Chapter 7 Sorting (Part II)

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1 Data Structure t Chapter 7 Sorting (Part II) Angela Chih-Wei i Tang Department of Communication Engineering National Central University Jhongli, Taiwan 2010 Spring

2 Outline Heap Max/min heap Insertion & Deletion Heap sort Radix sort LSD radix sort C.E., NCU, Taiwan Angela Chih-Wei Tang,

3 5.6.1 The Heap Abstract Data Type Definition A max (min) tree is a tree in which the key value in each node is no smaller (larger) than the key values in its children (if any). A max heap is a complete binary tree that is also a max tree. A min heap is a complete binary tree that is also a min tree. C.E., NCU, Taiwan Angela Chih-Wei Tang,

4 Figure 5.25: Sample Max Heaps C.E., NCU, Taiwan Angela Chih-Wei Tang,

5 Figure 5.26: Sample Min Heaps C.E., NCU, Taiwan Angela Chih-Wei Tang,

6 Outline Heap Max/min heap Insertion & Deletion Heap sort Radix sort LSD radix sort C.E., NCU, Taiwan Angela Chih-Wei Tang,

7 5.6.2 Priority Queues In a priority queue, the element to be deleted is the one with highest (or lowest) priority! An element with arbitrary priority can be inserted into the queue according to its priority. C.E., NCU, Taiwan Angela Chih-Wei Tang,

8 5.6.3 Insertion Into A Max Heap To implement insertion: i we need to go from an element to its parent Linked list? A parent field is required! Since a heap is a complete binary tree, we can use arrays! Lemma 5.3 allows us to locate easily the parent of any element! [1] 20 [2] 15? [3] 2 [4] 14 [5] 10 New node C.E., NCU, Taiwan Angela Chih-Wei Tang,

9 A Complete Binary Tree [1] 211 [2] [3] [4] 4 [5] 5 6 [6] Depth=3 C.E., NCU, Taiwan Angela Chih-Wei Tang,

10 Lemma 5.3: A complete Binary Tree If a complete binary tree with n nodes ( depth log 2 n 1 ) is represented sequentially, then for any node with index i, 1 i n, we have: (1) parent(i) is at i / 2 if i 1. If i 1, i is at the root and has no parent. (2) left_child(i) is at 2i if 2i n. If 2i n, then i has no left child. (3) right_child(i) is at 2i+1 if 2i 1 n. If 2i 1 n, then i has no right child. C.E., NCU, Taiwan Angela Chih-Wei Tang,

11 Fig. 5.28: Insertion Into A Max Heap (1/2) [1] 20 [1] 20 [2] 15 [3] 2 [2] 15 [3] 2 [4] 14 [5] 10 [4] 14 [5] 10 (a) Heap before insertion (b) Initial location of new node C.E., NCU, Taiwan Angela Chih-Wei Tang,

12 Fig. 5.28: Insertion Into A Max Heap (2/2) [1] Move Up! [1] 20 [1] 21 [2] 15 [3] 52 5 [2] 15 [3] [4] 14 [5] 10 [6] 2 [4] 14 [5] 10 [6] 2 (c) Insert 5 into heap (a) (d) Insert 21 into heap (a) C.E., NCU, Taiwan Angela Chih-Wei Tang,

13 The Declaration of A Heap /*maximum heap size+1 */ #define MAX_ELEMENTS 200 #define HEAP_FULL(n) (n==max_elements-1) #define HEAP_EMPTY(n) (!n) typdef struct { int key; /* other fields */ } element; element heap[max_elements]; int n=0; C.E., NCU, Taiwan Angela Chih-Wei Tang,

14 Program 5.13: Insertion into A Max Heap void insert_max_heap(element item, int *n) { /* insert item into a max heap of current size *n */ int i; if (HEAP_FULL(*n))( { fprintf(stderr, The heap is full. \n ); exit(1); } i = ++(*n); while ((i!= 1) && (item.key ey > heap[i/2].key)) ey)) { heap[i] = heap[i/2]; i/=2; } heap[i]=item; } C.E., NCU, Taiwan Angela Chih-Wei Tang,

15 Figure 5.29: Deletion from A Max Heap [1] 20 removed [1] 10 [2] 15 [3] 2 [4] [5] (a) Heap structure [1] 15 [2] 15 [3] 2 [4] 14 (b) 10 inserted at the root [2] 14 [3] 2 [4] 10 (c) Final heap Move Down! C.E., NCU, Taiwan 15

16 Program 5.14: Deletion from A Max Heap (1/2) element delete_max_heap(int *n) { /* delete element with the highest key from the heap */ int parent, child; element item, temp; if (HEAP_EMPTY(*n)) { fprintf(stderr, The heap is empty\n ); exit(1); } /* save value of the element with the highest key */ Item = heap[1]; /* use last element in heap to adjust heap */ temp = heap[(*n)--]; parent = 1; child = 2; C.E., NCU, Taiwan Angela Chih-Wei Tang,

