Funddamentals of data structures

Fundamentals of
Data Structure

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Data Structures

• "Once you succeed in writing the programs for complicated
algorithms, they usually run extremely fast. The computer
doesn't need to understand the algorithm, it’s task is only to run
the programs.“
• There are a number of facets to good programs: they must
– run correctly
– run efficiently
– be easy to read and understand
– be easy to debug and
– be easy to modify.


Data Structure (Cont.)

What is Data Structure ?
• A scheme for organizing related pieces of information

• A way in which sets of data are organized in a particular system

• An organised aggregate of data items

• A computer interpretable format used for storing, accessing, transferring
and archiving data

• The way data is organised to ensure efficient processing: this may be in
lists, arrays, stacks, queues or trees

Data structure is a specialized format for organizing and storing
data so that it can be be accessed and worked with in
appropriate ways to make an a program efficient

Data Structures (Cont.)

• Data Structure = Organised Data + Allowed Operations
There are two design aspects to every data structure:
the interface part
The publicly accessible functions of the type. Functions like
creation and destruction of the object, inserting and removing
elements (if it is a container), assigning values etc.

the implementation part :
Internal implementation should be independent of the interface.
Therefore, the details of the implementation aspect should be
hidden out from the users.


Collections

•Programs often deal with collections of items.
•These collections may be organised in many ways and use
many different program structures to represent them, yet, from
an abstract point of view, there will be a few common
operations on any collection.

create Create a new collection

add Add an item to a collection

delete Delete an item from a collection

find Find an item matching some criterion in the collection

destroy Destroy the collection


Analyzing an Algorithm
• Simple statement sequence
s1; s2; …. ; sk
– Complexity is O(1) as long as k is constant
• Simple loops
for(i=0;i<n;i++) { s; } where s is O(1)
– Complexity is n O(1) or O(n)
• Loop index doesn’t vary linearly
h = 1;
while ( h <= n ) { s; h = 2 * h;}
– Complexity O(log n)
• Nested loops (loop index depends on outer loop index)
for(i=0;i<n;i++) f
for(j=0;j<n;j++) { s; }
– Complexity is n O(n) or O(n2)


Arrays

An Array is the simplest form of implementing a collection
• Each object in an array is called an array element
• Each element has the same data type (although they may have different
values)
• Individual elements are accessed by index using a consecutive range of
integers
One Dimensional Array or vector
int A[10];
for ( i = 0; i < 10; i++)
A[i] = i +1;

A[0] A[1] A[2] A[n-2] A[n-1]

1 2 3 N-1 N


Arrays (Cont.)

Multi-dimensional Array
A multi-dimensional array of dimension n (i.e., an n-dimensional array or simply n-
D array) is a collection of items which is accessed via n subscript expressions. For
example, in a language that supports it, the (i,j) th element of the two-dimensional
array x is accessed by writing x[i,j].

Column
0 1 2 3 4 5 6 7 8 9 10 j n
0
1
R
2
O
: : : : : : : : : : : : : : :
W
i x

m

Arrays (Cont.)

Array : Limitations

• Simple and Fast but must specify size during construction
• If you want to insert/ remove an element to/ from a fixed position in the
list, then you must move elements already in the list to make room for
the subsequent elements in the list.
• Thus, on an average, you probably copy half the elements.
• In the worst case, inserting into position 1 requires to move all the
elements.
• Copying elements can result in longer running times for a program if
insert/ remove operations are frequent, especially when you consider the
cost of copying is huge (like when we copy strings)
• An array cannot be extended dynamically, one have to allocate a new
array of the appropriate size and copy the old array to the new array

Linked Lists
• The linked list is a very flexible dynamic data structure:
items may be added to it or deleted from it at will
– Dynamically allocate space for each element as needed
– Include a pointer to the next item
– the number of items that may be added to a list is limited only by the
amount of memory available
Linked list can be perceived as connected (linked) nodes
Each node of the list contains
• the data item
• a pointer to the next node
• The last node in the list contains a NULL pointer to indicate that it is the
end or tail of the list.

Data Next

object

Linked Lists (Cont.)
• Collection structure has a pointer to the list head
– Initially NULL
The variable (or handle)
• Add first item
which represents the list
– Allocate space for node is simply a pointer to the
– Set its data pointer to object node at the head of the
list.
– Set Next to NULL
– Set Head to point to new node

Collection
node
Head
Data Next Tail

object

Linked Lists (Cont.)
• Add a node
– Allocate space for node
– Set its data pointer to object
– Set Next to current Head
– Set Head to point to new node
Collection

Head

node node
Data Next Data Next

object2 object

Linked Lists - Add implementation
• Implementation
struct t_node {
void *item;
Recursive type definition -
struct t_node *next;
C allows it!
} node;
typedef struct t_node *Node;
struct collection {
Node head;
……
};
int AddToCollection( Collection c, void *item ) {
Node new = malloc( sizeof( struct t_node ) );
new->item = item;
new->next = c->head;
c->head = new; Error checking, asserts
return TRUE; omitted for clarity!
}

Linked Lists - Find implementation
• Implementation
void *FindinCollection( Collection c, void *key ) {
Node n = c->head;
while ( n != NULL ) {
if ( KeyCmp( ItemKey( n->item ), key ) == 0 ) {
return n->item;
n = n->next;
}
return NULL;
}

Add time Constant - independent of n
Search time Worst case - n

• A recursive implementation is also possible!

