CPCS204-15-Graphdatastructureinjava.pptx

Data Structures - I
“No Bonus” marks for this course anymore
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Dr. Jonathan (Yahya) Cazalas Dr. Muhammad Umair Ramzan
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O my Lord! Expand for me my chest [with
assurance] and ease for me my task and untie the
knot from my tongue that they may understand
my speech. (Quran 20 : 25-28)

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Graph
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Introduction

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Graph - Introduction
• Arrays
• Linked List
• Stacks
• Queues
• Tree
• Graph
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• Definition
oA graph is a non-linear and non-hierarchical
data structure that organizes data elements,
called vertices, by connecting them with links,
called edges
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edges (weight)
node or vertex

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• A graph G = (V,E) is composed of:
oV: set of vertices (nodes)
oE: set of edges (arcs) connecting the vertices V
• An edge e = (u,v) is a pair of vertices
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V= {a,b,c,d,e}
E=
{(a,b),(a,c),(a,d),(b,e),(c,d),(c,e),(d,e)}

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Graph Examples - nasair flight network
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Graph Examples – Computer Network
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Comcast
Regional Network
Intel UNT
Charter

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Graph - Application
• Traveling Salesman
• Find the shortest path that connects all
cities without a loop.
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Start

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Graph - Types
• Undirected Graph
oEdge has no
orientation
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• Directed Graph
oEdge has oriented
vertex

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Graph
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Terminologies

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Terminologies - Sub graph
• Subset of vertices and edges
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B
C
D
A
B
C
A
B
C
A
C
B
A
B
C A

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Terminologies –Path
• A path is a walk from one vertex to any other
vertex.
• The vertices traversed to reach to a vertex
from an other
oABCD is a path
oADC
oABDA
oBCDA
oABDCBA
oABCA
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B
C
D
A
path

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Terminologies – Simple Path
• A simple path is a path such that all vertices
are distinct
oABCD is a simple path
oADC
oABDA
oBCDA
oABDCBA
oABCA
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B
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D
A

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Terminologies – Cycle
• A cycle is a path that starts and ends at the
same point. For undirected graph, the
edges are distinct.
oCBDC is a Cycle
oABCD
oABDA
oBCDB
oABDCBA
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B
C
D
A

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Graph – Connected / Unconnected
• Connected Graph
oAll vertices are
connected
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• Unconnected Graph
oOne or more
vertices are not
linked

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Terminologies – DAG
• Directed Acyclic Graph (DAG) : directed
graph without cycle
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Terminologies – Weighted Graph
• Weighted graph: a graph with numbers
assigned to its edges
ocost, distance, travel time, etc.
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0
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20
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Terminologies – Complete Graph
• There is a direct edge between every two
vertices
• Total edges E = N(N-1)/2
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Terminologies – Sparse Graph
• There is a very small number of edges in
the graph
• Total edges E = N-1
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Graph
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Representation

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Graph - Representation
• Adjacency Matrix
• Adjacency List
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Representation – Adjacency Matrix
• Assume N nodes in graph
• Use Matrix A[0…N-1][0…N-1]
oif vertex i and vertex j are adjacent in graph,
A[i][j] = 1,
o otherwise A[i][j] = 0
oif vertex i has a loop, A[i][i] = 1
oif vertex i has no loop, A[i][i] = 0
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0
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A[i][j] 0 1 2 3
0 0 1 1 0
1 1 0 1 1
2 1 1 0 1
3 0 1 1 0
So, Matrix A =
0 1 1 0
1 0 1 1
1 1 0 1
0 1 1 0
Undirected Graph

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A[i][j] 0 1 2 3
0 0 1 1 1
1 0 0 0 1
2 0 0 0 1
3 0 0 0 0
0
1
3
2
So, Matrix A =
0 1 1 1
0 0 0 1
0 0 0 1
0 0 0 0
Directed Graph

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A[i][j] 0 1 2 3
0 0 20 10 1
1 20 0 0 5
2 10 0 0 4
3 1 5 4 0
0
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4
5
So, Matrix A =
0 20 10 1
20 0 0 5
10 0 0 4
1 5 4 0
Weighted Graph

