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is some constant c for which t = c nlgn If it takes t = 31 milliseconds to sort n = 200 elements, then (31) = c (200) lg(200), so c = (31 milliseconds)/(200 lg(200)) = 00155/lg(200) Then, for n = 40,000, t = c n lg n = (00155/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 = 124 s 143 144 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 141 on page 257, is insensitive to input.

The Multidata software provided a way for operators to digitize representations of metal shields called blocks, which are used to protect healthy, delicate tissue from exposure to x-rays The doctors prescribed the use of various blocks for individual patients The machine operators entered descriptions of the blocks into the software, and the Multidata software then calculated the amount of radiation to deliver When investigating the cause of the radiation overexposures, investigators found that operators could enter data about the blocks in such a way that the software would misunderstand the digitized representation, mistaking holes in the pattern of blocks for the block itself Multidata later recognized the problem as the.

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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 146 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.

// If numbers remain only in list_B, move those // to the sorted list while list_B is not empty sorted_list[index] <-- list_B[1] discard list_B[1] index <-- index + 1 // Return the sorted list return sorted_list The performance of merge is related to the lengths of the lists on which it operates, the total number of items being merged The real work of the routine is in moving the appropriate elements of the original lists into the sorted list Since the total number of such moves is equal to the sum of the numbers in the two lists, merge has a theta of nA + nB, or (nA + nB), where nA + nB is equal to the sum of the numbers in the two lists The merge_sort will use the merge routine, but first the merge_sort will divide the problem up into smaller and smaller sorting tasks.

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:.

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

Then merge_sort will reassemble the small sorted lists into one fully sorted list In fact, merge_sort divides the list of numbers until each sublist consists of a single number, which can be considered a sorted list of length 1 Then the merge_sort uses the merge procedure to join the sorted sublists The technique used by merge_sort to divide the problem into subproblems is called recursion The merge_sort repeatedly calls itself until the recursion bottoms out with lists whose lengths are one Then the recursion returns, reassembling the numbers in sorted order as it does Here is pseudocode for the merge sort It takes the list of numbers to be sorted, and it returns a sorted list of those numbers.

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