IT Data Structures Chapter 3

7/25/2019 IT Data Structures Chapter 3

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dvanced Data Structures

Chapter III:

Data Structures


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Contents

PointersTrees

- Binary Trees

- Traversals- Binary Search Trees

- AVL Trees

- M-way Search Trees


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Static vs. Dynamic Allocation

Static Allocation:- Binds memory space to variables at compile time.- Deals with variables that are created statically (compile time).

Examples:

i nt i , j ;

char s[ 10] ;

f l oat k;

doubl e d;


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Static vs. Dynamic Allocation

Dynamic Allocation:- Binds memory spaces to variables during run-time.- Deals with data structures whose length is only determinedduring run-time.

- Deals with variables that are created and disposed duringexecution.

Example: Pointers


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Pointer Variable

A variable used only to store the memory address ofanother variable.

Used to access dynamic variables

Value of a pointer variable-address


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Pointer Variable

Example:

Charptr at addr Dynamic Char Var at

07770 Address 0122222

Charptr at addr Dynamic Char Var at

07770 Address 0122222

0122222 A

A


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Pointer Variable

Declaration:

Syntax: *

where:

is any user defined name. is any simple C data type

Examples:

i nt *i ;f l oat *f ;

char *c;


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Pointer Variable

Another way:

Syntax:

typedef *

typedef Example:

t ypedef i nt *poi nt er ;

mai n( ){ poi nt er i ;

.

.

}


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Pointer Variable

Operations:

Function malloc()

- syntax: malloc()- uses stdlib.h

- the function call returns a pointer to enough

storage for an object of bytes. The storageis not initialized.


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Pointer Variable

Example:

mai n( )

{

i nt *p;

.

.

p = mal l oc( si zeof ( i nt ) ) ;

}


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Pointer Variable

The execution of malloc() would obtain a piece ofmemory (dynamic variable or node) from the systemadequate to store an integer value and assigns itsaddress to the pointer p.

Accessing a dynamic variable:

Syntax: *

the thing pointed to by


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Pointer Variable

Example:

p = mal l oc( si zeof ( i nt ) ) ;

*p = 10;

pr i nt f ( %d, *p) ;

Gar bage

10

p

p


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Pointer Variable

Function free() Destroy or de-allocate a dynamic variable to some other

data.

Making the allocated storage for the object pointed to by the

available once again to the system. Syntax: free()

Example:

f r ee( p) ;


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Pointer Variable

Example:

10p *p = 10

pGar bage

Af t er execut i ng f r ee( p)

nul lp P = NULL


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Dynamic Data Structures

We do not use pointers and dynamic variables bythemselves. Like structures that can be grouped tobecome array of records, pointers and dynamicvariables are normally grouped into something called

asLink List.

Basic link list singly linked list


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Singly Linked List

A sequence of dynamic variables or nodes that arelinked together

Declaration:

st r uct

{

;

.

.

;

}

Note: one of the fields must be a pointer type that would point to astructure that is of the same type as the one that contains it.


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Singly Linked List

Example:

st r uct node

{

i nt dat a;

st r uct node *l i nk;

}

dat a l i nk


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Singly Linked List

Note:

The linked list terminates with a node whose link field value isNULL

NULL is a predefined constant pointer which means, pointingto nothing.

1 22 30

f i r s t


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Singly Linked List

Declaration:#i ncl ude

st r uct node

{

i nt dat a;st r uct node *l i nk;

};

t ypedef st r uct node nodet ype;

t ypedef nodet ype *poi nt er ;


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Singly Linked List

voi d cr eat e_si ngl e( poi nt er *f i r st )

{ i nt num; poi nt er x, p;*f i r st = NULL;

do { scanf ( %d, &num) ;

i f ( num ! = - 1)

{ x = mal l oc( si zeof ( nodet ype) ) ;

x - > dat a = num;

x - > l i nk = NULL;

i f ( *f i r st = NULL) *f i r st = x;

el se p - > l i nk = x;

p = x;}

}whi l e( num ! = - 1) ;

}

mai n( )

{ poi nt er f i r st ;

cr eat e_si ngl e( &f i r st ) ; }


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Trees

Finite set of one or more nodes such that:

There is a specially designated node called root

The remaining nodes are partitioned into n>=0disjoints sets T1, T2, Tn, where each set is a tree

called subtrees of the root.


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Trees

Terminology:

Parent: immediate root of a node

Siblings: children of the same parent

Ancestors of a node: all nodes along the path from the rootto that node

Descendants of a node: all nodes of the subtrees of a node

Degree of a node: number of subtrees of a node

Root node: no parents or ancestors Leaf/terminal node: no children or descendants node with a

degree = 0

Nonleaf/nonterminal node: has at least one child or

descendant node with degree 0


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Trees

Terminology:

Degree of a tree = max[degree of nodes]

Level/generation of a node: 1 + length of its path from the

root Height/depth of a tree: total number of levels

Weight of a tree: number of leaf nodes

Forest: set of disjoint trees


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Binary Trees

Finite set of nodes which is either empty or

consisting of a root with at most two branches todisjoint binary trees called the left subtree and theright subtree.

