CS252 Algorithms
Wednesday, 17 April 2024

+ Questions

+ What's next? algorithmic paradigms
- greedy algorithms (weeks 4 & 5)
    - shortest paths
    - minimum spanning trees
- divide-and-conquer algorithms (week 6)
- dynamic programming (weeks 7-8)
- network flows (week 9)

+ Greedy algorithms
- Greedy is an adjective
- K&T Ch 4:
    "It is hard, if not impossible, to define precisely what
     is meant by a greedy algorithm. An algorithm is greedy if
     it builds up a solution in small steps, choosing a decision
     at each step myopically to optimize some underlying criterion."

+ Interval scheduling
- The problem
    In? Request to use the shared resource
        R = {[start_time_1, finish_time_1], [s2,f2]... }
    Out?
        A = subset of R = {...}
            non-overlapping elements of R with the maximal size (# of requests)
        (notably, what are we optimizing here?)
            size of A

    Why are we looking at this problem?
        - a greedy approach works
        - it's satisfying (elegant, simple alg., and it works! the proof is nice)
        - several obvious greedy approaches fail

- Greedy options
    - :) grab an interval from R that works with A (no overlaps) and
        has the earliest finish time
    - :( grab an interval that starts as soon as possible
    - :( grab the smallest interval

- Counter-examples

- Our algorithm

- Correctness proof
    - "staying ahead of optimal"

- Implementation and runtime

+ Related problems
- Maximize total length of intervals in A
- Maximize weights
- Minimize maximum lateness (start times are min start times)
- Minimize total lateness
- All of those, but with multiple copies of the resource
- ...

+ Exchange argument
- Problem: interval scheduling to minimize max lateness