M3_Non Linear Data Structures

Resmi N.G.

References:

Data Structures and Algorithms:

Alfred V. Aho, John E. Hopcroft, Jeffrey D. Ullman

A Practical Approach to Data Structures and Algorithms: Sanjay Pahuja

Syllabus� Non Linear Structures - Graphs - Trees - Graph and Tree

implementation using array and Linked List -Binary trees

- Binary tree traversals - pre-order, in-order and postorder

- Threaded binary trees – Binary Search trees - AVL trees -

B trees and B+ trees - Graph traversals - DFS, BFS

shortest path - Dijkstra’s algorithm, Minimum spanning

tree - Kruskal Algorithm, Prim’s algorithm

10/25/2012 2CS 09 303 Data Structures - Module 3

Trees

10/25/2012 CS 09 303 Data Structures - Module 3 3

Trees� A tree is a collection of nodes, one of which is designated

as the root, along with a relation (parenthood).

� Represents hierarchical relationship.

� A node can be of any type.


Trees

� A tree can be recursively defined as :

� A single node by itself is a tree. This node is called the root ofthe tree.

� A tree is a finite set of one or more nodes such that there is aspecially designated node called the root and the remainingnodes are partitioned into n>=0 disjoint sets T1, T2, …,Tn,where each of these sets is a subtree.

� Every node except the root has a parent node associated to it.


TREE TERMINOLGIES

� Root :Specially designed first node

� Degree of a node: Number of subtrees of a node

� Degree of a Tree :Maximum degree of any node in a given

tree.

� Terminal node/Leaf node: Node with degree zero.


� Non terminal node : Any node (except root) whosedegree is non-zero.

� Siblings: Children nodes of same parent node.

� Level : If a node is at level n, then it’s children will beat level n+1. Root is at level 0, next immediate islevel 1 etc..

� Path: If n1,n2,…nk be the node sequence of a tree suchthat ni is the parent of ni+1 for 1<=i<k, then thissequence is called a path from node n1 to node nk.


� Length of path :One less than the number of nodes in a

path.

� Ancestor/descendant of a node: If there is a path from

node a to node b, then a is an ancestor of b, and b is

descendant of a.

� Proper Ancestor/Proper Descendant : An ancestor or

descendant of node ,other than the node itself is called

proper ancestor or proper descendant respectively.


� Height of a node :Length of the longest path from the

node to a leaf.

� Height of a tree: Height of the root.

� Depth of a node : Length of the unique path from root

to that node.

� Depth of a tree :One more than the maximum level of

any node in a given tree.


Order of nodes� Children of a node are usually ordered from left-to-right.

� These are two distinct ordered trees.


a

b c

a

c b

� If a and b are siblings, and a is to the left of b, then all the

descendants of a are to the left of all descendants of b.

� A tree in which the order of nodes is ignored is referred to

as unordered tree.


Tree Traversal (Orderings)� Preorder

� Inorder

� Postorder

� These orderings are recursively defined as:

� If a tree t is null (with no nodes), then the empty list is the

preorder, inorder and postorder listing of T.

� If T consists of a single node, then that node itself is the

preorder, inorder and postorder listing of T.


� Otherwise, let T be a tree with root n and subtrees T1, T2,

…, Tk.

� …

� The preorder listing of the nodes of T is the root n of T

followed by the nodes of T1 in preorder, then the nodes of

T2 in preorder, and so on, upto the nodes of Tk in

preorder.


n

T1 T2 Tk

� The inorder listing of the nodes of T is the nodes of T1 in

inorder, followed by the root n of T , followed by the

nodes of T2, …, Tk in inorder.

� The postorder listing of the nodes of T is the nodes of T1

in postorder, then the nodes of T2 in postorder, and so on,

upto Tk, all followed by node n.


Preorder Procedure

� Procedure PREORDER (n : node)

� begin

� (1) list n;

� (2) for each child c of n, if any, in order from the left do

� PREORDER(c)

� end;



a

b c

d e f g

a


a

b c

d e f g

a b


a

b c

d e f g

a b d


a

b c

d e f g

a b d e


a

b c

d e f g

a b d e c


a

b c

d e f g

a b d e c f


a

b c

d e f g

a b d e c f g

� A, B, D, H, I, E, C, F, J, G, K


Postorder Procedure

� Procedure POSTORDER (n : node)

� begin

� (1) for each child c of n, if any, in order from the left do

� POSTORDER(c)

� (2) list n;

� end;



b c

d e f g

d

a


b c

d e f g

d e

a


b c

d e f g

d e b

a


b c

d e f g

a

d e b f


b c

d e f g

a

d e b f g


b c

d e f g

a

d e b f g c


b c

d e f g

a

d e b f g c a

� H, I, D, E, B, J, F, K, G, C, A


Inorder Procedure� Procedure INORDER (n : node)

� begin

� if n is a leaf then

� list n;

� else begin

� INORDER (leftmost child of n);

� list n;

� for each child c of n, except for the leftmost, in

� order from the left do

� INORDER(c)

� end

� end;



b c

d e f g

d

a


b c

d e f g

a

d b


b c

d e f g

a

d b e


b c

d e f g

a

d b e a


b c

d e f g

a

d b e a f


b c

d e f g

a

d b e a f c


b c

d e f g

a

d b e a f c g

� H, D, I, B, E, A, J, F, C, K, G


Labeled Trees and Expression Trees� Every leaf is labeled by an operand and consists of that

operand alone.

� Every interior node n is labeled by an operator.

� Suppose, n is labeled by a binary operator, #, and that the

left child represents expression E1 and the right child E2.

Then, n represents (E1) # (E2).



+ +

a b a

*

n1 = (a + b) * (a + c)

c

n3 = (a + c)n2 = (a + b)

n4 = (a)

n5 = (b) n6 = (a)

n7 = (c)

� The preorder listing of labels in an expression tree gives

the prefix form of the expression, where the operator

precedes its left and right operands.

� The prefix expression for (E1)#(E2), with # a binary

operator, is #P1P2 , where P1 and P2 are the prefix

expressions for E1 and E2 respectively.


� The postorder listing of labels in an expression tree gives

the postfix form of the expression, where left and right

operands precede the operator.

� The postfix expression for (E1)#(E2), with # a binary

operator, is P1P2#, where P1 and P2 are the postfix

expressions for E1 and E2 respectively.

