AVL Trees 1. 2 Outline Background Define balance Maintaining balance within a tree –AVL trees...

AVL Trees

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AVL Trees

AVL Trees

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Outline

Background

Define balance

Maintaining balance within a tree– AVL trees

– Difference of heights

– Rotations to maintain balance

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Background

From previous lectures:– Binary search trees store linearly ordered data

– Best case height: (ln(n))

– Worst case height: O(n)

Requirement:– Define and maintain a balance to ensure (ln(n)) height

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Prototypical Examples

These two examples demonstrate how we can correct for imbalances: starting with this tree, add 1:

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This is more like a linked list; however, we can fix this…

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Promote 2 to the root, demote 3 to be 2’s right child, and 1 remains the left child of 2

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The result is a perfect, though trivial tree

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Alternatively, given this tree, insert 2

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Again, the product is a linked list; however, we can fix this, too

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Promote 2 to the root, and assign 1 and 3 to be its children

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The result is, again, a perfect tree

These examples may seem trivial, but they are the basis for the corrections in the next data structure we will see: AVL trees

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AVL Trees

We will focus on the first strategy: AVL trees– Named after Adelson-Velskii and Landis

Balance is defined by comparing the height of the two sub-trees of a node– An empty tree has height –1

– A tree with a single node has height 0

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AVL Trees

A binary search tree is said to be AVL balanced if:– The difference in the heights between the left and right sub-trees is at

most 1, and

– Both sub-trees are themselves AVL trees

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AVL Trees

AVL trees with 1, 2, 3, and 4 nodes:

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AVL Trees

Here is a larger AVL tree (42 nodes):

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AVL Trees

The root node is AVL-balanced:– Both sub-trees are of height 4:

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All other nodes (e.g., AF and BL) are AVL balanced– The sub-trees differ in height by at most one

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Height of an AVL Tree

By the definition of complete trees, any complete binary search tree is an AVL tree

Thus an upper bound on the number of nodes in an AVL tree of height h a perfect binary tree with 2h + 1 – 1 nodes– What is an lower bound?

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Let F(h) be the fewest number of nodes in a tree of height h

From a previous slide:F(0) = 1

F(1) = 2

F(2) = 4

Can we find F(h)?

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The worst-case AVL tree of height h would have:– A worst-case AVL tree of height h – 1 on one side,

– A worst-case AVL tree of height h – 2 on the other, and

– The root node

We get: F(h) = F(h – 1) + 1 + F(h – 2)

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This is a recurrence relation:

The solution?

11)2F()1F(

12

01

)F(

hhh

h

h

h

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Use Maple:> rsolve( {F(0) = 1, F(1) = 2,

F(h) = 1 + F(h - 1) + F(h - 2)}, F(h) );

> asympt( %, h );

3 5

10

1

2

1

2

5

2

h

1

2

3 5

10

1

2

5

2

h 2 5

2

1 5

h

5 ( )1 5

2 5

2

1 5

h

5 ( )1 51

( )3 5 5 ( )1 5 h

5 ( ) 1 5 2h1

1

5

e( )h I

( ) 5 3 5 2h

( )1 5 ( )1 5 h

1 5

2

h

c

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This is approximately

F(h) ≈ 1.8944 h

where ≈ 1.6180 is the golden ratio– That is, F(h) = (h)

Thus, we may find the maximum value of h for a given n:

log( n / 1.8944 ) = log( n ) – 1.3277

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In this example, n = 88, the worst- and best-case scenarios differ in height by only 2

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If n = 106, the bounds on h are:– The minimum height: log2( 106 ) – 1 ≈ 19

– the maximum height : log( 106 / 1.8944 ) < 28

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Maintaining Balance

To maintain AVL balance, observe that:– Inserting a node can increase the height of a tree by at most 1

– Removing a node can decrease the height of a tree by at most 1

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Maintaining Balance

Insert either 15 or 42 into this AVL tree

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Maintaining Balance

Inserting 15 does not change any heights

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Maintaining Balance

Inserting 42 changes the height of two of the sub-trees

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Maintaining Balance

To calculate changes in height, the member function must run in O(1) time

Our implementation of height is O(n):

template <typename Comp>int Binary_search_node<Comp>::height() const { return max( ( left() == 0 ) ? 0 : 1 + left()->height(), ( right() == 0 ) ? 0 : 1 + right()->height() ); }

