CS252 Algorithms
Wednesday, 22 May 2024

+ Friday is Honors Convo day
- class 9:30-10:20
- honors convo 3-4

+ Questions?
- Homework
- Exam
- Other

+ Bellman-Ford question from Monday
- Textbook: grow paths backwards from target node t
- Alternative (and more like Dijkstra): grow paths
    forwards from source node s
- Understanding how the two M[k,u] recurrences and charts
    can help you deepen your understanding not just of
    shortest paths but also dynamic programming and
    recursion generally


+ Network flows, continued

Network
- directed graph G = (V,E); edge capacities c_e >= 0
- source s, no incoming edges
- sink t, no outgoing edges

A "flow" f on the network
- f: E -> R
- 0 <= f(e) <= c_e for all e
- f_in(v) = f_out(v) for all v != s or t

"Value" of a flow
- v(f) = sum_{e going out of s} f(e)

"Augmenting path" for a flow
- path from s to t
- "bottleneck" of the path

     b = min_{e in path} (c_e - f(e))

  that is, b is the "minimum available capacity" of all the edges in the path
- **IF b > 0**, this path can be used to "augment" v(f) by adding b to all the edges
  on the path. That's why it's called an augmenting path.

Computing maximum-value flow (Ford-Fulkerson algorithm)
1. set f(e) = 0 for all e
2. find an augmenting path with bottleneck b > 0
    (if none exists, go to 5)
3. for all edges e in the path, set f(e) = f(e) + b
4. go to 2
5. done

How do you do step 2?
- Compute the "residual graph" G_f
- Use BFS or DFS to find a path from s to t in G_f