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CSC 301.01, Class 29: Minimum spanning trees

Overview

  • Preliminaries
    • Notes and news
    • Upcoming work
    • Extra credit
    • Questions
  • Minimum spanning trees
  • Examples
  • Designing an MST algorithm
  • Prim’s algorithm and Kruskal’s algorithm
  • Efficiency
  • Proofs of correctness

News / Etc.

  • Sorry about the temrporarily broken Web site. The Bootstrap style I used changed unexpectedly. (I hear it broke many of the Course webs in the department.)
  • Sorry that I could not be in class on Friday.
  • 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.
  • No, grading is not done. Thursday, Friday, Saturday, and half of Sunday were booked. I spent six hours grading, but that’s not enough.
    • Weekdays are booked 8:30-5:00 (and it’s preregistration)
    • Tonight: Write draft of your exam. Write 151 project description. (Skip concert.)
    • Tuesday: CSC 151 exam grading.
    • Wednesday: CSC 151 exam grading.
    • Thursday: (Maybe) CSC 301 exam grading.
  • I have more on the agenda for today than I expect we will cover.
    Wednesday’s class will catch the overflow.

Upcoming work

  • Assignment 8 due Wednesday at 10:30 p.m. (All written problems!)

Extra credit (Academic/Artistic)

  • Leyla McCalla Trio, Nov. 6, 7:30 p.m. Herrick
  • Animated Films, Tuesday, Nov. 7, 11:00 a.m., Faulconer
  • CS Table (Computer-Aided Gerrymandering), Tuesday, Nov. 7, noon, Day Dining Room
  • Crip Technoscience, Disabled People as Makers and Knowers, Wednesday, Nov. 8, 4:15 p.m., JRC 101.

Extra credit (Peer)

  • VR club Sundays at 8pm in ???.
  • Not-Pub Quiz Wednesday at 9pm in Bobs. Free snacks.

Extra Credit (Misc)

  • Pioneer Weekend. Register by Nov. 8.

Other good things

Questions

Minimum spanning trees

Given a non-negative weighted non-directed connected graph, G(V,E), build a new connected graph G(V,E’), s.t.

  • E’ is a subset of E.
  • If G(V,F) is connected and F is a subset of E, the sum of the weights in F is less than or equal to the sum of the weights in E’.

Why is it called a minimum spanning tree rather than a minimum spanning graph?

  • If you have a graph, you can just cut out one edge from a cycle and its still connected.

Examples

Here’s a graph! (Picture on the board.)

Vertices: A, B, C, D, E, F, G
Edges: AB(6), AC(7), AE(8), BD(9), CD(1), CE(1), CF(2), DF(5),
EF(10), EG(14), FG(8).

Designing an MST algorithm

How might you approach the problem? (Once we’ve come up with some approaches, we’ll assign approaches to different groups.)

  • Divide and conquer
  • Greed
    • Removing largest
    • Selecting smallest
  • Use a variant of the shortest-path algorithm

Approaches, Revisited

Greed: Remove largest

unmark all edges
while (we still have cycles)
  let e be the larget unmarked edge
  if (removing e disconnects the graph)
    mark e
  else
    remove e
  end if
end while

Greed: Add smallest

E' = { }
Etmp = E
while (G(V,E') is not complete)
  let e be smallest edge in Etmp
  Etmp = Etmp - e
  if (G(V,E'+{e}) is cycle free)
    E' = E' + {e}
  end if
end while

Can we break these?

  • Not in five minutes of work.

Prim’s algorithm and Kruskal’s algorithm

Efficiency

Proofs of correctness