Data Structures and Algorithms Course’s slides: Hierarchical data structures algis.

Data Structuresand

Algorithms

Course’s slides: Hierarchical data structures

www.mif.vu.lt/~algis

Linear access time of linked lists is prohibitive

Does there exist any simple data structure for which the running time of most operations (search, insert, delete) is O(log N)?

A tree is a collection of nodes

The collection can be empty

(recursive definition) If not empty, a tree consists of a distinguished node r (the root), and zero or more nonempty subtrees T1, T2, ...., Tk, each of whose roots are connected by a directed edge from r

Some Terminologies

Child and parent Every node except the root has one parent A node can have an arbitrary number of children

Leaves Nodes with no children

Sibling nodes with same parent

Some Terminologies

Path Length

number of edges on the path

Depth of a node length of the unique path from the root to that node The depth of a tree is equal to the depth of the deepest

Height of a node length of the longest path from that node to a leaf all leaves are at height 0 The height of a tree is equal to the height of the root

Ancestor and descendant Proper ancestor and proper descendant

Example: UNIX Directory

Binary Trees A tree in which no node can have more

than two children

The depth of an “average” binary tree is considerably smaller than N, eventhough in the worst case, the depth can be as large as N – 1.

Example: Expression Trees

Leaves are operands (constants or variables) The other nodes (internal nodes) contain operators Will not be a binary tree if some operators are not binary

Tree traversal

Used to print out the data in a tree in a certain order

Pre-order traversal

Print the data at the root

Recursively print out all data in the left subtree

Recursively print out all data in the right subtree

Preorder, Postorder and Inorder

Preorder traversal

node, left, right prefix expression

++a*bc*+*defg

Postorder traversal

left, right, node postfix expression

abc*+de*f+g*+ Inorder traversal

left, node, right. infix expression

a+b*c+d*e+f*g

Binary Trees

Possible operations on the Binary Tree ADTparent

left_child, right_child

sibling

root, etc

ImplementationBecause a binary tree has at most two children, we can keep direct pointers to them

Compare: Implementation of a general tree

Binary Search TreesStores keys in the nodes in a way so that searching, insertion and deletion can be done efficiently.Binary search tree property

For every node X, all the keys in its left subtree are smaller than the key value in X, and all the keys in its right subtree are larger than the key value in X

Binary Search Trees

A binary search treeNot a binary search tree

Binary search trees

Average depth of a node is O(log N); maximum depth of a node is O(N)

Two binary search trees representing the same set:

Searching BST

If we are searching for 15, then we are done. If we are searching for a key < 15, then we should

search in the left subtree. If we are searching for a key > 15, then we should

search in the right subtree.

Inorder traversal of BST

Print out all the keys in sorted order

Inorder: 2, 3, 4, 6, 7, 9, 13, 15, 17, 18, 20

findMin/findMax

Return the node containing the smallest element in the tree

Start at the root and go left as long as there is a left child. The stopping point is the smallest element

Similarly for findMax

Time complexity = O(height of the tree)

Insert

Proceed down the tree as you would with a find

If X is found, do nothing (or update something)

Otherwise, insert X at the last spot on the path traversed

Delete

When we delete a node, we need to consider how we take care of the children of the deleted node.

This has to be done such that the property of the search tree is maintained.

Delete

Three cases:(1) the node is a leaf

Delete it immediately(2) the node has one child

Adjust a pointer from the parent to bypass that node

Delete

(3) the node has 2 children replace the key of that node with the

minimum element at the right subtree delete the minimum element

Has either no child or only right child because if it has a left child, that left child would be smaller and would have been chosen. So invoke case 1 or 2

AVL Trees - Lecture 8 25 12/26/03

Binary search tree – best time

All BST operations are O(d), where d is tree depth

minimum d is for a binary tree with N nodes

What is the best case tree? What is the worst case tree?

So, best case running time of BST operations is O(log N)

Nlogd 2

Binary Search Tree - Worst Time

Worst case running time is O(N)

What happens when you insert elements in ascending order? Insert: 2, 4, 6, 8, 10, 12 into an

empty BST Problem: Lack of “balance”:

compare depths of left and right subtree

Unbalanced degenerate tree

Balanced and unbalanced BST

5 71 3

Is this “balanced”?

