Quick Sort Algorithm

1. Chapter 11 Sorting

2. Quick Sort

3. Basic Idea 假設 n elements 的中間值 在正中間 叫 pivot (樞紐) 如何使pivot以左的都小於pivot 而pivot以右的都大pivot? 我是李二二 我是張一二 凡我跑過 凡我跑過 皆小於 pivot皆大於 pivot pivot swap 50 n/2 elements > pivot n/2 elements < pivot 下一回合 n/4 elements 直到 n = 1 時間 O(log n)

4. Big O issue • 上頁假設 n elements 的中間值剛好位於正中間 但這是不大可能的 • 實務上 可任取三個值 以其中間值為pivot 或取最左值為 pivot(如下頁演算法) 最差情況是 取了最小值為 pivot 則問題從 n變 n-1而非 n/2 則需 O(n) 時間

5. 快速排序法 Quick Sort 如果 a 為空或 left>=right 則結束 left right 令 a[left] 為pivot樞紐(即3) pivot • Quicksort example j i n-2 2 0 index 1 n-1 令 i 為left+1, j為right a 3 8 1 4 2 1. while a[i] < pivot, i向右移 2. while a[j] >= pivot, j向左移 3. if (i<j), a[i] 和 a[j] 互換 3 8 1 4 2 repeat until (i>j) left right right left a[left] 和 a[j] 互換 1 2 3 4 8 quickSort(a, left, j-1) quickSort(a, j+1, right) result 1 2 3 4 8

6. Quick Sort Algorithm quickSort(int[] a, int left, int right){ int pivot = a[left]; int i = left+1, j = right; do{ while( a[i] < pivot ){ i++; } while( a[j] >= pivot ){ j++; } if ( i < j ){ swap(a[i],a[j]) } } while( i <= j) swap(a[left], a[j]); quickSort(a, left, j-1); quickSort(a, j+1, right); }// end of quickSort()

7. Quick Sort /** * @author 胡升瑞 * This class implement the Quick Sort. * Have these method : QuickSort, swap, print. * Have parameter : array. */ public void quickSort(int[] array, int low, int high) { if (array == null || array.length == 0){return;} if (low >= high){return;} //pick the pivot int pivot = array[low]; //make left < pivot and right > pivot int i = low, j = high; while (i <= j) { while (array[i] < pivot) {i++;} while (array[j] > pivot) {j--;} if (i <= j) {swap(array,i,j); i++; j--;}} } //recursively sort two sub parts if (low < j){quickSort(array, low, j);} if (high > i){quickSort(array, i, high);} }

8. Quick Sort(cont.) /*Print the array of this object*/ public void print(int[] x) { for (int a : x) System.out.print(a + " "); System.out.println(); } /** * This methd is used to swap two position of value in one array * @param input Input array * @param a The first position to swap * @param b The second position to swap * @return Return the swaped array */ public int[] swap(int[] input, int a, int b) { int temp = input[a]; input[a] = input[b]; input[b] = temp; return input; }

9. Quick Sort(cont.) /** * The main method, program start from here * @param args */ public static void main(String[] args) { int[] x = { 3, 8, 1, 4, 2 }; print(x); int low = 0; int high = x.length - 1; quickSort(x, low, high); print(x); }

10. Bubble Sort

11. 2 2 4 4 n-1 index 4 4 2 3 n-2 1 1 3 2 1 3 3 1 1 0 第1回合 a[1]到定位 第n-1回合 a[n-1]到定位 最大氣泡 最小氣泡 4 3 2 1

12. 每回合處理 a[0 … i]使較大氣泡浮上 (i 從 n-1到1) (if a[j]>a[j+1] then swap(較大氣泡浮上) j從0到i-1 bubbleSort (int[n] a) { for(i=n-1;i>=1;i--) for(j=0;j<=i-1;j++) if a[j]>a[j+1] swap(a[j],a[j+1]); } // end of bubbleSort() swap(int a, int b) { int temp; temp=a; a=b; b=temp; } // end of swap()

13. Bubble.java /** * @author黃柏昇 * This class implement the Bubble sort. * Have these method : sort, swap, print. * Have parameter : array. */ /** * Do the sorting job * @return The array have been sorted */ publicvoid bubbleSort(int[] array) { for(inti = array.length-1; i >= 1; i--) { for(intj = 0; j <= i-1; j++) { if(array[j] > array[j+1]) array = swap(array, array[j],array[j+1]); } } returnarray; }