� The inorder listing of labels in an expression tree gives

the infix expression.


The ADT Tree Operations� PARENT (n, T) : Returns the parent of node n in tree T.

� If n is the root, it returns NULL.

� LEFTMOST_CHILD (n, T) : Returns the leftmost child of node

n in tree T.

� It returns NULL if n is a leaf.

� RIGHT_SIBLING (n, T) : Returns the right sibling of node n in

tree T, defined to be that node m with same parent p as n such

that m lies immediately to the right of n in the ordering of

children of p.


� LABEL (n, T) : Returns the label of node n in tree T.

� CREATEi (v, T1, T2, …, Ti) : For each value of i = 0, 1,

2, … CREATEi makes a new node r with label v and gives

it i children, which are the roots of trees T1, T2, …, Ti, in

order from the left. The tree with root r is returned.

� ROOT (T) : Returns the node that is the root of tree T, or

returns NULL T is null tree.

� MAKENULL (T) : makes T the null tree.


IMPLEMENTATION OF TREE� Array Representation

� Linked List representation


IMPLEMENTATION OF TREES� Array Representation (Parent Representation)

Uses the property of trees that each node has a unique

parent.

�Uses a linear array A where A[i] =j, if node j is parent of

node i, and A[i]=0,if node i is the root.

�It supports LABEL operator, where L be an array with

L[i], the Label of node i.


� With this representation, the parent of a node can be found

in constant time.

� Limitations:

� Lacks child-of information.

� Given a node n, it is expensive to determine the children

of n, or the height of n.

� Parent pointer representation does not specify the order of

the children of a node.


� Definition

type

node = integer;

TREE = array[1..MAXNODES] of node;


� Right Sibling Operation

function RIGHT_SIBLING (n:node; T:Tree): node;

{returns right sibling of node n in tree T}

Var i,parent:node ;

begin

parent:=T[n];

for i = n+1 to maxnodes do

{search for node after n with same parent}

if T[i] = parent then

return(i);

return(0);{null node will be returned if no right sibling is ever found}

End;(RIGHT_SIBLING}


Linked List representation Of Tree� Way of representing trees where a list of children is

formed for each node.

� Header : An array of header cells, indexed by nodes.

� Each header points to a linked list of nodes.

� Elements on the list headed by header[i] are the childrenof node i.


� Definition

� Type

� node = integer;

� LIST = {appropriate definition for list of nodes};

� position = {appropriate definition for positions in

lists};

� TREE = record

� header : array[1..maxnodes] of LIST;

� labels : array[1..maxnodes] of

labeltype;

� root : node;

� End;


� LEFTMOST-CHILD Operation

function LEFTMOST_CHILD (n:node ;T:Tree): node;

{returns the leftmost child of node n of tree T}

Var L : LIST {list of n’s children}

begin

L:= T.header[n];

if EMPTY(L) then {n is a leaf}

return(0)

else

return (RETRIEVE(FIRST(L),L);

End;{LEFTMOST_CHILD}


Definition of cellspace

For PARENT Operation

T.header[n] points directly to the first cell of the list.

Var

cellspace :array[1..maxnodes] of record

node :integer;

next :integer;

end;


� PARENT Operation

function PARENT(n:node ;T:Tree): node;

{returns the parent of node n of tree T}

Var p: node; {runs through possible parents of n}

i: position; {runs down the list of p’s children}

begin

for p:=1 to maxnodes do begin

i := T.header[p];

while i <> 0 do { see if n is among children of p}

if cellspace[i].node = n then

return(p)


else

i:= cellspace[i].next;

end;

return(0); {returns null node if parent not found}

end;{PARENT}


� Shortcoming

� Inability to create large trees from smaller trees using

CREATEi operator.



� Definition Of Node Space

Var

nodespace :array[1..maxnodes] of record

label:labeltype;

header: integer;

{cursor to cellspace}

end;

Leftmost-Child, Right-Sibling

Representation of a Tree

� Definition Of Cell Space

Var

cellspace :array[1..maxnodes] of record

label:labeltype;

leftmost_child :integer;

right_sibling: integer;

end;


Binary Trees� Definition

A binary tree is a tree data structure in which each node

has at most two child nodes, usually distinguished as "left

child" and "right child".

A binary tree can be defined as :

(1) either an empty tree, or

(2) a tree in which every node has either no children, a left

child, a right child, or both a left and a right child.



1

2

3 4

5

1

2

3 4

5

Two Distinct Binary Trees

1

2

3 4

5

An Ordinary Tree

� Left-Skewed: If a binary tree has only left subtree, it is

called left-skewed.

� Right-Skewed: If a binary tree has only right subtree, it is

called right-skewed.


� In a binary tree a degree of every node is maximum two.

� Binary tree with n nodes has exactly (n-1) edges.

� A full binary tree is a tree in which every node other than

the leaves has two children.

� A complete binary tree is a binary tree in which every

level, except possibly the last, is completely filled, and all

nodes are as far left as possible.


Complete Binary Tree


Full Binary Tree


� A full binary tree of depth k has 2k -1 nodes, k>=0.

Binary Tree Representations

� Array Representation

�Array stores nodes

�Nodes accessed sequentially

�Root node index starts with 1.

�SIZE of array=(2^d)-1 where d=depth of tree


Binary Tree Representations

� Pointer Based Implementation

�Binary Tree linked list declaration as

type

node = record

leftchild : node;

rightchild: node;

parent : node;

end;


Binary Trees: Pointer Based

Implementation� CREATION Algorithm

Function create (lefttree, righttree: node): node;

Var root : node;

begin

new ( root);

root .leftchild := lefttree;

root .rightchild := righttree;

root .parent:=0;

lefttree .parent := root;

righttree .parent:=root;

return (root)

end;{create}


� INSERTION Algorithm

Function insert(root: node,digit:number): node;

Var root : node;

begin

If root = NULL then begin

new(root);

root .leftchild:=NULL;

root .rightchild:=NULL;

root .data := digit;

count := count+1;

end;{if}


else if count %2 = 0 then begin

root .leftchild :=insert(root .leftchild, digit);

else

root .rightchild := insert(root .rightchild, digit);

end;{elseif}

return(root);

end;{insert}


Binary Trees Operations

� TRAVERSAL

�Pre-order(Node-left-right)

�In-Order(Left-Node-Right)

