CS252 Algorithms
Monday, 20 May 2024

+ Exam prep
- See Layla's guide
- Topics
    - Divide and conquer
    - Solving recurrences
        - Master Theorem
        - Find pattern & prove by induction
    - Dynamic programming
        - the general process
        - wire cutting
        - knapsack
        - Bellman-Ford
    - Network flows
        - what's a flow?
        - Ford-Fulkerson
        - cuts & network flows
    - Asymptotics, as in Exams 1 & 2
        - definitions of O, Ω, and ϴ
        - simple arguments from definitions
            - e.g. problem 2 from exam 2
        - know runtimes of data structures
            and algs we have studied

+ Network flows
    (concepts & mechanisms today and W; peek at applications F)

- water metaphor (capacity = liters/sec)
    - given an example network, how much water
        can you let through each pipe per second
        to maximize the flow at the sink t
        without overwhelming any of the junctions?

- directed graph G, so (u,v) means an edge u -> v
- edge capacities c_e / c_{uv} where e = (u,v)
- source (s) & sink (t)
- f: E -> R^{>=0}, f(e) <= c_e

- f_in(v) = sum_{e = (u,v) in E} f(e)
- f_out(v) = sum_{e = (v,u) in E} f(e)

- a "flow" is a function f as above, where
    - f_in(v) = f_out(v) for every v != s or t
    - f_in(s) = 0 = f_out(t) (true because there are no in-edges at s
        or out-edges at t)
    - 0 <= f(e) <= c_e for every edge e

- the "value of a flow" = sum_{e = (s,u) in E} f(e)

- REMEMBER: c_e is not the same thing as f(e)
    0 <= f(e) <= c_e, and f has some extra properties

---- how to compute all this? ----

- augmenting path in G
- residual graph G_f
- augmenting path in G_f
- how would you find an augmenting path?

- s-t cuts
- min cut = max flow