CS102 Sorting - Part 2
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1 CS102 Sorting - Part 2 Prof Tejada 1
2 Types of Sorts Incremental Approach Bubble Sort, Selection Sort, Insertion Sort, etc. Work slowly toward solution one step at a time Generally iterative in nature Divide and Conquer Approach Merge Sort, Quick Sort, etc. Divide: Break the problem into smaller subproblems Conquer: Solve the subproblems recursively Combine: Recombine subproblems to create a solution 3
3 Merge Sort Divide Divide the N-element sequence to be sorted into two subsequences of N/2 elements each Conquer Sort the two subsequences recursively using merge sort Combine Merge the two sorted subsequences to produce the a single sorted result. Repeat. 4
4 First pass through the list Merge Sort
5 Merge Sort Divide Divide
6 Merge Sort
7 Merge Sort Divide Divide Divide Divide
8 Merge Sort
9 Merge Sort Divide Divide Divide Divide Divide Divide Divide Divide
10 Merge Sort
11 Merge Sort
12 Merge Sort Merge Merge Merge Merge Merge Merge Merge Merge
13 Merge Sort Merge Merge Merge Merge Merge Merge Merge Merge
14 Merge Sort
15 Merge Sort Merge Merge Merge Merge
16 Merge Sort
17 Merge Sort Merge Merge
18 Merge Sort
19 Merge Sort What functions will we need to do merge sort? Need a function to split list in half Need a function to merge two sorted lists Most of the work in the merge sort algorithm is done in the "merge" method Precondition: Given two sorted arrays Postcondition: Need a single, merged sorted array How does it work? Please note that Merge Sort is not an in-place sorting algorithm It uses an output buffer whose size is on the order of the size of the input buffer, i.e., O(n) extra space Merge Sort may not be practical for some applications 20
20 Merge Sort (Merge Algorithm) Left Sublist Right Sublist 21
21 Merge Sort (Merge Algorithm) Right is smaller Copy right to output Left Sublist Right Sublist 22
22 Merge Sort (Merge Algorithm) Left Sublist Right Sublist 23
23 Merge Sort (Merge Algorithm) Left Sublist Right Sublist 24
24 Merge Sort (Merge Algorithm) 3 Left is smaller Copy left to output Left Sublist Right Sublist 25
25 Merge Sort (Merge Algorithm) Left Sublist Right Sublist 26
26 Merge Sort (Merge Algorithm) Left Sublist Right Sublist 27
27 Merge Sort (Merge Algorithm) 3 5 Left is smaller Copy left to output Left Sublist Right Sublist 28
28 Merge Sort (Merge Algorithm) Left Sublist Right Sublist 29
29 Merge Sort (Merge Algorithm) Left Sublist Right Sublist 30
30 Merge Sort (Merge Algorithm) Right is smaller Copy right to output Left Sublist Right Sublist 31
31 Merge Sort (Merge Algorithm) Left Sublist Right Sublist 32
32 Merge Sort (Merge Algorithm) Left Sublist Right Sublist 33
33 Merge Sort (Merge Algorithm) Left is smaller Copy left to output Left Sublist Right Sublist 34
34 Merge Sort (Merge Algorithm) Left Sublist Right Sublist 35
35 Merge Sort (Merge Algorithm) Left Sublist Right Sublist 36
36 Merge Sort (Merge Algorithm) Right is smaller Copy right to output Left Sublist Right Sublist 37
37 Merge Sort (Merge Algorithm) Left Sublist Right Sublist 38
38 Merge Sort (Merge Algorithm) Left Sublist Right Sublist 39
39 Merge Sort (Merge Algorithm) Right is smaller Copy right to output Left Sublist Right Sublist 40
40 Merge Sort (Merge Algorithm) Left Sublist Right Sublist 41
41 Merge Sort (Merge Algorithm) Left Sublist Right Sublist Copy all the remaining items from the left sublist to the output 42
42 Merge Sort (Merge Algorithm) Left Sublist Right Sublist Copy all the remaining items from the left sublist to the output Done 43
43 Merge Sort void merge(vector<int>& numbers, int start, int mid, int end) { vector<int> left; //not shown: copy start to mid vector<int> right; //not shown: copy mid+1 to end int leftindex=0, rightindex=0; } for(int i=start; i <= end; i++) { int leftvalue = left[leftindex]; int rightvalue = right[rightindex]; if(leftvalue <= rightvalue) { numbers[i] = leftvalue; leftindex++; } else { numbers[i] = rightvalue; rightindex++; } } 44
44 Merge Sort void mergesort(vector<int>& numbers, int start, int end) { //stop when list is zero/one element if(start < end) { int mid = (start+end)/2; } } mergesort(numbers, start, mid); //sort left half mergesort(numbers, mid+1, end); //sort right half merge(numbers, start, mid, end); //combine void mergesorthelper(vector<int>& numbers) { mergesort(numbers, 0, numbers.size()-1); } 45