Approaches to balancing trees

Don't balance

May end up with some nodes very deep

Strict balance

The tree must always be balanced perfectly

Pretty good balance

Only allow a little out of balance

Adjust on access

Self-adjusting

Balancing binary search trees

Many algorithms exist for keeping binary search trees balanced

Adelson-Velskii and Landis (AVL) trees (height-balanced trees)

Splay trees and other self-adjusting trees

B-trees and other multiway search trees

Perfect balance

Want a complete tree after every operation

tree is full except possibly in the lower right

This is expensive

For example, insert 2 in the tree on the left and then rebuild as a complete tree

Insert 2 &complete tree

6 91 4

AVL - good but not perfect balance

AVL trees are height-balanced binary search trees

Balance factor of a node

height(left subtree) - height(right subtree) An AVL tree has balance factor calculated at

every node

For every node, heights of left and right subtree can differ by no more than 1

Store current heights in each node

Height of an AVL tree

N(h) = minimum number of nodes in an AVL tree of height h.

N(0) = 1, N(1) = 2 Induction

N(h) = N(h-1) + N(h-2) + 1 Solution (recall Fibonacci analysis)

N(h) > h ( 1.62)h-1

Height of an AVL Tree

N(h) > h ( 1.62)

Suppose we have n nodes in an AVL tree of height h.

n > N(h) (because N(h) was the minimum)

n > h hence log n > h (relatively well balanced tree!!)

h < 1.44 log2n (i.e., Find takes O(logn))

Node Heights

height of node = hbalance factor = hleft-hright

empty height = -1

height=2 BF=1-0=1

Tree A (AVL) Tree B (AVL)

Node heights after insert 7

height of node = hbalance factor = hleft-hright

empty height = -1

balance factor 1-(-1) = 2

Tree A (AVL) Tree B (not AVL)

Insert and rotation in AVL trees

Insert operation may cause balance factor to become 2 or –2 for some node

only nodes on the path from insertion point to root node have possibly changed in height

So after the Insert, go back up to the root node by node, updating heights

If a new balance factor (the difference hleft-hright) is 2 or –2, adjust tree by rotation around the node

Single Rotation in an AVL Tree

Let the node that needs rebalancing be .

There are 4 cases: Outside Cases (require single rotation) : 1. Insertion into left subtree of left child of . 2. Insertion into right subtree of right child of . Inside Cases (require double rotation) : 3. Insertion into right subtree of left child of . 4. Insertion into left subtree of right child of .

The rebalancing is performed through four separate rotation algorithms.

Insertions in AVL trees

Consider a validAVL subtree

AVL insertion: outside case

Inserting into Xdestroys the AVL property at node j

AVL Insertion: Outside Case

Do a “right rotation”

AVL Insertion: Outside Case

Do a “right rotation”

Single right rotation

“Right rotation” done!(“Left rotation” is mirror symmetric)

Outside Case Completed

AVL property has been restored!

AVL Insertion: Inside Case

Consider a validAVL subtree

Inserting into Y destroys theAVL propertyat node j

Does “right rotation”restore balance?

“Right rotation”does not restorebalance… now k isout of balance

Consider the structureof subtree Y… j

Y = node i andsubtrees V and W

h or h-1

We will do a left-right “double rotation” . . .

Double rotation : first rotation

left rotation complete

Double rotation : second rotation

Now do a right rotation

X V ZW

Double rotation : second rotation

right rotation complete

Balance has been restored

hh h or h-1

Implementation

balance (1,0,-1)

rightleft

No need to keep the height; just the difference in height, i.e. the balance factor; this has to be modified on the path of insertion even if you don’t perform rotations

Once you have performed a rotation (single or double) you won’t need to go back up the tree

Single Rotation

RotateFromRight(n : reference node pointer) {p : node pointer;p := n.right;n.right := p.left;p.left := n;n := p}

You also need to modify the heights or balance factors of n and p

Insert

Double Rotation

Implement Double Rotation in two lines.

DoubleRotateFromRight(n : reference node pointer) {????}

Insertion in AVL Trees

Insert at the leaf (as for all BST)

only nodes on the path from insertion point to root node have possibly changed in height

So after the Insert, go back up to the root node by node, updating heights

If a new balance factor (the difference hleft-hright) is 2 or –2, adjust tree by rotation around the node

Insert in BST

Insert(T : reference tree pointer, x : element) : integer {if T = null then T := new tree; T.data := x; return 1;//the links to //children are nullcase T.data = x : return 0; //Duplicate do nothing T.data > x : return Insert(T.left, x); T.data < x : return Insert(T.right, x);endcase}

