Self Balancing Trees Data Structures and Algorithms CSE 373 SP 18 - - PowerPoint PPT Presentation

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Sep 27, 2023 212 likes •431 views

Self Balancing Trees Data Structures and Algorithms CSE 373 SP 18 - KASEY CHAMPION 1 Warm Up What will the binary search tree look like if you insert nodes in the following order: 5, 8, 7, 10, 9, 4, 2, 3, 1 What is the pre-order traversal

SLIDE 1

Self Balancing Trees

Data Structures and Algorithms

CSE 373 SP 18 - KASEY CHAMPION 1

SLIDE 2

Warm Up

What will the binary search tree look like if you insert nodes in the following order: 5, 8, 7, 10, 9, 4, 2, 3, 1 What is the pre-order traversal order for the resulting tree?

Socrative:

www.socrative.com Room Name: CSE373 Please enter your name as: Last, First

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SLIDE 3

Implement Dictionary

Binary Search Trees allow us to:

quickly find what we’re looking for
add and remove values easily

Dictionary Operations: Runtime in terms of height, “h” get() – O(h) put() – O(h) remove() – O(h)

What do you replace the node with? Largest in left sub tree or smallest in right sub tree

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10 “foo” 7 “bar” 12 “baz” 9 “sho” 5 “fo” 15 “sup” 13 “boo” 8 “poo” 1 “burp”

SLIDE 4

Implementing Put and Remove

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SLIDE 5

Height in terms of Nodes

For “balanced” trees h ≈ logc(n) where c is the maximum number of children Balanced binary trees h ≈ log2(n) Balanced trinary tree h ≈ log3(n) Thus for balanced trees operations take Θ(logc(n))

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SLIDE 6

Unbalanced Trees

Is this a valid Binary Search Tree? Yes, but… We call this a degenerat enerate e tree ee For trees, depending on how balanced they are, Operations at worst can be O(n) and at best can be O(logn) How are degenerate trees formed?

insert(10)
insert(9)
insert(7)
insert(5)

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10 9 7 5

SLIDE 7

Measuring Balance

Measuring balance: For each node, compare the heights of its two sub trees Balanced when the difference in height between sub trees is no greater than 1

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10 8 15 7 12 18 8 7 7 8 7 9

SLIDE 8

Meet AVL Trees

AVL Trees es must satisfy the following properties:

binary trees: all nodes must have between 0 and 2 children
binary search tree: for all nodes, all keys in the left subtree must be smaller and all keys in the right subtree

must be larger than the root node

balanced: for all nodes, there can be no more than a difference of 1 in the height of the left subtree from the
right. Math.abs(height(left subtree) – height(right subtree)) ≤ 1

AVL stands for Adelson-Velsky and Landis (the inventors of the data structure)

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SLIDE 9

Is this a valid AVL tree?

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7 4 10 3 9 12 5 8 11 13 14 2 6 Is it…

Binary
BST
Balanced?

yes yes yes

SLIDE 10

Is this a valid AVL tree?

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6 2 8 1 7 12 4 9 10 13 11 3 5 Is it…

Binary
BST
Balanced?

yes yes no Height = 2 Height = 0

SLIDE 11

Is this a valid AVL tree?

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8 6 11 2 15 7

9 Is it…

Binary
BST
Balanced?

yes no yes 9 > 8 5

SLIDE 12

Implementing an AVL tree dictionary

Dictionary Operations: get() – same as BST containsKey() – same as BST put() - ??? remove() - ???

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Add the node to keep BST, fix AVL property if necessary Replace the node to keep BST, fix AVL property if necessary

1 2 3 Unbalanced! 2 1 3

SLIDE 13

Rotations!

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a b X c Y Z a b X Y Z c a b X Y Z Insert ‘c’ Unbalanced! Balanced!

SLIDE 14

Rotations!

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a b X c Y Z a b X Y Z c a b X Y Z Insert ‘c’ Unbalanced! Balanced!

SLIDE 15

Practice

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15 8 22 4 24 10 3 19 6 20 17 put(16); 16

SLIDE 16

Practice

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15 8 22 4 24 10 3 19 6 20 17 put(16); 16

SLIDE 17

So much can go wrong

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1 3 2 Unbalanced! 3 1 2 Rotate Left Unbalanced! Rotate Right 1 3 2 Unbalanced!

SLIDE 18

Two AVL Cases

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1 3 2 1 2 3 Line Case Solve with 1 rotation Kink Case Solve with 2 rotations 3 2 1

Rotate Right Parent’s left becomes child’s right Child’s right becomes its parent

Rotate Left Parent’s right becomes child’s left Child’s left becomes its parent 3 1 2 Rotate subtree left Rotate root tree right Rotate subtree right Rotate root tree left

SLIDE 19

Double Rotations 1

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a e W d Y Z a e X X Z Insert ‘c’ Unbalanced! d X Y c a d W Y Z X e c

SLIDE 20

Double Rotations 2

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a d W Y Z X e c Unbalanced! a d W Y Z X e c