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• Undirected graph
oadjacency matrix is symmetric
oA[i][j]=A[j][i]
• Directed graph
oadjacency matrix may not be symmetric
oA[i][j]A[j][i]
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A[i][j] 0 1 2 3
0 0 1 1 1
1 0 0 0 1
2 0 0 0 1
3 0 0 0 0
A[i][j] 0 1 2 3
0 0 1 1 0
1 1 0 1 1
2 1 1 0 1
3 0 1 1 0

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Representation – Adjacency List
• An array of lists
• The ith element of the array is a list of
vertices that connect to vertex i
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0
1
3
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0
1
2
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1 2 3
3
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Directed Graph

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• An array of lists
• The ith element of the array is a list of
vertices that connect to vertex i
28 Undirected Graph
0
1
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1 2 3
0 3
0 3
0 1 2

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• Extend each node with an addition field:
weight
29 Weighted Graph
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20 10
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1 10 2 20 3 1
0 10 3 4
0 20 3 5
0 1 1 4 2 5

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Representation – Comparisons
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Cost
Adjacency
Matrix
Adjacency
List
Given two vertices u and v:
find out whether u and v are
adjacent
O(1)
degree of
node
O(N)
Given a vertex u:
enumerate all neighbors of u
O(N)
degree of
node
O(N)
For all vertices:
enumerate all neighbors of
each vertex
O(N2)
Summation
s of all node
degree
O(E)

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Graph
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Traversal

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Graph - Traversal
• Depth First Search (DFS) Traversal
• Breadth First Search (BFS) Traversal
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Traversal - DFS
• Probing List is implemented as stack (LIFO)
• Example
oA’s neighbor: B, C, E
oB’s neighbor: A, C, F
oC’s neighbor: A, B, D
oD’s neighbor: E, C, F
oE’s neighbor: A, D
oF’s neighbor: B, D
oStart from vertex A
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A
B C E
F D

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• Initial State
oVisited Vertices { }
oProbing Vertices { A }
oUnvisited Vertices { A, B, C, D, E, F }
A
B
C E
F D
– A’s neighbor: B C E
– B’s neighbor: A C F
– C’s neighbor: A B D
– D’s neighbor: E C F
– E’s neighbor: A D
– F’s neighbor: B D
A
stack
Traversal - DFS

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• Pick a vertex from stack, it is A, mark it as
visited
• Find A’s first unvisited neighbor, push it into
stack
o Visited Vertices { A }
o Probing vertices { A, B }
o Unvisited Vertices { B, C, D, E, F }
A
B
C E
F D
B
A
stack
A
Traversal - DFS

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• Pick a vertex from stack, it is B, mark it as
visited
• Find B’s first unvisited neighbor, push it in
stack
o Visited Vertices { A, B }
o Probing Vertices { A, B, C }
o Unvisited Vertices { C, D, E, F }
C
B
A
stack
B
A
A
B
C E
F D
Traversal - DFS

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• Pick a vertex from stack, it is C, mark it as
visited
• Find C’s first unvisited neighbor, push it in
stack
o Visited Vertices { A, B, C }
o Probing Vertices { A, B, C, D }
o Unvisited Vertices { D, E, F }
stack
A
B
C E
F D
D
C
B
A
C
B
A
Traversal - DFS

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• Pick a vertex from stack, it is D, mark it as
visited
• Find D’s first unvisited neighbor, push it in
stack
o Visited Vertices { A, B, C, D }
o Probing Vertices { A, B, C, D, E }
o Unvisited Vertices { E, F }
stack
A
B
C E
F D
D
C
B
E
A
D
C
B
A
Traversal - DFS

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• Pick a vertex from stack, it is E, mark it as
visited
• Find E’s first unvisited neighbor, no vertex
found, Pop E
o Visited Vertices { A, B, C, D, E }
o Probing Vertices { A, B, C, D }
o Unvisited Vertices { F }
stack
A
B
C E
F D
D
C
B
A
D
C
B
E
A
Traversal - DFS

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visited
• Find D’s first unvisited neighbor, push it in
stack
o Visited Vertices { A, B, C, D, E }
o Probing Vertices { A, B, C, D, F}
o Unvisited Vertices { F }
stack
A
B
C E
F D
D
C
B
F
A
D
C
B
A
Traversal - DFS

