1. Chapter 28 Graphs and Applications Part1. Non-weighted Graphs CIS265/506 Cleveland State University – Prof. Victor Matos Adapted from: Introduction to Java Programming: Comprehensive Version, Eighth Edition by Y. Daniel Liang

2. Modeling Using Graphs 2

3. Weighted Graph Animation 3 www.cs.armstrong.edu/liang/animation/GraphLearningTool.html

4. Seven Bridges of Königsberg 4 The city, islands, and bridge could be abstracted as a graph holding vertices and edges (bridges). The problem was to find a walk through the graph that would begin and end at the same vertex crossing each edge exactly once. The city of Konigsberg is connected to two islands through seven bridges.

5. Seven Bridges of Königsberg 5 No solution!! Def. A path that traverses all bridges once and also has the same starting and ending is called an Eulerian circuit. Such a circuit exists if, and only if, the graph is connected, and there are no nodes of odd degree at all. A node degree is the count of edges adjacent to the node. Leonhard Euler, 1707

6. Basic Graph Terminology 6 1.What is a graph? 2.Directed vs. undirected graphs 3.Weighted vs. non-weighted graphs 4.Adjacent vertices 5.Incident 6.Degree 7.Neighbor 8.loop Leonhard Euler, 1707-1785

7. Basic Graph Terminology 7 1.What is a graph? A graph G(V, E) is a representation of a set of vertices V and edges E. Two vertices v1 and v2 in V are related iff there is and edge e 12 in E connecting v1 and v2. Leonhard Euler, 1707-1785

8. Basic Graph Terminology 8 1.What is a graph?

9. Basic Graph Terminology 9 2.Node, vertice, vertex 3.Parallel edge 4.Loops 5.Complete graph 6.Simple graph (connected, undirected, no loops, no parallel weighted)

10. Basic Graph Terminology 10 7.Spanning tree T of a connected, undirected graph G is a tree composed of all the vertices and some (or perhaps all) of the edges of G. A graph may have several Spanning trees !

11. Java: Representing Graphs 11 Representing Vertices Representing Edges: Edge Array Edge Objects (POJO) Adjacency Matrices Adjacency Lists

12. Representing Vertices 12 String[ ] vertices = {“Seattle“, “San Francisco“, … }; City[ ] vertices = {city0, city1, … }; public class City { private String cityName; private int population;... } List vertices; A vertex could be an object of any type.

13. Representing Edges: Edge Array 13 int[][] edges = {{0, 1}, {0, 3} {0, 5}, {1, 0}, {1, 2}, … }; There is an edge between city0 and city1

14. Representing Edges: Edge Object public class Edge { int v1, v2; public Edge(int v1, int v2) { this.v1 = v1; this.v 2 = v2; } List list = new ArrayList (); list.add( new Edge(0, 1) ); list.add( new Edge(0, 3) ); …

15. Representing Edges: Adjacency Matrix 15 int[][] adjacencyMatrix = { {0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0}, // Seattle {1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0}, // San Francisco {0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0}, // Los Angeles {1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0}, // Denver {0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0}, // Kansas City {1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0}, // Chicago {0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0}, // Boston {0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0}, // New York {0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1}, // Atlanta {0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1}, // Miami {0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1}, // Dallas {0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0} // Houston }; Entry adjacencyMatrix[i][j] is 1 if there is an edge connecting city-i and city-j

16. Representing Edges: Adjacency List 16 List [ ] neighbors = new LinkedList [12];

17. Modeling Graphs 17 In a style similar to the Java Collection Framework we will define 1.an interface named Graph that contains all common operations of graphs and 2.an abstract class named AbstractGraph that partially implements the Graph interface. 3.Many concrete graphs may be added to the design. For example, we will define such graphs named UnweightedGraph and WeightedGraph.

18. Modeling Graphs 18

19. Modeling Graphs 19

20. Graph Traversals - Trees 20  Depth-first search (DFS) and breadth-first search (BFS)  Both traversals result in a spanning tree, which can be modeled using a class.

21. Depth-First Search (DFS) 21 1. In the case of a tree, DFS search starts from the root (similar to pre-order navigation). 2. In a graph, the search can start from any vertex. dfs(vertex v) { visit v; for each neighbor w of v if (w has not been visited) { dfs(w); }

22. DFS Depth-First Search Example 22 Begin at node 0

23. DFS Depth-First Search Example 23

24. Applications of the DFS 24 1.Detecting whether a graph is connected. Search the graph starting from any vertex. If the number of vertices searched is the same as the number of vertices in the graph, the graph is connected. Otherwise, the graph is not connected 2.Detecting / finding a path between two vertices. 3.Finding all connected components. A connected component is a maximal connected subgraph in which every pair of vertices are connected by a path. 4.Detecting / Finding a cycle in a graph.

25. Breadth-First Search (BFS) 25 With breadth-first traversal of a tree, the nodes are visited level by level. 1. First the root is visited, 2. then all the children of the root, 3. then the grandchildren of the root from left to right, 4. and so on.