Insert in AVL trees

Insert(T : reference tree pointer, x : element) : {if T = null then {T := new tree; T.data := x; height := 0; return;}case T.data = x : return ; //Duplicate do nothing T.data > x : Insert(T.left, x); if ((height(T.left)- height(T.right)) = 2){ if (T.left.data > x ) then //outside case T = RotatefromLeft (T); else //inside case T = DoubleRotatefromLeft (T);} T.data < x : Insert(T.right, x); code similar to the left caseEndcase T.height := max(height(T.left),height(T.right)) +1; return;}

Example of Insertions in an AVL Tree

Insert 5, 40

Example of Insertions in an AVL Tree

Now Insert 45

Single rotation (outside case)

Imbalance35 45

Now Insert 34

Double rotation (inside case)

Imbalance

Insertion of 34

1 25 340

AVL Tree Deletion

Similar but more complex than insertion

Rotations and double rotations needed to rebalance

Imbalance may propagate upward so that many rotations may be needed.

Arguments for AVL trees:

1. Search is O(log N) since AVL trees are always balanced.2. Insertion and deletions are also O(logn)3. The height balancing adds no more than a constant factor to the

speed of insertion.

Arguments against using AVL trees:4. Difficult to program & debug; more space for balance factor.5. Asymptotically faster but rebalancing costs time.6. Most large searches are done in database systems on disk and use

other structures (e.g. B-trees).7. May be OK to have O(N) for a single operation if total run time for

many consecutive operations is fast (e.g. Splay trees).

Pros and Cons of AVL Trees

Double Rotation Solution

DoubleRotateFromRight(n : reference node pointer) {RotateFromLeft(n.right);RotateFromRight(n);}

Outline

Balanced Search Trees• 2-3 Trees• 2-3-4 Trees• Red-Black Trees

Why care about advanced implementations?

Same entries, different insertion sequence:

Not good! Would like to keep tree balanced.

2-3 Trees each internal node has either 2 or 3 children all leaves are at the same level

Features

2-3 Trees with Ordered Nodes2-node 3-node

• leaf node can be either a 2-node or a 3-node

Example of 2-3 Tree

Traversing a 2-3 Treeinorder(in ttTree: TwoThreeTree)

if(ttTree’s root node r is a leaf)visit the data item(s)

else if(r has two data items){

inorder(left subtree of ttTree’s root)visit the first data iteminorder(middle subtree of ttTree’s root)visit the second data iteminorder(right subtree of ttTree’s root)

}else{

inorder(left subtree of ttTree’s root)visit the data iteminorder(right subtree of ttTree’s root)

Searching a 2-3 treeretrieveItem(in ttTree: TwoThreeTree,

in searchKey:KeyType,out treeItem:TreeItemType):boolean

if(searchKey is in ttTree’s root node r){

treeItem = the data portion of rreturn true

}else if(r is a leaf)

return falseelse{

return retrieveItem( appropriate subtree,searchKey, treeItem)

What did we gain?

What is the time efficiency of searching for an item?

Gain: Ease of Keeping the Tree Balanced

Binary SearchTree

2-3 Tree

both trees afterinserting items39, 38, ... 32

Inserting ItemsInsert 39

insert in leafdivide leaf

and move middlevalue up to parent

result

insert in leaf

divide leafand move middlevalue up to parent

overcrowdednode

Inserting Items... still inserting 36

divide overcrowded node,move middle value up to parent,

attach children to smallest and largest

result

Inserting ItemsAfter Insertion of 35, 34, 33

Inserting so far

Inserting ItemsHow do we insert 32?

Inserting Items creating a new root if necessary tree grows at the root

Inserting ItemsFinal Result

Deleting ItemsDelete 70

Deleting ItemsDeleting 70: swap 70 with inorder successor (80)

Deleting ItemsDeleting 70: ... get rid of 70

Deleting ItemsResult

Deleting ItemsDeleting 100

Deleting ItemsResult

Deleting ItemsDeleting 80 ...

Deleting ItemsFinal Result

comparison withbinary search tree

Deletion Algorithm 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 Ielse

try to redistribute nodes from siblings (see next slide)if not possible, merge node (see next slide)

Deleting item I:

Deletion Algorithm II

A sibling has 2 items: redistribute item

between siblings andparent

No sibling has 2 items: merge node move item from parent

to sibling

Redistribution

Merging

Deletion Algorithm III

Internal node n has no item left redistribute

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

Redistribution

Merging

Deletion Algorithm IVIf merging process reaches the root and root is without item delete root

Operations of 2-3 Trees

all operations have time complexity of log n

2-3-4 Trees• similar to 2-3 trees• 4-nodes can have 3 items and 4 children

4-node

2-3-4 Tree example

2-3-4 Tree: InsertionInsertion 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)

2-3-4 Tree: InsertionInserting 60, 30, 10, 20, 50, 40, 70, 80, 15, 90, 100

2-3-4 Tree: InsertionInserting 60, 30, 10, 20 ...

