1. CS212: Data Structures and Algorithms
Graphs

2. Outline Adjacency Matrix Adjacency list Linked list Depth First
Graphs (Basic Concepts)
Implementations of Graphs
Adjacency Matrix
Adjacency list
Linked list
Graph Traversals
Depth First
Breadth First

3. Graphs
A graph G = (V, E) consists of a set of vertices, V, and a set of edges, E, Each edge is a pair (v, w), where v, w  V
Edges are sometimes referred to as arcs
If the pair (V, W) is ordered, then the graph is directed (Diagraphs)
Vertex w is adjacent to v, if and only if (v,w)  E
For undirected graph, edge (v, w) or (w, v) => w is adjacent to v and v is adjacent to w
Sometimes an edge (v, w), has a third component, (v, w, c), known as either a weight or a cost

4. Graphs
A path in a graph is a sequence of vertices w1, w2, w3, …… wn such that (wi, wi+1)  E for 1<=i<n
The length is the number of edges, equals to n-1
A path from a vertex to itself with length 0
Path(v,v) is referred to as a loop

5. Undirected Graphs Examples1 2 3 4
(Connected)
Vertices {1, 2, 3, 4}
Edges {(1,2), (1, 3), (2, 3), (2, 4), (3, 4)} 1 2 3 4
Path = {(1, 2),(2,4), (4, 3)}

6. Undirected Graphs Examples1 2 3 4
Tree (Graphs without Cycle) 1 2 3 4
Disconnected

7. Graphs
A Simple Path: When all vertices are distinct, except that the first and the last could be the same
A cycle in a directed graph is a path of length at least 1, such that w1 = wn
This cycle is simple if the path is simple
A path u, v, u in an undirected graph should not be considered as a cycle, because u, v and v, u are the same edge
In a directed graph, these are different
A directed graph is acyclic if it has no cycles (D.A.G)

8. Graphs
An undirected graph is connected, if there is a path from every vertex to every other vertex
A directed graph with this property is called strongly connected
If a directed graph is not strongly connected, but the underlying graph (without directions) is connected, then the graph is weakly connected
A complete graph is a graph in which there is an edge between every pair of vertices

9. Directed Graphs Examples
Strongly Connected
Directed Cycle
Weakly Connected

10. Graphs Applications Airport System: Airport vertices and flights edges
Examples of a real life situation that can be modeled by graphs are
Airport System: Airport vertices and flights edges
Traffic flow: Each street intersection represents a vertex, and each street is an edge

11. Implementations of Graphs
Static Implementation
Uses a two-dimensional array of integers
Also called Adjacency Matrix Implementation
Adjacency List Implementation
Uses a one-dimensional array of pointers
Linked List Implementation
Uses a linked list of pointers
Provides a complete dynamic implementation

12. 1. Static Implementation of Graph2 3 4 5 6 7 1 2 5 3 7 6 4

13. 1. Static Implementation of Graph
class SGraph {
private:
int noOfVertices;
bool **aMatrix;
public:
SGraph(int nv) { //creates adjacency matrix to represent graph
noOfVertices = nv;
//Dynamically creates rows of two dimensional array
aMatrix = new bool*[noOfVertices];
//Dynamically creates columns of two dimensional array
for(int i = 0; i<noOfVertices; i++)
aMatrix[i] = new bool[noOfVertices]; }
Class continues on next slide…

14. 1. Static Implementation of Graph
void insertGraph() //inserts graph in adjacency matrix {
char choice;
for(int i = 0; i<noOfVertices; i++)
for(int j = 0; j<noOfVertices; j++) {
cout<<"is there an edge b/w vertex"<<i<<"and"<<j;
cin>>choice;
if(choice == 'y' || choice == 'Y')
aMatrix[i][j] = 1;
else
aMatrix[i][j] = 0; } };

15. 2. Adjacency List Implementation
For each vertex keep a list of all adjacent vertices 1 2 3 4 5 6 7 1 2 5 3 7 6 4