Linked Lists - Delete implementation
• Implementation

void *DeleteFromCollection( Collection c, void *key ) {
Node n, prev;
n = prev = c->head;
while ( n != NULL ) {
if ( KeyCmp( ItemKey( n->item ), key ) == 0 ) {
prev->next = n->next;
return n;
}
prev = n;
n = n->next; head
}
return NULL;
}

Linked Lists - Variations
• Simplest implementation By ensuring that the tail of the list is
always pointing to the head, we can
– Add to head
build a circularly linked list
– Last-In-First-Out (LIFO) semantics head is tail->next
• Modifications LIFO or FIFO using ONE pointer

– First-In-First-Out (FIFO)
head
– Keep a tail pointer
struct t_node { tail
void *item;
struct t_node *next;
} node;
struct collection {
Node head, tail;
};

Linked Lists - Doubly linked
• Doubly linked lists
– Can be scanned in both directions Applications requiring
both way search
struct t_node {
Eg. Name search in
void *item;
telephone directory
struct t_node *prev,
*next;
} node;

struct collection {
Node head, tail;
}; head prev prev prev

tail

Binary Tree
• The simplest form of Tree is a Binary Tree
Note the
– Binary Tree Consists of
recursive
• Node (called the ROOT node) definition!
• Left and Right sub-trees
• Both sub-trees are binary trees
• The nodes at the lowest levels of the tree (theIn an ordered binary tree
ones with no sub-
trees) are called leaves the keys of all the nodes in
• the left sub-tree are less
than that of the root
• the keys of all the nodes
in the right sub-tree are
greater than that of the
root,
• the left and right sub-
trees are themselves
ordered binary trees.
Each sub-tree
is itself
a binary tree

Binary Tree (Cont.)
• If A is the root of a binary tree and B is the root
of its left/right subtree then A
o A is the father of B
o B is the left/right son of A
B C
• Two nodes are brothers if they are left and right
sons of the same father
D
E
• Node n1 is an ancestor of n2 (and n2 is descendant
of n1) if n1 is either the father of n2 or the father of F G
some ancestor of n2

• Strictly Binary Tree: If every nonleaf node in a binary tree has non empty left
and right subtrees
• Level of a node: Root has level 0. Level of any node is one more than the level
of its father
• Depth: Maximum level of any leaf in the tree
A binary tree can contain at most 2l nodes at level l
Total nodes for a binary tree with depth d = 2d+1 - 1

Binary Tree - Implementation

struct t_node {
void *item;
struct t_node *left;
struct t_node *right;
};


struct t_collection {
Node root;
……
};

Binary Tree - Implementation
• Find

extern int KeyCmp( void *a, void *b );
/* Returns -1, 0, 1 for a < b, a == b, a > b */

void *FindInTree( Node t, void *key ) { Less,
if ( t == (Node)0 ) return NULL; search
switch( KeyCmp( key, ItemKey(t->item) ) ) { left
case -1 : return FindInTree( t->left, key );
case 0: return t->item;
case +1 : return FindInTree( t->right, key );
}
} Greater,
search right
void *FindInCollection( collection c, void *key ) {
return FindInTree( c->root, key );
}

Binary Tree - Performance
• Find
– Complete Tree

– Height, h
• Nodes traversed in a path from the root to a leaf
– Number of nodes, h
• n = 1 + 21 + 22 + … + 2h = 2h+1 - 1
• h = floor( log2 n )

Binary Tree - Traversing
Traverse: Pass through the tree, enumerating each node once

• PreOrder (also known as depth-first order)
1. Visit the root
2. Traverse the left subtree in preorder
3.Traverse the right subtree in preorder
• InOrder (also known as symmetric order)
1. Traverse the left subtree in inorder
2. Visit the root
3. Traverse te right subtree in inorder
• PostOrder (also known as symmetric order)
1. Traverse the left subtree in postorder
2. Traverse the right subtree in postorder
3. Visit the root

Binary Tree - Applications
• A binary tree is a useful data structure when two-way decisions must be
made at each point in a process

– Example: Finding duplicates in a list of numbers

• A binary tree can be used for representing an expression containing
operands (leaf) and operators (nonleaf node).

