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1 The merge sort main sort() method, the array is already sorted if its length is 0 or 1 Otherwise, the postcondition of the three-parameter sort() method guarantees that the array will be sorted after that method returns because the entire array is passed to it That postcondition is the same as the postcondition of the merge() method, which is invoked last, so it remains to verify that the merge() method s postcondition will be true The merge() method s postcondition follows from its loop invariant, because when that loop has finished, the tmp[] array is sorted and that is copied back into a[] in the same order So it remains to verify the loop invariant for all k < q - p Suppose the invariant is false for some k, that is, tmp[0.k) is not sorted Then there must be some x and y in tmp[0..

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By using the divide-and-conquer strategy, the merge sort obtains an O(n lgn) run time, a significant improvement over the O(n2) times spent by the previous sorting algorithms The strategy is 1 Split the sequence into two subsequences 2 Sort each subsequence separately 3 Merge the two subsequences back together The merge sort does the first step in the simplest balanced way possible: It splits the sequence at its middle If the first step is done in other ways, we obtain different sorting algorithms The divide-and-conquer strategy is also used in the binary search (page 31) The simplest unbalanced way to split the sequence is to put all but the last element in the first subsequence, leaving only the last element in the second subsequence This produces the recursive version of the insertion sort (See Problem 1422 on page 277.

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k), where x was copied into tmp[] before y but x > y We may assume without loss of generality that x was copied from the left half of a[] and y from the right half, as Figure 142 The merge sort shown in Figure 142 Thus, x = a[r] and y = a[s] for some indexes r and s such that p r < i and m s < j Now the two halves of a[] are each already separately sorted Then for every element z in a[m.s], z a[s] But a[s] = y < x Therefore, every element z in a[m.s] must have been copied into tmp[] before x was, because this assignment.

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) Another unbalanced way to split the sequence is to put the largest element alone in the second subsequence, leaving all the other elements in the first subsequence This produces the recursive version of the selection sort (See Problem 1418 on page 276) Not that this makes the merge step 3 trivial: Merely append the largest element to the end of the first subsequence A fourth way to split the sequence is to partition it so that every element in the first subsequence is less than every the element in the second subsequence This condition of course is true in the previous case that led to the recursive selection sort However, if we can obtain this property together with having the two subsequences the same size, then we obtain a new O(n lgn) algorithm, called the quick sort.

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The input x ( n ) is completely unknown, and we would like to recover x ( n ) from y ( n ) . The following procedure is proposed for recovering part of x ( n ) from y(n).

tmp[k++] = ( a[i]<=a[j] a[i++] : a[j++] ); always copies the smaller element first. But that means that a[s] was copied into tmp[] before x. But a[s] = y, which was assumed to have been copied into tmp[] after x. This contradiction

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