CS252 Algorithms
Friday, 5 April 2024

+ Today
- Gale-Shapley proof
    - the proof itself
    - what their proof style says for your proofs
- Your questions about homework
- How to approach using G-S in varied contexts
- Later today
    - Zoom office hours, same link, usual time (3A 12-1)
    - grader feedback for PS#1
    - solutions for PS#1
    - PS#3
- Monday: moving on to graphs



+ Gale-Shapely correctness proof [variation: #m < #w]

- Terminates after at most n^2 iterations  [#m * #w]
    (when do you need a termination proof?)
    one proposal per iteration; no repeated proposals; n^2 possible proposals

- S is a matching
    - def of "matching"?
        A matching S is a set of ordered pairs, each
        from M × W, with the property that each member of M
        and each member of W appears in *at most one* pair in S.

    - why can't one women be engaged to > 1 man?
    - why can't one man be engaged to > 1 woman?
        analyze the two locations in the algorithm where a
        (m,w) pair can get added to S

- S is a perfect matching [variation #m < #w; replace this with "all men are engaged"]
    - def of "perfect"?
        A perfect matching S' is a matching with the property that
        each member of M and each member of W appears in exactly one
        pair in S'
    - is this one obvious?
    - Only case to consider
        there's a free m who has already made all n of his proposals
        but this means all women are still engaged (since they've all
            received at least one proposal, from m)
        who are all those w's engaged to? n-1 men? ... contradiction

- S is stable
    - review the definition of "stable"
    - by contradiction, with two cases
        assume there's (m1, w1) and (m2, w2) in S
            where m1 prefers w2 to w1, and w2 prefers m1 to w2

        case 1: m1 proposed to w2 in the past
            then m2 proposed to w2 later
            but then m2 and w2 wouldn't have gotten engaged because
                w2 only switches engagements when she hears a proposal
                from somebody she prefers, so this couldn't happen this way.

        case 2: m1 never proposed to w2
            but m1 proposed to w1, who he dis-prefers to w2, but
            that's not the order in which men are proposing,
            so that couldn't happen either

        so our assumption must be false, so S is stable
    
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+ Questions

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+ Variants
- what if there are more women than men, and men are still the proposers
    - can you change the theorem statement and proof
         yes--this isn't too bad for this scenario

    - can you modify the input data and output data instead?
        we start #w with #m-length pref lists
                 #m with #w-length pref lists
        k = #w - #m, add k fake men
            fake men's preference lists?
                randomize
            women's pref lists need to include the fake men:
                stick them at the bottom of each woman's list
        run G-S
            the women engaged to fake men are now just unmarried at the

- same, but more men than women
    - modify the proof?
    - modify input and output to use regular G-S

- #m == #w, but preference lists can have ties

- CS Match (# students != # classes; classes have multiple slots; etc.)
    - use the "transform the input" approach
        - a class with a 28-person cap becomes 28 distinct classes
        - preference list from each student is supplemented by adding
            all the courses they didn't list in a lowest-priority tie
            at the bottom
        - do whatever you did to deal with preference ties in the previous scenario
    - who proposes? (I'm pretty sure the students are the proposers; I'll ask)
    - what do the classes' preference lists look like?

- etc.