Traversal of the tree will result in infix, prefix or postfix forms of expression

Two binary trees are MIRROR SIMILAR if they are both empty or if they are
nonempty, the left subtree of each is mirror similar to the right subtree

General Tree
• A tree is a finite nonempty set of elements in which one
element is called the ROOT and remaining element partitioned
into m >=0 disjoint subsets, each of which is itself a tree
• Different types of trees – binary tree, n-ary tree, red-black tree,
AVL tree

A Hierarchical Tree

Heaps
Heaps are based on the notion of a complete tree
A binary tree is completely full if it is of height, h, and has 2h+1-1 nodes.
• A binary tree of height, h, is complete iff
– it is empty or
– its left subtree is complete of height h-1 and its right subtree is completely full of height
h-2 or
– its left subtree is completely full of height h-1 and its right subtree is complete of height
h-1.
• A complete tree is filled from the left:
– all the leaves are on
– the same level or two adjacent ones and
– all nodes at the lowest level are as far to the left as possible.
• A binary tree has the heap property iff
– it is empty or
– the key in the root is larger than that in either child and both subtrees have the heap
property.

Heaps (Cont.)
• A heap can be used as a priority queue:
• the highest priority item is at the root and is trivially extracted. But if the
root is deleted, we are left with two sub-trees and we must efficiently re-
create a single tree with the heap property.
• The value of the heap structure is that we can both extract the highest
priority item and insert a new one in O(logn) time.
Example:
A deletion will remove the T
at the root

Heaps (Cont.)
To work out how we're going to maintain
the heap property, use the fact that a
complete tree is filled from the left. So that
the position which must become empty is
the one occupied by the M. Put it in the
vacant root position.

This has violated the condition that the root
must be greater than each of its children.
So interchange the M with the larger of its
children.

The left subtree has now lost the heap
property. So again interchange the M with
the larger of its children.

We need to make at most h interchanges of
a root of a subtree with one of its children to

Heaps (Cont.)
Addition to a Heap
To add an item to a heap, we follow the reverse procedure.
Place it in the next leaf position and move it up.
Again, we require O(h) or O(logn) exchanges.

Comparisons

Arrays Linked List Trees
Simple, fast Simple Still Simple
Inflexible Flexible Flexible
Add O(1) O(1) O(log n)
O(n) inc sort sort -> no adv
Delete O(n) O(1) - any O(log n)
O(n) - specific
Find O(n) O(n) O(log n)
O(logn) (no bin search)
binary search

Queues
Queues are dynamic collections which have some concept of order
• FIFO queue
– A queue in which the first item added is always the first one out.
• LIFO queue
– A queue in which the item most recently added is always the
first one out.
• Priority queue
– A queue in which the items are sorted so that the highest priority
item is always the next one to be extracted.

Queues can be implemented by Linked Lists

Stacks
• Stacks are a special form of collection
with LIFO semantics

• Two methods
– int push( Stack s, void *item );
- add item to the top of the stack
– void *pop( Stack s );
- remove most recently pushed item from the top of the stack

• Like a plate stacker

• Other methods
int IsEmpty( Stack s );
Determines whether the stack has anything in it
void *Top( Stack s );
Return the item at the top without deleting it

Stack (Cont.)
• Stack very useful for Recursions
• Key to call / return in functions &
procedures

function f( int x, int y) {
int a;
if ( term_cond ) return …;
a = ….;
return g( a );
}

function g( int z ) {
int p, q;
p = …. ; q = …. ;
return f(p,q);
} Context
for execution of f

Searching
Computer systems are often used to store large amounts of data
from which individual records must be retrieved according to
some search criterion. Thus the efficient storage of data to
facilitate fast searching is an important issue
Things to consider
– the average time
– the worst-case time and
– the best possible time.
• Sequential Searches
– Time is proportional to n

– We call this time complexity O(n)

– Both arrays (unsorted) and linked lists

Binary Search
• Sorted array on a key
• first compare the key with the item in the
middle position of the array
• If there's a match, we can return
immediately.
• If the key is less than the middle key,
then the item sought must lie in the
lower half of the array
• if it's greater then the item sought must
lie in the upper half of the array
• Repeat the procedure on the lower (or
upper) half of the array - RECURSIVE

Time complexity O(log n)

