CSC 301.01, Class 30: Miniumum spanning trees, continued
- Notes and news
- Upcoming work
- Extra credit
- Note on algorithm design strategies
- Prim’s algorithm and Kruskal’s algorithm
- Proofs of correctness
News / Etc.
- I expect that most upper-level CS classes will fill this semester. You help the department better consider stress points and how to address them if you register earlier, rather than later.
- Some interesting courses to consider
- Wilson short course: Human Centered Design for Global Social Transformation
- Wilson short course: Leadership in a Future of Automation and Income Inequality.
- ENG 295-01. Lighting the Page: Digital Methods in Literary Studies. (ENG-120 prereq)
- HIS 295-01. Digital Methods in Historical Studies. (HIS-100 prereq)
Extra credit (Academic/Artistic)
- Crip Technoscience, Disabled People as Makers and Knowers, Wednesday, Nov. 8, 4:15 p.m., JRC 101.
- Workshop by Sanjay Khanna ‘85 “Khanna can pitch. Can you?” Tonight at 7pm in ARH 120. Sign up at https://grinnell.co1.qualtrics.com/jfe/form/SV_8B7YaJkixNUBmjH
Extra credit (Peer)
- Pub-free Quiz, TONIGHT
- Virtual Food (and real food) event at VR club Saturday 6-8 in DLab
- VR club Sundays at 8pm in DLAB.
Extra Credit (Misc)
Other good things
- I can’t apply the topological sort algorithm on a graph with a cycle.
- Don’t use a bootleg copy of the book.
- What do you mean by “describe a graph”?
- Give enough explanation that someone could draw/make it.
- Can every DAG be expanded into a tree by repeating nodes?
- I believe so. But it can be exponentially larger.
A note on algorithm design strategies
- Divide and conquer: Divide the input in half (approximately). Solve
the problem for either or both halves. Combine the solutions into a
a solution for the overall problem.
- Binary search
- Merge sort
- Quick sort
- “Greed” as a strategy
- Given a group of choices, choose one that is obviously largest/smallest
- If you’ve chosen greed as a strategy, you should decide how you
are going to be greedy.
- What’s the set of values you choose among?
- Do you choose largest/smallest
- One more strategy: Exhaustive search
- Advantage: Correct
- Disadvantage: Expensive
- Soon: Dynamic programming (fancy “caching”)
Greedy approaches: Prim’s algorithm, Kruskal’s algorithm, and …
- Prim’s: Throw away all the edges, pick a vertex, and repeatedly add
lowest weight edge that expands the tree of vertices reachable from
that vertex (and does not create a cycle).
- Choose smallest
- Set: Neighboring edges from the tree
- Q: How do we know if we’ve made a cycle?
- Breadth-first exploration or depth-first exploration.
- Expensive O(n+m)
- In the typical implementation of Prim’s, we mark nodes as we go. That means “check for cycle” is O(1). Requires O(n) work at the beginning to unmark all of the vertices.
- Kruskal’s Throw away all the edges, repeatedly add the lowest weight
edge that does not cause a cycle. (Alternately: That connects two
- Choose smallest from among all remaining edges
- May have to do a more expensive operation to check if we are connected two disconnected components.
- Other: Repeatedly throw away the largest edge unless it disconnects
- Choose largest from among all remaining edges
- Requires a connectivity check: O(n+m)
Proofs of correctness
Suppose that we want to prove that one of these algorithms is correct. What kinds of proof techniques might we try?
- Induction - Often useful in CS (if it works for a graph of size N, it should work for a graph of size N+1) (strong or weak)
- Contradiction - Assume it doesn’t work. Show that leads to a logical inconsistency. “Suppose G(V,N’) is not an MST …”
- Constructive. Building up from ground zero. (Invariants)
- Prove the contraposative.
- Reduce to known fact/proof.
Let’s try proving that Kruskal’s algorithm is correct.
- Suggested technique: Contradiction.
- Suppose Kruskal’s algorithm is not correct.
- That means that there exists a graph for which Kruskal’s generates an incorrect MST.
- Look at the MST for that graph.
- There exists an edge in the Kruskal Tree that is not in the MST.
- What happens when we add it to the real MST? That creates a cycle. The Kruskal edge must have a lower weight than one other edge in the cycle. (O/w Kruskal would not have chosen it, since it would have considered all of the others first.)
- We can replace the heigher weight edge with the Kruskal edge and get a more minimal MST. There aren’t “more minmal” MSTs.
- Therefore our initial assumption was wrong.
And Kruskal’s algorithm is correct.
+---+ | | +---+