Advanced Implementations of Tables: Balanced Search Trees and - PowerPoint PPT Presentation

Advanced Implementations of Tables: Balanced Search Trees and Hashing

Balanced Search Trees � Binary search tree operations such as insert, delete, retrieve, etc. depend on the length of the path to the desired node � The path length can vary from log 2 (n+1) to O(n) depending on how balanced or unbalanced the tree is � The shape of the tree is determined by the values of the items and the order in which they were inserted Binary searching & introduction to trees CMPS 12B, UC Santa Cruz 2

Examples 10 40 20 20 60 30 40 50 70 10 30 50 Can you get the same tree with 60 different insertion orders? 70 Binary searching & introduction to trees CMPS 12B, UC Santa Cruz 3

2-3 Trees � Each internal node has two or three children � All leaves are at the same level � 2 children = 2-node, 3 children = 3-node Binary searching & introduction to trees CMPS 12B, UC Santa Cruz 4

2-3 Trees (continued) � 2-3 trees are not binary trees (duh) � A 2-3 tree of height h always has at least 2 h -1 nodes � i.e. greater than or equal to a binary tree of height h � A 2-3 tree with n nodes has height less than or equal to log 2 ( n +1) � i.e less than or equal to the height of a full binary tree with n nodes Binary searching & introduction to trees CMPS 12B, UC Santa Cruz 5

Definition of 2-3 Trees � T is a 2-3 tree of height h if � T is empty (height 0), OR � T is of the form r T L T R � Where r is a node that contains one data item and T L and T R are 2-3 trees, each of height h -1, and the search key of r is greater than any in T L and less than any in T R , OR Binary searching & introduction to trees CMPS 12B, UC Santa Cruz 6

Definition of 2-3 Trees (continued) � T is of the form r T L T M T R � Where r is a node that contains two data items and T L , T M , and T R are 2-3 trees, each of height h -1, and the smaller search key of r is greater than any in T L and less than any in T M and the larger search key in r is greater than any in T M and smaller than any in T R . Binary searching & introduction to trees CMPS 12B, UC Santa Cruz 7

Placing Data in a 2-3 tree 1. A 2-node must contain a single data item whose value is � greater than any in its left subtree, and � smaller than any in its right subtree � 2. A 3-mode must contain two data items, the smaller of which is � greater than any in its left subtree, and � smaller than any in its middle subtree, and � the greater of which is � greater than any in its middle subtree, and � smaller than any in its right subtree � 3. A leaf may contain either one or two data items Binary searching & introduction to trees CMPS 12B, UC Santa Cruz 8

2-3 Node Code import searchkeys.*; public class Tree23Node { private KeyedItem smallitem; private KeyedItem largeItem; private Tree23Node leftChild; private Tree23Node middleChild; private Tree23Node rightChild; } Binary searching & introduction to trees CMPS 12B, UC Santa Cruz 9

Traversing a 2-3 Tree � Just like a binary tree: preorder, inorder, postorder 50 90 20 70 120 150 10 30 40 60 80 100 110 130 140 160 Binary searching & introduction to trees CMPS 12B, UC Santa Cruz 10

Searching a 2-3 tree � Same efficiency as a balanced binary search tree: O(logn) � BUT, easy to keep balanced, unlike binary search trees � This means that as you insert elements, the balance is easily maintained, and worst case performance remains O(logn) Binary searching & introduction to trees CMPS 12B, UC Santa Cruz 11

Inserting 39 Into A 2-3 Tree 50 30 70 90 10 20 60 80 160 40 Binary searching & introduction to trees CMPS 12B, UC Santa Cruz 12

Inserting 38 50 30 70 90 10 20 60 80 160 39 40 Binary searching & introduction to trees CMPS 12B, UC Santa Cruz 13

Inserting 38 (continued) 50 30 70 90 10 20 60 80 160 38 39 40 Binary searching & introduction to trees CMPS 12B, UC Santa Cruz 14

Inserting 37 50 30 39 70 90 10 20 60 80 160 38 40 Binary searching & introduction to trees CMPS 12B, UC Santa Cruz 15

Inserting 36 50 30 39 70 90 10 20 60 80 160 37 38 40 Binary searching & introduction to trees CMPS 12B, UC Santa Cruz 16

Inserting 36 (continued) 50 30 39 70 90 10 20 60 80 160 36 37 38 40 Binary searching & introduction to trees CMPS 12B, UC Santa Cruz 17

