1. CIS121-204 – Fall 2007 Lab 12 – Last Lab Tuesday, December 4, 2007

2. Course Evaluation Importance of Evaluation They tell me to read this. Penn Engineering takes teaching quality very seriously. Therefore, please take the time to give a candid and serious response to each of the questions on this form. USE A NUMBER 2 PENCIL. The information you supply is used in three ways: (a) by the instructor, to help improve the quality of the course in subsequent years; (b) by the Department and School to evaluate the quality of instruction for purposes of rewarding excellent instructors; and (c) FOR UNDERGRADUATES, by the Penn Course Review, a student publication providing advice to students. It is especially important that you take the time to give written comments. Remember that your comments are anonymous and that these forms are taken directly to the department office by a designated student; the instructor receives the information only after the grades for the course have been submitted.

3. Graph Graph G=(V,E) consists of a set of vertices V and a set of edges E, where E is a subset of VxV. vertex: singular vertices: plural G is simple if it contains no self loops or parallel edges. Multigraph is graph with parallel edges.

4. Graph: Example 1 Graph 1 is a multigraph with a self loop at vertex 4. Vertex 0 is adjacent to vertices 1 and 2. Graph 1 is not connected; it has two connected components: {0,1,2,3} and {4}. Edges of Graph 1 are weighted because each edge has a “weight.” Graph 1 is undirected because edges have no directions. 2 43 01 Graph 1 V={0,1,2,3,4} E={(0,1),(0,2),(0,2),(1,3), (2,3),(4,4)} Note that G1 is a multigraph, so E is a multiset (having duplicate elements). 2550 2007 14 11 2547 100

5. Graph: Example 2 Graph 2 is a directed graph because edges have directions. Graph 2 is simple because there is at most one edge in the same direction between any two vertices. Edges of Graph 2 are unweighted—all edges have the same “weight,” say 1. Graph 2 is connected because for any two vertices u and v, there is a path from u to v or from v to u. If for any two vertices u and v, there is a path from u to v and from v to u, the graph is strongly connected. 1 2 5 3 6 0 4 Graph 2

6. Adjacency Lists If there is an edge from u to v, then v is in the adjacency list of u. What’s the adjacency lists of this graph? The first few: 0: {2} 1: {0,4} … 1 2 5 3 6 0 4

7. Single-Source Shortest Paths Minimize the “cost” (or length) to go from a “source” vertex to any other vertex. The length of shortest path from a vertex to itself is zero. If there is no path from u to v, then the length of shortest path from u to v is infinity. Weighted graph: CIS320—Dijkstra’s Algorithm Unweighted graph: Now.

8. Single-Source Shortest Path on Unweighted, Directed Graphs Input: G and source vertex u Starting from u, try to expand the set reachable vertices. Have a queue of discovered but unprocessed vertices. For each element in the queue: If a new vertex w is discovered, then know the shortest path from u to w. Put w in the queue If an old vertex is encountered, do nothing. Let’s do an example.

9. SSSP on Unweighted Graph: Example Want to find the length of shortest paths from source vertex 0. Assign the initial lengths: 0 for vertex 0 and infinity otherwise Enqueue 0 because we already “discovered” vertex 0 but has not processed it. That’s all for the first step. 1 2 5 3 6 0 4 Queue: [0] 0∞ ∞∞ ∞ ∞∞

10. SSSP on Unweighted Graph: Example (cont.) Dequeue: 0 Length: 0 Consider the neighbor of 0 2: undiscovered Update length. Enqueue 2. 1 2 5 3 6 0 4 Queue: [2] 0∞ ∞1 ∞ ∞∞

11. SSSP on Unweighted Graph: Example (cont.) Dequeue: 2 Length: 1 Consider the neighbor of 2 0: discovered Do nothing. 3: undiscovered Update length. Enqueue 3. 5: undiscovered Update length. Enqueue 5. 1 2 5 3 6 0 4 Queue: [3,5] 0∞ ∞1 2 2∞

12. SSSP on Unweighted Graph: Example (cont.) Dequeue: 3 Length: 2 Consider the neighbor of 3 1: undiscovered Update length. Enqueue 1. 4: undiscovered Update length. Enqueue 4. 1 2 5 3 6 0 4 Queue: [5,1,4] 03 31 2 2∞

13. SSSP on Unweighted Graph: Example (cont.) Dequeue: 5 Length: 2 Consider the neighbor of 5 6: undiscovered Update length. Enqueue 6. 1 2 5 3 6 0 4 Queue: [1,4,6] 03 31 2 23

14. SSSP on Unweighted Graph: Example (cont.) Dequeue: 1 Length: 3 Consider the neighbor of 1 0: discovered Do nothing. 4: discovered Do nothing. 1 2 5 3 6 0 4 Queue: [4,6] 03 31 2 23

15. SSSP on Unweighted Graph: Example (cont.) Dequeue: 4 Length: 3 Consider the neighbor of 4 None. Done. 1 2 5 3 6 0 4 Queue: [6] 03 31 2 23

16. SSSP on Unweighted Graph: Example (cont.) Dequeue: 6 Length: 3 Consider the neighbor of 6 3: discovered Do nothing. 4: discovered Do nothing. 5: discovered Do nothing. 1 2 5 3 6 0 4 Queue: [] 03 31 2 23

17. SSSP on Unweighted Graph: Example (cont.) Queue is empty. Done Done. 1 2 5 3 6 0 4 Queue: [] 03 31 2 23

18. SSSP on Unweighted Graph: Algorithm Array sp of length |V| initialized as infinity sp[u]=0 //the shortest path from u to u has length 0 Queue Q Q.enqueue(u) while Q is not empty s=Q.dequeue() for each v such that (s,v) is an edge if sp[s]+1<sp[v] sp[v]=sp[s]+1 Q.enqueue(v) return sp

19. Takeaways from CIS121 Using a correct data structure saves time. Three steps in writing a program: Designing Implementing Testing You should spend most of your time in the first and last steps. Eclipse… please. Real programmers don’t have a life???

20. What to do after CIS121? You probably want to see my handwriting on the whiteboard for the last time in this course… … because I will use a chalkboard for the review session.

21. Time’s Up. Thanks for the good time we have had. Hope you enjoy the section. I always do. See you in the review session and the final exam. Reminder: Final Exam Date:Thursday, December 13, 2007 Time:9-11AM Place:Skirkanich Hall Auditorium Good luck. See you around. Feel free to (and please) say hi.