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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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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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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
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int a[] = { 44, 88, 55, 99, 66, 33, 22, 88, 77 }
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by each of the following algorithms: a. The quick sort b. The heap sort c. The bubble sort d. The selection sort e. The insertion sort f. The merge sort 14.12 Modify the bubble sort so that it sorts the array in descending order. 14.13 Modify the bubble sort so that it is smart enough to terminate as soon as the array is sorted. 14.14 Prove Theorem 14.1 on page 257. 14.15 Prove Theorem 14.2 on page 257. 14.16 The shaker sort is the same as the bubble sort except that it alternates bubbling up and down the array. Implement the shaker sort, and determine whether it is more efficient than the straight insertion sort. 14.17 Modify the selection sort (Algorithm 14.2 on page 257) so that it uses the smallest element of {si . . . sn 1} in step 2. 14.18 Rewrite the selection sort recursively. 14.19 Prove Theorem 14.3 on page 258. 14.20 Prove Theorem 14.4 on page 258. 14.21 Modify the insertion sort so that it sorts the array indirectly. This requires a separate index array whose values are the indexes of the actual data elements. The indirect sort rearranges the index array, leaving the data array unchanged.
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14.22 Rewrite the insertion sort recursively. 14.23 Prove Theorem 14.5 on page 259. 14.24 Prove Theorem 14.6 on page 259. 14.25 Prove Theorem 14.7 on page 259. 14.26 Modify the quick sort so that it selects its pivot as the last element instead of the first element of the subsequence. 14.27 Modify the quick sort so that it selects its pivot as the median of the first, middle, and last elements. 14.28 Modify the quick sort so that it reverts to the insertion sort when the array size is below 8. 14.29 Since the heap sort runs in O(n lgn) time, why isn t it always preferred over the quick sort, which runs in O(n2) in the worst case 14.30 Since the heap sort runs in O(n lgn) time and requires no extra array space, why isn t it always preferred over the merge sort, which requires duplicate array space 14.31 Prove Theorem 14.15 on page 269. 14.32 Here is the Las Vegas sort, as applied to sorting a deck of cards: 1. Randomly shuffle the cards. 2. If the deck is not sorted, repeat step 1. Derive the complexity function for this sorting algorithm.
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Answers to Review Questions
14.1 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 14.3 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. 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. 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.
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