... 50, 40 ...

2-3-4 Tree: InsertionInserting 50, 40 ...

... 70, ...

2-3-4 Tree: InsertionInserting 70 ...

... 80, 15 ...

2-3-4 Tree: InsertionInserting 80, 15 ...

... 90 ...

... 100 ...

2-3-4 Tree: Insertion Procedure

Splitting 4-nodes during Insertion

Splitting a 4-node whose parent is a 2-node during insertion

Splitting a 4-node whose parent is a 3-node during insertion

2-3-4 Tree: DeletionDeletion 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)

2-3-4 Tree: DeletionTurning 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

10 20 40

10 30 40

"rotation"

2-3-4 Tree: DeletionTurning 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)

merging10 30 40

2-3-4 Tree: Deletion Practice

Delete 32, 35, 40, 38, 39, 37, 60

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 betweenoriginal 2-nodes and 2-nodes that represent 3- and 4-nodes

Red-Black Representation of 4-node

Red-Black Representation of 3-node

Red-Black Tree Example

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

Splitting a 4-node that is a root

Splitting a 4-node whose parent is a 2-node

Splitting a 4-node whose parent is a 3-node

Motivation for B-Trees

So far we have assumed that we can store an entire data structure in main memory

What if we have so much data that it won’t fit?

We will have to use disk storage but when this happens our time complexity fails

The problem is that Big-Oh analysis assumes that all operations take roughly equal time

This is not the case when disk access is involved

Motivation (cont.)

Assume that a disk spins at 3600 RPM

In 1 minute it makes 3600 revolutions, hence one revolution occurs in 1/60 of a second, or 16.7ms

On average what we want is half way round this disk – it will take 8ms

This sounds good until you realize that we get 120 disk accesses a second – the same time as 25 million instructions

In other words, one disk access takes about the same time as 200,000 instructions

It is worth executing lots of instructions to avoid a disk access

Motivation (cont.)

Assume that we use an Binary tree to store all the details of people in Canada (about 32 million records)

We still end up with a very deep tree with lots of different disk accesses; log2 20,000,000 is about 25, so this takes about 0.21 seconds (if there is only one user of the program)

We know we can’t improve on the log n for a binary tree

But, the solution is to use more branches and thus less height!

As branching increases, depth decreases

Definition of a B-tree

A B-tree of order m is an m-way tree (i.e., a tree where each node may have up to m children) in which:

1. the number of keys in each non-leaf node is one less than the number of its children and these keys partition the keys in the children in the fashion of a search tree

2. all leaves are on the same level

3. all non-leaf nodes except the root have at least m / 2 children

4. the root is either a leaf node, or it has from two to m children

5. a leaf node contains no more than m – 1 keys

The number m should always be odd

An example B-Tree

51 6242

55 60 7064 9045

1 2 4 7 8 13 15 18 25

27 29 46 48 53

A B-tree of order 5 containing 26 items

Note that all the leaves are at the same level

Suppose we start with an empty B-tree and keys arrive in the following order:1 12 8 2 25 6 14 28 17 7 52 16 48 68 3 26 29 53 55 45

We want to construct a B-tree of order 5

The first four items go into the root:

To put the fifth item in the root would violate condition 5

Therefore, when 25 arrives, pick the middle key to make a new root

Constructing a B-tree

1281 2

Constructing a B-tree

Add 25 to the tree

1 12 8 2 25 6 14 28 17 7 52 16 48 68 3 26 29 53 55 45

1281 2 25

Exceeds Order. Promote middle and split.

Constructing a B-tree (contd.)

6, 14, 28 get added to the leaf nodes:

1 12 8 2 25 6 14 28 17 7 52 16 48 68 3 26 29 53 55 45

1 2 25

1 2 2561 2 2814

Adding 17 to the right leaf node would over-fill it, so we take the middle key, promote it (to the root) and split the leaf

1 12 8 2 25 6 14 28 17 7 52 16 48 68 3 26 29 53 55 45

2 2561 2 2814 2817

7, 52, 16, 48 get added to the leaf nodes

1 12 8 2 25 6 14 28 17 7 52 16 48 68 3 26 29 53 55 45

2561 2 2814

7 5216 48

Adding 68 causes us to split the right most leaf, promoting 48 to the root

1 12 8 2 25 6 14 28 17 7 52 16 48 68 3 26 29 53 55 45

7621 161412 52482825 68

Adding 3 causes us to split the left most leaf

1 12 8 2 25 6 14 28 17 7 52 16 48 68 3 26 29 53 55 45

7621 161412 25 28 52 683 7

1 12 8 2 25 6 14 28 17 7 52 16 48 68 3 26 29 53 55 45

Add 26, 29, 53, 55 then go into the leaves

481783

1 2 6 7 52 6825 28161412 26 29 53 55

Add 45 increases the trees level

1 12 8 2 25 6 14 28 17 7 52 16 48 68 3 26 29 53 55 45

481783

29282625 685553521614126 71 2 45

Exceeds Order. Promote middle and split.

