1. Graph Implementations
Chapter 29
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Contents
An Overview of Two Implementations
The Adjacency Matrix
The Adjacency List
Vertices and Edges
Specifying the Class Vertex
The Inner Class Edge
Implementing the Class Vertex
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Contents
An Implementation of the ADT Graph
Basic Operations
Graph Algorithms
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Objectives
Describe adjacency matrix
Describe adjacency list
Specify and implement classes that represent vertices and edges of a graph
Implement ADT graph by using adjacency lists
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Adjacency Matrix
For a graph of n vertices
Has n rows, n columns
Rows and columns numbered 0 to n -1
Each row and column corresponds to vertex of graph
Note:
Sparse graph wastes space
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Figure 29-1 (a) An unweighted, directed graph and (b) its adjacency matrix
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Question 1Consider the graph in Figure 28-4b of the previous chapter. Number the vertices from 0 through 3, starting at the vertex in the upper left corner and moving in a clockwise direction. What adjacency matrix represents this graph?
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Question 1 Consider the graph in Figure 28-4b of the previous chapter. Number the vertices from 0 through 3, starting at the vertex in the upper left corner and moving in a clockwise direction. What adjacency matrix represents this graph?
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The Adjacency List
Represents only those edges that originate from vertex
Space not reserved for edges that do not exist
Use less memory than corresponding adjacency matrix
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10. Figure 29-2 Adjacency lists for the directed graph in Figure 29-1a
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11. Figure 29-2 Adjacency lists for the directed graph in Figure 29-1a
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Question 2 What adjacency lists represent the graph described in Question 1?
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Question 2 What adjacency lists represent the graph described in Question 1?
2. Vertex 0 references the list 1, 2, 3.
Vertex 1 references the list 0, 2, 3.
Vertex 2 references the list 0, 1, 3.
Vertex 3 references the list 0, 1, 2.
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14. Specifying the Class Vertex
Need a way to identify vertices
Operations that mark a vertex as visited
Adjacency list indicates its neighbors
Path operations:
Get
Set
Test
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15. Specifying the Class Vertex
Note interface, Listing 29-1
Inner class Edge, Listing 29-2
Class Vertex, Listing 29-3
Private class neighborIterator, Listing 29-4
Note: Code listing files
must be in same folder
as PowerPoint files
for links to work
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Question 3 Given the interface VertexInterface and the class Vertex , write Java statements that create the vertices and edge s for the following directed, weighted graph. This graph contains three vertices— A , B, and C —and four edges, as follows: A → B, B → C , C → A, A → C . These edges have the weights 2, 3, 4, and 5, respectively.
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Question 3 Given the interface VertexInterface and the class Vertex , write Java statements that create the vertices and edge s for the following directed, weighted graph. This graph contains three vertices— A , B, and C —and four edges, as follows: A → B, B → C , C → A, A → C . These edges have the weights 2, 3, 4, and 5, respectively.
3. VertexInterface<String> vertexA = new Vertex<String>("A");
VertexInterface<String> vertexB = new Vertex<String>("B");
VertexInterface<String> vertexC = new Vertex<String>("C");
vertexA.addEdge(vertexB, 2.0);
vertexB.addEdge(vertexC, 3.0);
vertexC.addEdge(vertexA, 4.0);
vertexA.addEdge(vertexC, 5.0);
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18. Implementation of the ADT Graph
With either implementation
Need container for the graph’s vertices
Could be list or dictionary
Depending on how object is identified
We will use dictionary
Identify objects with a string
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19. Class DirectedGraph, Listing 29-5
Figure 29-3 (a) A directed graph and (b) its implementation using adjacency lists
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Question 4 Create an instance of the class DirectedGraph for the graph described in Question 3.
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Question 4 Create an instance of the class DirectedGraph for the graph described in Question 3. 4.
DirectedGraph<String> myGraph = new DirectedGraph<String>();
myGraph.addVertex("A");
myGraph.addVertex("B");
myGraph.addVertex("C");
myGraph.addEdge("A", "B", 2.0);
myGraph.addEdge("B", "C", 3.0);
myGraph.addEdge("C", "A", 4.0);
myGraph.addEdge("A", "C", 5.0);
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Figure 29-4 The performance of basic operations of the ADT graph when implemented by using adjacency lists
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Graph Algorithms
Breadth-first traversal, Listing 29-A
Depth-first traversal left as exercise
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Question 5 Write Java statements that display the vertices in a breadth-first traversal of the graph that you created in Question 4, beginning with vertex A.
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Question 5 Write Java statements that display the vertices in a breadth-first traversal of the graph that you created in Question 4, beginning with vertex A. 5.
QueueInterface<String> bfs = myGraph.getBreadthFirstTraversal("A");
while (!bfs.isEmpty())
System.out.print(bfs.dequeue() + " ");
System.out.println();
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Graph Algorithms
Breadth-first traversal, Listing 29-A
Depth-first traversal left as exercise
Shortest path, Listing 29-B
Implementation of method getCheapestPath for weighted graph left as an exercise
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Question 6 Write Java statements that display the vertices in the shortest path from vertex A to vertex C for the graph that you created in Question 4. Also, display the length of this path.
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Question 6 Write Java statements that display the vertices in the shortest path from vertex A to vertex C for the graph that you created in Question 4. Also, display the length of this path. 6.
StackInterface<String> path = new LinkedStack<String>();
int pathLength = myGraph.getShortestPath("A", "C", path);
System.out.println("The shortest path from A to C has length " + pathLength +
" and passes through the following vertices:");
while (!path.isEmpty())
System.out.print(path.pop() + " ");
System.out.println();
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End
Chapter 29
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