�Post-Order(Left-Right-Node)


� Recursive TRAVERSAL

�Pre-order Algorithm Steps:

(1) Visit the root node

(2) Traverse the left subtree in pre-order

(3) Traverse the right subtree in pre-order



�Pre-order Algorithm

procedure preorder (root : node)

begin

if root <> NULL begin

write (root .data);

preorder(root .lchild);

preorder(root .rchild);

end{if}

end;{preorder}



�In-order Algorithm Steps:

(1) Traverse the left subtree in inorder


(3) Traverse the right subtree in inorder



�Inorder Algorithm

procedure inorder(root : node)

begin


inorder(root .lchild);

write (root .data);

inorder(root .rchild);

end{if}

end;{inorder}



�Postorder Algorithm Steps:

(1) Traverse the left subtree in postorder

(2) Traverse the right subtree in postorder




�Postorder Algorithm

procedure postorder(root : node)

begin


postorder(root .lchild);

postorder(root .rchild);

write (root .data);

end{if}

end;{preorder}


Binary Tree Vs General Tree

�Binary Tree : May be empty

Tree : cannot be empty

�Binary Tree : Exactly 2 subtrees

Tree : Any number of subtrees

�Binary Tree : Ordered

Tree : Unordered


Binary Search Tree


Binary Search Tree Operations

� Inserting a node

• Searching a node

• Deleting a node

�


BST INSERTION


� procedure INSERT (x: elementtype, var A: Set));

� {add x to set A}

� Begin

� if A = NIL then begin

� new (A);

� A . element := x;

� A . leftchild := NIL;

� A . rightchild := NIL;

� end

� else if x < A . element then

� INSERT (x, A .leftchild);

� else if x > A . element then

� INSERT (x, A .rightchild);

� {if x = A . element then, do nothing; x is already in

the set}

� End; {INSERT}


BST SEARCH


� function MEMBER (x: elementtype, var A : SET) : boolean;

� {returns true if x is in A, false otherwise}

� Begin

� if A = NIL then

� return (false)

� else if x = A .element then

� return (true)

� else if x < A .element then

� return (MEMBER (x, A .leftchild))

� else {x > A .element}

� return (MEMBER (x, A .rightchild))

� End; {MEMBER}

BST DELETION

� (1) Deleting leaf node

� - Just delete the leaf node

� (2) Deleting a node with a single child (either a left

child or a right child)

� - Replace the node with its left (or right) child.

� (3) Deleting a node with both left and right child

� - Replace the node to be deleted with its inorder

successor (with smallest value in its right subtree) and

then delete the node.


BST DELETION


� Procedure DELETE (x: elementtype, var A : SET);

� {remove x from set A}

� Begin

� if A <> NIL then

� if x < A .element then

� DELETE (x, A .leftchild)

� else if x > A .element then

� DELETE(x, A .rightchild))

� {if we reach here, x is at the node pointed to

by A}


� else if A .leftchild=NIL and A .rightchild=NIL then

� A := NIL; {delete the leaf holding x}

� else if A .leftchild=NIL then {A has only right child}

� A := A .rightchild;

� else if A .rightchild=NIL then {A has only left child}

� A := A .leftchild;

� else {both children are present}

� A .element := DELETEMIN (A .rightchild);

� End; {DELETE}


� Function DELETEMIN (var A : SET) : elementtype;

� {returns and removes the smallest element from set A}

� Begin

� if A .leftchild = NIL then begin

� {A points to the smallest element}

� DELETEMIN := A .element;

� A := A .rightchild);

� {replace the node pointed to by A by its right child}

� end

� else {the node pointed to by A has a left child}

DELETEMIN := DELETEMIN (A .leftchild);

� End; {DELETEMIN}


(1) Deleting leaf node with value 13


(2) Deleting node with single (right)child (with value 16)

Replace the node with its right child (with value 20).


Replace the node with its inorder successor and delete the node.

(3) Deleting node with left and right child (with value 5)

BST Traversal

� Inorder : 3 5 6 7 10 12 13 15 16 18 20 23 (sorted in

ascending order)

� Preorder : 15 5 3 12 10 6 7 13 16 20 18 23

� Postorder : 3 7 6 10 13 12 5 18 23 20 16 15


Deleting a node with two children from

a BST

� Rule 1: Find the largest node of left subtree (inorder

predecessor).

� Rule 2: Find the smallest node of right subtree(inorder

successor).

� A node's in-order successor is the left-most child of its

right subtree, and a node's in-order predecessor is the

right-most child of its left subtree.



Rule 2: Find the smallest node of right subtree.

Rule 1: Find the largest node of left subtree.

Node to be deleted : node with value 7.

The triangles represent subtrees of arbitrary size.

Binary Search trees Vs Arrays� Advantages:

� Complexity of searching: O (log2N)

� Better insertion time: O (log2N) Vs O(N)

� Better deletion time

� Disadvantage:

� BST requires more memory space to store the two pointer

references to left and right child for each data element.


Application of BST� Sorting:

� We can sort the data by reading it, item by item, andconstructing a BST as we go.

� The inorder traversal of a BST gives the elements in ascendingorder.

� A sorted array can be produced from a BST by traversing thetree in inorder and inserting each element sequentially into thearray as it is visited.

� Time Complexity -?????


Threaded Trees� Binary trees have a lot of wasted space: each of the leaf

nodes has 2 null pointers.

� We can use these pointers to help us in inorder traversals.

� We have the pointers that reference the next node in aninorder traversal; called threads.

� To know whether a pointer is an actual link or a thread, aboolean variable can be maintained for each pointer.


Threaded Tree Example

8

75

3

11

13

1

6

9


Threaded Binary Trees� A binary tree is threaded according to a particular traversal

order.

� A binary tree is threaded by making all right child

pointers that would normally be null point to the inorder

successor of the node, and all left child pointers that

would normally be null point to the inorder predecessor of

the node.


� Types:

� Single Threaded: each node is threaded towards either the

inorder predecessor or successor.

� Double threaded: each node is threaded towards both the

inorder predecessor and successor.



Threads are references to the predecessors and successors of the node

according to an inorder traversal. Inorder of the threaded tree is

ABCDEFGHI.


Inorder: DBAEC

Threaded Tree Traversal� We start at the leftmost node in the tree, print it, and

follow its right thread.

� If we follow a thread to the right, we output the node and

continue to its right.

