CS252 Algorithms
Wednesday, 3 April 2024

+ Questions
- Hash tables? Off limits all term.


+ Slack
- Threads & notifications
- Recommended: skim everything, don't skip the threads
- Should I use @channel?

+ O(m + n)
- Searching tip: O, Ω, ϴ are called "Landau notation" (or "Bachmann–Landau")

- Big-oh with multiple variables

   f(x,y) = O(g(x,y)) if ∃ C > 0, N > 0 st
        x >= N and y >= N ==> f(x,y) <= Cg(x,y)

- What does that mean for an arguably two-variable situation like Problem 1?

+ Data structures; let's look at
- Arrays
- Linked lists
    - Singly-linked
    - Doubly-linked
    - With and without tail pointer
- Stacks
    - Implemented with an array
        (variables? push and pop operations?)
    - Implemented with a linked list
        (variables? single/double/tail? push and pop operations?)
- Queues
    - Implemented with an array
        (variables? add and remove operations?)
    - Implemented with a linked list
        (variables? single/double/tail? add and remove operations?)

+ Gale-Shapely correctness proof, super-short summary
- Termination
- Perfect match (n pairs, no repeated m or w)
- Stable match

+ Gale-Shapely correctness proof, longer summary
- (p7, 1.1) After getting one proposal, a woman remains engaged
- (p7, 1.3) GS terminates after at most n^2 iterations
- (p8, 1.4) If man m is free, then there is a woman w
    that has not been proposed to by m
- (p8, 1.5) At the end, the set S is a perfect matching
- (p8, 1.6) At the end, the set S is a stable matching

+ Try a variant
- what if there are more women than men, and men are still the proposers

+ Other stuff of interest
- regardless of the order of the men selected in the loop,
    you get the same matching (whoa!)

- once you understand the core theorem, there are tons of variants
    - allow preference ties ("indifference")
    - allow different sized sets
    - allow a single "job" to hire multiple "workers" (CS match)
    - ...

- Reading proofs is not like other reading

- Correctness requires that an algorithm terminates

- Small examples (plus tinkering) help you get the feel of an algorithm


=====
- Perfect match
  S is always a matching
  terminates w/ free man? Nope.
- Stable
  Contradiction:
    Suppose \exists (m,w), (m',w') \in S
      where m likes w' better than w and w' likes m better than m'
    Then m proposed to w' earlier than he proposed to w. What happened then?
    case 1: m proposed to w' before m' did; w wouldn't have switched for m',
      so this couldn't be it
    case 2: m proposed to w' after m' did; w would have switched,
      so this couldn't be it
    Contradiction.