lec35.md

Sorting & Algorithmic Bounds

$$\log{(N!)} \in \Theta(N \log{N})$$

• $$\log{(N!)}$$ asymptotically same function as $$N \log{N}$$

Puppy, Cat, Dog, Walrus

• Decision tree not complete binary tree → round up to next integer tree height/depth

Generalized

• Tree needs at least $$\lceil \log{(N!)} \rceil$$ levels to have $$N!$$ leaves

Reducing

• Sorting = specific solution to general problem of puppy, cat, dog → any lower bound on difficulty of puppy, cat, dog must ALSO apply to sorting

Optimality

• Sorting via sequential (non-parallel) comparisons → can't do better than $$\Theta(N \log{N})$$

Random Numbers

• Recording random stuff = expensive → usually rely on pseudorandomness instead (overflow via repeated multiplication on starting seed value)