17 Program 5.14: Deletion from A Max Heap (2/2) } while (child <= *n) { /* find the larger child of the current parent */ if (child < *n) && (heap[child].key < heap[child+1].key) child++; if (temp.key >= heap[child].key) break; /* move to the next lower level */ heap[parent] = heap[child]; parent = child; child *=2; } heap[parent] = temp; return item; C.E., NCU, Taiwan Angela Chih-Wei Tang,

18 Outline Heap Max/min heap Insertion & Deletion Heap sort Radix sort LSD radix sort C.E., NCU, Taiwan Angela Chih-Wei Tang,

19 Program 7.14: Heap Sort void heapsort(element list[], int n) /* perform a heapsort on the array */ { int i, j; element temp; for (i=n/2; i>0; i--) adjust(list, i, n); for (i=n-1; i>0; i--) SWAP(list[1], list[i+1], temp); adjust(list, 1, i); } C.E., NCU, Taiwan Angela Chih-Wei Tang,

20 Program 7.13: Adjusting A Max Heap void adjust(element list[], int root, int n) /* adjust the binary tree to establish the heap */ { int child, rootkey; element temp; rootkey =list[root].key; t] child=2*root; while (child <=n) { if ((child<n) && (list[cihld].key<list[child+1].key)) child++; if (rootkey>list[child].key) /*compare root and max. child*/ break; else { list[child/2]=list[child]; [ ]; /*move to parent */ child*=2; } } list[child/2]=temp; } Angela Chih-Wei Tang,

21 Converting An Array Into A Max Heap (Fig & 7.12) Input list : (26, 5, 77, 1, 61, 11, 59, 15, 48, 19) [1] 26 [1] 77 [2] 5 [3] 77 [2] 61 [3] 59 [4] 1 [5] [4] [5] [6] [7] [6] [7] [8] [9] [10] [8] [9] [10] (b) Max heap following 1st (a) Input array for loop of heapsort C.E., NCU, Taiwan Angela Chih-Wei Tang,

22 Fig. 7.13: Heap Sort Example (1/2) [1] 61 [1] 59 [2] 48 [3] 59 [2] 48 [3] 26 [4] 15 [5] [4] [5] [6] [7] [6] [7] 5 1 [8] [9] [10] [8] [9] [10] (a) (b) C.E., NCU, Taiwan Angela Chih-Wei Tang,

23 Fig. 7.13: Heap Sort Example (2/2) [1] 48 [1] 26 [2] 19 [3] 26 [2] 19 [3] 11 [4] 15 [5] [4] [5] [6] [7] [6] [7] [8] [9] [10] [8] [9] [10] (c) (d) C.E., NCU, Taiwan Angela Chih-Wei Tang,

24 Outline Heap Max/min heap Insertion & Deletion Heap sort Radix sort LSD radix sort C.E., NCU, Taiwan Angela Chih-Wei Tang,

25 7.8 Radix Sort How about sorting records that have several keys but not only one? An example: 2 keys per record K 0 [Suit]: < < < 1 K [Face value]: J Q K A Least Significant ifi Digit it (LSD) sort: following the sort on a key, the piles are put together to obtain a single pile which is then sorted on the next least significant key. The process is continued until the pile is sorted on the most significant key. LSD indicates only the order in which the keys are sorted but not how each key is to be sorted! C.E., NCU, Taiwan Angela Chih-Wei Tang,

26 Fig. 7.15: Arrangement of Cards after First Pass of LSD Sort C.E., NCU, Taiwan Angela Chih-Wei Tang,

27 Fig. 7.16: Simulation of radix_sort (1/3) list[1] list[2] list[3] list[4] list[5] list[6] list[7] list[8] list[9] list[10] e[0] e[1] e[2] e[3] e[4] e[5] e[6] e[7] e[8] e[9] f[0] f[1] f[2] f[3] f[4] f[5] f[6] f[7] f[8] f[9] list[1] list[2] list[3] list[4] list[5] list[6] list[7] list[8] list[9] list[10] C.E., NCU, Taiwan Angela Chih-Wei Tang,

28 Fig. 7.16: Simulation of radix_sort (2/3) list[1] list[2] list[3] list[4] list[5] list[6] list[7] list[8] list[9] list[10] e[0] e[1] e[2] e[3] e[4] e[5] e[6] e[7] e[8] e[9] f[0] f[1] f[2] f[3] f[4] f[5] f[6] f[7] f[8] f[9] list[1] list[2] list[3] list[4] list[5] list[6] list[7] list[8] list[9] list[10] C.E., NCU, Taiwan Angela Chih-Wei Tang,

29 Fig. 7.16: Simulation of radix_sort (3/3) list[1] list[2] list[3] s[3] list[4] list[5] list[6] list[7] list[8] list[9] list[10] e[0] e[1] e[2] e[3] e[4] e[5] e[6] e[7] e[8] e[9] f[0] f[1] f[2] f[3] f[4] f[5] f[6] f[7] f[8] f[9] list[1] list[2] list[3] list[4] list[5] list[6] list[7] list[8] list[9] list[10] C.E., NCU, Taiwan Angela Chih-Wei Tang,

