Graph Data Structures

1. Graph Data Structures 1843 ORD SFO 802 1743 337 1233 LAX DFW Graphs

2. Vertices and edges are positions store elements Accessor methods endVertices(e): an array of the two endvertices of e opposite(v, e): the vertex opposite of v on e areAdjacent(v, w): true iff v and w are adjacent replace(v, x): replace element at vertex v with x replace(e, x): replace element at edge e with x Update methods insertVertex(o): insert a vertex storing element o insertEdge(v, w, o): insert an edge (v,w) storing element o removeVertex(v): remove vertex v (and its incident edges) removeEdge(e): remove edge e Iterable collection methods incidentEdges(v): edges incident to v vertices(): all vertices in the graph edges(): all edges in the graph Main Methods of the Graph ADT Graphs

3. Adjacency Matrix Structure a v b • Edge list structure • Augmented vertex objects • Integer key (index) associated with vertex • 2D-array adjacency array • Reference to edge object for adjacent vertices • Null for non nonadjacent vertices • The “old fashioned” version just has 0 for no edge and 1 for edge u w 2 w 0 u 1 v b a Graphs

4. Adjacency List Structure a v b • Incidence sequence for each vertex • sequence of references to edge objects of incident edges • Augmented edge objects • references to associated positions in incidence sequences of end vertices u w w u v b a Graphs

5. Performance Graphs

6. Applications of BFS and DFS CSE 2011 Winter 2011

7. Some Applications of BFS and DFS • BFS • To find the shortest path from a vertex s to a vertex v in an unweighted graph • To find the length of such a path • To find out if a graph contains cycles • To construct a BSF tree/forest from a graph • DFS • To find a path from a vertex s to a vertex v. • To find the length of such a path. • To construct a DSF tree/forest from a graph.

8. Testing for Cycles

9. Testing for Cycles • Method isCyclic(v) returns true if a directed graph(with only one component) contains a cycle, and returns false otherwise. else return true; return false;

10. Finding Cycles • To output the cycle just detected, use info in prev[ ]. • NOTE: The code above applies only to directed graphs. • Homework: Explain why that code does not work for undirected graphs.

11. Finding Cycles in Undirected Graphs • To detect/find cycles in an undirected graph, we need to classify the edges into 3 categories during program execution: • unvisited edge: never visited. • discovery edge: visited for the very first time. • cross edge: edge that forms a cycle. • Code fragment 13.10, p. 623. • When the BFS algorithm terminates, the discovery edges form a spanning tree. • If there exists a cross edge, the undirected graph contains a cycle.

12. BFS Algorithm (in textbook) • The algorithm uses a mechanism for setting and getting “labels” of vertices and edges AlgorithmBFS(G, s) L0new empty sequence L0.insertLast(s) setLabel(s, VISITED) i  0 while Li.isEmpty() Li +1new empty sequence for all v  Li.elements() for all e  G.incidentEdges(v) ifgetLabel(e) = UNEXPLORED w opposite(v,e) if getLabel(w) = UNEXPLORED setLabel(e, DISCOVERY) setLabel(w, VISITED) Li +1.insertLast(w) else setLabel(e, CROSS) i  i +1 AlgorithmBFS(G) Inputgraph G Outputlabeling of the edges and partition of the vertices of G for all u  G.vertices() setLabel(u, UNEXPLORED) for all e  G.edges() setLabel(e, UNEXPLORED) for all v  G.vertices() ifgetLabel(v) = UNEXPLORED BFS(G, v) Breadth-First Search

13. L0 A L1 B C D E F Example unexplored vertex A visited vertex A unexplored edge discovery edge cross edge L0 L0 A A L1 L1 B C D B C D E F E F Breadth-First Search

14. L0 L0 A A L1 L1 B C D B C D L2 E F E F L0 L0 A A L1 L1 B C D B C D L2 L2 E F E F Example (2) Breadth-First Search

15. L0 L0 A A L1 L1 B C D B C D L2 L2 E F E F Example (3) L0 A L1 B C D L2 E F Breadth-First Search

16. Computing Spanning Trees

17. Trees • Tree: a connected graph without cycles. • Given a connected graph, remove the cycles  a tree. • The paths found by BFS(s) form a rooted tree (called a spanning tree), with the starting vertex as the root of the tree. BFS tree for vertex s = 2 What would a level-order traversal of the tree tell you?