Binary Search Implementation
static void *bin_search( collection c, int low, int high, void *key ) {
int mid;
if (low > high) return NULL; /* Termination check */
mid = (high+low)/2;
switch (memcmp(ItemKey(c->items[mid]),key,c->size)) {
case 0: return c->items[mid]; /* Match, return item found */
case -1: return bin_search( c, low, mid-1, key); /* search lower half */
case 1: return bin_search( c, mid+1, high, key ); /* search upper half */
default : return NULL;
}
}

void *FindInCollection( collection c, void *key ) {
/* Find an item in a collection
Pre-condition:
c is a collection created by ConsCollection
c is sorted in ascending order of the key
key != NULL
Post-condition: returns an item identified by key if one exists, otherwise returns NULL */

int low, high;
low = 0; high = c->item_cnt-1;
return bin_search( c, low, high, key );
}

Binary Search vs Sequential Search
• Find method Logs
– Sequential search Base 2 is by far
the most common
• Worst case time: c1 n in this course.
Assume base 2
– Binary search unless otherwise
noted!
• Worst case time: c2 log2n Compare n with logn

60

50
Small
problems - 40 n
we’re not Large 4 log n
Time

30
interested! problems -
n

20 we’re
log2n interested
10
Binary search in this gap!
More complex 0

Higher constant factor 0 10 20 30 40 50 60
n

Sorting
A file is said to be SORTED on the key if i < j implies that k[i]
preceeds k[j] in some ordering of the keys

Different types of Sorting

• Exchange Sorts
• Bubble Sort
• Quick Sort
• Insertion Sorts

• Selection Sorts
• Binary Tree Sort
• Heap Sort

Insertion Sort
First card is already sorted
With all the rest,
‚Scan back from the end until you find the first
card larger than the new one O(n)
—Move all the lower ones up one slot O(n)
„insert it O(1) ‚
For n cards Complexity O(n2)
A K 10 5 J 4 2 Q 9
o
9 „

Bubble Sort
Bubble Sort
• From the first element
– Exchange pairs if they’re out of order
– Repeat from the first to n-1
– Stop when you have only one element to check
/* Bubble sort for integers */
#define SWAP(a,b) { int t; t=a; a=b; b=t; }

void bubble( int a[], int n ) { Outer loop n iterations
int i, j;
for(i=0;i<n;i++) { /* n passes thru the array */
Inner loop
/* From start to the end of unsorted part */
n-1, n-2, n-3, … , 1 iterations
for(j=1;j<(n-i);j++) {
/* If adjacent items out of order, swap */
if( a[j-1]>a[j] ) SWAP(a[j-1],a[j]); O(1) statement
}
}
}
Overall O(n2)

Partition Exchange or Quicksort
• Example of Divide and Conquer algorithm
• Two phases
– Partition phase
• Divides the work into half

< pivot pivot > pivot
– Sort phase
• Conquers the halves!
< pivot > pivot

< p’ p’ > p’ pivot < p” p” > p”

quicksort( void *a, int low, int high ) {
int pivot;
if ( high > low ) /* Termination condition! */ {
pivot = partition( a, low, high );
quicksort( a, low, pivot-1 );
quicksort( a, pivot+1, high );
}
}

Heap Sort
Heaps also provide a means of sorting:
• construct a heap,
• add each item to it (maintaining the heap property!),
• when all items have been added, remove them one by one
(restoring the heap property as each one is removed).
• Addition and deletion are both O(logn) operations. We need to
perform n additions and deletions, leading to an O(nlogn)
algorithm
• Generally slower

Comparisons of Sorting
• The Sorting Repertoire

– Insertion O(n2) Guaranteed
– Bubble O(n2) Guaranteed
– Heap O(n log n) Guaranteed
– Quick O(n log n) Most of the time!
O(n2)
– Bin O(n) Keys in small range
O(n+m)
– Radix O(n) Bounded keys/duplicates
O(nlog n)

Hashing
• A Hash Table is a data structure that associates each element
(e) to be stored in our table with a particular value known as a
key (k)

• We store item’s (k,e) in our tables

• Simplest form of a Hash Table is an Array

• A bucket array for a hash table is an array A of size N, where
each cell of A is thought of as a bucket and the integer N
defines the capacity of the array,

Bucket Arrays
• If the keys (k) associated with each element (e) are well distributed in
the range [0, N-1] this bucket array is all that is needed.

• An element (e) with key (k) is simply inserted into bucket A[k].

• So A[k] = (Item)(k, e);

• Any bucket cells associated with keys not present, stores a
NO_SUCH_KEY object.

• If keys are not unique, that is there exists element key pairs (e1, k) and
(e2, k) we will have two different elements mapped to the same bucket.

• This is known as a collision. And we will discuss this later.

• We generally want to avoid such collisions.

Direct Access Table
• If we have a collection of n elements whose keys are unique
integers in (1,m), where m >= n,
then we can store the items in a direct address table, T[m],
where Ti is either empty or contains one of the elements of our
collection.
• Searching a direct address table is clearly an O(1) operation:
for a key, k, we access Tk,
• if it contains an element, return it,
• if it doesn't then return a NULL.

