lec33.md

Basic Sorts, Quick Sort

Insertion Sort Runtime

Insertion Sort on Almost Sorted Arrays

• Insertion sort runtime can be bounded by # of inversions (# of out-of-order pairs)
  • Each swap fixes 1 inversion → total runtime = # of swaps = # of inversions
  • Runtime proportional to how much work needed to be done

Insertion Sort Sweet Spots

• Almost sorted = small # of inversions → linear # of inversions out of possible $$N^{2}$$
• Insertion sort runtime always $$\Theta(N + K)$$, $$K$$ = # of inversions

Sorts So Far

• Heapsort = flavor of selection sort → still sorts by finding smallest/largest item each time

Quicksort

Context

• $$D$$ = # of entries in dictionary
• Binary search on tape $$\Theta(N)$$ b/c physical movement about halved each time → $$\frac{N}{2} + \frac{N}{4} + \cdots + 1 = N - 1$$

Partition Sort (Quicksort)

• Given array, partition on pivot → after partitioning, pivot in sorted position (invariant)
  • Partitioning = single item sort

Runtime/Performance

• Quicksort runtime non-deterministic, depends on where pivot lands
• Best case, pivot always lands in middle

Sorting Summary

• Counting inversions takes $$\Theta(N \log{N})$$ time