� If we follow a link to the right, we go to the leftmost node,

print it, and continue.


117

Threaded Tree Traversal

8

75

3

11

13

1

6

9

Start at leftmost node, print it

Output1

10/25/2012 CS 09 303 Data Structures - Module 3

118


8

75

3

11

13

1

6

9

Follow thread to right, print node

Output13


119


8

75

3

11

13

1

6

9

Follow link to right, go to leftmost node and print

Output135


120


8

75

3

111

6

9


Output1356


13

121


8

75

3

1

6

9


Output13567


11

13

122


8

75

3

11

13

1

6

9


Output135678


123


8

75

3

1

6

9


Output1356789


11

13

124


8

75

3

11

13

1

6

9


Output135678911


125


8

75

3

1

6

9


Output13567891113


11

13

Advantages

� The traversal operation is faster than that of its unthreadedversion, because with threaded binary tree non-recursiveimplementation is possible which can run faster.

� We can efficiently determine the predecessor and successornodes starting from any node (no stack required).

� Any node is accessible from any other node. Threads areusually upward whereas links are downward. Thus, in athreaded tree, one can move in either direction and nodes are infact circularly linked.


Limitations of BSTs� BSTs can become highly unbalanced (In worst case, time

complexity for BST operations become O(n)).


root

A

C

F

M

Z

Height-balanced Binary Search

Trees� A self-balancing (or height-balanced) binary search tree

is any binary search tree that automatically keeps itsheight (number of levels below the root) small afterarbitrary item insertions and deletions.

� A balanced tree is a BST whose every node above the lastlevel has non-empty left and right subtree.

� Number of nodes in a complete binary tree of height h is2h+1 – 1. Hence, a binary tree of n elements is balanced if:

2h – 1 < n <= 2h+1 - 1


AVL Trees� Named after Russian Mathematicians: G.M. Adelson-

Velskii and E.M. Landis who discovered them in 1962.

� An AVL tree is a binary search tree which has the

following properties:

� The sub-trees of every node differ in height by at most

one.

� Every sub-tree is an AVL tree.



� Implementations of AVL tree insertion rely on adding an

extra attribute, the balance factor to each node.

� Balance factor (bf) = HL - HR

� An empty binary tree is an AVL tree.

� A non-empty binary tree T is an AVL tree iff :

� | HL - HR | <= 1

� For an AVL tree, the balance factor, HL - HR , of a node

can be either 0, 1 or -1.

� The balance factor indicates whether the tree is:

� left-heavy (the height of the left sub-tree is 1 greater than

the right sub-tree),

� balanced (both sub-trees are of the same height) or

� right-heavy (the height of the right sub-tree is 1 greater

than the left sub-tree).



+1

0

00

0

0

+1

00

-1

2

-1

+10

0

AVL TreeAVL Tree

0 Not an AVL Tree

� Insertion is similar to that of a BST.

� After inserting a node, it is necessary to check each of the

node's ancestors for consistency with the rules of AVL.

� If after inserting an element, the balance of any tree is

destroyed then a rotation is performed to restore the

balance.


AVL Tree Insertion


� For each node checked, if the balance factor remains −1,

0, or +1 then no rotations are necessary.

� However, if balance factor becomes less than -1 or greater

than +1, the subtree rooted at this node is unbalanced.

� Theorem: When an AVL tree becomes unbalanced after an

insertion, exactly one single or double rotation is required

to balance the tree.

� Let A be the root of the unbalanced subtree.

� There are four cases which need to be considered.

� Left-Left (LL) rotation

� Right-Right (RR) rotation

� Left-Right (LR) rotation

� Right-Left (RL) rotation


� LL Rotation: Inserted node is in the left subtree of left

subtree of node A.

� RR Rotation: Inserted node is in the right subtree of right

subtree of node A.

� LR Rotation: Inserted node is in the right subtree of left

subtree of node A.

� RL Rotation: Inserted node is in the left subtree of right

subtree of node A.


� LL Rotation: New element 2 is inserted in the left subtree ofleft subtree of A , whose bf becomes +2 after insertion.

� To rebalance the tree, it is rotated so as to allow B to be the

root with BL and A to be its left subtree and right child

respectively, and BR and AR to be the left and right subtrees ofA.


6

4

A

B

0

+1

AR

BRBL

6

4

A

B

+2

AR

BRBL 2

+1

0


6

4

B

ARBR

BL 2

0

0 A0

A.Leftchild = B.rightchild

B.Rightchild = A

� RR Rotation: New element 10 is inserted in the right subtree ofright subtree of A , whose bf becomes -2 after insertion.

� To rebalance the tree, it is rotated so as to allow B to be theroot with A as its left child and BR as its right subtree, and ALand BL as the left and right subtrees of A respectively.


6

8

A

B0

-1

AL

BRBL

6

A-2

8

B

-1

AL

10BL

BR


10

8

B

BRAL

A 6

0

00

A.Rightchild = B.leftchild

B.leftchild = A

BL

� Left-Left case and Left-Right case:

� If the balance factor of P is 2, then the left subtree outweighsthe right subtree of the given node, and the balance factor ofthe left child L must be checked. The right rotation with P asthe root is necessary.

� If the balance factor of L is +1, a single right rotation (with Pas the root) is needed (Left-Left case).

� If the balance factor of L is -1, two different rotations areneeded. The first rotation is a left rotation with L as the root.The second is a right rotation with P as the root (Left-Rightcase).


� Right-Right case and Right-Left case:

� If the balance factor of P is -2 then the right subtree outweighsthe left subtree of the given node, and the balance factor of theright child (R) must be checked. The left rotation with P as theroot is necessary.

� If the balance factor of R is -1, a single left rotation (with P asthe root) is needed (Right-Right case).

� If the balance factor of R is +1, two different rotations areneeded. The first rotation is a right rotation with R as the root.The second is a left rotation with P as the root (Right-Leftcase).


AVL Tree Deletion� If the node is a leaf or has only one child, remove it.

� Otherwise, replace it with either the largest in its left subtree (in order predecessor) or the smallest in its right subtree (in order successor), and remove that node.

� The node that was found as a replacement has at most onesub tree.

� After deletion, retrace the path back up the tree (parent ofthe replacement) to the root, adjusting the balance factorsas needed.