Analysis of Bucket Arrays
• Drawback 1: The Hash Table uses O(N) space which is not
necessarily related to the number of elements n actually present in
our set.

• If N is large relative to n, then this approach is wasteful of space.

• Drawback 2: The bucket array implementation of Hash Tables
requires key values (k) associated with elements (e) to be unique
and in the range [0, N-1], which is often not the case.

Hash Functions
• Associated with each Hash Table is a function h, known as a
Hash Function.

• This Hash Function maps each key in our set to an integer in
the range [0, N-1]. Where N is the capacity of the bucket array.

• The idea is to use the hash function value, h(k) as an index into
our bucket array.

• So we store the item (k, e) in our bucket at A[h(k)]. That is
A[h(k)] = (Item)(k, e);

• Unfortunately, finding a perfect hashing function is not always
possible. Let's say that we can find a hash function, h(k),
which maps most of the keys onto unique integers, but maps a
small number of keys on to the same integer. If the number of
collisions (cases where multiple keys map onto the same
integer), is sufficiently small, then hash tables work quite well
and give O(1) search times.

Handling the collisions
• In the small number of cases, where multiple keys map to the
same integer, then elements with different keys may be stored
in the same "slot" of the hash table. It is clear that when the
hash function is used to locate a potential match, it will be
necessary to compare the key of that element with the search
key. But there may be more than one element which should be
stored in a single slot of the table. Various techniques are used
to manage this problem:
• chaining,
• overflow areas,
• re-hashing,
• using neighbouring slots (linear probing),
• quadratic probing,
• random probing,

Chaining
• Chaining
One simple scheme is to chain all collisions in lists attached to
the appropriate slot. This allows an unlimited number of
collisions to be handled and doesn't require a priori knowledge
of how many elements are contained in the collection. The
tradeoff is the same as with linked lists versus array
implementations of collections: linked list overhead in space
and, to a lesser extent, in time.

Rehashing
• Re-hashing schemes use a second hashing operation when
there is a collision. If there is a further collision, we re-hash
until an empty "slot" in the table is found. The re-hashing
function can either be a new function or a re-application of the
original one. As long as the functions are applied to a key in
the same order, then a sought key can always be located.

• h(j)=h(k), so the next hash function,
h1 is used. A second collision occurs,
so h2 is used.

Overflow
• Divide the pre-allocated table into two sections: the primary
area to which keys are mapped and an area for collisions,
normally termed the overflow area.
• When a collision occurs, a slot in the overflow area is used for
the new element and a link from the primary slot established as
in a chained system. This is essentially the same as chaining,
except that the overflow area is pre-allocated and thus possibly
faster to access. As with re-hashing, the maximum number of
elements must be known in advance, but in this case, two
parameters must be estimated: the optimum size of the primary
and overflow areas.
• design systems with
multiple overflow tables

Graph
A graph consists of a set of nodes (or vertices) and a set of arcs (or edges)
Graph G = Nodes {A,B, C} Arcs {(A,C), (B,C)}
Terminology :
• V = Set of vertices (or nodes)
• |V| = # of vertices or cardinality of V (in usual terminology |V| = n)
• E = Set of edges, where an edge is defined by two vertices
• |E| = # of edges or cardinality of E
• A Graph, G is a pair G = (V, E)

Labeled Graphs:
We may give edges and vertices
labels. Graphing applications often
require the labeling of vertices Edges
might also be numerically labeled.
For instance if the vertices represent
cities, the edges might be labeled to
represent distances.

Graph Terminology
Directed (or digraph) & Undirected Graphs

A directed graph is one in which every edge (u, v) has a direction,
so that (u, v) is different from (v, u). In an undirected graph, there
is no distinction between (u, v) and (v, u). There are two possible
situations that can arise in a directed graph between vertices u
and v.
• i) only one of (u, v) and (v, u) is present.
• ii) both (u, v) and (v, u) are present.
•An edge (u, v) is said to be
directed from u to v if the pair (u, v)
is ordered with u preceding v.
E.g. A Flight Route
•An edge (u, v) is said to be
undirected if the pair (u, v) is not
ordered
E.g. Road Map

Graph Terminology
• Two vertices joined by an edge are called the end vertices or
endpoints of the edge.
• If an edge is directed its first endpoint is called the origin and
the other is called the destination.
• Two vertices are said to be adjacent if they are endpoints of
the same edge.
• The degree of a vertex v, denoted deg(v), is the number of
incident edges of v.
• The in-degree of a vertex v, denoted indeg(v) is the number of
incoming edges of v.
• The out-degree of a vertex v, denoted outdeg(v) is the number
of outgoing edges of v.

Graph Terminology

• Two vertices joined by an edge are called the end vertices or
endpoints of the edge.