16. 2. Adjacency List Implementation
class DGraph {
private:
int noOfVertices;
SinglyLinkedList<int> *aList;
public:
//creates adjacency list to represent graph
DGraph(int nv)
noOfVertices = nv;
aList = new SinglyLinkedList<int>[noOfVertices]; }
Class continues on next slide…

17. 2. Adjacency List Implementation
//inserts graph in adjacency list
void insertGraph()
{ for(int i = 0; i<noOfVertices; i++)
{ int vertexNo;
cout<<"Enter a number of vertex adjacent to vertex" <<i<<" Enter -1 to terminate";
cin>>vertexNo;
while(vertexNo != -1)
{ aList[i].addToTail(vertexNo); } };

18. Implementations of Graphs
In most real life situations, the vertices have names, which are known at compile time, instead of numbers
We have to provide a mapping of names to numbers using a hash table

19. Graph Traversals
Traversing a graph consists of visiting each vertex only once
We can traverse the graph by two ways:
Depth first
Breadth first

20. Depth First Graph Traversal
Depth-first search is a generalization of preorder traversal for trees
It is basically a Last visited, First explored strategy
Let V be the last vertex visited and N1, N2, …, Nk are adjacent to V
A depth-first traversal will:
visit N1, then
proceed to traverse all the unvisited successors of N1, then
proceed to traverse the remaining adjacent vertices of V in similar fashion

21. Depth First Graph Traversal
Declare an array visited of type bool and initialize it to false
Each index of this array indicates weather a particular vertex is visited or not

22. Depth First Graph Traversal
//Algorithm for Depth First Traversal
void depthFirstTraversal(int TotalVertices) {
bool *visited = new int[TotalVertices];
for(int i = 0; i < TotalVertices; i++)
visited[i] = false;
while there is a vertex v which is not visited
DFT(v, visited) }
Algorithm Continues on next slide…

23. Depth First Graph Traversal
//Algorithm for DFT
void DFT(Vertex v, bool visited[ ])
{ Stack s;
s.push(v);
while(!s.isempty())
{ v = s.pop();
if(visited[v] == false)
{ visited[v] = true;
for each vertex u adjacent to v
if(visited[u] == false)
s.push(u); }

24. Breadth First Graph Traversal
It is basically a level order traversal
Let V be the last node visited and N1, N2, …, Nk are adjacent to V
A breadth-first traversal will
visit N1, then N2, and so forth through Nk, then
proceed to traverse all the unvisited vertices adjacent to N1, then
traverse the unvisited vertices adjacent to N2,…Nk in similar fashion

25. Breadth First Graph Traversal
//Algorithm for Breadth First Traversal
breadthFirstTraversal(Vertex v) {
Add V to Queue & mark it
While (Queue not Empty) {
Remove Vertex from Queue & call it w
for (each unvisited vertex u adjacent to w) mark u visited
add u in Queue }

26. Breadth First Graph TraversalC D B E F G I H

27. Breadth First Graph TraversalC
Queue B A C D B E F G I H

28. Breadth First Graph Traversal
Dqueue C B A C D B E F G I H

29. Breadth First Graph Traversal
Enqueu D D A C D B E F G I H

30. Breadth First Graph Traversal
Dqueue B D A C D B E F G I H

31. Breadth First Graph Traversal
Enqueue E, I, H E I H A C D B E F G I H

32. Breadth First Graph Traversal
Dqueue D E I H A C B D E I H F G

33. Breadth First Graph Traversal
Dqueue E I H A C B D E I H F G

34. Breadth First Graph Traversal
Enqueue F, G H F G A C D B E F G I H

35. Breadth First Graph Traversal
Dqueue I H F G A C D B E F G I H

36. Breadth First Graph Traversal
Dqueue I H F G A C D B E F G I H
H already in queue i.e. visited

37. Breadth First Graph Traversal
Dqueue H F G A C B D E I H F G

38. Breadth First Graph Traversal
Dqueue F G A C B D E I H F G

39. Breadth First Graph Traversal
Dqueue G A C B D E I H F G