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Balanced Search Trees and Hashing Balanced Search Trees 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 log2(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 CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 2 Examples 10 40 20 20 60 30 40 10 30 50 70 50 Can you get the same tree with 60 different insertion orders? 70 CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 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 CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 4 2-3 Trees (continued) 2-3 trees are not binary trees (duh) A 2-3 tree of height h always has at least 2h-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 log2(n+1) i.e less than or equal to the height of a full binary tree with n nodes CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 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 TL TR Where r is a node that contains one data item and TL and TR are 2-3 trees, each of height h-1, and the search key of r is greater than any in TL and less than any in TR, OR CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 6 Definition of 2-3 Trees (continued) T is of the form r TL TM TR Where r is a node that contains two data items and TL, TM, and TR are 2-3 trees, each of height h-1, and the smaller search key of r is greater than any in TL and less than any in TM and the larger search key in r is greater than any in TM and smaller than any in TR. CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 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 CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 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; } CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 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 CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 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) CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 11 Inserting 39 Into A 2-3 Tree 50 30 70 90 10 20 40 60 80 160 CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 12 Inserting 38 50 30 70 90 10 20 39 40 60 80 160 CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 13 Inserting 38 (continued) 50 30 70 90 10 20 38 39 40 60 80 160 CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 14 Inserting 37 50 30 39 70 90 10 20 38 40 60 80 160 CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 15 Inserting 36 50 30 39 70 90 10 20 37 38 40 60 80 160 CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 16 Inserting 36 (continued) 50 30 39 70 90 10 20 36 37 38 40 60 80 160 CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 17 Inserting 36 (continued) 50 30 37 39 70 90 10 20 36 38 40 60 80 160 CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 18 Inserting 75 37 50 30 39 70 90 10 20 36 38 40 60 80 160 CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 19 Inserting 77 37 50 30 39 70 90 10 20 36 38 40 60 75 80 160 CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 20 Inserting 77 (continued) 37 50 30 39 70 90 10 20 36 38 40 60 75 77 80 160 CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 21 Inserting 77 (continued) 37 50 30 39 70 77 90 10 20 36 38 40 60 75 80 160 CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 22 Inserting 77 (continued) 37 50 77 30 39 70 90 10 20 36 38 40 60 75 80 160 CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 23 Inserting 77 (continued) 50 37 77 30 39 70 90 10 20 36 38 40 60 75 80 160 CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 24 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 CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 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 CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 26 Redistribute I P L S L S P CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 27 Merge I L S S L CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 28 Redistribute II P L S L S P a b c d a b c d CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 29 Merge II L S S L c a b a b c CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 30 Delete S L S L a b c a b c CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 31 2-3 Trees: Results Slightly more complicated than binary search trees, BUT 2-3 Trees are always balanced Every operation takes O(logn) CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 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 CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 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) CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 34 Red-Black Representation of a 4-Node M S M L S L a b c d a b c d CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 35 Red-Black Representation of a 3-Node L S c S L a b a b c S a L b c CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 36 2-3-4 Tree 37 50 30 35 39 70 90 10 20 32 33 34 36 38 40 60 80 160 CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 37 Equivalent Red-Black Tree 37 30 50 20 35 39 90 10 33 36 38 40 70 100 32 34 60 80 CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 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; } CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 39 Searching and Traversing Red-Black Trees Red-Black trees are binary search trees Just search them the same way you would any other binary search tree Inserting Split 4-nodes on the way down by changing paired red child references to black Insert into a leaf CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 40 Splitting a 4-node root M M S L S L a b c d a b c d CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 41 Splitting a 4-node P P M e M e S L S L a b c d a b c d CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 42 Splitting a 4-node P P a a M M S L S L b c d e b c d e CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 43 Splitting a 4-node P P M Q M Q e f S L e f S L a b c d a b c d CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 44 Splitting a 4-node Q P P f M Q M e S L e f S L a b c d a b c d CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 45 Splitting a 4-node P M a Q P Q M f a S L f S L b c d e b c d e CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 46 AVL Trees Insert in the appropriate spot Rotate as necessary to restore balance Rotate your way back up the tree Every operation is O(logn) CMPS 12B, UC Santa Cruz Binary searching & introduction to trees 47 Hashing Trees allow for efficient table operations, but What if you want O(1) behavior? What if time is much more critical than space? Basic idea: Array Same for insert, lookup, delete index Search Hash Number Key Function in 0..n-1 “Hash Table”
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