Lower bound for comparison based sorting algorithms

The problem of sorting can be viewed as following.

Input: A sequence of n numbers <a₁, a₂, . . . , a_n>.
Output: A permutation (reordering) <a‘₁, a‘₂, . . . , a‘_n> of the input sequence such that a‘₁ <= a‘₂ ….. <= a‘_n.

A sorting algorithm is comparison based if it uses comparison operators to find the order between two numbers.  Comparison sorts can be viewed abstractly in terms of decision trees. A decision tree is a full binary tree that represents the comparisons between elements that are performed by a particular sorting algorithm operating on an input of a given size. The execution of the sorting algorithm corresponds to tracing a path from the root of the decision tree to a leaf. At each internal node, a comparison a_i <= a_j is made. The left subtree then dictates subsequent comparisons for a_i <= a_j, and the right subtree dictates subsequent comparisons for a_i > a_j. When we come to a leaf, the sorting algorithm has established the ordering. So we can say following about the decison tree.

1) Each of the n! permutations on n elements must appear as one of the leaves of the decision tree for the sorting algorithm to sort properly.

2) Let x be the maximum number of comparisons in a sorting algorithm. The maximum height of the decison tree would be x. A tree with maximum height x has at most 2^x leaves.

After combining the above two facts, we get following relation.

  
      n!  <= 2^x

 Taking Log on both sides.
      log2(n!)  <= x

 Since log2(n!)  = Θ(nLogn),  we can say
      x = Ω(nLog2n)

Therefore, any comparison based sorting algorithm must make at least nLog₂n comparisons to sort the input array, and Heapsort and merge sort are asymptotically optimal comparison sorts.

References:
Introduction to Algorithms, by Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest and Clifford Stein

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