14. Bubble.java(continued) /* Print the array of this object*/ publicvoid print() { for(inti = 0; i < array.length; i++) System.out.print(array[i] + " "); System.out.println(); } /** * This methd is used to swap two position of value in one array * @param input Input array * @param a The first position to swap * @param b The second position to swap * @return Return the swaped array */ publicint[] swap(int[] input, inta, intb) { inttemp = input[a]; input[a] = input[b]; input[b] = temp; returninput; } publicint[] getArray() { returnarray; } publicvoid setArray(int[] array) { this.array = array; }

15. Bubble.java(continued) /** * The main method, program start from here * @param args */ publicstaticvoid main(String[] args) { int[] x = { 2, 4, 1, 3 }; print(x); bubbleSort(x); print(x); }

16. Selection Sort

17. index 0 1 n-2 n-1 4 2 3 1 min 2 4 3 1 第0回合 min 第1回合 4 2 1 3 min 4 第n-2回合 3 1 2 3 4 1 2

18. 每回合處理 a[i … n-1](i 從 0 到 n-2) select min，與 a[ i ] swap selectionSort ( int[n] a) { for( i = 0, i <= n-2, i++){ /*select min */ min = i ; for ( j = i, j <= n-1, j++) if a[ j + 1] < a[min] min=j+1; swap ( a[min], a[ i ]); } } // end of selectionSort()

19. Selection.java /** * @author曾繼禾 * This class implement the selection sort. * Have these method : sort, swap, print. * Have parameter : array. */ /** * Do the sorting job * @return The array have been sorted */ publicint[] selectionSort(int[] array) { for(int i = 0; i < array.length; i++) { int min = i; for(int j = i; j < array.length; j++) { if(array[j] < array[min]) min = j; } array = swap(array, min, i); } returnarray; }

20. Selection.java(continued) /** * Print the array of this object */ publicvoid print() { for(int i = 0; i < array.length; i++) System.out.print(array[i] + " "); System.out.println(); } /** * This methd is used to swap two position of value in one array * @param input Input array * @param a The first position to swap * @param b The second position to swap * @return Return the swaped array */ publicint[] swap(int[] input, int a, int b) { int temp = input[a]; input[a] = input[b]; input[b] = temp; return input; } publicint[] getArray() { returnarray; } publicvoid setArray(int[] array) { this.array = array; } }

21. Main.java /** * @author曾繼禾 * * This is main class, using Selection object to do selection sort * * Have method : main */ /** * The main method, program start from here * @param args */ publicstaticvoid main(String[] args) { int[] x = { 3, 1, 4, 2 }; print(x); selectionSort(x); print(x); }

22. Insertion Sort

23. index 1 n-2 n-1 0 3 1 4 2 unsorted未排序 sorted 排序好 4 2 1 3 第1回合 2 4 3 第二回合 1 2 1 4 3 第 n-1回合 1 2 3 4

24. 每回合處理未排序的a[i … n-1](i 從 1到 n-1) 把a[i] insert 到左邊排序好的適當位置 (if a[j]<a[j-1] then swap else 已到適當位置 不做了 而 j 從 i 到1) insertionSort (int[n] a){ for(i=1;i<=n-1;i++) for(j=i;j>=1;j--) if a[j]<a[j-1] swap(a[j],a[j-1])else break; } // end of insertionSort() swap(int a, int b) { int temp; temp=a;a=b;b=temp; }

25. Insertion.java /** * @author黃柏昇 * This class implement the Insertion sort. * Have these method : sort, swap, print. * Have parameter : array. */ /** * Do the sorting job * @return The array have been sorted */ publicvoid insertionSort(int[] array) { for(inti = 1; i <= array.length-1; i++) { for(intj = i; j >= 1; j--) { if(array[j] < array[j-1]) array = swap(array, array[j],array[j-1]); } } returnarray; }

26. Insertion.java(continued) /* Print the array of this object*/ publicvoid print() { for(inti = 0; i < array.length; i++) System.out.print(array[i] + " "); System.out.println(); } /** * This methd is used to swap two position of value in one array * @param input Input array * @param a The first position to swap * @param b The second position to swap * @return Return the swaped array */ publicint[] swap(int[] input, inta, intb) { inttemp = input[a]; input[a] = input[b]; input[b] = temp; returninput; } publicint[] getArray() { returnarray; } publicvoid setArray(int[] array) { this.array = array; }

27. Insertion.java(continued) /** * The main method, program start from here * @param args */ publicstaticvoid main(String[] args) { int[] x = { 3, 1, 4, 2 }; print(x); insertionSort(x); print(x); }