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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 14.22 on page 277.) 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 14.18 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. THE QUICK SORT The quick sort is like the merge sort: It is recursive, it requires an auxiliary function with several loops, and it runs in O(n lg n) time. But in most cases it is quicker than the merge sort. The quick sort works by partitioning the array into two pieces separated by a single element x that is greater than or equal to every element in the left piece and less than or equal to every element in the right piece. This guarantees that the single element x, called the pivot element, is in its correct position. Then the algorithm proceeds, applying the same method to the two pieces separately. This is naturally recursive and very quick. Algorithm 14.6 The Quick Sort (Precondition: s = {sp . . . sq 1} is a sequence of q p ordinal values.) (Postcondition: The entire sequence s is sorted.) 1. If q p > 1, do steps 2 5. 2. Apply Algorithm 14.7 to s, obtaining the pivot index m. 3. (Invariant: the pivot element sm is in its correct sorted position.) 4. Apply the quick sort to {s0, s1, . . . , sm 1}. 5. Apply the quick sort to {sm+1, si+2, . . . , sn 1}. Algorithm 14.7 Partition (Precondition: s = {sp . . . sq 1} is a sequence of q p ordinal values.) (Postcondition: Return m, where p m < q and si sm sj for p i 1. Set x = s p (the pivot element). 2. Set i = p and j = q.
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Repeat steps 4 7 while i < j. Decrement j until either sj < x or j = i. If j > i, copy sj into si . Increment i until either si > x or i = j. If j > i, copy sj into si . Copy x into sj.
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