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CSC 301.01, Class 30: Miniumum spanning trees, continued

Overview

  • Preliminaries
    • Notes and news
    • Upcoming work
    • Extra credit
    • Questions
  • Note on algorithm design strategies
  • Prim’s algorithm and Kruskal’s algorithm
  • Proofs of correctness
  • Efficiency

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)

Upcoming work

Extra credit (Academic/Artistic)

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

Questions

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 disconnected components)
    • 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 the graph.
    • 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.

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Efficiency

Prim’s

Kruskal’s