A

B

A

B


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Binary Trees

2i-1 : maximum number of nodes on level l of a binary

tree.

2k-1: maximum number of nodes in a binary tree ofheigh k.

Skewed binary tree: tree that either have 0 leftsubtrees or 0 right subtrees.

Full binary tree of depth k: a binary tree of depth kwith 2k-1 nodes.


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Tree Representations

Using Arrays

- static/sequential representation

0

1 2

3 4 5 6

913

Array size needed will be 2k-1, k is the height of a given tree


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Tree Representations

Using linked lists

1

12 15

5 8 25


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Binary Search Tree

A binary tree:

All identifiers in the left subtree are less than theidentifier in the root.

All identifiers in the right subtree are greater thanthe identifier in the root.

The left and right subtrees are also binary searchtrees.


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Binary Search Tree

Examples:

10

8 15

13 204

K

B Q

E

D J


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Binary Search Tree

Algorithm: Insertion

If the tree is empty, then insert the new item in thetrees root node.

If the roots key matches the new items key thenskip insertion.

If the new items key is smaller than the roots key,then insert the new item in the roots left subtree.

Otherwise insert the new item in the roots rightsubtree.


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Binary Search Tree

Algorithm: Searching

If the tree is empty, then the target key is not inthe tree.

If the target key is found in the root item then thetarget key is found in the root item.

If the target key is smaller than the roots key thensearch the left subtree, otherwise search the right

subtree.


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Tree Traversal

Traversals:

Preorder: Data Left Tree Right Tree

Inorder Left Tree- Data Right Tree

Postorder Left Tree Right Tree - Data


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Tree Traversal

Give the preorder, inorder and postorder traversals:Tree #1 Tree #2

C E G

F

DB

A

D

F H Z

GE

B C

A


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Counting Trees

Create the corresponding counting trees with the

following traversals:1. Pre: IAMHEDBCFL

Post: EHDMALFCBI

In: AHEMDICFLB2. Pre: ABDGCEHIF

In: DGBAHEICF

3. Post: CBFEGDAIn: CBAEFDG

4. Post: FABG/+CD - ^*

In: F/AGB*+^C-D


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AVL Trees

Introduced by Adelson-Velskii and Landis in 1962

Height-balanced binary search trees

Binary search trees in which the heights of thesubtrees of each node differ by at most one level.

Insertion of deletion of nodes causes the search treeto be rebalanced.


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AVL Trees

Insertion

1. Insert the node at the proper point.2. Starting from the point of insertion, take the first node whose

subtree heights differ by more than one level as A.

3.

If the new node is inserted in the left subtree of A, take theleft child of A as B; if inserted in the right subtree of A, takethe right child of A as B.

4. If the new node is inserted in the left subtree of B, take the left

child of B as C; if inserted in the right subtree of B, take theright child of B as C.

5. Take the inorder traversal of A, B, and C.

6. Take the middle node as the parent of the two other nodes.

7. Adjust other nodes if necessary.


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AVL Trees

Deletion

1. Delete the node.2. If the deleted node is a non-terminal node, the immediate

predecessor (immediate successor) replaces the deleted node.

3. Starting from the deepest level, take the first node whose

subtree heights differ by more than one level as A.4. Take the right child of A as B if it has more descendants than

the left child; otherwise, take the left child of A as B.

5. Take the right child of B as C if it has more descendants than

the left child; otherwise, take the left child of B as C. If equalprioritize LL over LR or RR over RL.

6. Take the inorder traversal of A, B, and C.

7. Take the middle node as the parent of the two other nodes.

8. Adjust other nodes.

9. Rebalance if needed.


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AVL Trees

Examples: Create AVL trees with the following insertion and

deletion operations. Each number is independent of the othernumbers.

1. Ins: M, T, O, Q, S, R, W, V, Y, U, C, G, E, X

2. Ins: 10, 20, 30, 40, 50, 60, 703. Ins: 13, 20, 15, 17, 19, 18, 23, 22, 25, 21, 8, 5, 6, 4, 3, 2

Del: 13, 22, 19, 6, 18

4. Ins: 15, 20, 25, 10, 30, 35, 23, 28, 27, 22, 29, 33

Del: 10

Ins: 24, 21, 26

Del: 22, 23, 21, 28, 27, 20, 25, 26


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AVL Trees14

20

7

19

6721181542

351793

5

1161

Del: 35, 14, 5, 11, 15, 17, 20Ins: 14, 10, 13, 28

Del: 1, 67, 9Ins: 35, 15, 20


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M-Way Search Trees

#keys values in a given file = # of nodes that will be

needed by an AVL tree 1M records 1M nodes are needed

Solution: To lessen the number of nodes, is byallowing the nodes to store more than one key value.This leads to shorter in height and smaller number ofcomparisons.

M-way or Multiway Search Trees: A kind of searchtree in which each node can store 1 to M-1 values 1 to

M links to subtrees.