• If an edge is directed its first endpoint is called the origin and the
other is called the destination.

• Two vertices are said to be adjacent if they are endpoints of the same
edge.

Graph Terminology

a B
b h
A d D F j
e i
c
C g

f
E

Graph Terminology

a B
b h
A d D F j
e i
c
C g
Vertices A and B f
are endpoints of edge a
E

Graph Terminology

a B
b h
A d D F j
e i
c
C g
Vertex A is the f
origin of edge a
E

Graph Terminology

a B
b h
A d D F j
e i
c
C g
Vertex B is the f
destination of edge a
E

Graph Terminology

a B
b h
A d D F j
e i
c
C g
Vertices A and B are f
adjacent as they are
endpoints of edge a E

Graph Terminology

• An edge is said to be incident on a vertex if the vertex is one of the
edges endpoints.

• The outgoing edges of a vertex are the directed edges whose origin
is that vertex.

• The incoming edges of a vertex are the directed edges whose
destination is that vertex.

Graph Terminology

a V
b h
U d X Z j
e i
c
W g Edge 'a' is incident on vertex V
Edge 'h' is incident on vertex Z
f Edge 'g' is incident on vertex Y
Y

Graph Terminology

a V
b h
U d X Z j
e i
c
W g The outgoing edges of vertex W
are the edges with vertex W as
f origin {d, e, f}
Y

Graph Terminology

a V
b h
U d X Z j
e i
c
W g The incoming edges of vertex X
are the edges with vertex X as
f destination {b, e, g, i}
Y

Graph Terminology

• The degree of a vertex v, denoted deg(v), is the number of incident
edges of v.

• The in-degree of a vertex v, denoted indeg(v) is the number of
incoming edges of v.

• The out-degree of a vertex v, denoted outdeg(v) is the number of
outgoing edges of v.

Graph Terminology

a V
b h
U d X Z j
e i
c
W g The degree of vertex X
is the number of incident
f edges on X.
deg(X) = ?
Y

Graph Terminology

a V
b h
U d X Z j
e i
c
W g The degree of vertex X
is the number of incident
f edges on X.
deg(X) = 5
Y

Graph Terminology

a V
b h
U d X Z j
e i
c
W g The in-degree of vertex X
is the number of edges that
f have vertex X as a destination.
indeg(X) = ?
Y

Graph Terminology

a V
b h
U d X Z j
e i
c
W g The in-degree of vertex X
f have vertex X as a destination.
indeg(X) = 4
Y

Graph Terminology

a V
b h
U d X Z j
e i
c
W g The out-degree of vertex X
f have vertex X as an origin.
outdeg(X) = ?
Y

Graph Terminology

a V
b h
U d X Z j
e i
c
W g The out-degree of vertex X
f have vertex X as an origin.
outdeg(X) = 1
Y

Graph Terminology

• Path:

» Sequence of alternating vertices and edges
» begins with a vertex
» ends with a vertex
» each edge is preceded and followed by its endpoints
• Simple Path:

» A path where all where all its edges and vertices are
distinct

Graph Terminology

V P1
a b h
U d X Z j
e i
c
W g We can see that P1 is
a simple path.
f
P1 = {U, a, V, b, X, h, Z}
Y

Graph Terminology

a V
b h
U d X Z j
e i
c
W g

f
P2 is not a simple path
as not all its edges and Y
vertices are distinct.
P2 = {U, c, W, e, X, g, Y, f, W, d, V}

Graph Terminology

• Cycle:

» Circular sequence of alternating vertices and
edges.
» Each edge is preceded and followed by its
endpoints.

• Simple Cycle:

» A cycle such that all its vertices and edges are
unique.

Graph Terminology

a V
b h
U d X Z j
e i
c
W g

f
Y
Simple cycle
{U, a, V, b, X, g, Y, f, W, c}

Graph Terminology

a V
b h
U d X Z j
e i
c
W g

f
Y
Non-Simple Cycle
{U, c, W, e, X, g, Y, f, W, d, V, a}

Graph Representation
Adjacency Matrix Implementation
A |V| × |V| matrix of 0's and 1's. A 1 represents a connection or an edge.
Storage = |V|² (this is huge!!)

For a non-directed graph there will always be symmetry along the top left to bottom
right diagonal. This diagonal will always be filled with zero's. This simplifies Coding.
Coding is concerned with storing the graph
in an efficient manner.
One way is to take all the bits from the
adjacency matrix and concatenate
them to form a binary string.

For undirected graphs, it suffices to concatenate the bits of the upper right triangle of
the adjacency matrix. Graph number zero is a graph with no edges.

