COSC 2P03 Week 5 1

Representation of an AVL Node

class AVLNode{ < declarations for info stored in node, e.g. int info; >

AVLnode left; AVLnode right; int height;

int height(AVLNode T) { return T == null? -1 : T.height; } …};

COSC 2P03 Week 5 2

AVL insertAVLNode insert(AVLNode T, AVLNode newNode){ if(T == null) T = newNode; else if(newNode.info < T.info) // insert in left subtree { T.left = insert(T.left, newNode); if(height(T.left) - height(T.right) == 2) if(newNode.info < T.left.info) //left subtree of T.left T = rotateWithLeftChild(T); else //right subtree of T.left T = doubleWithLeftChild(T); } else // insert in right

subtree { T.right = insert(T.right, newNode); if(height(T.right) - height(T.left) == 2) if(newNode.info > T.right.info) // right subtree of

T.right T = rotateWithRightChild(T); else // left subtree of T.right T = doubleWithRightChild(T); } T.height = max(height(T.left), height(T.right)) + 1; return T;}

COSC 2P03 Week 5 3

Single rotation – left-left• Insertion in left subtree of k2.left caused an imbalance

at k2: need to rebalance from k2 down.

AVLNode rotateWithLeftChild(AVLNode k2){ AVLNode k1 = k2.left; k2.left = k1.right; k1.right = k2; k2.height = max(height(k2.left,

height(k2.right)) + 1; k1.height = max(height(k1.left),

k2.height) + 1; return k1;}

• Note: right-right case is symmetric to above (left↔right).

COSC 2P03 Week 5 4

Double rotation – right-left

• Insertion in right subtree of left child caused imbalance at k3: need to rebalance from k3 down.

AVLNode doubleWithLeftChild(AVLNode k3){ k3.left = rotateWithRightChild(k3.left); return rotateWithLeftChild(k3);}

• Note: left-right case is symmetric to above (left↔right).

B Trees (section 4.7 of textbook)

• Commonly used for organizing data on disk– Disk access time (time to read or write a block) dominates

cost– Very important to minimize number of disk accesses

• Some notes on terminology:– This textbook uses the name B tree, but elsewhere they are

known as B+ trees– In this textbook, the order of a B tree is the maximum

number of children per node. – Elsewhere, order refers to the minimum number of

children in index nodes other than the root.

COSC 2P03 Week 5 5

B tree definitions

A B-tree of order M is an M-ary tree such that:

1. Data items are only in the leaves

2. Non-leaf (index) nodes store up to M-1 keys: – key i determines the smallest possible key in subtree i+1.

3. The root is either a leaf or has between 2 and M children

Every node other than the root is at least half-full:

4. All non-leaf nodes (except root) have at least M/2 children

5. All leaves are at the same depth and have between L/2 and L records.

COSC 2P03 Week 5 6

B tree and Binary Search Tree comparison• Binary search trees: nodes have 0, 1 or 2 children and 1 key• B-trees: non-leaf nodes have up to M children: a node with d

keys has d+1 children• Binary search trees: data is stored in both leaf and non-leaf

nodes, and for any given index node N:– N’s left subtree contains only items with keys < N’s key– N’s right subtree contains only items with keys > N’s key

• B-trees: non-leaf nodes contain up to M-1 keys (k1, …, kM-1):

– Subtree to left of k1 contains only items with keys < k1

– Subtree between ki and ki+1 contains only items with keys <ki+1 and ≥ki

– Subtree to right of kM-1 contains only items with keys ≥kM-1COSC 2P03 Week 5 7

COSC 2P03 Week 5 8

B-tree ExampleM=5 and L=3

100 150 200

10 30 50 110 120 130 160 180 220 240 260

3 10 30 50 100 110 120 130 150 160 180 200 220 240 2607 15 35 80 105 115 125 140 155 165 190 210 225 250 270

20 90 170 230 275

COSC 2P03 Week 5 9

B-tree example: insert 40100 150 200

10 30 50 110 120 130 160 180 220 240 260

3 10 30 50 100 110 120 130 150 160 180 200 220 240 2607 15 35 80 105 115 125 140 155 165 190 210 225 250 270

20 40 90 170 230 275

COSC 2P03 Week 5 10

B-tree example: insert 70100 150 200

10 30 50 80 110 120 130 160 180 220 240 260

3 10 30 50 80 100 110 120 130 150 160 180 200 220 240 2607 15 35 70 90 105 115 125 140 155 165 190 210 225 250 270

20 40 170 230 275

COSC 2P03 Week 5 11

B-tree example: insert 2530 100 150 200

10 20 50 80 110 120130 160 180 220 240 260

3 10 20 30 50 80 100 110 120 130 150 160 180 200 220 240 2607 15 25 35 70 90 105 115 125 140 155 165 190 210 225 250 270

40 170 230 275

COSC 2P03 Week 5 12

B-tree example: insert 23530 100 150 200

10 20 50 80 110 120130 160 180 220 230 240260

3 10 20 30 50 80 100 110 120 130 150 160 180 200 220 230 240 2607 15 25 35 70 90 105 115 125 140 155 165 190 210 225 235 250 270

40 170 275

COSC 2P03 Week 5 13

B-tree example: insert 280

30 100 200 240

10 20 50 80 110120130 160180 220230 260275

3 10 20 30 50 80 100 110 120 130 150 160 180 200 220 230 240 260 2757 15 25 35 70 90 105 115 125 140 155 165 190 210 225 235 250 270 280

40 170

150

COSC 2P03 Week 5 14

Heaps

A heap is a binary tree that satisfies all of the following properties:

• Structure property: It is a complete binary tree

• Heap-order property: Every node satisfies the heap condition:– The key of every node n must be smaller than (or equal

to) the keys of its children, i.e. n.info n.left.info and n.info n.right.info