18. Computing a BFS Tree • Use BFS on a vertex BFS( v ) with array prev[ ] • The paths from source s to the other vertices form a tree

19. Computing Spanning Forests

20. Computing a BFS Forest • A forest is a set of trees. • A connected graph gives a tree (which is itself a forest). • A connected component also gives us a tree. • A graph with k components gives a forest of k trees.

21. Example A graph with 3 components P Q N L R O M s D E C A F B K G H

22. Example of a Forest We removed the cycles from the previous graph. P Q N L R O M s D E C A forest with 3 trees A F B K G H

23. Computing a BFS Forest • Use BFS method on a graph BFSearch( G ), which calls BFS( v ) • Use BFS( v ) with array prev[ ]. • The paths originating from v form a tree. • BFSearch( G ) examines all the components to compute all the trees in the forest.

24. Applications of DFS • Is there a path from source s to a vertex v? • Is an undirected graph connected? • Is a directed graph strongly connected? • To output the contents (e.g., the vertices) of a graph • To find the connected components of a graph • To find out if a graph contains cycles and report cycles. • To construct a DSF tree/forest from a graph

25. Finding Cycles Using DFS • Similar to using BFS. • For undirected graphs, classify the edges into 3 categories during program execution: unvisited edge, discovery edge, and back (cross) edge. • Code Fragment 13.1, p. 613. • If there exists a back edge, the undirected graph contains a cycle.

26. DFS Algorithm • The algorithm uses a mechanism for setting and getting “labels” of vertices and edges AlgorithmDFS(G, v) Inputgraph G and a start vertex v of G Outputlabeling of the edges of G in the connected component of v as discovery edges and back edges setLabel(v, VISITED) for all e  G.incidentEdges(v) ifgetLabel(e) = UNEXPLORED w opposite(v,e) if getLabel(w) = UNEXPLORED setLabel(e, DISCOVERY) DFS(G, w) else setLabel(e, BACK) AlgorithmDFS(G) Inputgraph G Outputlabeling of the edges of G as discovery edges and back edges for all u  G.vertices() setLabel(u, UNEXPLORED) for all e  G.edges() setLabel(e, UNEXPLORED) for all v  G.vertices() ifgetLabel(v) = UNEXPLORED DFS(G, v) Depth-First Search

27. A B D E C A A B D E B D E C C Example unexplored vertex A visited vertex A unexplored edge discovery edge back edge Depth-First Search

28. A A A B D E B D E B D E C C C A B D E C Example (cont.) Depth-First Search

29. DFS Tree Resulting DFS-tree. Notice it is much “deeper” than the BFS tree. Captures the structure of the recursive calls: • when we visit a neighbor w of v, we add w as child of v • whenever DFS returns from a vertex v, we climb up in the tree from v to its parent

30. Applications – DFS vs. BFS • What can BFS do and DFS can’t? • Finding shortest paths (in unweighted graphs) • What can DFS do and BFS can’t? • Finding out if a connected undirected graph is biconnected • A connected undirected graph is biconnected if there are no vertices whose removal disconnects the rest of the graph. • Application in computer networks: ensuring that a network is still connected when a router/link fails.

31. A graph that is not biconnected

32. L0 A L1 B C D L2 E F DFS vs. BFS A B C D E F DFS BFS

33. Final Exam • Final Exam • Sunday, April 17, 10:00-13:00 • Materials: • All lectures notes and corresponding sections in the textbook from the beginning to today’s lecture • Homework questions • Assignments 1 to 5