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Maintaining Balance

Introduce a member variable

int tree_height;

The member function is now:

template <typename Comp>int BinarySearchNode<Comp>::height() const { return tree_height; }

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Maintaining Balance

Only insert and erase may change the height– This is the only place we need to update the height

– These algorithms are already recursive

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Insert

template <typename Comp>void AVL_node<Comp>::insert( const Comp & obj ) { if ( obj < element ) { if ( left() == 0 ) { left_tree = new AVL_node<Comp>( obj ); tree_height = 1; } else { left()->insert( obj ); tree_height = max( tree_height, 1 + left()->height() ); } } else if ( obj > element ) { // ... }}

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Maintaining Balance

Using the same, tree, we mark the stored heights:

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Maintaining Balance

Again, consider inserting 42– We update the heights of 38 and 44:

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Maintaining Balance

If a tree is AVL balanced, for an insertion to cause an imbalance:– The heights of the sub-trees must differ by 1

– The insertion must increase the height of the deeper sub-tree by 1

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Maintaining Balance

Starting with our sample tree

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Maintaining Balance

Inserting 23:– 4 nodes change height

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Maintaining Balance

Insert 9– Again 4 nodes change height

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Maintaining Balance

In both cases, there is one deepest node which becomes unbalanced

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Maintaining Balance

Other nodes may become unbalanced– E.g., 36 in the left tree is also unbalanced

We only have to fix the imbalance at the lowest point

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Maintaining Balance

Inserting 23– Lowest imbalance at 17

– The root is also unbalanced

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Maintaining Balance

A quick rearrangement solves all the imbalances at all levels

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Maintaining Balance

Insert 31– Lowest imbalance at 17

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Maintaining Balance

A simple rearrangement solves the imbalances at all levels

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Maintaining Balance: Case 1

Consider the following setup– Each blue triangle represents a tree of height h

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Insert a into this tree: it falls into the left subtree BL of b

– Assume BL remains balanced

– Thus, the tree rooted at b is also balanced

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However, the tree rooted at node f is now unbalanced– We will correct the imbalance at this node

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We will modify these three pointers– At this point, this references the unbalanced root node f

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Specifically, we will rotate these two nodes around the root:– Recall the first prototypical example

– Promote node b to the root and demote node f to be the right child of b

AVL_node<Comp> *pl = left(), *BR = left()->right();

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This requires the address of node f to be assigned to the right_tree member variable of node b

AVL_node<Comp> *pl = left(), *BR = left()->right();pl->right_tree = this;

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Assign any former parent of node f to the address of node b

Assign the address of the tree BR to left_tree of node f

AVL_node<Comp> *pl = left(), *BR = left()->right();pl->right_tree = this;ptr_to_this = pl;left_node = BR;

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The nodes b and f are now balanced and all remaining nodes of the subtrees are in their correct positions

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Additionally, height of the tree rooted at b equals the original height of the tree rooted at f – Thus, this insertion will no longer affect the balance of any ancestors all

the way back to the root

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Alternatively, consider the insertion of c where b < c < f into our original tree

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Assume that the insertion of c increases the height of BR

– Once again, f becomes unbalanced

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Unfortunately, the previous rotation does not fix the imbalance at the root of this subtree: the new root, b, remains unbalanced

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Unfortunately, this does not fix the imbalance at the root of this subtree

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Re-label the tree by dividing the left subtree of f into a tree rooted at d with two subtrees of height h – 1

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Now an insertion causes an imbalance at f– The addition of either c or e will cause this

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We will reassign the following pointersAVL_node<Comp> *pl = left(), *plr = left()->right(), *LRL = left()->right()->left(), *LRR = left()->right()->right();

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Specifically, we will order these three nodes as a perfect tree– Recall the second prototypical example

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To achieve this, b and f will be assigned as children of the new root dAVL_node<Comp> *pl = left(),

*plr = left()->right(), *LRL = left()->right()->left(), *LRR = left()->right()->right();plr->left_tree = pl;plr->right_tree = this;

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We also have to connect the two subtrees and original parent of fAVL_node<Comp> *pl = left(), *plr = left()->right(), *LRL = left()->right()->left(), *LRR = left()->right()->right();plr->left_tree = pl;plr->right_tree = this;ptr_to_this = lrptr;pl->right_tree = LRL;left_tree = LRR;