� The retracing can stop if the balance factor becomes −1 or

+1 indicating that the height of that subtree has remained

unchanged.

� If the balance factor becomes 0 then the height of the

subtree has decreased by one and the retracing needs to

continue.

� If the balance factor becomes −2 or +2 then the subtree is

unbalanced and needs to be rotated to fix it.


Reference� Sanjay Pahuja, ‘A Practical Approach to Data Structures

and Algorithms’, First Ed. 2007.

� For Height Balanced Trees: AVL, B-Trees, refer:

� Pg. 292 – 296, 301 – 315


B-Trees� B-tree is a tree data structure that is a generalization of a

binary search tree.ie; A node can have more than twochildren.

� It is also called Balanced M-way tree or balanced sort tree.

� A node of the tree may contain many records or keys andpointers to children.

� Used in external sorting.

� It is not a binary tree.


Properties of B-tree of order M

(1) Each node(except the root and leaf) has maximum of M

children and a minimum of ceil(M/2) children and for root, any

number from 2 to maximum.

(2) Each node has one fewer key than children with a

maximum of M-1 keys.



(3) Keys are arranged in a defined order within the node.

All keys in the subtree to the left of a key are predecessors of

the key and those to the right are successors of the key.

(4) All leaves are on the same level. ie. There is no

empty subtree above the level of the leaves.

B-Tree Insertion(1) Search and find the position for insertion.

(2) Add the key to the node if the node can accommodate it.

(3) If not, ie. if a new key is to be inserted into a full node,the node is split into two and the key with median value isinserted in parent node.

Continue splitting upward, if required, until the root isreached.

If parent node is the root node, and has to be split, a new rootis created and the tree grows taller by one level.


B-Tree Deletion(1) Search and find the key to be deleted.

(2) If the key is in a terminal node, the key along with

appropriate pointer is deleted.

(3) If the key is not in a terminal node, it is replaced by a

copy of its successor(key with next higher value).



(4) If on deleting the key, the new node size is lower than

the minimum, an underflow occurs.

(a) If either of adjacent siblings contains more than

minimum number of keys, the central key is chosen from

the collection:

� contents of node with less than minimum number of keys,

� more than minimum number of keys, and

� the separating key from parent node.

� This key is written back to parent; the left and right halves

are written back to siblings.

� (b) If none of the adjacent siblings contains more than

minimum number of keys, concatenation is used.

The node is merged with its adjacent sibling and the

separating key from its parent.


B-Tree of Order 5 Example

� All internal nodes have at least ceil(5 / 2) = ceil(2.5) = 3 children (and hence at least 2 keys), other than the root node.

�

� The maximum number of children that a node can have is 5 (so that 4 is the maximum number of keys).

� All nodes other than the root must have a minimum of 2 keys.


B-Tree Order 5 Insertion

Insert C N G A H E K Q M F W L T Z

D P R X Y S

� Originally we have an empty B-tree of order 5.

� The first 4 letters get inserted into the same node


� When we try to insert the H, we find no room in this node, so

we split it into 2 nodes, moving the median item G up into a

new root node.


C N G A H E K Q M F W L T Z

D P R X Y S

� Inserting E, K, and Q proceeds without requiring any splits.



D P R X Y S

� Inserting M requires a split.



D P R X Y S

� The letters F, W, L, and T are then added without any split.



D P R X Y S

� When Z is added, the rightmost leaf must be split. The

median item T is moved up into the parent node.



D P R X Y S

� The insertion of D causes the leftmost leaf to be split. Dhappens to be the median key and so it is moved up into theparent node.

� The letters P, R, X, and Y are then added without any split.



D P R X Y S

� Finally, when S is added, the node with N, P, Q, and Rsplits, sending the median Q up to the parent.

� The parent node is full, so it splits, sending the median Mup to form a new root node.



D P R X Y S

B-Tree Order 5 Deletion

� Initial B-Tree


Delete H � Since H is in a leaf and the leaf has more than minimum

number of keys, we just remove it.


Delete T� Since T is not in a leaf, we find its successor (the next item in

ascending order), which happens to be W.

� Move W up to replace the T. ie; What we really have to do isto delete W from the leaf .


B+ Trees� Variant of the original B-tree in which all records are stored in

the leaves and all leaves are linked sequentially.

� The B+ tree is used as an indexing method in relational

database management systems.

� All keys are duplicated in the leaves.

� This has the advantage that all the leaves are linked together

sequentially, and hence the entire tree may be scanned without

visiting the higher nodes at all.


� The B + Tree consists of two types of nodes:

� (1) internal nodes and (2) leaf nodes

� • Internal nodes point to other nodes in the tree.

� • Leaf nodes point to data in the database using data pointers.

� Leaf nodes also contain an additional pointer, called the

sibling pointer, which is used to improve the efficiency of

certain types of search.


� The B + -Tree is a balanced tree because every path from the

root node to a leaf node is the same length.

� A balanced tree is one in which all searches for individual

values require the same number of nodes to be read from the

disc.

� Order of a B + Tree

� The order of a B + Tree is the number of keys and pointers that

an internal node can contain.

� An order size of m means that an internal node can contain m-1

keys and m pointers.


Insertion in B+ Tree


Insert sequence : 5, 8, 1, 7, 3, 12, 9, 6

Order: 3

Empty Tree

The B+Tree starts as a single leaf node. A leaf node consists

of one or more data pointers and a pointer to its right sibling.

This leaf node is empty.


Inserting Key Value 5

To insert a key, search for the location where the key has to

be inserted.

Here, the B+Tree consists of a single leaf node, L1, which is

empty. Hence, the key value 5 must be placed in leaf node L1.



Again, search for the location where key value 8 is to be

inserted. This is in leaf node L1. There is room in L1; so insert

the new key.



Searching for where the key value 1 should appear also results

in L1 but L1 is now full as it contains the maximum two records.


L1 must be split into two nodes. The first node will contain the

first half of the keys and the second node will contain the second

half of the keys.


We now require a new root node to point to each of these nodes.

We create a new root node and promote the rightmost key from

node L1.


Insert Key Value 7

Search for the location where key 7 is to be inserted, that is,

L2. Insert key 7 into L2.


Insert Key Value 3

Search for the location where key 3 is to be inserted. That is

L1. But, L1 is full and must be split.


The rightmost key in L1, i.e. 3, must now be promoted up the

tree.

