Introduction to Algorithms, Second Edition

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The University of Texas at Austin Comparison & Integer Sorting Department of Computer Sciences Professor Vijaya Ramachandran Lecture 21 CS357: ALGORITHMS, Spring 2006 Lower bound on comparison-based sorting There are several algorithms that sort n elements in O ( n log n ) time using only comparisons between pairs of input elements in order to obtain their relative ordering. These include MergeSort and Heapsort and also Quicksort (expected time when randomized). An algorithm that sorts n elements by only making comparisons between pairs of input elements in order to deduce their relative order will be called a comparison-based algorithm . We will now show that any comparison-based algorithm for sorting must make Ω( n log n ) comparisons and hence must have Ω( n log n ) running time. Let A be such an algorithm. As described in the textbook, we will capture the comparisons it makes on every input of length n in terms of a binary decision tree . Thus the decision tree represents all comparisons made by the algorithm on every input of length n , but ignores data movement and other aspects of the algorithm. The leaves of this decision tree represent all possible outputs for an input of length n
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