Data Structures Balanced Trees CSCI 2720 Spring 2007

Data Structures Balanced Trees CSCI 2720 Spring 2007

Outline q Balanced Search Trees • 2 -3 Trees • 2 -3 -4 Trees • Red-Black Trees CSCI 2720 Slide 2

Why care about advanced implementations? Same entries, different insertion sequence: Not good! Would like to keep tree balanced. CSCI 2720 Slide 3

2 -3 Trees Features Ø each internal node has either 2 or 3 children Ø all leaves are at the same level CSCI 2720 Slide 4

2 -3 Trees with Ordered Nodes 2 -node 3 -node • leaf node can be either a 2 -node or a 3 -node CSCI 2720 Slide 5

Example of 2 -3 Tree CSCI 2720 Slide 6

Traversing a 2 -3 Tree inorder(in tt. Tree: Two. Three. Tree) if(tt. Tree’s root node r is a leaf) visit the data item(s) else if(r has two data items) { inorder(left subtree of tt. Tree’s root) visit the first data item inorder(middle subtree of tt. Tree’s root) visit the second data item inorder(right subtree of tt. Tree’s root) } else { inorder(left subtree of tt. Tree’s root) visit the data item inorder(right subtree of tt. Tree’s root) } CSCI 2720 Slide 7

Searching a 2 -3 Tree retrieve. Item(in tt. Tree: Two. Three. Tree, in search. Key: Key. Type, out tree. Item: Tree. Item. Type): boolean if(search. Key is in tt. Tree’s root node r) { tree. Item = the data portion of r return true } else if(r is a leaf) return false else { return retrieve. Item( appropriate subtree, search. Key, tree. Item) } CSCI 2720 Slide 8

What did we gain? What is the time efficiency of searching for an item? CSCI 2720 Slide 9

Gain: Ease of Keeping the Tree Balanced Binary Search Tree both trees after inserting items 39, 38, . . . 32 2 -3 Tree CSCI 2720 Slide 10

Inserting Items Insert 39 CSCI 2720 Slide 11

Inserting Items Insert 38 insert in leaf divide leaf and move middle value up to parent result CSCI 2720 Slide 12

Inserting Items Insert 37 CSCI 2720 Slide 13

Inserting Items Insert 36 insert in leaf divide leaf and move middle value up to parent overcrowded node CSCI 2720 Slide 14

Inserting Items. . . still inserting 36 divide overcrowded node, move middle value up to parent, attach children to smallest and largest result CSCI 2720 Slide 15

Inserting Items After Insertion of 35, 34, 33 CSCI 2720 Slide 16

Inserting so far CSCI 2720 Slide 17

Inserting so far CSCI 2720 Slide 18

Inserting Items How do we insert 32? CSCI 2720 Slide 19

Inserting Items creating a new root if necessary tree grows at the root CSCI 2720 Slide 20

Inserting Items Final Result CSCI 2720 Slide 21

Deleting Items Delete 70 70 80 CSCI 2720 Slide 22

Deleting Items Deleting 70: swap 70 with inorder successor (80) CSCI 2720 Slide 23

Deleting Items Deleting 70: . . . get rid of 70 CSCI 2720 Slide 24

Deleting Items Result CSCI 2720 Slide 25

Deleting Items Delete 100 CSCI 2720 Slide 26

Deleting Items Deleting 100 CSCI 2720 Slide 27

Deleting Items Result CSCI 2720 Slide 28

Deleting Items Delete 80 CSCI 2720 Slide 29

Deleting Items Deleting 80. . . CSCI 2720 Slide 30

Deleting Items Deleting 80. . . CSCI 2720 Slide 31

Deleting Items Deleting 80. . . CSCI 2720 Slide 32

Deleting Items Final Result comparison with binary search tree CSCI 2720 Slide 33

Deletion Algorithm I Deleting item I: 1. Locate node n, which contains item I 2. If node n is not a leaf swap I with inorder successor deletion always begins at a leaf 3. If leaf node n contains another item, just delete item I else try to redistribute nodes from siblings (see next slide) if not possible, merge node (see next slide) CSCI 2720 Slide 34

Deletion Algorithm II Redistribution A sibling has 2 items: redistribute item between siblings and parent Merging No sibling has 2 items: merge node move item from parent to sibling CSCI 2720 Slide 35

Deletion Algorithm III Redistribution Internal node n has no item left redistribute Merging Redistribution not possible: merge node move item from parent to sibling adopt child of n If n's parent ends up without item, apply process recursively CSCI 2720 Slide 36

