Copyright 1998 by Addison-Wesley Publishing Company 147. Chapter 15. Stacks and Queues

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1 Copyright 1998 by Addison-Wesley Publishing Company 147 Chapter 15 Stacks and Queues

2 Copyright 1998 by Addison-Wesley Publishing Company 148 tos (-1) B tos (1) A tos (0) A A tos (0) How the stack routines work: empty stack, push(a), push(b), pop

3 Copyright 1998 by Addison-Wesley Publishing Company 149 makeempty( ) size = 0 enqueue(a) size = 1 back front back A front enqueue(b) size = 2 A front back B dequeue( ) size = 1 back B front dequeue( ) size = 0 back front Basic array implementation of the queue

4 Copyright 1998 by Addison-Wesley Publishing Company 150 After 3 enqueues size = 3 enqueue(f) size = 4 back F back C D E front C D E front back dequeue( ) F D E size = 3 front back dequeue( ) F E size = 2 front back dequeue( ) F size = 1 front Array implementation of the queue with wraparound

5 Copyright 1998 by Addison-Wesley Publishing Company 151 topofstack D C B A Linked list implementation of the stack

6 Copyright 1998 by Addison-Wesley Publishing Company 152 front back A B C D Linked list implementation of the queue

7 Copyright 1998 by Addison-Wesley Publishing Company 153 back Before... back After... X enqueue operation for linked-list-based implementation

8 Copyright 1998 by Addison-Wesley Publishing Company 154 Chapter 16 Linked Lists

9 Copyright 1998 by Addison-Wesley Publishing Company 155 frontoflist A B C D Basic linked list

10 Copyright 1998 by Addison-Wesley Publishing Company 156 A B current tmp x Insertion into a linked list: create new node (tmp), copy in x, set tmp s next reference, set current s next reference

11 Copyright 1998 by Addison-Wesley Publishing Company 157 A X B current Deletion from a linked list

12 Copyright 1998 by Addison-Wesley Publishing Company 158 A B C header Using a header node for the linked list

13 Copyright 1998 by Addison-Wesley Publishing Company 159 header Empty list when header node is used

14 Copyright 1998 by Addison-Wesley Publishing Company 160 A B head tail Doubly linked list

15 Copyright 1998 by Addison-Wesley Publishing Company 161 head tail Empty doubly linked list

16 Copyright 1998 by Addison-Wesley Publishing Company A B... c a X d b Insertion into a doubly linked list by getting new node and then changing references in order indicated

17 Copyright 1998 by Addison-Wesley Publishing Company 163 A B C D first Circular doubly linked list

18 Copyright 1998 by Addison-Wesley Publishing Company 164 Chapter 17 Trees

19 Copyright 1998 by Addison-Wesley Publishing Company 165 A B C D E F G H I J K A tree

20 Copyright 1998 by Addison-Wesley Publishing Company 166 root... T 1 T 2 T 3 T k Tree viewed recursively

21 Copyright 1998 by Addison-Wesley Publishing Company 167 A B C D E F G H I J K First child/next sibling representation of tree in Figure 17.1

22 Copyright 1998 by Addison-Wesley Publishing Company 168 mark* books* courses*.login dsaa* ecp* ipps* cop3223* cop3530* ch1 ch2 ch1 ch2 ch1 ch2 syl syl Unix directory

23 Copyright 1998 by Addison-Wesley Publishing Company 169 mark books courses.login dsaa ecp ipps cop3223 cop3530 ch1 ch2 ch1 ch2 ch1 ch2 syl syl The directory listing for tree in Figure 17.4

24 Copyright 1998 by Addison-Wesley Publishing Company 170 mark* (1) books* (1) courses* (1).login (2) dsaa* (1) ecp* (1) ipps* (1) cop3223* (1) cop3530* (1) ch1 (9) ch2 (7) ch1 (4) ch2 (6) ch1 (3) ch2 (8) syl (2) syl (3) Unix directory with file sizes

25 Copyright 1998 by Addison-Wesley Publishing Company 171 ch1 9 ch2 7 dsaa 17 ch1 4 ch2 6 ecp 11 ch1 3 ch2 8 ipps 12 books 41 syl 2 cop syl 3 cop courses 8.login 2 mark 52 Trace of the size method

26 Copyright 1998 by Addison-Wesley Publishing Company a * a - d d b c b c Uses of binary trees: left is an expression tree and right is a Huffman coding tree

