CSC 207.02 2019S, Class 39: Shortest paths in graphs

Overview

• Preliminaries
  • Notes and news
  • Upcoming work
  • Extra credit
  • Questions
• Sample final
• The shortest path problem
• Shortest paths in unweighted graphs
• Shortest paths in weighted graphs
• Dijkstra’s algorithm
• Lab

Preliminaries

News / Etc.

• Please sit with your partner from Monday.
• While we do not have labs or readings for Friday, you are expected to show up (not least, so that you can fill out the EOCE).
• I’m working on getting grades to you as soon as I can. I’ll try to have all grades in by Tuesday.
• Review sessions for final: TBD, probably Wednesday evening.
• DO NOT FILL IN THE END-OF-COURSE EVALUATIONS UNTIL FRIDAY’S CLASS
  • If you’ve done so already, please speak with me asap.

Upcoming work

• No more readings.
• No more lab writeups.
• No more homework assignments.
• Final exam: 9am or 2pm, Thursday or Friday of finals week.
  • Let me know which of the four times you plan to take the final.

Extra credit

Extra credit (Academic/Artistic)

• Tonight Gridshock documentary, 7pm, Wednesday, Strand

Extra credit (Peer)

• Track and Field, Friday and Saturday at St. Norbert (?)

Extra credit (Wellness)

Extra credit (Wellness, Regular)

• Any organized exercise. (See previous eboards for a list.)
• 60 minutes of some solitary self-care activities that are unrelated to academics or work. Your email reflection must explain how the activity contributed to your wellness.
• 60 minutes of some shared self-care activity with friends. Your email reflection must explain how the activity contributed to your wellness.

Extra credit (Misc)

Other good things

• Tonight Digital Music Making class presentation, Wednesday 7:30 pm. Sebring-Lewis. (Rap, noise, pop, more.)
• Today 4:30 p.m., Choreography, Flanagan

Questions

Sample final

The Sample Final is now posted. You can find it by going to Current > Exam. We’ll quickly talk through the policies.

How will you know that the exam is a reasonable length?

Faith?

I may try doing it and seeing how long it takes.

How many types of problems are there?

I will choose five of the six types mentioned on the sample final. (There will almost certainly be a “draw” problem, a “draw graph and run algorithm” problem, and an asymptotic analysis problem.)

Will we get stuff back soon?

Yes.

Can the tests be sketchier code than the algorithms?

Yes.

Will we get some pre-written code?

It depends on the problem.

The shortest path problem

Given a graph and two vertices, source and sink, find the shortest path from source to sink, where the length of a path is computed from the edges in the path.

If there is no path from source to sink, there is no shortest path. Different variants handle that issue in different ways.

Shortest paths in unweighted graphs

In an unweighted graph, the length of the path is the number of edges in the path.

Think/Pair/Share: How can you find the shortest path in an unweighted graph? (Don’t say “Dijkstra’s algorithm”.)

• Brute force
  • Find every non-looping path from source to sink.
  • Take the shortest
• Breadth first search

Shortest paths in weighted graphs

In a weighted graph, the length of the path is most frequently the sum of the weights of the edges in the path.

• There are other versions, such as the product of the weights, or the maximum weight, or …

Consider the two following approaches:

• If “we can use the algorithm”, explain why.
• If “we can’t use the algorithm”, give a counter-example.

Think/Pair/Share: Can we use the algorithm we used for unweighted graphs? Why or why not?

• Yes, if we convert the weighted graph to an unweighted graph using the technique of “replace an edge of weight w by w edges of weight 1”
• Important moral: Instead of transforming the algorithm, transform the data.
• Is this a good approach?
  • Yes, it’s clever. We like clever things.
  • The new graph may be much larger. It’s expensive to build the graph.
  • We’ve turned an O(m+n) algorithm into a O(sum-weights) algorithm.
  • May put us further from what we are modeling.
  • Won’t work for real weights.
  • Harder to update the graph (e.g., changing weights).

Think/Pair/Share: Can we use the obvious “greedy” algorithm: Take the smallest-weight edge from start, then the smallest-weight edge from there that does not cause a cycle, then the smallest-weight edge from there that does not cause a cycle, …. Why or why not?

• Easy to find counter-examples

Dijkstra’s algorithm

Key insight: Instead of finding the shortest path from source to sink, we find the shortest path from source to every vertex in the graph.

We’ll do an example to think through it.

Lab