Inserting 36 (continued) 50 30 37 39 70 90 36 38 40 10 20 60 80 160 Binary searching & introduction to trees CMPS 12B, UC Santa Cruz 18

Inserting 75 37 50 30 39 70 90 36 38 10 20 40 60 80 160 Binary searching & introduction to trees CMPS 12B, UC Santa Cruz 19

Inserting 77 37 50 30 39 70 90 36 38 40 60 10 20 75 80 160 Binary searching & introduction to trees CMPS 12B, UC Santa Cruz 20

Inserting 77 (continued) 37 50 30 39 70 90 36 38 40 60 10 20 75 77 80 160 Binary searching & introduction to trees CMPS 12B, UC Santa Cruz 21

Inserting Into A 2-3 Tree � Insert into the leaf node in which the search key belongs � If the leaf has two values, stop � If the leaf has three values, split the node into two nodes with the smallest and largest values, and � Push the middle value into the parent node � Continue with the parent node until either � you push a value into a node that had only one value, or � you create a new root node Binary searching & introduction to trees CMPS 12B, UC Santa Cruz 25

Deleting From A 2-3 Tree � The inverse of inserting � Delete the value (in a leaf), then � Merge empty nodes � If necessary, delete empty root Binary searching & introduction to trees CMPS 12B, UC Santa Cruz 26

Redistribute I P L S L S P Binary searching & introduction to trees CMPS 12B, UC Santa Cruz 27

Merge I L S S L Binary searching & introduction to trees CMPS 12B, UC Santa Cruz 28

Redistribute II P L S L S P � a � b � c � d � a � b � c � d Binary searching & introduction to trees CMPS 12B, UC Santa Cruz 29

Merge II L S S L � c � a � b � a � b � c Binary searching & introduction to trees CMPS 12B, UC Santa Cruz 30

Delete S L S L � a � b � c � a � b � c Binary searching & introduction to trees CMPS 12B, UC Santa Cruz 31

2-3 Trees: Results � Slightly more complicated than binary search trees, BUT � 2-3 Trees are always balanced � Every operation takes O(logn) Binary searching & introduction to trees CMPS 12B, UC Santa Cruz 32

2-3-4 Trees � Slightly less complicated than 2-3 Trees � Each node can contain 1–3 values and have 1–4 children � Inserting � Split 4-nodes on the way down � Insert into leaf � Deleting: � Only delete from 3-node or 4-node Binary searching & introduction to trees CMPS 12B, UC Santa Cruz 33

Red-Black Trees � 2-3 Trees are always balanced � O(logn) time for all operations � 2-3-4 Trees are always balanced � O(logn) time for all operations, and � Insertion and deletion can be done in a single pass from root to leaf � But, require slightly more storage per node � Red-Black Trees have the advantage of 2-3-4 trees, without the overhead � Represent the 2-3-4 Tree as a binary tree with colored references (red or black) Binary searching & introduction to trees CMPS 12B, UC Santa Cruz 34

Red-Black Representation of a 4-Node M S M L S L � a � b � c � d � a � b � c � d Binary searching & introduction to trees CMPS 12B, UC Santa Cruz 35

Red-Black Representation of a 3-Node L � c S S L � a � b � a � b � c S � a L � b � c Binary searching & introduction to trees CMPS 12B, UC Santa Cruz 36

2-3-4 Tree 37 50 30 35 70 90 39 10 20 32 33 34 36 38 40 60 80 160 Binary searching & introduction to trees CMPS 12B, UC Santa Cruz 37

Equivalent Red-Black Tree 37 30 50 20 35 39 90 10 33 36 38 40 70 100 32 34 60 80 Binary searching & introduction to trees CMPS 12B, UC Santa Cruz 38

Red-Black Tree Node public class RBTreeNode { public static final int RED = 0; public static final int BLACK = 1; private KeyedItem item; private RBTreeNode leftChild; private RBTreeNode rightChild; private int leftColor; private int rightColor; } Binary searching & introduction to trees CMPS 12B, UC Santa Cruz 39

Advanced Implementations of Tables: Balanced Search Trees and - PowerPoint PPT Presentation

Advanced Implementations of Tables: Balanced Search Trees and Hashing Balanced Search Trees Binary search tree operations such as insert, delete, retrieve, etc. depend on the length of the path to the desired node The path length can

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