Deletion Algorithm IV If merging process reaches the root and root is without item delete root CSCI 2720 Slide 37

Operations of 2 -3 Trees all operations have time complexity of log n CSCI 2720 Slide 38

2 -3 -4 Trees • similar to 2 -3 trees • 4 -nodes can have 3 items and 4 children 4 -node CSCI 2720 Slide 39

2 -3 -4 Tree Example CSCI 2720 Slide 40

2 -3 -4 Tree: Insertion procedure: • similar to insertion in 2 -3 trees • items are inserted at the leafs • since a 4 -node cannot take another item, 4 -nodes are split up during insertion process Strategy • on the way from the root down to the leaf: split up all 4 -nodes "on the way" insertion can be done in one pass (remember: in 2 -3 trees, a reverse pass might be necessary) CSCI 2720 Slide 41

2 -3 -4 Tree: Insertion Inserting 60, 30, 10, 20, 50, 40, 70, 80, 15, 90, 100 CSCI 2720 Slide 42

2 -3 -4 Tree: Insertion Inserting 60, 30, 10, 20. . . 50, 40. . . CSCI 2720 Slide 43

2 -3 -4 Tree: Insertion Inserting 50, 40. . . 70, . . . CSCI 2720 Slide 44

2 -3 -4 Tree: Insertion Inserting 70. . . 80, 15. . . CSCI 2720 Slide 45

2 -3 -4 Tree: Insertion Inserting 80, 15. . . 90. . . CSCI 2720 Slide 46

2 -3 -4 Tree: Insertion Inserting 90. . . 100. . . CSCI 2720 Slide 47

2 -3 -4 Tree: Insertion Inserting 100. . . CSCI 2720 Slide 48

2 -3 -4 Tree: Insertion Procedure Splitting 4 -nodes during Insertion CSCI 2720 Slide 49

2 -3 -4 Tree: Insertion Procedure Splitting a 4 -node whose parent is a 2 -node during insertion CSCI 2720 Slide 50

2 -3 -4 Tree: Insertion Procedure Splitting a 4 -node whose parent is a 3 -node during insertion CSCI 2720 Slide 51

2 -3 -4 Tree: Deletion procedure: • similar to deletion in 2 -3 trees • items are deleted at the leafs swap item of internal node with inorder successor • note: a 2 -node leaf creates a problem Strategy (different strategies possible) • on the way from the root down to the leaf: turn 2 -nodes (except root) into 3 -nodes deletion can be done in one pass (remember: in 2 -3 trees, a reverse pass might be necessary) CSCI 2720 Slide 52

2 -3 -4 Tree: Deletion Turning a 2 -node into a 3 -node. . . Case 1: an adjacent sibling has 2 or 3 items "steal" item from sibling by rotating items and moving subtree 30 50 20 50 40 10 20 30 40 10 "rotation" 25 25 CSCI 2720 Slide 53

2 -3 -4 Tree: Deletion Turning a 2 -node into a 3 -node. . . Case 2: each adjacent sibling has only one item "steal" item from parent and merge node with sibling (note: parent has at least two items, unless it is the root) 30 50 50 40 10 10 30 40 merging 25 35 CSCI 2720 Slide 54

2 -3 -4 Tree: Deletion Practice Delete 32, 35, 40, 38, 39, 37, 60 CSCI 2720 Slide 55

Red-Black Tree • binary-search-tree representation of 2 -3 -4 tree • 3 - and 4 -nodes are represented by equivalent binary trees • red and black child pointers are used to distinguish between original 2 -nodes and 2 -nodes that represent 3 - and 4 -nodes CSCI 2720 Slide 56

Red-Black Representation of 4 -node CSCI 2720 Slide 57

Red-Black Representation of 3 -node CSCI 2720 Slide 58

Red-Black Tree Example CSCI 2720 Slide 59

Red-Black Tree Example CSCI 2720 Slide 60

Red-Black Tree Operations Traversals same as in binary search trees Insertion and Deletion analog to 2 -3 -4 tree need to split 4 -nodes need to merge 2 -nodes CSCI 2720 Slide 61

Splitting a 4 -node that is a root CSCI 2720 Slide 62

Splitting a 4 -node whose parent is a 2 -node CSCI 2720 Slide 63

Splitting a 4 -node whose parent is a 3 -node CSCI 2720 Slide 64

Splitting a 4 -node whose parent is a 3 -node CSCI 2720 Slide 65

Splitting a 4 -node whose parent is a 3 -node CSCI 2720 Slide 66