45 Merge Sort Merge Sort What s the best case scenarios? What s the Big O for this case? What s the worst case scenario? What s the Big O for this case? What s the overall Big O? What are the problems with merge sort? 46
46 Merge Sort Merge Sort What s the best case scenarios? What s the Big O for this case? None, O(n log n) What s the worst case scenario? What s the Big O for this case? None, O(n log n) What s the overall Big O? O(n log n) What are the problems with merge sort? 47
47 Merge Sort The Good Stable Sort (generally) Maintains original ordering of equal elements Fairly easy to code Useful for lists that must be accessed sequentially e.g. Linked Lists Sorting in place is really difficult The Bad O(N) additional space complexity if not in place There are better in-place sorts Lots of copying data 48
48 Extra Material Sorting Algorithm Animations What do sorting algorithms sound like? CSCI 102 YouTube channel 49
49 Quick Sort Divide Pick an element in the array and call it Split the rest of the array into two subarrays: One array containing all numbers <= One array containing all numbers > Conquer Sort the two new arrays recursively using quick sort Combine The arrays are sorted in place. No recombination necessary. 50
50 Quick Sort Divide Pick an element in the array and call it We will simply use the last element Split the rest of the array into two subarrays: One array containing all numbers <= One array containing all numbers > Conquer Sort the two new arrays recursively using quick sort Combine The arrays are sorted in place. No recombination necessary. 51
51 Quick Sort The idea behind the Divide part
52 Quick Sort The idea behind the Divide part ) Choose the last element in the array as the pivot It doesn t move (until the very end) So, we will pin it where it s at for now 53
53 Quick Sort The idea behind the Divide part ) Choose the last element in the array as the pivot 2) Identify all the numbers smaller than the pivot (in blue) Identify all the numbers larger than the pivot (in orange) 54
54 Quick Sort The idea behind the Divide part ) Choose the last element in the array as the pivot 2) Identify all the numbers smaller than the pivot (in blue) Identify all the numbers larger than the pivot (in orange) 3) Move all the numbers smaller than the pivot to the left Move all the numbers larger than the pivot to the right 55
55 Quick Sort The idea behind the Divide part ) Choose the last element in the array as the pivot 2) Identify all the numbers smaller than the pivot (in blue) Identify all the numbers larger than the pivot (in orange) 3) Move all the numbers smaller than the pivot to the left Move all the numbers larger than the pivot to the right 4) Move the pivot between the blue and the orange numbers 56
56 Quick Sort The idea behind the Divide part ) Choose the last element in the array as the pivot 2) Identify all the numbers smaller than the pivot (in blue) Identify all the numbers larger than the pivot (in orange) 3) Move all the numbers smaller than the pivot to the left Move all the numbers larger than the pivot to the right 4) Move the pivot between the blue and the orange numbers The pivot is now at the correct position in the final sort order 57
57 Quick Sort The idea behind the Divide part final sort order QuickSort this QuickSort this 1) Choose the last element in the array as the pivot 2) Identify all the numbers smaller than the pivot (in blue) Identify all the numbers larger than the pivot (in orange) 3) Move all the numbers smaller than the pivot to the left Move all the numbers larger than the pivot to the right 4) Move the pivot between the blue and the orange numbers The pivot is now at the correct position in the 58
58 How to do it? Quick Sort
59 Quick Sort How to do it? Step (1) is easy 60
60 Quick Sort How to do it? Step (1) is easy Steps (2) and (3) needs to be done together 61
61 Quick Sort How to do it? Step (1) is easy Steps (2) and (3) needs to be done together Step (4) is done in a slightly different way Swap the pivot and the first element in the orange section 62
62 How to do it? Quick Sort QuickSort this QuickSort this Step (1) is easy Steps (2) and (3) needs to be done together Step (4) is done in a slightly different way Swap the pivot and the first element in the orange section Doesn t really change the property of QuickSort 63
