CS252 Algorithms
Wednesday, 27 March 2024

+ Questions

+ Mathematical definitions of efficiency
- O, Ω, ϴ
    (and exactness vs. sloppiness)
    (log2 vs. ln vs. log10...)
- worst-case analysis
- average-case analysis

+ Let's try it.

    Let f(n) = n^2 + 3n + 5
    Prove: f(n) = O(n^2)

    Need to find: c > 0, n0 > 0 such that
        n >= n0 ==> f(n) <= cn^2

    Let c = 9 and n0 = 1
    Then:
          n >= n0 ==> n >= 1
                  ==> n^2 + 3n + 5 <= n^2 + 3n^2 + 5n^2 = 9n^2
                  ==> f(n) = n^2 + 3n + 5 <= 9 n^2
                  ==> f(n) <= cn^2

    Therefore f is O(n^2)

(scratch paper) n >= 1 ==> n^2 >= n

    Prove: f(n) = Ω(n^2) [DO IT FOR YOURSELF]

    Let g(n) = 3n^2
    Prove: f(n) = Ω(g(n))


+ ChatGPT/Claude/CoPilot/Gemini/...
- Are they useful? For what?
- What do you think the rules should be?
- Citation and collaboration

    Goal: learning
    Constraint: fairness (including economic fairness)


+ Notation (and Jeff's preference, which we're ignoring)
- Is there a difference between f and f(n)?
- What kind of object is O(f(n))?
- is, =, ∈

+ Miscellaneous notes
- n^2, log(n), and bad intuition
- what does O(1) mean? (walk through the definition)
- what about that absolute value in the ChatGPT proof?

+ Examples: named or otherwise familiar algorithms for each

  O(1)
  O(log n)
  O(n log n)
  O(n)
  O(n^2)

  O(n^3)
  O(sqrt(n))
  O(log log n)
  O(2^n)