L1 was pointing to key 5 in B1. Therefore, all the key values in

B1 to the right of and including key 5 are moved one position to

the right.


Insert Key Value 12

Search for the location where key 12 is to be inserted, L2.

Try to insert 12 into L2 but L2 is full and it must be split.

As before, we must promote the rightmost value of L2 but

B1 is full and so it must be split.


Now the tree requires a new root node, so we promote the

rightmost value of B1 into a new node.


Insert Key Value 9

Search for the location where key value 9 is to be inserted, L4.

Insert key 9 into L4.


Insert Key Value 6

Key value 6 should be inserted into L2 but it is full.

Therefore, split it and promote the appropriate key value.


Deletion in B+ Tree

Deletion sequence: 9, 8, 12.


Delete Key Value 9

First, search for the location of key value 9, L4.

Delete 9 from L4. L4 is not less than half full and the tree is correct.


Delete Key Value 8

Search for key value 8, L5.

Deleting 8 from L5 causes L5 to underflow, that is, it becomes

less than half full.


Redistribute some of the values from L2. This is possible

because L2 is full and half its contents can be placed in L5.

As some entries have been removed from L2, its parent B2 must

be adjusted to reflect the change.


Deleting Key Value 12

Deleting key value 12 from L4 causes L4 to underflow.

However, because L5 is already half full we cannot redistribute

keys between the nodes.

L4 must be deleted from the index and B2 adjusted to reflect the

change.

B+ Trees� Reference:

� http://www.mec.ac.in/resources/notes/notes/ds/bplus.htm


GRAPH

• Graph G = (V,E) is a collection of vertices and edges.

• (1) Set of vertices - V

• (2) Set of Edges - E

� where V ={ V1,V2,V3……..}

� E ={ e1,e2,e3……..}


GRAPH TERMINOLOGIES

• Directed Graph (Digraph) :A graph where nodes are

connected with directed edges.

• Arrow head is at vertex called head and other end is

called tail.


• LOOP: If an edge is incident from and into the same

vertex.

• ADJACENT VERTICES :Two vertices are adjacent if

they are joined by an edge.

• ISOLATED VERTEX : If there is no edge incident with

the vertex.


• ISOMORPHIC: Two graphs are said to be isomorphic if equal

number of vertices , edges and also corresponding images

exist.

• SUBGRAPH :Let G = (V,E) be a graph. Then, G1 = (V1,E1)

is a subgraph, if V1 is a subset of V and E1 is a subset of E.

• SPANNING / INDUCED SUBGRAPH : A subgraph that

contains all vertices of G.


• DEGREE: The number of edges incident on a vertex.

• WEIGHTED GRAPH :A graph in which every edge is

assigned some weight or value.

• PATH: A sequence of vertices.

• SIMPLE PATH : Path in which first and last vertices are

distinct.

• CYCLE : Length of path must be minimum 1 and begins and

ends at same vertex.


• LENGTH OF PATH: Number of edge arcs on the path.

• LABELED DIGRAPH :A graph in which each arc or

vertex has an associated label.

• CONNECTED GRAPH : Graph in which there exists a

path from any vertex to any other vertex. Otherwise, it is

said to be disconnected.

• {Disconnected graph contains components.}


REPRESENTATION OF GRAPH

1) Sequential Representation

� � Adjacency matrix representation

� � Incident matrix representation

2) Linked list representation


ADJACENCY MATRIX REPRESENTATION

� Order of Adjacency matrix = Number of vertices *

number of vertices.

� For a directed and undirected graph, adjacency matrix

conditions are:

Aij = 1 { if there is an edge from Vi to Vj }

= 0 { if there is no edge from Vi to Vj }

� For a weighted graph adjacency matrix conditions are:

Aij = Wij { if there is an edge from Vi to Vj,

where Wij is the weight. }

= -1 { if there is no edge from Vi to Vj }


INCIDENT MATRIX REPRESENTATION

� Order of Incident matrix = Number of vertices * number

of edges

� For an undirected graph, incident matrix conditions are:

Aij = 1 { if an edge ej is incident on vertex Vi}

= 0 {otherwise }

� For a directed graph, incident matrix conditions are:

Aij = 1 {if an edge ej is going outward from vertex Vi}

= -1{if an edge ej is incident on vertex Vi}

= 0 {otherwise}


LINKED LIST REPRESENTATION

� For a directed and undirected graph, store all the vertices

of graph in a list and each adjacent vertex as linked list

node.

� For a weighted graph, linked list node will contain an

extra field - weight.


Example� Suppose the adjacency matrix for a graph is:

�

� The corresponding adjacency list representation is:

� 1 -> 1 -> 2 -> 3 -> 4

2 -> 1

3 -> 2 -> 4

4 -> 2 -> 310/25/2012 CS 09 303 Data Structures - Module 3 212

1 2 3 4

1 1 1 1 1

2 1 0 0 0

3 0 1 0 1

4 0 1 1 0

Operations on Graphs� FIRST (v) : returns the index for the first vertex adjacent

to v.

� NEXT(v,i ) : returns the index after index i for the vertices

adjacent to v.

� VERTEX ( v, i ) : returns the vertex with index i among

the vertices adjacent to v.


(1) FIRST (V) :

Var A : array [1..n,1..n] of boolean

Function FIRST (v: integer ):integer;

Var i : integer ;

Begin

for i:= 1 to n do

if A [v, i] then

return (i);

return (0); { if we reach here v has no adjacent vertex}

End; {FIRST }


(2) NEXT(v,i ) :

Function NEXT (v: integer , i : integer ):integer;

Var j : integer ;

Begin

for j:= i+1 to n do

if A [v, j] then

return (j);

return (0);

End; {NEXT }

(3) VERTEX ( v, i ) : returns the vertex with index i

among the vertices adjacent to v.


(4)TRAVERSAL OF ADJACENT VERTICES OF V

i :=FIRST(v);

while i <> NULL do begin

w : = VERTEX (v,i);

{ some action on w }

i:=NEXT(v,i);

end; { while }


(5)CREATING A GRAPH

Steps

(1) Input total number of vertices in a graph, say n.

(2) Allocate memory dynamically for the vertices to

store in list array.

(3) Input the first vertex and vertices through which it

has edge by linking node from list array through

nodes.

(4) Repeat the process incrementing the list array to add

other vertices and edges.

