_small_index > too_big_index, go to 1. 5. Swap data[too_small_index] and data[pivot_index] 2 4 10 12 13 50 57 63 100pivot_index = 0 [0] [1] [2] [3] [4] [5] [6] [7] [8] too_big_index too_small_index
- 18. 1. While data[too_big_index]<= data[pivot] ++too_big_index 2. While data[too_small_index] > data[pivot] --too_small_index 3. If too_big_index < too_small_index swap data[too_big_index] and data[too_small_index] 4. While too_small_index > too_big_index, go to 1. 5. Swap data[too_small_index] and data[pivot_index] 2 4 10 12 13 50 57 63 100pivot_index = 0 [0] [1] [2] [3] [4] [5] [6] [7] [8] > data[pivot]<= data[pivot]
- 19. Quicksort Analysis Assumethat keys are random, uniformly distributed. Best case running time: O(n log2n) Worst case running time? Recursion: 1. Partition splits array in two sub-arrays: • one sub-array of size 0 • the other sub-array of size n-1 1. Quicksort each sub-array Depth of recursion tree?
- 20. Quicksort Analysis Assumethat keys are random, uniformly distributed. Best case running time: O(n log2n) Worst case running time? Recursion: 1. Partition splits array in two sub-arrays: • one sub-array of size 0 • the other sub-array of size n-1 1. Quicksort each sub-array Depth of recursion tree? O(n)
- 21. Quicksort Analysis Assumethat keys are random, uniformly distributed. Best case running time: O(n log2n) Worst case running time? Recursion: 1. Partition splits array in two sub-arrays: • one sub-array of size 0 • the other sub-array of size n-1 1. Quicksort each sub-array Depth of recursion tree? O(n) Number of accesses per partition?
- 22. Quicksort Analysis Assumethat keys are random, uniformly distributed. Best case running time: O(n log2n) Worst case running time? Recursion: 1. Partition splits array in two sub-arrays: • one sub-array of size 0 • the other sub-array of size n-1 1. Quicksort each sub-array Depth of recursion tree? O(n) Number of accesses per partition? O(n)
- 23. Quicksort Analysis Assumethat keys are random, uniformly distributed. Best case running time: O(n log2n) Worst case running time: O(n2 )!!!
- 24. Quicksort Analysis Assumethat keys are random, uniformly distributed. Best case running time: O(n log2n) Worst case running time: O(n2 )!!! What can we do to avoid worst case?
- 25. Improved Pivot Selection Pickmedian value of three elements from data array: data[0], data[n/2], and data[n-1]. Use this median value as pivot.
- 26. Improving Performance of Quicksort Improved selection of pivot. For sub-arrays of size 3 or less, apply brute force search: Sub-array of size 1: trivial Sub-array of size 2: if(data[first] > data[second]) swap them Sub-array of size 3: left as an exercise.