27 Copyright 1998 by Addison-Wesley Publishing Company 173 root X t1.root t2.root Result of a naive merge operation

28 Copyright 1998 by Addison-Wesley Publishing Company 174 root T1.root oldroot oldt1.root X T2.root Aliasing problems in the merge operation; T1 is also the current object

29 Copyright 1998 by Addison-Wesley Publishing Company 175 S L S R Recursive view used to calculate the size of a tree: S T = S L + S R +1

30 Copyright 1998 by Addison-Wesley Publishing Company 176 H L +1 H L H R H R +1 Recursive view of node height calculation: H T =max(h L +1, H R +1 )

31 Copyright 1998 by Addison-Wesley Publishing Company Preorder, postorder, and inorder visitation routes

32 Copyright 1998 by Addison-Wesley Publishing Company 178 a b c d e a 0 b 0 a 1 b 1 a 1 d 0 b 2 a 1 d 1 b 2 a 1 d 2 b 2 a 1 b 2 a 1 a 1 d b c 0 a 2 e 0 c 1 a 2 e 1 c 1 a 2 e 2 c 1 a 2 c 1 a 2 c 2 a 2 a 2 e c a Stack states during postorder traversal

33 Copyright 1998 by Addison-Wesley Publishing Company 179 Chapter 18 Binary Search Trees

34 Copyright 1998 by Addison-Wesley Publishing Company Two binary trees (only the left tree is a search tree)

35 Copyright 1998 by Addison-Wesley Publishing Company Binary search trees before and after inserting 6

36 Copyright 1998 by Addison-Wesley Publishing Company Deletion of node 5 with one child, before and after

37 Copyright 1998 by Addison-Wesley Publishing Company Deletion of node 2 with two children, before and after

38 Copyright 1998 by Addison-Wesley Publishing Company 184 X X X S L S R S L S R S L S R K < S L + 1 K == S L + 1 K > S L + 1 Using the size data field to implement findkth

39 Copyright 1998 by Addison-Wesley Publishing Company 185 Balanced tree on the left has a depth of log N; unbalanced tree on the right has a depth of N 1

40 Copyright 1998 by Addison-Wesley Publishing Company Binary search trees that can result from inserting a permutation 1, 2, and 3; the balanced tree in the middle is twice as likely as any other

41 Copyright 1998 by Addison-Wesley Publishing Company Two binary search trees: the left tree is an AVL tree, but the right tree is not (unbalanced nodes are darkened)

42 Copyright 1998 by Addison-Wesley Publishing Company 188 H H 1 S H 1 S H 2 H 2 Minimum tree of height H

43 Copyright 1998 by Addison-Wesley Publishing Company 189 k 2 k 1 k 1 k 2 A B C A B C Single rotation to fix case 1

44 Copyright 1998 by Addison-Wesley Publishing Company 190 k 1 k C A B 2 6 k A k B C Single rotation fixes AVL tree after insertion of 1

45 Copyright 1998 by Addison-Wesley Publishing Company 191 k 2 k 1 k 1 k 2 A B C A B C Symmetric single rotation to fix case 4

46 Copyright 1998 by Addison-Wesley Publishing Company 192 k 2 k 1 k 1 k 2 P Q R P Q R Single rotation does not fix case 2

47 Copyright 1998 by Addison-Wesley Publishing Company 193 k 3 k 2 k 1 k 1 k 3 k2 A B C D A B C D Left-right double rotation to fix case 2

48 Copyright 1998 by Addison-Wesley Publishing Company 194 k 1 k 3 A D k 2 B C 5 k 1 k k A B C D Double rotation fixes AVL tree after insertion of 5

49 Copyright 1998 by Addison-Wesley Publishing Company 195 k 1 k 2 k 3 k 1 k 3 k 2 A B C D A B C D Left-right double rotation to fix case 3

50 Copyright 1998 by Addison-Wesley Publishing Company 196 A red-black tree is a binary search tree with the following ordering properties: 1. Every node is colored either red or black. 2. The root is black. 3. If a node is red, its children must be black. 4. Every path from a node to a null reference must contain the same number of black nodes. Red-black tree properties

51 Copyright 1998 by Addison-Wesley Publishing Company Example of a red-black tree; insertion sequence is 10, 85, 15, 70, 20, 60, 30, 50, 65, 80, 90, 40, 5, 55)

52 Copyright 1998 by Addison-Wesley Publishing Company 198 G P P S X G X C D E A B C S A B D E If S is black, then a single rotation between the parent and grandparent, with appropriate color changes, restores property 3 if X is an outside grandchild

