Key Comparison Sorts

Theorem. Any comparison sort requires Ω(n log n) comparisons in the worst case.

Proof.

• Use decision tree for the algorithm
• Binary tree with node for each comparison, leaf for each solution
• Each leaf represents a permutation of the input range
• Sort of a particular data set is represented by a descending path to a leaf
• Length of this path = number of comparisons made by the sort
• n! leaves
• Depth >= log₂ n!
• log₂ n! >= Ω(n log n) by Stirlings formula
  See Narrative for elementary proof.