CS252 Algorithms
Friday, 24 May 2024

+ Today is Honors Convo day
- class 9:30-10:20
- office hours 10:30-11:20 (3A)
- honors convo 3:00-4:00

+ Questions

+ Playing with another flow network
- See the slides

====== We didn't do this today ======
====== (this is an outline of the key ideas in 7.2) ======

+ Network flows

Assume
- G = (V,E)
- s in V, no in-coming edges
- t in V, no out-going edges
- integer capacity c_e > 0 for every e in E

Definitions
- f : E -> R
- f^out(v) = sum_{(v,u) in E} f(v,u)    <-- v is a node
- f^in(v) = sum_{(u,v) in E} f(u,v)     <-- v is a node
- f is a "flow" if
    - for any e in E, 0 <= f(e) <= c_e
    - for any v in V != s or t, f_in(v) = f_out(v)
- v(f) = "value" of the flow = sum_{(s,u) in E} f(s,u)

- G_f is the residual graph (not gonna define it here)

- (A,B) is an "s-t cut" of G if
    - s in A
    - t in B
    - B = V - A

- c(A,B)   = sum_{(u,v) in E, u in A, v in B} f(u,v)    <-- A is a set of nodes
- f^out(A) = sum_{(u,v) in E, u in A, v in B} f(u,v)    <-- A is a set of nodes
- f^in(A)  = sum_{(u,v) in E, u in B, v in A} f(u,v)    <-- A is a set of nodes
- c(A,B)   = sum_{(u,v) in E, u in A, v in B}

Theorems (throughout, f is any flow, (A,B) is any s-t cut)
- v(f) = f^out(A) - f^in(A)
- v(f) <= c(A,B)
- If v(f) = c(A,B), then f is a maximum flow
- If G_f has no s-t paths, then f is a maximum flow
    (and A = {v in V | an s-v path exists in G_f} gives you a cut for which v(f) = c(A,B))

That last one says "Ford-Fulkerson gives you a maximum flow
Max flow = min cut