53 Copyright 1998 by Addison-Wesley Publishing Company 199 G X P S P G A X D E A B C S B C D E If S is black, then a double rotation involving X, the parent, and the grandparent, with appropriate color changes, restores property 3 if X is an inside grandchild

54 Copyright 1998 by Addison-Wesley Publishing Company 200 G P P S X G X C D E A B C S A B D E If S is red, then a single rotation between the parent and grandparent, with appropriate color changes, restores property 3 between X and P

55 Copyright 1998 by Addison-Wesley Publishing Company 201 X X C1 C2 C1 C2 Color flip; only if X s parent is red do we continue with a rotation

56 Copyright 1998 by Addison-Wesley Publishing Company Color flip at 50 induces a violation; because it is outside, a single rotation fixes it

57 Copyright 1998 by Addison-Wesley Publishing Company Result of single rotation that fixes violation at node 50

58 Copyright 1998 by Addison-Wesley Publishing Company Insertion of 45 as a red node

59 Copyright 1998 by Addison-Wesley Publishing Company 205 P P X T X T Deletion: X has two black children, and both of its sibling s children are black; do a color flip

60 Copyright 1998 by Addison-Wesley Publishing Company 206 P T X T P R R X Deletion: X has two black children, and the outer child of its sibling is red; do a single rotation

61 Copyright 1998 by Addison-Wesley Publishing Company 207 P R X T P T R X Deletion: X has two black children, and the inner child of its sibling is red; do a double rotation

62 Copyright 1998 by Addison-Wesley Publishing Company 208 X P T T X' P C B C B C X' B X is black and at least one child is red; if we fall through to next level and land on a red child, everything is good; if not, we rotate a sibling and parent

63 Copyright 1998 by Addison-Wesley Publishing Company 209 The level of a node is One if the node is a leaf The level of its parent, if the node is red One less than the level of its parent, if the node is black 1. Horizontal links are right pointers (because only right children may be red). 2. There may not be two consecutive horizontal links (because there cannot be consecutive red nodes). 3. Nodes at level 2 or higher must have two children. 4. If a node does not have a right horizontal link, then its two children are at the same level. AA-tree properties

64 Copyright 1998 by Addison-Wesley Publishing Company AA-tree resulting from insertion of 10, 85, 15, 70, 20, 60, 30, 50, 65, 80, 90, 40, 5, 55, 35

65 Copyright 1998 by Addison-Wesley Publishing Company 211 P X P X A B C A B C skew is a simple rotation between X and P

66 Copyright 1998 by Addison-Wesley Publishing Company 212 R X R G X G A B A B split is a simple rotation between X and R; note that R s level increases

67 Copyright 1998 by Addison-Wesley Publishing Company After inserting 45 into sample tree; consecutive horizontal links are introduced starting at After split at 35; introduces a left horizontal link at After skew at 50; introduces consecutive horizontal nodes starting at 40

68 Copyright 1998 by Addison-Wesley Publishing Company After split at 40; 50 is now on the same level as 70, thus inducing an illegal left horizontal link After skew at 70; this introduces consecutive horizontal links at After split at 30; insertion is complete

69 Copyright 1998 by Addison-Wesley Publishing Company When 1 is deleted, all nodes become level 1, introducing horizontal left links

70 Copyright 1998 by Addison-Wesley Publishing Company 216 Five-ary tree of 31 nodes has only three levels

71 Copyright 1998 by Addison-Wesley Publishing Company B-tree of order 5

72 Copyright 1998 by Addison-Wesley Publishing Company 218 A B-tree of order M is an M-ary tree with the following properties: 1. The data items are stored at leaves. 2. The nonleaf nodes store up to M 1 keys to guide the searching; key i represents the smallest key in subtree i The root is either a leaf or has between 2 and M children. 4. All nonleaf nodes (except the root) have between M 2 and M children. 5. All leaves are at the same depth and have between L 2 and L children, for some L. B-tree properties

73 Copyright 1998 by Addison-Wesley Publishing Company B-tree after insertion of 57 into tree in Figure 18.70

74 Copyright 1998 by Addison-Wesley Publishing Company Insertion of 55 in B-tree in Figure causes a split into two leaves

75 Copyright 1998 by Addison-Wesley Publishing Company Insertion of 40 in B-tree in Figure causes a split into two leaves and then a split of the parent node

76 Copyright 1998 by Addison-Wesley Publishing Company B-tree after deletion of 99 from Figure 18.73