Graph Applications
Some of the applications of graphs are :
• Networks (computer, cities ....)
• Maps (any geographic databases )
• Graphics : Geometrical Objects
• Neighborhood graphs
• Flow Problem
• Workflow

Reachability

• Reachability: Given two vertices 'u' and 'v' of a directed graph G,
we say that 'u' reaches 'v' if G has a directed path from 'u' to 'v'.

• That is 'v' is reachable from 'u'.

• A directed graph is said to be strongly connected if for any two
vertices 'u' and 'v' of G, 'u' reaches 'v'.

Graphs

• Depth First Search

• Breadth First Search

• Directed Graphs (Reachability)

• Application to Garbage Collection in Java

• Shortest Paths

• Dijkstra's Algorithm

Depth First Search
Algorithim DFS()
Input graph G
Output labeling of the edges of G
as discovery edges and back edges

for all u in G.vertices()
setLabel(u, Unexplored)

for all e in G.incidentEdges()
setLabel(e, Unexplored)

for all v in G.vertices()
if getLabel(v) = Unexplored
DFS(G, v).

Algorithm DFS(G, v)
Input graph G and a start vertex v of G
Output labeling of the edges of G as
discovery edges and back edges

setLabel(v, Visited)

for all e in G.incidentEdges(v)

if getLabel(e) = Unexplored
w <--- opposite(v, e)

if getLabel(w) = Unexplored
setLabel(e, Discovery)
DFS(G, w)
else
setLabel(e, BackEdge)

Depth First Search

A Unexplored Vertex

A Visited Vertex

Unexplored Edge
Discovery Edge
Back Edge

Start At Vertex A

A

B D E

C

A
Discovery Edge

B D E

C

A

B D E

Visited Vertex B
C

A

B D E

Discovery Edge
C

A

B D E

C

Visited Vertex C

A

B D E

C

Back Edge

A

B D E

C

Discovery Edge

A

B D E

C
Visited Vertex D

A

B D E

C
Back Edge

A

B D E

C Discovery Edge

A

B D E

C Visited Vertex E

A Discovery Edge

B D E

C

A B C D

E F G H

I J K L

M N O P

Breadth First Search
Algorithm BFS(G)
Input graph G
Output labeling of the edges
and a partitioning of the
vertices of G

for all u in G.vertices()
setLabel(u, Unexplored)

for all e in G.edges()
setLabel(e, Unexplored)

for all v in G.vertices()
if getLabel(v) = Unexplored
BFS(G, v)

Algorithm BFS(G, v)
L0 <-- new empty list
L0.insertLast(v)
setLabel(v, Visited)
i <-- 0

while(¬Li.isEmpty())
Li+1 <-- new empty list
for all v in G.vertices(v)
for all e in G.incidentEdges(v)
if getLabel(e) = Unexplored
w <-- opposite(v)
if getLabel(w) = Unexplored
setLabel(e, Discovery)
setLabel(w, Visited)
Li+1.insertLast(w)
else
setLabel(e, Cross)
i <-- i + 1

Start Vertex A
Create a sequence L0
A insert(A) into L0

B C D

E F

Start Vertex A
while L0 is not empty
L0 A create a new empty list L1

B C D

E F

Start Vertex A
for each v in L0 do
L0 A get incident edges of v

B C D

E F

Start Vertex A if first incident edge is unexplored
get opposite of v, say w
L0 A if w is unexplored
set edge as discovery

B C D

E F

Start Vertex A
set vertex w as visited
L0 A and insert(w) into L1

L1 B C D

E F

Start Vertex A get next incident edge
L0 A

L1 B C D

E F

Start Vertex A
if edge is unexplored
L0 A we get vertex opposite v
say w,
if w is unexplored

L1 B C D

E F

Start Vertex A
if w is unexplored
L0 set edge as discovery
A

L1 B C D

E F

Start Vertex A
set w as visited and
L0 add w to L1
A

L1 B C D

E F

Start Vertex A
continue in this fashion
L0 A until we have visited all
incident edge of v

L1 B C D

E F

Start Vertex A
as L0 is now empty
L0 A we continue with list L1

L1 B C D

E F

Start Vertex A
as L0 is now empty
L0 A we continue with list L1

L1 B C D

L2 E F

Start Vertex A

L0 A

L1 B C D

L2 E F

Weighted Graphs

• In a weighted graph G, each edge 'e' of G has associated with it,
a numerical value, known as a weight.

• Edge weights may represent distances, costs etc.

• Example: In a flight route graph the weights associated with each
graph edge could represent the distances between airports.

Shortest Paths

• Given a weighted graph G and two vertices 'u' and 'v' of G, we
require that we find a path between 'u' and 'v' that has a minimum
total weight between 'u' and 'v' also known as a (shortest path).

• The length of a path is the sum of the weights of the paths edges.