26. BFS Breadth-First Search Algorithm 26 bfs(vertex v) { create an empty queue for storing vertices to be visited; add v into the queue; mark v visited; while the queue is not empty { dequeue a vertex, say u, from the queue process u; for each neighbor w of u if w has not been visited { add w into the queue; mark w visited; }

27. BFS Breadth-First Search Example 27 Queue: 0 Queue: 1 2 3 Queue: 2 3 4 isVisited[0] = true isVisited[1] = true isVisited[2] = true isVisited[3] = true isVisited[4] = true

28. BFS Breadth-First Search Example 28

29. Applications of the BFS 29 1.Detecting whether a graph is connected. A graph is connected if there is a path between any two vertices in the graph. 2.Detecting whether there is a path between two vertices. 3.Finding a shortest path between two vertices. You can prove that the path between the root and any node in the BFS tree is the shortest path between the root and the node 4.Finding all connected components. A connected component is a maximal connected subgraph in which every pair of vertices are connected by a path. 5.Detecting /Finding whether there is a cycle in the graph. 6.Testing whether a graph is bipartite. A graph is bipartite if the vertices of the graph can be divided into two disjoint sets such that no edges exist between vertices in the same set.

30. Programming Excersice 30 Represent the following graph. Traverse using DFS and BFS

31. BFS & DFS Programming Exercise 31 public static void main(String[] args) { Graph graph = new Graph (); graph.addVertice("V0"); graph.addVertice("V1"); graph.addVertice("V2"); graph.addVertice("V3"); graph.addVertice("V4"); graph.addVertice("V5"); graph.addEdge(0,1); graph.addEdge(1,2); graph.addEdge(1,3); graph.addEdge(1,4); graph.addEdge(4,5); graph.addEdge(2,5); graph.addEdge(3,5); graph.showData(); System.out.println("\nDFS >>> "); graph.clearVisitedMarkers(); graph.dfs(0); System.out.println("\nBFS >>> "); graph.clearVisitedMarkers(); graph.bfs(0); }

32. BFS & DFS Programming Exercise 32 public class Graph { private ArrayList vertices; private ArrayList > neighbors; private ArrayList visitedMarkers; public Graph() { vertices = new ArrayList (); neighbors = new ArrayList >(); visitedMarkers = new ArrayList (); } public void addVertice(E newVertice){ vertices.add(newVertice); visitedMarkers.add(false); neighbors.add(new ArrayList () ); } public void addEdge( int v1, int v2){ ArrayList adjacent1 = neighbors.get(v1); if (!adjacent1.contains(v2)) neighbors.get(v1).add(v2); ArrayList adjacent2 = neighbors.get(v2); if (!adjacent2.contains(v1)) neighbors.get(v2).add(v1); }

33. BFS & DFS Programming Exercise 33 public void markVerticeAsVisited(int v){ visitedMarkers.set(v, true); } public void clearVisitedMarkers(){ for(int i=0; i<vertices.size(); i++){ visitedMarkers.set(i, false); } public void showData() { for (int i=0; i<vertices.size(); i++){ System.out.printf("\nVertice[%d]= %s", i, vertices.get(i)); } for (int i=0; i < vertices.size(); i++){ System.out.printf("\nneighbors[%d]: ", i); ArrayList adjacent = neighbors.get(i); for (int j=0; j<adjacent.size(); j++){ System.out.printf(" %d,", adjacent.get(j) ); } }//showData

34. 34 public void dfs(int v) { visitedMarkers.set(v, true); for(int n : neighbors.get(v)){ if (!visitedMarkers.get(n)){ System.out.printf("[v%d-v%d] ", v, n); dfs(n); } }//dfs BFS & DFS Programming Exercise

35. BFS & DFS Programming Excersice 35 public void bfs(int v){ ArrayList queue = new ArrayList (); clearVisitedMarkers(); visitedMarkers.set(v, true); System.out.printf("BFS >>> starting at: v%d \n", v); queue.add(v); while(queue.size() > 0){ int currentDequeuedVertice = queue.remove(0); ArrayList adjacents = neighbors.get(currentDequeuedVertice); for(int i=0; i<adjacents.size(); i++){ int adjacentNode = adjacents.get(i); if (!visitedMarkers.get(adjacentNode)){ System.out.printf("[v%d-v%d] ", currentDequeuedVertice, adjacentNode); queue.add(adjacentNode); visitedMarkers.set(adjacentNode, true); }//if }//for }//while }//bfs }

36. Case: The Nine Tail Problem 36 1. Nine coins are placed in a three by three matrix with some face up and some face down. 2. A legal move is to take any coin that is face up and reverse it, together with the coins adjacent to it (this does not include coins that are diagonally adjacent). 3. Your task is to find the minimum number of the moves that lead to all coins face down.

37. Case: The Nine Tail Problem 37

38. Finding Eulerian Paths 38 An Eulerian path traverses each edge of a connected graph G exactly once. It exists only if there are no more than two odd nodes. 1. Pick an odd vertex to start. 2. From that vertex move to a node using an “unvisited” edge. 3. Mark that edge as “visited” (see note) 4. Try to repeat 2-4 until all edges have been traversed Note: DFS algorithm marks nodes as visited. More than two odd nodes ⟶ no Euler Path

39. The Hamiltonian Path Problem 39 Def. A Hamiltonian path in a graph is a path that visits each vertex in the graph exactly once. Def. A Hamiltonian cycle is a cycle that visits each vertex in the graph exactly once and returns to the starting vertex. Notes: Problem is NP-Complete Many applications such as: Scheduling the Traveling Salesman Itinerary.