Lower bound for comparison based sorting algorithms

The problem of sorting can be viewed as following.

Input: A sequence of n numbers <a1, a2, . . . , an>.
Output: A permutation (reordering) <a1, a2, . . . , an> of the input sequence such that a1 <= a2 ….. <= 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 ai <= aj is made. The left subtree then dictates subsequent comparisons for ai <= aj, and the right subtree dictates subsequent comparisons for ai > aj. 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.

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

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