CIS121-204 – Fall 2007

Lab 12 – Last LabTuesday, December 4, 2007

Course EvaluationImportance 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.

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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: singularvertices: plural

G is simple if it contains no self loops or parallel edges.

Multigraph is graph with parallel edges.

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

0 1

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

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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.

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0

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Graph 2

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}…

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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.

Single-Source Shortest Pathon 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.

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.

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Queue:

[0]

0 ∞

∞∞ ∞

∞ ∞

SSSP on Unweighted Graph: Example (cont.)

Dequeue: 0 Length: 0 Consider the

neighbor of 0 2: undiscovered

Update length. Enqueue 2.

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Queue:

[2]

0 ∞

∞1 ∞

∞ ∞

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.

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Queue:

[3,5]

0 ∞

∞1 2

2 ∞

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.

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Queue:

[5,1,4]

0 3

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2 ∞

SSSP on Unweighted Graph: Example (cont.)

Dequeue: 5 Length: 2 Consider the

neighbor of 5 6: undiscovered

Update length. Enqueue 6.

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Queue:

[1,4,6]

0 3

31 2

2 3

SSSP on Unweighted Graph: Example (cont.) Dequeue: 1 Length: 3 Consider the

neighbor of 1 0: discovered

Do nothing. 4: discovered

Do nothing.

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Queue:

[4,6]

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31 2

2 3

SSSP on Unweighted Graph: Example (cont.)

Dequeue: 4 Length: 3 Consider the

neighbor of 4 None. Done.

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Queue:

[6]

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31 2

2 3

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.

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Queue:

[]

0 3

31 2

2 3

SSSP on Unweighted Graph: Example (cont.)

Queue is empty. Done Done.

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Queue:

[]

0 3

31 2

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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

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???

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.

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.