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How does the bubble sort perform on: a. An array that is already sorted b. An array that is sorted in reverse order How does the selection sort perform on: a. An array that is already sorted b. An array that is sorted in reverse order How does the merge sort perform on: a. An array that is already sorted b. An array that is sorted in reverse order How does the quick sort perform on: a. An array that is already sorted b. An array that is sorted in reverse order How does the heap sort perform on: a. An array that is already sorted b. An array that is sorted in reverse order

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141 The bubble sort is so slow because it operates only locally Each element moves only one position at a time For example, the element 99 in Example 143 on page 256 is moved by six separate calls to the swap() function to be put into its correct position at a[8] The run time is nearly proportional to the number of comparisons made That number is n(n 1)/2 For every positive integer n, n(n 1)/2 < n2, so n(n 1)/2 = O(n2) Thus, O(n2) is the complexity function The O(n) sorting algorithms (radix sort and bucket sort) are slower than the O(n lg n) sorting algorithms (merge sort, quick sort, and heap sort) because, although their running time is proportional to n, the constant of proportionality is large because of large overhead.

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After the five-point DFT of each row in the data array is computed. the resulting complex array is multiplied by the array of twiddle factors:

The last step then involves computing the three-point DFT of each column. This produces the output array X ( k ) , which is

The bubble sort, the selection sort, and the insertion sort are all O(n2) algorithms. Which is the fastest and which is the slowest among them

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For both the radix sort and the bucket sort, each iteration requires copying all the elements into a list of queues or arrays and then copying them back The merge sort, quick sort, and bucket sort all use the divide-and-conquer strategy The bubble sort, selection sort, insertion sort, merge sort, and quick sort work as well on linked lists as on arrays The quick sort and bucket sort are significantly slower in the worst case The insertion sort, shell sort, and radix sort are significantly faster in the best case Recursion carries the overhead of many recursive method invocations The quick sort implements the divide-and-conquer strategy: first it performs its O(lgn) partitioning of the sequence, and then it recursively sorts each of the two pieces independently The merge sort implements the divide-and-conquer strategy but in the reverse order: It makes its two recursive calls first before performing its O(lgn) merge.

The connections between the three- and five-point DFTs are shown in the following figure, along with the eight twiddle factors:

(h) Using the prime factor algorithm with N I = 5 and N2 = 3, we set A = N 2 = 3 and B = N , = 5. Then, with C = N ~ ( ( N , ' ) ) N ,= 6 and D = N ~ ( ( N ; ' ) ) ~= 10, we have the following index mappings for n and k : ,

14.10 The merge sort, the quick sort, and the heap sort are all O(n lg n) algorithms. Which is the fastest and which is the slowest among them 14.11 Trace by hand the sorting of this array

Both algorithms do O(n) amount of work O(lgn) times thus obtaining O(n lgn) complexity The merge sort is best for sorting linked lists and external files The quick sort reverts to the selection sort in the worst case, when the sequence is already sorted..

The interconnections between the five- and three-point DFTs are the same as in the mixed-radix algorithm. However, there are no twiddle factors, and the ordering of the input and output arrays is different. The 15-point prime fuctor algorithm is diagrammed in the figure below.

The savings with the prime factor algorithm over the mixed-radix FFT are the eight complex multiplies by the twiddle factors.

int a[] = { 44, 88, 55, 99, 66, 33, 22, 88, 77 }

14.2 14.3

Let x ( n ) be a sequence of length 1024 that is to be convolved with a sequence h ( n ) of length L. For what values of L is it more efficient to perform the convolution directly than it is to perform the convolution by taking the inverse DFT of the product X (k)H ( k ) and evaluating the DFTs using a radix-2 FFT algorithm

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