The linear worst-case algorithm is a classic result even though it is impractical

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98 A Lower Bound for Sorting

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Although we have O(N log N ) algorithms for sorting, it is not clear that this is as good as we can do In this section we prove that any algorithm for sorting that uses only comparisons requires Q ( N log N) comparisons (and hence time) in the worst case In other words, any algorithm that sorts by using element comparisons must use at least roughly N log N comparisons for some input sequence We can use a similar technique to show that this condition holds on average

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Any comparisonbased sorting algorithm must use roughly N log N comparisons on average and in the worst case

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

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Internal selection method that makes recursive calls Uses median-of-three partitioning an6 a cutoff of 10 Places the kth smallest item in ark-11 a is an array of Comparable items low is the left-most index of the subarray high is the right-most index of the subarray k is the desired rank (1 is minimum) in the entire array template <class Comparable> void quickselect( vector<Comparable> & a, int low, int high, int k ) i if( low + CUTOFF > high ) insertionsort( a, low, high ) ; else

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/ / Sort low, middle, high int middle = ( low + high ) / 2 ; if( a[ middle I < a[ low I ) swap ( a [ low ] , a [ middie I ) if( a[ high ] < a[ low I ) swap ( a [ low ] , a [ high I ) ; if ( a [ high 1 < a [ middle 1 1 swap ( a [ middle ] , a [ high I

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/ / Place pivot at position high Comparable pivot = a[ middle I ; swap ( a [ middle I , a[ high - 1 I )

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/ / Begin partitioning int i, j; for( i = low, j = high - 1;

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while( a[ ++i ] < pivot i { 1 while( pivot < a[ --j I ) i I if( i < j ) swap( a[ i I , a[ j I 1 ; else break ;

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swap( a[ i 1 , a[ high

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/ / Restore pivot

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/ / Recurse; only this part changes if( k <= i ) quickselect( a, low, i - 1 , k ) ; else if( k > i + 1 ) quickselect( a, i + 1 , high, k ) ;

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Quickselect with median-of-three partitioning and cutoff for small arrays

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The proofs are abstract; we the worst-case lower bound

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Must every sorting algorithm work by using comparisons The answer is no However, algorithms that do not involve the use of general comparisons are likely to work only for restricted types, such as integers Although we may often need to sort only integers (see Exercise 916), we cannot make such sweeping assumptions about the input of a general-purpose sorting algorithm We may assume only the given-namely, that, because the items need to be sorted, any two items can be compared Next, we prove one of the most fundamental theorems in computer science, as Theorem 93 Recall first that the product of the first N positive integers is N! The proof is an existence proof, which is somewhat abstract It shows that some bad input must always exist

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Any algorithin that sorts hx using element co~nparisons only tnust use at least [log ( N ! ) ]comparisons for some input sequence