(5) Exit


(6)SEARCHING & DELETING FROM A GRAPH

Steps

(1) Input an edge to be searched.

(2) Search for an initial vertex of edge in list arrays by

incrementing the array index.

(3) Once it is found, search through linked list for the

terminal vertex of the edge.

(4) If found, display “The edge is present in the graph”.

(5) Then delete the node where the terminal vertex is

found and rearrange the linked list.

(6) Exit.


(7)TRAVERSING A GRAPH

• Breadth First Search (BFS )

(1) Input the vertices of the graph and its edges G=(V,E )

(2) Input the source vertex and mark it as visited.

(3) Add the source vertex to queue .

(4) Repeat step 5 and 6 until the queue is empty. ( i.e,

front >rear )

(5) Pop the front element of queue and display it.

(6) Add the vertices, which is neighbor to just popped

element, if it is not in the queue .

(7) Exit.


• Breadth First Search (BFS ) ALGORITHM

� Procedure bfs (V)

� {bfs visits all vertices adjacent to V using BFS}

� Var

� Q : QUEUE of vertex ;

� x,y : vertex;

� begin

� mark[v] := visited;

� ENQUEUE(V,Q);


• while not EMPTY (Q) do begin

• x:= FRONT (Q);

• DEQUEUE(Q);

• for each vertex y adjacent to x do

• if mark[y] = unvisited then begin

• mark[y] := visited;

• ENQUEUE(y,Q);

• end {if}

• end;{while }

• End; {bfs}

�


Example


• Depth First Search (DFS ) ALGORITHM

� STEPS

� (1) Input the vertices and edges of graph G=(V,E).

� (2) Input the source vertex and assign it to the variable S.

� (3) Push the source vertex to the stack.

� (4) Repeat steps 5 and 6 until the stack is empty.

� (5) Pop the top element of stack & display it.

� (6) Push the vertices adjacent to just popped element if it

is not in the stack & is displayed (i.e. not visited).

� (7) exit


• Depth First Search (DFS ) ALGORITHM

� ALGORITHM

� Procedure DFS(V: Vertex );

� Var W : vertex;

� begin

(1) mark(V) := visited;

(2) for each vertex W on L(V) do

(3) if mark[w]=unvisited then

(4) DFS(W);

� end; {DFS}

� L(V) is the adjacency list.


Tree Searches� A tree search starts at the root

and explores nodes from there,looking for a goal node (a nodethat satisfies certain conditions,depending on the problem)

L M N O P

G

Q

H JI K

FED

B C

A


Depth First Search� A depth-first search (DFS)

explores a path all the way to a

leaf before backtracking and

exploring another path.

� For example, after searching A,

then B, then D, the search

backtracks and tries another

path from B.

� Node are explored in the order

A B D E H L M N I O P C F G

J K Q

� N will be found before J

L M N O P

G

Q

H JI K

FED

B C

A


Breadth First Search� A breadth-first search (BFS)

explores nodes nearest the root

before exploring nodes further

away.

� For example, after searching A,

then B, then C, the search

proceeds with D, E, F, G.

� Node are explored in the order

A B C D E F G H I J K L M N

O P Q

� J will be found before NL M N O P

G

Q

H JI K

FED

B C

A


How to do DFS?� Put the root node on a stack;

while (stack is not empty) do begin

remove a node from the stack;

if (node is a goal node)

� return success;

put all children of node onto the stack;

end

return failure;

� At each step, the stack contains a path of nodes from the root.

� The stack must be large enough to hold the longest possible

path, that is, the maximum depth of search.


How to do BFS?� Put the root node on a queue;

while (queue is not empty) do begin

remove a node from the queue;

if (node is a goal node)

� return success;

put all children of node onto the queue;

end

return failure;

� Just before starting to explore level n, the queue holds all the

nodes at level n-1.

� In a typical tree, the number of nodes at each level increases

exponentially with the depth.


Greedy Algorithms


Minimum Spanning Tree/Minimum

Cost Spanning Tree (MST)

� Spanning Tree for a graph G = (V,E) is a subgraph G 1=

(V 1 ,E1 ) of G that contains all the vertices of G.

• The vertex set V1 is same as that of graph G.

• The edge set E1 is a subset of G.

• There is no cycle.

� A graph can have many different spanning trees.


� A weighted tree is one in which each edge is assigned a

weight.

� Weight or cost of a spanning tree is the sum of weights of

its edges.

� A Minimum Spanning Tree or Minimum-Weight

Spanning Tree is a spanning tree with weight less than

or equal to the weight of every other spanning tree.


� This figure shows there may be more than one minimum

spanning tree in a graph.

� In the figure, the two trees below the graph are two possibilities

of minimum spanning tree of the given graph.


MST Algorithms� Kruskal’s Algorithm

� Finds an MST for a connected weighted undirected graph.

� If the graph is not connected, then it finds a minimum

spanning forest (a minimum spanning tree for each

connected component).

� Prim’s Algorithm

� Finds an MST for a connected weighted undirected graph.

� Prim's algorithm requires the graph to be connected.


� Kruskal’s algorithm works by growing the minimum spanning

tree one edge at a time, adding the lowest cost edge that does

not create a cycle.

� It starts with each vertex as a separate tree and merges these

trees together by repeatedly adding the lowest cost edge that

merges two distinct subtrees (i.e. does not create a cycle).


KRUSKAL’s ALGORITHM

- It builds the MST in forest (A forest is a disjoint union oftrees.).

� Initially, each vertex is in its own tree in forest.

� Then, the algorithm considers each edge in order by increasingweight.

� If an edge (u, v) connects two different trees, then (u, v) isadded to the set of edges of the MST, and the two trees connectedby the edge (u, v) are merged into a single tree.

� On the other hand, if the edge (u, v) connects two vertices inthe same tree, then edge (u, v) is discarded.


KRUSKAL’s ALGORITHM

� Kruskal(G) - Informal Algorithm

� Sort the edges in order of increasing weight

� count = 0

� while (count < n-1) do

� get next edge (u,v)

� if (component (u) <> component(v))

� add edge to T

� component(u) = component(v)

� end

� end

� end


� KRUSKAL’s ALGORITHM:OPERATIONS REQUIRED

• DELETEMIN : deletes edge of minimum cost from a

PRIORITY QUEUE .

• MERGE (A,B,C ): merge components A and B in C and

call the result A or B arbitrarily.

