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14.14 14.15 14.16 14.17 14.18 14.19
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The quick sort is best for sorting large arrays of primitive types. The selection sort can be seen as a sort-on-output process: Insert the elements into an array as they are given, and then repeatedly select out the next largest element. The insertion sort can be seen as a sorton-input process: Repeatedly insert each element into its correct ordered position in an array, and then remove them in their array order. So the selection sort inserts the elements into the array in O(n) time and removes them in O(n 2 ), while the insertion sort inserts the elements into the array in O(n2) time and removes them in O(n). Both result in an O(n2) algorithm. The heap sort can be seen as a partial-sort-on-input-and-partial-sort-on-output process: Insert the elements into an array maintaining the (partially sorted) heap property, and then repeatedly select the first (which is the smallest) element and restore the heap property. Both the insertion process and the removal process have the same O(n lgn) running time, resulting in a total O(n lgn) running time. The Java API uses the merge sort to implement its Arrays.sort() methods for arrays of objects, and it uses the quick sort to implement its Arrays.sort() methods for arrays of primitive types. The shell sort, quick sort, and heap sort are unstable. The merge sort, radix sort, and bucket sort require extra array storage. The bubble sort, selection sort, insertion sort, merge sort, and quick sort work as well on external files of records. The shell sort, merge sort, quick sort, and bucket sort all would run significantly faster on a parallel computer. Matching the algorithms with their graphical output is shown in Figure 14.8.
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Figure 14.8 Sorting algorithms in motion
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14.1 The O(n2) algorithm should take: a. 12.4 milliseconds (4 times as long) to run on the 400-element array. b. 124 seconds (40,000 times as long) to run on the 40,000-element array. That s about 2 minutes. This answer can be computed algebraically as follows. The running time t is proportional to n2, so there is some constant c for which t = c n2. If it takes t = 3.1 milliseconds to sort n = 200 elements, then (3.1 milliseconds) = c (200 elements)2, so c = (3.1 milliseconds)/(200 elements)2 = 0.0000775 milliseconds/element2. Then, for n = 40,000, t = c n2 = (0.0000775 milliseconds/ element2) (40,000 elements)2 = 124,000 milliseconds = 124 seconds. The O(n lg n) algorithm should take 1.24 seconds (400 times as long) to run on the 40,000-element array. This answer can be computed algebraically. The running time t is proportional to nlgn, so there
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is some constant c for which t = c nlgn. If it takes t = 3.1 milliseconds to sort n = 200 elements, then (3.1) = c (200) lg(200), so c = (3.1 milliseconds)/(200 lg(200)) = 0.0155/lg(200). Then, for n = 40,000, t = c n lg n = (0.0155/lg(200))(40,000 lg(40,000)) = 620 (lg(40,000)/lg(200)). Now 40,000 = 2002, so lg(40,000) = lg(2002) = 2 lg 200. Thus, lg(40,000)/lg(200) = 2, so t = 620 2 milliseconds = 1240 milliseconds = 1.24 s. 14.3 14.4 The insertion sort has its worst performance on an array that is sorted in reverse order, because each new element inserted requires all of the elements on its left to be shifted one position to the right. The bubble sort, as implemented in Algorithm 14.1 on page 257, is insensitive to input. That means that it will execute the same number n(n 1)/2 of comparisons regardless of the original order of the elements in the array. So it doesn t matter whether the array is already sorted or whether it is sorted in reverse order; it is still very slow. The selection sort is also insensitive to input: It takes about the same amount of time to sort arrays of the same size, regardless of their initial order. The merge sort is also insensitive to input: It takes about the same amount of time to sort arrays of the same size, regardless of their initial order. The quick sort is quite sensitive to input. As implemented in Algorithm 14.6 on page 263, the quick sort will degrade into an O(n2) algorithm in the special cases where the array is initially sorted in either order. That is because the pivot element will always be an extreme value within its subarray, so the partitioning splits the subarray very unevenly, thereby requiring n steps instead of lg n. The heap sort is a little sensitive to input, but not much. The heapify() function may require fewer than lg n iterations. The bubble sort is slower than the selection sort, and the insertion sort (in most cases) is a little faster. The merge sort is slower than the heap sort, and the quick sort (in most cases) is faster. a. Trace of the quick sort:
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a[0] 44 22 a[1] 88 33 a[2] 55 44 a[3] 99 77 55 a[4] 66 a[5] 33 55 77 a[6] 22 88 a[7] 88 a[8] 77 99
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b. Trace of the heap sort:
a[0] 44 99 77 88 44 88 22 77 33 66 22 55 33 44 33 22 a[1] 88 99 44 88 a[2] 55 a[3] 99 88 a[4] 66 a[5] 33 a[6] 22 a[7] 88 a[8] 77
44 99 77 88
77 66 44 22
44 88 22 77 33 66 55
22 44 33
c. Trace of the bubble sort:
a[0] 44 a[1] 88 55 a[2] 55 88
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