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MIT6_042JS10_lec11_sol

# MIT6_042JS10_lec11_sol - Massachusetts Institute of...

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Massachusetts Institute of Technology 6.042J/18.062J, Spring ’10 : Mathematics for Computer Science February 26 Prof. Albert R. Meyer revised February 21, 2010, 1418 minutes Solutions to In-Class Problems Week 4, Fri. Problem 1. The table below lists some prerequisite information for some subjects in the MIT Computer Science program (in 2006). This defines an indirect prerequisite relation, , that is a strict partial order on these subjects. 18 . 01 6 . 042 18 . 01 18 . 02 18 . 01 18 . 03 6 . 046 6 . 840 8 . 01 8 . 02 6 . 001 6 . 034 6 . 042 6 . 046 18 . 03 , 8 . 02 6 . 002 6 . 001 , 6 . 002 6 . 003 6 . 001 , 6 . 002 6 . 004 6 . 004 6 . 033 6 . 033 6 . 857 (a) Explain why exactly six terms are required to finish all these subjects, if you can take as many subjects as you want per term. Using a greedy subject selection strategy, you should take as many subjects as possible each term. Exhibit your complete class schedule each term using a greedy strategy. Solution. It helps to have a diagram of the direct prerequisite relation: 18.01 8.01 6.001 6.046 6.002 8.02 18.03 6.042 18.02 6.034 6.004 6.840 6.033 6.857 6.003 There is a -chain of length six: 8 . 01 8 . 02 6 . 002 6 . 004 6 . 033 6 . 857 Creative Commons 2010, Prof. Albert R. Meyer .

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2 Solutions to In-Class Problems Week 4, Fri. So six terms are necessary, because at most one of these subjects can be taken each term. There is no longer chain, so with the greedy strategy you will take six terms. Here are the subjects you take in successive terms. 1: 6.001 8.01 18.01 2: 6.034 6.042 8.02 18.02 18.03 3: 6.002 6.046 4: 6.003 6.004 6.840 5: 6.033 6: 6.857 (b) In the second term of the greedy schedule, you took five subjects including 18.03. Identify a set of five subjects not including 18.03 such that it would be possible to take them in any one term (using some nongreedy schedule). Can you figure out how many such sets there are? Solution. We’re looking for an antichain in the relation that does not include 18.03. Every such antichain will have to include 18.02, 6.003, 6.034. Then a fourth subject could be any of 6.042, 6.046, and 6.840. The fifth subject could then be any of 6.004, 6.033, and 6.857. This gives a total of nine antichains of five subjects. (c) Exhibit a schedule for taking all the courses —but only one per term. Solution. We’re asking for a topological sort of . There are many. One is 18.01, 8.01, 6.001, 18.02, 6.042, 18.03, 8.02, 6.034, 6.046, 6.002, 6.840, 6.004, 6.003, 6.033, 6.857. (d) Suppose that you want to take all of the subjects, but can handle only two per term. Exactly how many terms are required to graduate? Explain why.
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MIT6_042JS10_lec11_sol - Massachusetts Institute of...

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