• FIND (v, C) : returns the name of the component of C of

which vertex v is a member. This operation will be used

to determine whether the two vertices of an edge are in

the same or in different components.

• INITIAL (A, v, C ) : makes A the name of the component

in C containing only one vertex, namely v.


Procedure Kruskal (V : Set of vertex; E : Set of edge ) :Var T :

Set of edge ) ;

Var

ncomp : integer ; { current number of components }

edges : PRIORITYQUEUE ;{ the set of edges }

components :MFSET ;

{the set V grouped into a MERGE- FIND set of components}

u,v : vertex;

e : edge ;

nextcomp : integer ; {name for new component }

ucomp, vcomp :integer ;{component names }


Begin

MAKENULL(T);

MAKENULL(edges);

nextcomp := 0;

ncomp := number of members of V;

for v in V do begin

{ initialize a component to contain one vertex of V }

nextcomp:= nextcomp+1;

INITIAL(nextcomp, v, components );

end; {for}

for e in E do {initialize priority queue of edges }

INSERT(e, edges);


while ncomp > 1 do begin

e:= DELETEMIN(edges);

let e = (u,v);

ucomp := FIND(u, components);

vcomp:= FIND(v, components);

if ucomp <> vcomp then begin

{e connects two different components}

MERGE (ucomp , vcomp, components );

ncomp := ncomp -1;

INSERT (e ,T);

end ;{if ]

end; {while }

End; {kuruskal }


PRIM’s ALGORITHM for MST

� Prim's algorithm starts with an arbitrary vertex v and

``grows'' a tree from it, repeatedly finding the lowest-cost

edge that will link some new vertex into this tree.


Algorithm (Informal)� Start with a tree which contains only one node.

� Identify a node (outside the tree) which is closest to the

tree and add the minimum weight edge from that node to

some node in the tree and incorporate the additional node

as a part of the tree.

� If there are less than n – 1 edges in the tree, go to 2.


PRIM’S ALGORITHM� Procedure Prim (G: graph; var T: set of edges);

� {Prim constructs a minimum-cost spanning tree T for G}.

� Var

� U: set of vertices;

� u, v : vertex;

� Begin

� T:= Φ;

� U:= {1};


� while U<>V do begin

� let (u,v) be a lowest cost edge such that

� u is in U and v is in V-U;

� end

� End; {Prim}


PRIM’s ALGORITHM FOR MST� Procedure Prim (C :array [1..n ,1..n] of real );

� {Prim prints the edges of MST for a graph with vertices

{1,2,3….n} and cost matrix C on edges }

� Var

� LOWCOST :array[1..n] of real;

� CLOSEST :array[1..n] of integer;

� i, j, k, min :integer;

� {i and j are indices .During a scan of the LOWCOST

array, k is the index of the CLOSEST vetex found so far and

min := LOWCOST[k] }


� begin

� for i:= 2 to n do begin

� {initialize with only vertex 1 in the set U }

� LOWCOST[i] :=C [1,i] ;

� CLOSEST[i]:=1;

� end;{for }

�


� For i:= 2 to n do begin

� {find the CLOSEST vertex k outside of U to some vertex in

U }

� Min:= LOWCOST[2];

� k:= 2;

� For j:=3 to n do

� If LOWCOST[i] < min then begin

� Min := LOWCOST[j];

� k := j;

� end;

� Writeln(k, CLOSEST[k]); {print edge }

� LOWCOST[k] := infinity; {k is added to U }


� For j:=2 to n do { adjust costs to U }

� If (C[k,j] < LOWCOST[j] ) and

(LOWCOST[j] < infinity ) then begin

� LOWCOST[j] := C[k,j];

� CLOSEST[j]:= k;

� End; {if }

� End {for }

� End ;{Prim}

� Where CLOSEST[i] � Gives vertex in U that is currently closest to vertex i in V-U.

� LOWCOST[i] � Cost of the edge (i,CLOSEST[i])



Start with the tree < { v1 }, { } >. Vertex v4 is closest to tree . . .


v3 is closest to tree . . .


v2 and v5 are closest to tree, pick v5, say . .


v6 is closest to tree . . .


v2 is closest (and only remaining) vertex . . .

SHORTEST PATH Algorithms for Directed

Graphs


� Single Source Shortest Paths Algorithm – to determine

the cost of the shortest path from the source to every other

vertex in V.

� Dijkstra’s Algorithm

� All-Pairs Shortest Paths Algorithm – to find for each

ordered pair of vertices (v,w), the shortest path from v to

w.

� Floyd’s Algorithm

Dijkstra’s Algorithm� The algorithm works by maintaining a set S of vertices

whose shortest distance from the source is already known.

� Initially, S contains only the source vertex.

� At each step, we add to S a remaining vertex v whose

distance from the source is as short as possible.

� Assuming that all arcs have nonnegative costs, we can

always find a shortest path from the source to v that passes

only through vertices in S.


� At each step of the algorithm, we use an array D to record

the length of the shortest path to each vertex.

� Once S includes all vertices, all paths are shortest paths, so

D will hold the shortest distance from the source to each

vertex.



� Procedure Dijikstra

� {Dijikstra computes the cost of shortest paths from vertex 1 to every vertex of a directed graph}

� Begin

� S := {1};

� For i :=2 to n do

� D[i] :=C[1,i]; {Initialize D }

� For i:=1 to n-1 do begin

� Choose a vertex w in V-S such that D[w] is minimum ;

� add w to S;

� For each vertex v in V-S do

� D[v]:= min( D[v], D[w] + C[w,v]);

� end;{for}

End;{Dijikstra}

FLOYD’S ALGORITHM�

Procedure Floyd(Var A :array[1..n ,1..n] of real;

C :array[1..n,1..n] of real );

{Floyd computes shortest path matrix A given arc cost matrix C}

� Var i, j, k : integer;

� Begin

� For i := 1 to n do

� For j := 1 to n do

� A[i, j] := C[i, j];

� For i:=1 to n do

� A[i, i] := 0;

�


� For k := 1 to n do

� For i:=1 to n do

� For j:=1 to n do

� If A[i, k] + A[k, j] < A[i,j] then

� A[i,j] := A[i, k] + A[k, j];

� End; {Floyd }


Thank You


M3_Non Linear Data Structures

Documents

Transcript of M3_Non Linear Data Structures