
  So worst-case comparisons are quadratic.

  ### Worst-case recursion depth

  Depth becomes n (a chain), so auxiliary stack space is Θ(n) (if not optimized).

  ———

  ### Space complexity

  - In-place partition implementations use O(1) extra array storage.
  - But recursion uses stack:
      - average/balanced: O(log n)
      - worst case: O(n)

  ———

  ### Why quicksort is still popular

  Despite worst-case O(n^2), it is often fastest in practice because:

  - good cache locality,
  - low constant factors,
  - in-place behavior,
  - randomized or median-of-three pivot selection usually avoids pathological splits.

  Many libraries use introsort (quicksort + fallback to heapsort) to guarantee O(n log n) worst-case
  while keeping quicksort speed on average.


› Write a very detailed analysis of the P vs NP problem including all known complexity classes


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