Dijkstra's Algorithm
• The distance of a vertex v from a vertex s is the length of a shortest
path between s and v

• Dijkstra’s algorithm computes the distances of all the vertices from
a given start vertex s

Assumptions:

 the graph is connected

 the edges are undirected

 the edge weights are nonnegative

Dijkstra's Algorithm
• We grow a “cloud” of vertices, beginning with s and eventually covering all the vertices

• We store with each vertex v a label d(v) representing the distance of v from s in the
subgraph consisting of the cloud and its adjacent vertices

At each step

 We add to the cloud the vertex u outside the cloud with the smallest distance label,
d(u)

 We update the labels of the vertices adjacent to u

Edge Relaxation

• Consider an edge e = (u,z) such that

 u is the vertex most recently added to the cloud
 z is not in the cloud

The relaxation of edge e updates distance d(z) as follows:
d(z) ← min{d(z),d(u) + weight(e)}

8 A 4
2
B 7 1 D
C
2 3 9 5
E F

Add starting vertex
to cloud.
8 A(0) 4
2
B 7 1 D
C
2 3 9 5
E F

We store with each vertex
v a label d(v) representing
the distance of v from s
8 A(0) 4
in the subgraph consisting
2 of the cloud and its adjacent
B 7 1 D vertices.
C
2 3 9 5
E F

We store with each vertex
v a label d(v) representing
the distance of v from s
8 A(0) 4 in the subgraph consisting
of the cloud and its adjacent
2
7 1 D(4)
vertices.
B(8)
C(2)

2 3 9 5
E F

At each step we add to the cloud
the vertex outside the cloud
8 A(0) 4 with the smallest
distance label d(v).
2
B(8) 7 1 D(4)
C(2)

2 3 9 5
E F

We update the vertices adjacent
to v.
d(v) = min{d(z), d(v) + weight(e)}
8 A(0) 4
2
B(8) 7 1 D(3)
C(2)

2 3 9 5
E(5) F(11)

with the smallest
distance label d(v).
8 A(0) 4
2
B(8) 7 1 D(3)
C(2)

2 3 9 5
E(5) F(11)

to v.
d(v) = min{d(z), d(v) + weight(e)}.
8 A(0) 4
2
B(8) 7 1 D(3)
C(2)

2 3 9 5
E(5) F(8)

with the smallest
8 A(0) 4 distance label d(v).

2
B(8) 7 1 D(3)
C(2)

2 3 9 5
E(5) F(8)

to v.
d(v) = min{d(z), d(v) + weight(e)}.
8 A(0) 4
2
B(7) 7 1 D(3)
C(2)

2 3 9 5
E(5) F(8)

with the smallest
8 A(0) 4 distance label d(v).

2
B(7) 7 1 D(3)
C(2)

2 3 9 5
E(5) F(8)

7 7
B C

2 3 3
2
2 D
A E F

1 2 2
6
4 H
G

7 7
B C

2 3 3
2
2 D
A(0) E F

1 2 2
6
4 H
G

Insert

7 7
B(2) C

2 3 3
2
2 D
A(0) E F

1 2 2
6
4 H
G(6)

Update

7 7
B(2) C

2 3 3
2
2 D
A(0) E F

1 2 2
6
4 H
G(6)

Insert

7 7
B(2) C(9)

2 3 3
2
2 D
A(0) E(4) F

1 2 2
6
4 H
G(6)

Update

7 7
B(2) C(9)

2 3 3
2
2 D
A(0) E(4) F

1 2 2
6
4 H
G(6)

Insert

7 7
B(2) C(9)

2 3 3
2
2 D
A(0) E(4) F(6)

1 2 2
6
4 H
G(5)

Update

7 7
B(2) C(9)

2 3 3
2
2 D
A(0) E(4) F(6)

1 2 2
6
4 H
G(5)

Insert

7 7
B(2) C(9)

2 3 3
2
2 D
A(0) E(4) F(6)

1 2 2
6
4 H(9)
G(5)

Update

7 7
B(2) C(9)

2 3 3
2
2
A(0) E(4) F(6) D

1 2 2
6
4 H(9)
G(5)

Insert

7 7
B(2) C(9)

2 3 3
2
2 D
A(0) E(4) F(6)

1 2 2
6
4 H(8)
G(5)

Update

7 7
B(2) C(9)

2 3 3
2
2 D
A(0) E(4) F(6)

1 2 2
6
4 H(8)
G(5)

Insert

7 7
B(2) C(9)

2 3 3
2
2 D(10
A(0) E(4) F(6) )

1 2 2
6
4 H(8)
G(5)

Update

7 7
B(2) C(9)

2 3 3
2
2 D(10
A(0) E(4) F(6) )

1 2 2
6
4 H(8)
G(5)

Insert

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