63 Quick Sort How to do steps (2) and (3) together?
64 Quick Sort How to do steps (2) and (3) together? Unsorted 4 Keep array split into 3 sections: Numbers less than or equal to the pivot (<= 4, in blue) Numbers greater than the pivot (> 4, in orange) Numbers yet to be sorted 65
65 Quick Sort How to do steps (2) and (3) together? Unsorted 4 Keep array split into 3 sections: Numbers less than or equal to the pivot (<= 4, in blue) Numbers greater than the pivot (> 4, in orange) Numbers yet to be sorted 66
66 Quick Sort How to do steps (2) and (3) together? Less Than Unsorted 4 Keep array split into 3 sections: Numbers less than or equal to the pivot (<= 4, in blue) Numbers greater than the pivot (> 4, in orange) Numbers yet to be sorted 67
67 Quick Sort How to do steps (2) and (3) together? Less Than More Than Unsorted 4 Keep array split into 3 sections: Numbers less than or equal to the pivot (<= 4, in blue) Numbers greater than the pivot (> 4, in orange) Numbers yet to be sorted 68
68 Quick Sort How to do steps (2) and (3) together? Less Than More Than Unsorted 4 Keep array split into 3 sections: Numbers less than or equal to the pivot (<= 4, in blue) Numbers greater than the pivot (> 4, in orange) Numbers yet to be sorted 69
69 Quick Sort How to do steps (2) and (3) together? Less Than More Than Unsorted 4 Swap 70
70 Quick Sort How to do steps (2) and (3) together? Less Than More Than Unsorted 4 71
71 Quick Sort How to do steps (2) and (3) together? Less Than More Than Unsorted 4 Swap 72
72 Quick Sort How to do steps (2) and (3) together? Less Than More Than Unsorted 4 73
73 Quick Sort How to do steps (2) and (3) together? Less Than More Than Unsorted 4 74
74 Quick Sort How to do steps (2) and (3) together? Less Than More Than 4 75
75 Quick Sort How to do steps (2) and (3) together? Less Than More Than 4 Note: we only swap when the number is < pivot When the number is > pivot, we just move along and did nothing 76
76 Quick Sort Step (4) Less Than More Than 4 Swap 77
77 Quick Sort Step (4) Less Than More Than 8 78
78 Quick Sort Step (4) Less Than More Than 8 What if some other cell has the same value as the pivot? Does that cell belong to the blue or the orange section? 79
79 Quick Sort Step (4) Less Than More Than 8 What if some other cell has the same value as the pivot? Does that cell belong to the blue or the orange section? The answer is: it does not matter if it is done consistently Arbitrarily, we use: blue is for numbers pivot 80
80 Sort Recursively Quick Sort Sort Recursively
81 Sort Recursively Quick Sort Sort Recursively
82 Sort Recursively Quick Sort Sort Recursively Sort Recursively
83 Sort Recursively Quick Sort Sort Recursively Sort Recursively
84 Sort Recursively Quick Sort Sort Recursively Sort Recursively Sort Recursively
85 Sort Recursively Quick Sort Sort Recursively Sort Recursively Sort Recursively
86 All done! No merge! Quick Sort Sort Recursively Sort Recursively Sort Recursively
87 Quick Sort What functions will we need to do quick sort? Need a function to partition list into two sublists Need a function to sort a list Most of the work in the quick sort algorithm is done in the "partition" method Precondition: Given an array Postcondition: Choose a partition value X and split the array into two sublists: one with all values <= X and one with all values > X How does it work? 88
88 Quick Sort int partition(vector<int>& numbers, int start, int end) { int pivot = numbers[end]; //i is the index of the last number < pivot int i = start-1; } //j is the index of the first unsorted number for (int j=start; j < end; j++) { if(numbers[j] <= pivot) { i++; swap(numbers[i],numbers[j]); } } i++; swap(numbers[i],numbers[end]); //return the position of the pivot //this is the right place for the pivot return(i); 89
89 Quick Sort int quicksort(vector<int>& numbers, int start, int end) { //stop when list is zero/one element int i = start-1; //j is the index of the first unsorted number if (start < end) { //partition and return the pivot location int pivot_idx = partition(numbers,start,end); } } //recursively sort the sublists quicksort(numbers,start,pivot_idx-1); quicksort(numbers,pivot_idx+1,end); int quicksorthelper(vector<int>& numbers) { quicksort(numbers, 0, numbers.size()-1); } 90
90 Quick Sort Quick Sort What s the best case scenarios? What s the Big O for this case? What s the worst case scenario? What s the Big O for this case? What s the overall Big O? What are the problems with quick sort? 91
91 Quick Sort Quick Sort What s the best case scenarios? What s the Big O for this case? Every partition splits list in half, O(n log n) What s the worst case scenario? What s the Big O for this case? 2 Sorted or reverse sorted, O(n ) What s the overall Big O? 2 O(n ), but O(n log n) on the average What are the problems with quick sort? 2 1) certain inputs can be O(n ) 2) slow on small lists 92
92 Quick Sort What are some ways we could improve quick sort to get better performance? Randomize selection of partition Try to ensure a balanced partition Try to ensure no particular input can ever give worst case performance Sort small lists with a sort that works better on smaller lists (e.g. Insertion Sort) Avoid slow quicksort of small lists 93
93 Extra Material Sorting Robots Sorting Algorithm Animations What do sorting algorithms sound like? algorithms/musical_sorting_algorithms.html CSCI 102 YouTube channel 94
94 Comparison Sorts Big O of comparison sorts It is mathematically provable that comparison-based sorts can never perform better than O(n log n) So can we ever have a sorting algorithm that performs better than O(n log n)? 95
95 Comparison Sorts Big O of comparison sorts It is mathematically provable that comparison-based sorts can never perform better than O(n log n) So can we ever have a sorting algorithm that performs better than O(n log n)? Yes, but only if we know for sure that the input has some known characteristics 96
96 Sorting in Linear (O(n)) Time Counting Sort If you know that the input values range from 0 to K where K is small Radix Sort Sort numbers one bit (or digit) at a time starting with the least significant bit (or digit) to the most Sorting subroutine must be stable, i.e., if there is a tie among a few elements, their relative position in the list must not change Bucket Sort Divide all the numbers into buckets with non-overlapped ranges (n is the number of buckets) Sort each bucket (can use Bucket Sort) Concatenate the sorted buckets 97
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