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Now the tree rooted at d is balanced

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Again, the height of the root did not change

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Rotations: Summary

There are two symmetric cases to those we have examined:– insertions into either the

• left-left sub-tree or the right-right sub-tree

which cause an imbalance require a single rotation to balance the node

– insertions into either the• left-right sub-tree or the right-left sub-tree

which cause an imbalance require two rotations to balance the node

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Insert (Implementation)

template <typename Comp>

void AVL_node<Comp>::insert( const Comp & obj ) {

if ( obj < element ) {

if ( left() == 0 ) {

// cannot cause an AVL imbalance

left_tree = new AVL_node<Comp>( obj );

tree_height = 1;

} else {

left()->insert( obj );

if ( left()->height() - right()->height() == 2 ) {

// determine if it is a left-left or left-right insertion

// perform the appropriate rotation

}

}

} else if ( obj > element ) {

// ...

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Insertion (Implementation)

Comments:– All the rotation statements are (1)

– An insertion requires at most 2 rotations

– All insertions are still (ln(n))

– It is possible to tighten the previous code

– Aside• if you want to explicitly rotate the nodes A and B, you must also pass a

reference to the parent pointer as an argument:

insert( Object obj, AVL_node<Comp> * & parent )

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Example Rotations

Consider the following AVL tree

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Example Rotations

• Usually an insertion does not require a rotation

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Example Rotations

• Inserting 71 cause a right-right imbalance at 61

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Example Rotations

• A single rotation balances the tree

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Example Rotations

• The insertion of 46 causes a right-left imbalance at 42

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Example Rotations

• First, we rotate outwards with 43-48

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Example Rotations

• We balance the tree by rotating 42-43

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Example Rotations

• Inserting 37 causes a left-left imbalance at 42

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Example Rotations

• A single rotation around 38-42 balances the tree

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Example Rotations

• Inserting 54 causes a left-right imbalance at 48

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Example Rotations

• We start with a rotation around 48-43

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Example Rotations

• We balance the tree by rotating 48-55

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Example Rotations

• Finally, we insert 99, causing a right-right imbalance at the root 36

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Example Rotations

• A single rotation balances the tree• Note the weight has been shifted to the left sub-tree

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Erase

Removing a node from an AVL tree may cause more than one AVL imbalance– Like insert, erase must check after it has been successfully called on a

child to see if it caused an imbalance

– Unfortunately, it may cause h/2 imbalances that must be corrected• Insertions will only cause one imbalance that must be fixed

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Erase

Consider the following AVL tree

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Erase

Suppose we erase the front node: 1

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Erase

While its previous parent, 2, is not unbalanced, its grandparent 3 is– The imbalance is in the right-right subtree

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Erase

We can correct this with a simple rotation

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Erase

The node of that subtree, 5, is now balanced

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Erase

Recursing to the root, however, 8 is also unbalanced– This is a right-left imbalance

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Erase

Promoting 11 to the root corrects the imbalance

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Erase

At this point, the node 11 is balanced

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Erase

Still, the root node is unbalanced– This is a right-right imbalance

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Erase

Again, a simple rotation fixes the imbalance

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Erase

The resulting tree is now AVL balanced– Note, few erases will require one rotation, even fewer will require more

than one

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AVL Trees as Arrays?

We previously saw that:– Complete tree can be stored using an array using (n) memory

– An arbitrary tree of n nodes requires O(2n) memory

Is it possible to store an AVL tree as an array and not require exponentially more memory?

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Recall that in the worst case, an AVL tree of n nodes has a height at most

log(n) – 1.3277

Such a tree requires an array of size

2log(n) – 1.3277 + 1 – 1

We can rewrite this as

2–0.3277 nlog(2) ≈ 0.7968 n1.44

Thus, we would require O(n1.44) memory

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While the polynomial behaviour of n1.44 is not as bad as exponential behaviour, it is still reasonably sub-optimal when compared to the linear growth associated with node-based trees

Here we see n and n1.44 on [0, 1000]

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Summary

• In this topic we have covered:– Insertions and removals are like binary search trees

– With each insertion or removal there is at most one correction required• a single rotation, or• a double rotation;

which runs in O(1) time

– Depth is O( ln(n) )

∴ all operations are O( ln(n) )

AVL Trees 1. 2 Outline Background Define balance Maintaining balance within a tree –AVL trees...

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