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M-Way Search Trees

Example:

50

60 9030

70 80 12055 5610


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M-Way Search Trees

Types:

B-Tree Bayer, McCreight and Kaufman in 1972

Balanced M-Way search tree in which all leaf nodes are at

the same level Properties

An interior or nonleaf node with N values should contain N+1links

Root node contains at least 1 value and at most M-1 values

All non-root contain at least M/2 - 1 values and at most M-1values


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M-Way Search Trees

B-Tree:

Insertion:1. Insert the value in the leaf node where it should belong.

2. If the number of values in any node exceeds the maximum,

that is, the node overflows, the middle value is extracted andis moved to the parent node (if not existing, create one with anew label) and among the remaining values, the lower halfwill remain in the node and the upper half will be stored in a

new node with a new label. In case there are two middlevalues, extract the smaller value. Labeling new nodes is donefrom left to right from the lowest level going up.

3. Split the other nodes if necessary


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M-Way Search Trees

B-Tree:

Deletion:1. Delete the value.

2. If deletion was made from a nonterminal node, the immediate

predecessor replaces the deleted value. The immediatepredecessor of a node is the largest value in the nodes leftsubtree while the immediate successor of a node is thesmallest value in the nodes right subtree.

3. For each node that underflows, that is, a node where thenumber of values is less than the minimum do:

If the nearest right sibling has more than the minimum numberof values then


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M-Way Search Trees

a. Move the parent value into the underflowing node and movethe smallest value in the nearest right sibling into the parentnode.

b. Adjust other nodes if necessary

Else

if the nearest left sibling has more than the minimum number ofvalues then,a. Move the parent value into the underflowing node and movethe largest value in the nearest left sibling into the parent node.

b. Adjust other nodes if necessary

Elsea. Merge the values of the underflowing node, the values of thenearest right sibling (if non-existent, nearest left sibling), andtheir parent node into the leftmost node.

b. Adjust node if necessary


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M-Way Search Trees

B+Tree

Similar to B tree except that the interior or nonleaf nodescontain only keys and links; and leaf nodes which contain

keys and addresses are linked in key sequence to facilitaterapid sequential accessing

Properties similar to that of B Tree.


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M-Way Search Trees

B+ Tree:

Insertion:1. Insert the value in a leaf node where it should belong.

2. If the number of values in a leaf node overflows, the M/2 th

value is replicated in the parent node and among theremaining values, the firstM/2 values will remain in thenode while the remaining nodes will be stored in a new nodewith a new label.

3. If the number of values in a nonleaf overflows, splitting isdone in the sam way that is done with B-Trees.

4. Split the other nodes if necessary.


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M-Way Search Trees

B+ Tree:

Deletion:1. Delete the value from the leaf node.

2. Delete the value also from the nonterminals where it existsand replace it with a copy of the immediate predecessor.

3. For each underflowing node

if nonterminal then

follow the procedure for B-Trees

Elseif the nearest right sibling has more than the minimum then

a. move the smallest value in the nearest right sibling into theunderflowing node.

b. Update nonleaf nodes

c. Adjust other nodes if necessary


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M-Way Search Trees

Else

if the nearest left sibling has more than the minimum number ofvalues then

a. Move the largest value in the nearest left sibling into theunderflowing node

b. Update the nonleaf nodesc. Adjust otherother nodes

Else

a. Merge the values of the underflowing node with the values of

its nearest right sibling (if non-existent, nearest left sibling)maintaining the leftmost node label.

b. Update nonleaf nodes


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M-Way Search Trees

B*Tree Similar to B-Trees except that every node in a B* tree is at

least 2/3 full rather than half-full.

Idea is to reduce the frequency of splitting

When a node overflows, its nearest siblings are checked for

possible accommodation of an overflow value. If not possible, rather than it being split, a redistribution of

values takes place

Properties: Interior or nonleaf node with N values contains N+1 links The root node contains at least 1 value and at most 2(2M-2)/3

values

All non-root nodes contain at least (2M-1)/3 - 1 values and at

most M-1 values.


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M-Way Search Trees

B* Tree:

Insertion:1. Insert the value in the leaf node where it should belong.

2. If the root node overflows, splitting is done in the same way

as in B-Trees.3. For each non-root that overflows do

if the nearest left sibling is not full then

a. Move the parent value into the nearest left sibling andmove the smallest value in the overflowing node into theparent node

b. Adjust nodes if necessary

Else


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M-Way Search Trees

Else

if the nearest right sibling is not full thena. Move the parent value into the nearest right sibling andmove the largest value in the overflowing node into theparent node.

b. Adjust nodes if necessary.

Else

a. Take the values of the overflowing node and of its left

(right, if not existing) sibling and their parent value anddistribute then into three nodes with (2M-2)/3, (2M-1)/3 , and (2M)/3 values respectively. Old labels willremain with the leftmost nodes.

b. Adjust nodes if necessary.


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M-Way Search Trees

B* Tree:

Deletion:1. Delete the value

2. If deletion was made from a nonterminal node, then theimmediate predecessor replaces the deleted value.

3. For each underflowing node do:

check the nearest two siblings (nearest right sibling first) for avalue that can be acquired

if there exists such a value then

a. Rotate values as necessary

b. Adjust nodes if necessary

ElseMerge the values of the underflowing node and of its sibling and their

parent value into the leftmost node.

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IT Data Structures Chapter 3

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