1. Graph Operations And Representation

2. Sample Graph Problems Path problems. Connectedness problems.
Spanning tree problems.

3. Path Finding Path between 1 and 8. Path length is 20. 2 3 8 1 10 4 5 911 6 7
Vertices represent cities and edges represent highways. Edge weights are distances or driving times. Depending on the context, path length may either be the number of edges on the path or the sum of the weights of the edges on the path.
Path length is 20.

4. Another Path Between 1 and 82 3 8 10 1 4 5 9 11 6 7
Since a graph may have more than one path between two vertices, we may be interested in finding a path with a particular property. For example, find a path with minimum length
Path length is 28.

5. Example Of No Path 2 3 8 10 1 4 5 9 11 6 7
No path between 2 and 9.

6. Connected Graph Undirected graph.
There is a path between every pair of vertices.

7. Example Of Not Connected2 3 8 10 1 4 5 9 11 6 7

8. Connected Graph Example2 3 8 10 1 4 5 9 11 6 7
Determine whether an undirected graph is connected.

9. Connected Components 2 3 8 10 1 4 5 9 11 6 7

10. Connected Component A maximal subgraph that is connected.
Cannot add vertices and edges from original graph and retain connectedness.
A connected graph has exactly 1 component.

11. Not A Component 2 3 8 10 1 4 5 9 11 6 7
Determine connected components of an undirected graph

12. Communication Network2 3 8 10 1 4 5 9 11 6 7
Each edge is a link that can be constructed (i.e., a feasible link).

13. Communication Network Problems
Is the network connected?
Can we communicate between every pair of cities?
Find the components.
Want to construct smallest number of feasible links so that resulting network is connected.

14. Cycles And Connectedness2 3 8 10 1 4 5 9 11 6 7
Removal of an edge that is on a cycle does not affect connectedness.

15. Cycles And Connectedness2 3 8 1 10 4 5 9 11 6 7
Connected subgraph with all vertices and minimum number of edges has no cycles.

16. Tree Connected graph that has no cycles.
n vertex connected graph with n-1 edges.
In graph terminology, the term rooted tree is used to denote what we were earlier calling a tree (Chapter 12).

17. Spanning Tree
Subgraph that includes all vertices of the original graph.
Subgraph is a tree.
If original graph has n vertices, the spanning tree has n vertices and n-1 edges.

18. Minimum Cost Spanning Tree2 3 8 10 1 4 5 9 11 6 7
Tree cost is sum of edge weights/costs.

19. A Spanning Tree Spanning tree cost = 51. 2 4 3 8 8 1 6 10 2 4 5 4 4 39 8 11 5 6 2 7 6 7
Spanning tree cost = 51.

20. Minimum Cost Spanning Tree2 4 3 8 8 1 6 10 2 4 5 4 4 3 5 9 8 11 5 6 2
In the communication networks area, we are interested in finding minimum cost spanning trees. 7 6 7
Spanning tree cost = 41.

21. A Wireless Broadcast Tree2 4 3 8 8 1 6 10 2 4 5 4 4 3 5 9 8 11 5 6 2
Edge cost is power needed to reach a node. If vertex 1 broadcasts with power 2, only vertex 4 is reached. If it broadcasts with power 4, both 2 and 4 are reached. Min-broadcast rooted spanning tree is NP-hard. Cost of tree of previous slide becomes 26. 7 6 7
Source = 1, weights = needed power.
Cost = = 41.

22. Graph Representation Adjacency Matrix Adjacency Lists
Linked Adjacency Lists
Array Adjacency Lists

23. Adjacency Matrix 0/1 n x n matrix, where n = # of vertices
A(i,j) = 1 iff (i,j) is an edge 1 2 3 4 5 2 3 1 4 5 1 1 1 1 1 1 1 1 1 1

24. Adjacency Matrix Properties2 3 1 4 5
Diagonal entries are zero.
Adjacency matrix of an undirected graph is symmetric.
A(i,j) = A(j,i) for all i and j.

25. Adjacency Matrix (Digraph)2 3 1 4 5
Diagonal entries are zero.
Adjacency matrix of a digraph need not be symmetric.

26. Adjacency Matrix n2 bits of space
For an undirected graph, may store only lower or upper triangle (exclude diagonal).
(n-1)n/2 bits
O(n) time to find vertex degree and/or vertices adjacent to a given vertex.

27. Adjacency Lists
Adjacency list for vertex i is a linear list of vertices adjacent from vertex i.
An array of n adjacency lists.
aList[1] = (2,4)
aList[2] = (1,5)
aList[3] = (5)
aList[4] = (5,1)
aList[5] = (2,4,3) 2 3 1 4 5

28. Linked Adjacency Lists
Each adjacency list is a chain. 2 3 1 4 5
aList[1]
aList[5]
[2]
[3]
[4] 2 4 1 5 5 5 1 2 4 3
Array length n simply means we need an array with n spots. A direct implementation using a Java array would need n+1 spots, because spot 0 would not be utilized. However, by using spot 0 for vertex 1, spot 1 for vertex 2, and so on, we could get by with a Java array whose length is actually n.
Array Length = n
# of chain nodes = 2e (undirected graph)
# of chain nodes = e (digraph)

29. Array Adjacency Lists Each adjacency list is an array list. aList[1]2 3 1 4 5
aList[1]
aList[5]
[2]
[3]
[4]
Array Length = n
# of list elements = 2e (undirected graph)
# of list elements = e (digraph)

30. Weighted Graphs Cost adjacency matrix.
C(i,j) = cost of edge (i,j)
Adjacency lists => each list element is a pair (adjacent vertex, edge weight)

31. Number Of C++ Classes Needed
Graph representations
Adjacency Matrix
Adjacency Lists
Linked Adjacency Lists
Array Adjacency Lists
3 representations
Graph types
Directed and undirected.
Weighted and unweighted.
2 x 2 = 4 graph types
3 x 4 = 12 C++ classes

32. Abstract Class Graph template<class T> class graph { public:
// ADT methods come here
// implementation independent methods come here };

33. Abstract Methods Of Graph
// ADT methods
virtual ~graph() {}
virtual int numberOfVertices() const = 0;
virtual int numberOfEdges() const = 0;
virtual bool existsEdge(int, int) const = 0;
virtual void insertEdge(edge<T>*) = 0;
virtual void eraseEdge(int, int) = 0;
virtual int degree(int) const = 0;
virtual int inDegree(int) const = 0;
virtual int outDegree(int) const = 0;

34. Abstract Methods Of Graph
// ADT methods (continued)
virtual bool directed() const = 0;
virtual bool weighted() const = 0;
virtual vertexIterator<T>* iterator(int) = 0;
virtual void output(ostream&) const = 0;