Inserting into a B-Tree

Attempt to insert the new key into a leaf

If this would result in that leaf becoming too big, split the leaf into two, promoting the middle key to the leaf’s parent

If this would result in the parent becoming too big, split the parent into two, promoting the middle key

This strategy might have to be repeated all the way to the top

If necessary, the root is split in two and the middle key is promoted to a new root, making the tree one level higher

Exercise in Inserting a B-Tree

Insert the following keys to a 5-way B-tree:

3, 7, 9, 23, 45, 1, 5, 14, 25, 24, 13, 11, 8, 19, 4, 31, 35, 56

Answer to Exercise

Java Applet Source

Removal from a B-tree

During insertion, the key always goes into a leaf. For deletion we wish to remove from a leaf. There are three possible ways we can do this:

1 - If the key is already in a leaf node, and removing it doesn’t cause that leaf node to have too few keys, then simply remove the key to be deleted.

2 - If the key is not in a leaf then it is guaranteed (by the nature of a B-tree) that its predecessor or successor will be in a leaf -- in this case can we delete the key and promote the predecessor or successor key to the non-leaf deleted key’s position.

Removal from a B-tree (2)

If (1) or (2) lead to a leaf node containing less than the minimum number of keys then we have to look at the siblings immediately adjacent to the leaf in question:

3: if one of them has more than the min’ number of keys then we can promote one of its keys to the parent and take the parent key into our lacking leaf

4: if neither of them has more than the min’ number of keys then the lacking leaf and one of its neighbours can be combined with their shared parent (the opposite of promoting a key) and the new leaf will have the correct number of keys; if this step leave the parent with too few keys then we repeat the process up to the root itself, if required

Type #1: Simple leaf deletion

12 29 52

2 7 9 15 22 56 69 7231 43

Delete 2: Since there are enoughkeys in the node, just delete it

Assuming a 5-wayB-Tree, as before...

Note when printed: this slide is animated

Type #2: Simple non-leaf deletion

12 29 52

7 9 15 22 56 69 7231 43

Delete 52

Borrow the predecessoror (in this case) successor

Type #4: Too few keys in node and its siblings

12 29 56

7 9 15 22 69 7231 43

Delete 72Too few keys!

Join back together

Type #4: Too few keys in node and its siblings

7 9 15 22 695631 43

Type #3: Enough siblings

7 9 15 22 695631 43

Delete 22

Demote root key andpromote leaf key

Type #3: Enough siblings

297 9 15

695643

Exercise in Removal from a B-Tree

Given 5-way B-tree created by these data (last exercise):

3, 7, 9, 23, 45, 1, 5, 14, 25, 24, 13, 11, 8, 19, 4, 31, 35, 56

Add these further keys: 2, 6,12

Delete these keys: 4, 5, 7, 3, 14

Answer to Exercise

Java Applet Source

Analysis of B-Trees The maximum number of items in a B-tree of order

m and height h:

root m – 1level 1 m(m – 1)level 2 m2(m – 1). . .level h mh(m – 1)

So, the total number of items is(1 + m + m2 + m3 + … + mh)(m – 1) =[(mh+1 – 1)/ (m – 1)] (m – 1) = mh+1 – 1

When m = 5 and h = 2 this gives 53 – 1 = 124

Reasons for using B-Trees

When searching tables held on disc, the cost of each disc transfer is high but doesn't depend much on the amount of data transferred, especially if consecutive items are transferred

If we use a B-tree of order 101, say, we can transfer each node in one disc read operation

A B-tree of order 101 and height 3 can hold 1014 – 1 items (approximately 100 million) and any item can be accessed with 3 disc reads (assuming we hold the root in memory)

If we take m = 3, we get a 2-3 tree, in which non-leaf nodes have two or three children (i.e., one or two keys)

B-Trees are always balanced (since the leaves are all at the same level), so 2-3 trees make a good type of balanced tree

Data Structures and Algorithms Course’s slides: Hierarchical data structures algis.

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