30 Program 7.15: LSD Radix Sort (1/2) list_pointer radix_sort(list_pointer ptr) /*Radix Sort using a linked list */ { list_pointer front[radix_size], rear[radix_size]; int i, j, digit; for (i = MAX_DIGIT-1; i>=0; i--) /* numbers between 0 and 999*/ { #define MAX_DIGIT 3 for(j=0; j<radix_size;j++) #define RADIX_SIZE 10 front[j] = rear[j]=null; typdef struct t list_node *list_pointer; i t while(ptr) { typdef struct list_node { digit = ptr->key[i]; int key[max_digit]; if (!front[digit]) it]) list_pointer link; front[digit] = ptr; }; else } rear[digit] = ptr; ptr=ptr->link; rear[digit]->link = ptr; C.E., NCU, Taiwan Angela Chih-Wei Tang,

31 Program 7.15: LSD Radix Sort (2/2) /* reestablish the linked list for the next pass */ ptr = NULL; for (j=radix_size-1; j>=0; j--) if (front[j]) { rear[j]->link = ptr; ptr= front[j]; } } return ptr; } C.E., NCU, Taiwan Angela Chih-Wei Tang,

32 Appendix. A Steps from Fig to Fig (1/3) [1] 26 [1] 26 [2] 5 [3] 77 [2] 5 [3] 77 [4] 1 [5] [4] [5] [6] [7] [8] [9] [10] [8] [9] [10] [6] [7] (a) Input array (a.1) C.E., NCU, Taiwan Angela Chih-Wei Tang,

33 Appendix. A Steps from Fig to Fig (2/3) [1] 26 [1] 26 [2] 5 [3] 77 [2] 5 [3] [4] [5] [4] [5] [6] [7] [6] [7] [8] [9] [10] [8] [9] [10] (a.2) (a.3) C.E., NCU, Taiwan Angela Chih-Wei Tang,

34 Appendix. A Steps from Fig to Fig (3/3) [1] 26 [1] 77 [2] 61 [3] 77 [2] 61 [3] [4] [5] [4] [5] [6] [7] [6] [7] [8] [9] [10] [8] [9] [10] (a.4) (b) A max heap! But not sorted well! C.E., NCU, Taiwan Angela Chih-Wei Tang,

35 Appendix. B Steps of Fig (i=9) Adjust left subtree! [1] 5 [1] 61 [2] 61 [3] 59 [2] 49 [3] [4] [5] [4] [5] [6] [7] [6] [7] [8] [9] [10] [8] [9] 1. A[1] <---> A[10] (SWAP ) 2. A[1] <---> A[9] (Adjust ) 77 [10] (a) Output: 77 C.E., NCU, Taiwan Angela Chih-Wei Tang,

36 Appendix. B Steps of Fig (i=8) Adjust right subtree! [1] 1 [1] 59 [2] 49 [3] 59 [2] 49 [3] [4] [5] [4] [5] [6] [7] 5 61 [8] [9] [10] [8] [9] 1. A[1] <---> A[9] (SWAP ) 2. A[1] <---> A[8] (Adjust ) [6] [7] 77 [10] (b) Output: 77, 61 C.E., NCU, Taiwan Angela Chih-Wei Tang,

37 Appendix. B Steps of Fig (i=7) Adjust left subtree! [1] 5 [1] 49 [2] 49 [3] 26 [2] 19 [3] [4] [5] [4] [5] [6] [7] [6] [7] [8] [9] [10] [8] [9] 1. A[1] <---> A[8] (SWAP ) 2. A[1] <---> A[7] (Adjust ) 77 [10] (c) Output: 77, 61, 59 C.E., NCU, Taiwan Angela Chih-Wei Tang,

38 Appendix. B Steps of Fig (i=6) Adjust right subtree! [1] 1 [1] 26 [2] 19 [3] 26 [2] 19 [3] [4] [5] [4] [5] [6] [7] [6] [7] [8] [9] [10] [8] [9] 1. A[1] <---> A[7] (SWAP ) 2. A[1] <---> A[6] (Adjust ) 77 [10] (d) Output: 77, 61, 59, 49 C.E., NCU, Taiwan Angela Chih-Wei Tang,

SET 1C Binary Trees. 2. (i) Define the height of a binary tree or subtree and also define a height balanced (AVL) tree. (2)

SET 1C Binary Trees. 2. (i) Define the height of a binary tree or subtree and also define a height balanced (AVL) tree. (2) SET 1C Binary Trees 1. Construct a binary tree whose preorder traversal is K L N M P R Q S T and inorder traversal is N L K P R M S Q T 2. (i) Define the height of a binary tree or subtree and also define