Algorithm Analysis Advanced Data Structure. Chung-Ang University, Jaesung Lee

Transcription

1 Algorithm Analysis Advanced Data Structure Chung-Ang University, Jaesung Lee

2 Priority Queue, Heap and Heap Sort 2

3 Max Heap data structure 3

4 Representation of Heap Tree 4

5 Representation of Heap Tree 5

6 Representation of Heap Tree Let us consider the following elements arranged in the form of array as follows: 6

7 Representation of Heap Tree A heap tree represented using a single array looks as follows: 7

8 Operations on heap tree Insertion Deletion and Merging 8

9 Insertion into a heap tree 9

10 Insertion into a heap tree 10

11 Insertion into a heap tree The algorithm Max_heap_insert to insert a data into a max heap tree is as follows: 11

12 Insertion into a heap tree Insert 40, 80, 35, 90, 45, 50, 70 12

13 Insertion into a heap tree Insert 40, 80, 35, 90, 45, 50, 70 13

14 Insertion into a heap tree Insert 40, 80, 35, 90, 45, 50, 70 14

15 Insertion into a heap tree Insert 40, 80, 35, 90, 45, 50, 70 15

16 Insertion into a heap tree Insert 40, 80, 35, 90, 45, 50, 70 16

17 Insertion into a heap tree Insert 40, 80, 35, 90, 45, 50, 70 17

18 Insertion into a heap tree Insert 40, 80, 35, 90, 45, 50, 70 18

19 Insertion into a heap tree Insert 40, 80, 35, 90, 45, 50, 70 19

20 Deletion of a node from heap tree The algorithm for deleting root node is as follows: 20

21 Deletion of a node from heap tree And the algorithm for adjust function is as follows: 21

22 Deletion of a node from heap tree 22

23 Merging two heap trees 23

24 Applications of heap tree 24

25 Heap Sort (Step 1) Build a heap tree with the given set of data. (Step 2-a) Remove the top most item (the largest) and replace it with the last element in the heap. (Step 2-b) Re-heapify the complete binary tree. (Step 2-c) Place the deleted node in the output. (Step 3) Continue Step 2 until the heap tree is empty. 25

26 Heap Sort A Heap Sort Algorithm 26

27 Heap Sort A Heap Sort Algorithm 27

28 Heap Sort A Heap Sort Algorithm 28

29 Heap Sort Time complexity 29

30 Heap Sort Example 30

31 Heap Sort Example 31

32 Heap Sort Example 32

33 Heap Sort Example 33

34 Heap Sort Example 34

35 Heap Sort Example 35

36 Heap Sort Example 36

37 Heap Sort Priority queue implementation using heap tree 37

38 Heap Sort Priority queue implementation using heap tree 38

39 Priority Queue Binary Search Trees 39

40 Priority Queue Binary Tree Searching 40

41 Priority Queue Why use binary search trees? 41

42 Priority Queue Why use binary search trees? 42

43 Priority Queue Why use binary search trees? 43

44 Priority Queue Why use binary search trees? 44

45 Priority Queue Why use binary search trees? 45

46 Priority Queue Why use binary search trees? 46

47 Priority Queue Inserting nodes into a Binary Search Tree 47

48 Priority Queue Deleting nodes from a Binary Search Tree 48

49 Priority Queue Deleting nodes from a Binary Search Tree 49

50 Priority Queue Deleting nodes from a Binary Search Tree 50

51 Priority Queue Deleting nodes from a Binary Search Tree 51

52 Priority Queue Deleting nodes from a Binary Search Tree 52

53 Dictionary Dictionary 53

54 Dictionary Dictionary (Example) 54

55 Dictionary Dictionary (Example) 55

56 Dictionary Get operation Put operation Remove operation 56

57 Disjoint Set Operations Relation between Dictionary and Set A set is an unordered collection (possibly empty) of distinct items. We can implement a set as a bit vector over the universal set. We can implement a set with a list structure (with insertion constraints). A multiset or bag is a set without the uniqueness constraint. Basic operations of a multiset: Search, Insert, Delete. A basic data structure that accomplishes these operations is dictionary. Sometimes we need to dynamically partition some n-element set into a collection of disjoint sets. Sometimes we need to take the union or intersection of sets. 57

58 Disjoint Set Operations Set and Disjoint Set Disjoint Set Operations: Union and Find 58

59 Disjoint Set Operations Disjoint Set Union (example) = {1,7,8,9} = {2,5,10} S = {1,2,5,7,8,9,10} 59

60 Disjoint Set Operations Find (example) = {1,7,8,9} = {2,5,10} = 3,4,6 (4) = (5) = (7) = 60

61 Disjoint Set Operations Set Representation (example) = {1,7,8,9} = {2,5,10} = 3,4,6 61

62 Disjoint Set Operations Disjoint Union 62

63 Disjoint Set Operations Find 63

64 Disjoint Set Operations Union and Find Algorithms (example) For the following sets, the array representation is as shown below. 64

65 Disjoint Set Operations Algorithm for Union operation 65

66 Disjoint Set Operations Algorithm for Find operation 66

67 Disjoint Set Operations Analysis of SimpleUnion(i,j) and SimpleFind(i) For example, consider the sets: 67

68 Disjoint Set Operations Analysis of SimpleUnion(i,j) and SimpleFind(i) The sequence of Union operations results the degenerate tree as below. Time complexity is =. 68