Graphs

1. Graphs Chapter 29

2. Some Examples and Terminology Road Maps Airline Routes Mazes Course Prerequisites Trees Traversals Breadth-First Traversal Dept-First Traversal Topological Order Paths Finding a Path Shortest Path in an Unweighted Graph Shortest Pat in a Weighted Graph Java Interfaces for the ADT Graph Chapter Contents

3. Some Examples and Terminology • Vertices or nodes are connected by edges • A graph is a collection of distinct vertices and distinct edges • Edges can be directed or undirected • When it has directed edges it is called a digraph • A subgraph is a portion of a graph that itself is a graph

4. Road Maps Nodes Edges Fig. 29-1 A portion of a road map.

5. Road Maps Fig. 29-2 A directed graph representing a portion of a city's street map.

6. Paths • A sequence of edges that connect two vertices in a graph • In a directed graph the direction of the edges must be considered • Called a directed path • A cycle is a path that begins and ends at same vertex • Simple path does not pass through any vertex more than once • A graph with no cycles is acyclic

7. Weights • A weighted graph has values on its edges • Weights or costs • A path in a weighted graph also has weight or cost • The sum of the edge weights • Examples of weights • Miles between nodes on a map • Driving time between nodes • Taxi cost between node locations

8. Weights Fig. 29-3 A weighted graph.

9. Connected Graphs • A connected graph • Has a path between every pair of distinct vertices • A complete graph • Has an edge between every pair of distinct vertices • A disconnected graph • Not connected

10. Connected Graphs Fig. 29-4 Undirected graphs

11. Adjacent Vertices • Two vertices are adjacent in an undirected graph if they are joined by an edge • Sometimes adjacent vertices are called neighbors Fig. 29-5 Vertex A is adjacent to B, but B is not adjacent to A.

12. Airline Routes • Note the graph with two subgraphs • Each subgraph connected • Entire graph disconnected Fig. 29-6 Airline routes

13. Mazes Fig. 29-7 (a) A maze; (b) its representation as a graph

14. Course Prerequisites Fig. 29-8 The prerequisite structure for a selection of courses as a directed graph without cycles.

15. Trees • All trees are graphs • But not all graphs are trees • A tree is a connected graph without cycles • Traversals • Preorder, inorder, postorder traversals are examples of depth-first traversal • Level-order traversal of a tree is an example of breadth-first traversal • Visit a node • For a tree: process the node's data • For a graph: mark the node as visited

16. Trees Fig. 29-9 The visitation order of two traversals; (a) depth first; (b) breadth first.

17. Breadth-First Traversal • Algorithm for breadth-first traversal of nonempty graph beginning at a given vertex Algorithm getBreadthFirstTraversal(originVertex)vertexQueue = a new queue to hold neighborstraversalOrder = a new queue for the resulting traversal orderMark originVertex as visitedtraversalOrder.enqueue(originVertex)vertexQueue.enqueue(originVertex)while (!vertexQueue.isEmpty()){ frontVertex = vertexQueue.dequeue()while (frontVertex has an unvisited neighbor) { nextNeighbor = next unvisited neighbor of frontVertexMark nextNeighbor as visitedtraversalOrder.enqueue(nextNeighbor) vertexQueue.enqueue(nextNeighbor) }}return traversalOrder A breadth-first traversal visits a vertex and then each of the vertex's neighbors before advancing

18. Breadth-First Traversal Fig. 29-10 (ctd.) A trace of a breadth-first traversal for a directed graph, beginning at vertex A.

19. Depth-First Traversal • Visits a vertex, then • A neighbor of the vertex, • A neighbor of the neighbor, • Etc. • Advance as possible from the original vertex • Then back up by one vertex • Considers the next neighbor

20. Depth-First Traversal Fig. 29-11 A trace of a depth-first traversal beginning at vertex A of the directed graph in Fig. 29-10a.

21. Topological Order • Given a directed graph without cycles • In a topological order • Vertex a precedes vertex b whenever • A directed edge exists from a to b

22. Topological Order Fig. 29-12 Three topological orders for the graph of Fig. 29-8.

23. Topological Order Fig. 29-13 An impossible prerequisite structure for three courses as a directed graph with a cycle.

24. Topological Order • Algorithm for a topological sort Algorithm getTopologicalSort()vertexStack = a new stack to hold vertices as they are visitedn = number of vertices in the graphfor (counter = 1 to n){ nextVertex = an unvisited vertex whose neighbors, if any, are all visited Mark nextVertex as visitedstack.push(nextVertex)}return stack

25. Topological Order Fig. 29-14 Finding a topological order for the graph in Fig. 29-8.

26. Shortest Path in an Unweighted Graph Fig. 29-15 (a) an unweighted graph and (b) the possible paths from vertex A to vertex H.

27. Shortest Path in an Unweighted Graph Fig. 29-16 The graph in 29-15a after the shortest-path algorithm has traversed from vertex A to vertex H

28. Shortest Path in an Unweighted Graph Fig. 29-17 Finding the shortest path from vertex A to vertex H in the unweighted graph in Fig. 29-15a.

29. Shortest Path in an Weighted Graph Fig. 29-18 (a) A weighted graph and (b) the possible paths from vertex A to vertex H.

30. Shortest Path in an Weighted Graph • Shortest path between two given vertices • Smallest edge-weight sum • Algorithm based on breadth-first traversal • Several paths in a weighted graph might have same minimum edge-weight sum • Algorithm given by text finds only one of these paths

31. Shortest Path in an Weighted Graph Fig. 29-19 Finding the cheapest path from vertex A to vertex H in the weighted graph in Fig 29-18a.

32. Shortest Path in an Weighted Graph Fig. 29-20 The graph in Fig. 29-18a after finding the cheapest path from vertex A to vertex H.

33. Java Interfaces for the ADT Graph • Methods in the BasicGraphInterface • addVertex • addEdge • hasEdge • isEmpty • getNumberOfVertices • getNumberOfEdges • clear

34. Java Interfaces for the ADT Graph Operations of the ADT graph enable creation of a graph and answer questions based on relationships among vertices Fig. 29-21 A portion of the flight map in Fig. 29-6.