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• Pick a vertex from stack, it is F, mark it as
visited
• Find F’s first unvisited neighbor, no vertex
found, Pop F
o Visited Vertices { A, B, C, D, E, F }
o Probing Vertices { A, B, C, D}
o Unvisited Vertices { } stack
A
B
C E
F D
D
C
B
A
D
C
B
F
A
Traversal - DFS

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visited
• Find D’s first unvisited neighbor, no vertex
found, Pop D
o Probing Vertices { A, B, C }
A
B
C E
F D
C
B
A
D
C
B
A
Traversal - DFS

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• Pick a vertex from stack, it is C, mark it as
visited
• Find C’s first unvisited neighbor, no vertex
found, Pop C
o Probing Vertices { A, B }
A
B
C E
F D
B
A
C
B
A
Traversal - DFS

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• Pick a vertex from stack, it is B, mark it as
visited
• Find B’s first unvisited neighbor, no vertex
found, Pop B
o Probing Vertices { A }
A
B
C E
F D
A
B
A
Traversal - DFS

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• Pick a vertex from stack, it is A, mark it as
visited
• Find A’s first unvisited neighbor, no vertex
found, Pop A
o Probing Vertices { }
A
B
C E
F D
A
Traversal - DFS

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• Now probing list is empty
• End of Depth First Traversal
oVisited Vertices { A, B, C, D, E, F }
oProbing Vertices { }
oUnvisited Vertices { }
stack
A
B
C E
F D
Traversal - DFS

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• Probing List is implemented as queue
(FIFO)
• Example
oA’s neighbor: B C E
oB’s neighbor: A C F
oC’s neighbor: A B D
oD’s neighbor: E C F
oE’s neighbor: A D
oF’s neighbor: B D
oStart from vertex A
A
B
C E
F D
Traversal – Breadth First Search

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• Initial State
oVisited Vertices { }
oProbing Vertices { A }
oUnvisited Vertices { A, B, C, D, E, F }
A
B
C E
F D
A
queue
Traversal – BFS

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• Delete first vertex from queue, it is A, mark it as
visited
• Find A’s all unvisited neighbors, mark them as
visited, put them into queue
o Visited Vertices { A, B, C, E }
o Probing Vertices { B, C, E }
o Unvisited Vertices { D, F }
A
B
C E
F D
A
queue
B E
C
Traversal – BFS

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• Delete first vertex from queue, it is B, mark it as
visited
• Find B’s all unvisited neighbors, mark them as
o Visited Vertices { A, B, C, E, F }
o Probing Vertices { C, E, F }
o Unvisited Vertices { D }
A
B
C E
F D
B E
C
queue
C F
E
Traversal – BFS

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• Delete first vertex from queue, it is C, mark it as
visited
• Find C’s all unvisited neighbors, mark them as
o Visited Vertices { A, B, C, E, F, D }
o Probing Vertices { E, F, D }
o Unvisited Vertices { }
A
B
C E
F D
C F
E
queue
E D
F
Traversal – BFS

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• Delete first vertex from queue, it is E, mark it as
visited
• Find E’s all unvisited neighbors, no vertex found
o Probing Vertices { F, D }
A
B
C E
F D
E D
F
queue
F D
Traversal – BFS

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• Delete first vertex from queue, it is F, mark it as
visited
• Find F’s all unvisited neighbors, no vertex found
o Probing Vertices { D }
A
B
C E
F D
F D
queue
D
Traversal – BFS

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• Delete first vertex from queue, it is D, mark it as
visited
• Find D’s all unvisited neighbors, no vertex found
o Probing Vertices { }
A
B
C E
F D
D
queue
Traversal – BFS

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• Now the queue is empty
• End of Breadth First Traversal
oVisited Vertices { A, B, C, E, F, D }
oProbing Vertices { }
oUnvisited Vertices { }
A
B
C E
F D
queue
Traversal – BFS

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• Depth First Search(DFS)
oOrder of visited: A, B, C, D, E, F
• Breadth First Search (BFS)
oOrder of visited: A, B, C, E, F, D
A
B
C E
F D
Difference – DFS & BFS

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57
Allah (Alone) is Sufficient for us,
and He is the Best Disposer of
affairs (for us).(Quran 3 : 173)

CPCS204-15-Graphdatastructureinjava.pptx

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