Graph Data Structures: Implementation and Traversal Methods

1. COSC 2007Data Structures II Chapter 14 Graphs II

2. Topics • Graph implementation • Adjacency matrix • adjacency list • Traversal • 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. A B L C P R Implementation of Graphs • Adjacency Matrix • Example:

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

14. 2 3 8 1 4 5 9 10 6 7 11 Graph Search Methods • A search method starts at a given vertex v and visits/labels/marks every vertex that is reachable from v.

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) • 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. 2 2 3 8 1 1 4 5 9 10 6 7 11 Depth-First Search Example Start search at vertex 1.

19. Graph Traversal • 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 • n / 2