1. COSC 2007 Data Structures II
April 26, 2017
COSC 2007 Data Structures II
Chapter 14
Graphs II
Graphs

2. Topics Traversal Graph implementation Adjacency matrix adjacency list
Depth-first search
Breadth-first search

3. ADT Graphs
Insertion & deletion operations are somewhat different for graphs than for other ADTs in that they apply to either nodes or edges
Elements:
A graph consists of nodes and edges. Each node will contain one data element
Each node is uniquely identified by the key value of the element it contains
Structure:
An edge is a one-to-one relationship between a pair of distinct nodes
A pair of nodes can be connected by at most one edge, but any node can be connected to any collection of other nodes

4. ADT Graphs Operations: Create an empty graph
Determine whether a graph is empty
Determine the number of nodes in a graph
Determine the number of edges in a graph
Determine whether an edge exists between two nodes
Insert a node/ an edge
Delete a node/ an edge
Retrieve a node having a given search key
Traverse a graph

5. Implementation of Graphs
Two common implementations:
Adjacency matrix
Adjacency list
Adjacency Matrix (Incidence Matrix)
A graph containing n nodes can be represented by an n x n matrix of Boolean values or 1/0
M[i,j] = 1 (T) <==> If there is an edge between node i & node j of the graph.
0 (F) Otherwise

6. Implementation of Graphs
Adjacency Matrix
Example: A B L P C R A B L C R P T

7. Implementation of Graphs
Adjacency Matrix
The ith row and ith column are identical (Symmetric matrix)
Fewer than half the entries are needed, since each edge is recorded twice and the diagonal contains all zeros
Using a lower triangular matrix is possible, but it is not convenient
If the graph is directed, the full matrix, except the diagonal, is needed

8. Implementation of Graphs
Adjacency Lists
Keep an array of linked lists, one points to one LL
For each node, in the linked list, there is an edge list of the neighbors of that node
Any other possibility?

9. Implementation of Graphs
Maintain each adjacency list as a linked chain
Array h head nodes keeps track of the lists
h[i] points to the first node of adjacency list for vertex i G 1 2 3 4

10. Implementation of Graphs
Questions:

11. 2 3 1 4 5
Graph Traversal
Process each node in a graph exactly once (if & only if the graph is connected)
The order of visiting the nodes is not unique, because it depends on the representation

12. A B C F D E
Graph Traversal
An adjacency list representation of the graph could lead to many different orderings of the visit, depending on the sequence in which the edges are entered
If a graph contains a cycle, a graph-traversal algorithm can loop indefinitely, unless the visited nodes are marked

13. Graph Search Methods
A vertex u is reachable from vertex v iff there is a path from v to u. 2 3 1 4 5 8 9 10 6 7 11

14. COSC 2007
April 26, 2017
Graph Search Methods
A search method starts at a given vertex v and visits/labels/marks every vertex that is reachable from v. 2 3 8 10 1 4 5 9 11 6 7
Visit, mark, and label used as synonyms.
Graphs

15. Graph Search Methods Many graph problems solved using a search method
Path from one vertex to another
Is the graph connected?
Etc.
Commonly used search methods:
Depth-first search
Breadth-first search

16. Graph Traversal Depth First Search (DFS) Idea of DFS algorithm
Follow a path from a previous node as far as possible before moving to the next neighbor of the starting node
DFS uses a stack to put all nodes to be traversed (ready stack).
Idea of DFS algorithm
“Mark" a node as soon as we visit it
Try to move to an unmarked adjacent node
If there are no unmarked nodes at the present position, backtrack along the nodes already visited, until a node is reached that is adjacent to an unvisited node
Visit this node
Continue the process

17. Graph Traversal Depth First Search (DFS) Recursive dfs(in v:Vertex)
// Traverses a graph beginning at node v by using a
// depth-first search (Recursive version)
Mark v as visited
for (each unvisited node u adjacent to v)
dfs(u)

18. Depth-First Search Example2 3 8 10 1 4 5 9 11 6 7
Start search at vertex 1.

19. Graph Traversal dfs(v)
Depth First Search (DFS)
Iterative (Using a Stack)
dfs(v)
// Traverses a graph beginning at node v by using a DFS
s.CreateStack( )
s.push ( v) // Push v onto stack & mark it
Mark v as visited
while (!s.isEmpty( ) )
{ if (no unvisited nodes are adjacent to v on stack top)
s.pop () // Backtrack
else
{ Select an unvisited node u adjacent to the node on top of stack
s.push (u)
Mark u as visited
} // end if
} // end while

20. Graph Traversal Breadth First Search (BFS)
Look at all of the neighbors of a node first, then follow the paths from each of these nodes to its neighbors, and so on
Nodes that have been visited but not explored are kept in a (ready) queue, so that when we are ready to move to a node adjacent to the current node, we will be able to return to another node adjacent to the old current node after our move

21. Graph Traversal Breadth First Search (BFS) Iterative: Using Queue
bfs( v:Vertex)
// Traverses a graph beginning at node v by using a
// Breadth-first search
q.createQueue ( )
q.enqueue(v) // Add v to queue & mark it
Mark v as visited
while (!q.isEmpty ( ) )
{ q.dequeue(w)
for (each unvisited node u adjacent to w)
{ Mark u as visited
q.enqueue(u)
} // end for
} // end while

22. Breadth-First Search Visit start vertex and put into a FIFO queue.
Repeatedly remove a vertex from the queue, visit its unvisited adjacent vertices, put newly visited vertices into the queue.

23. Review A graph-traversal algorithm stops when ______.
it first encounters the designated destination vertex
it has visited all the vertices that it can reach
it has visited all the vertices
it has visited all the vertices and has returned to the vertex that it started from

24. Review
A ______ is the subset of vertices visited during a traversal that begins at a given vertex.
circuit
multigraph
digraph
connected component

25. Review An iterative DFS traversal algorithm uses a(n) ______. list
array
Queue
stack

26. Review An iterative BFS traversal algorithm uses a(n) ______. list
array
Queue
stack

27. Review A ______ is an undirected connected graph without cycles. tree
multigraph
digraph
connected component

28. Review
A connected undirected graph that has n vertices must have at least ______ edges. n
n – 1
n / 2
n * 2

29. Review
A connected undirected graph that has n vertices and exactly n – 1 edges ______.
cannot contain a cycle
must contain at least one cycle
can contain at most two cycles
must contain at least two cycles

30. Review A tree with n nodes must contain ______ edges. n n – 1 n – 2