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Theorem 14.19 The bucket sort runs in O(n) time. The algorithm has three parallel loops, each iterating n times. The last loop has an inner loop, but it averages only one iteration. The minimum() and maximum() methods also require n steps each. Hence the number of steps executed is proportional to 5n. Like the radix sort, the O(n) bucket sort is in practice much slower than the O(n lgn) sorting algorithms because of the substantial overhead costs. Review Questions
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14.1 14.2 14.3 14.4 Why is the bubble sort so slow The bubble sort makes n(n 1)/2 comparisons to sort n elements. How does it follow that its complexity function is O(n2) Why are the O(n) sorting algorithms (radix sort and bucket sort) slower than the O(n lg n) sorting algorithms (merge sort, quick sort, and heap sort) The merge sort applies the general method, known as divide and conquer, to sort an array. It divides the array into pieces and applies itself recursively to each piece. What other sorting algorithm(s) use this method Which sorting algorithms work as well on linked lists as on arrays Which sorting algorithms have a different worst case complexity than their average case Which sorting algorithms have a different best case complexity than their average case Why is the nonrecursive version of a recursive sorting algorithm usually more efficient How is the quick sort like the merge sort
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14.5 14.6 14.7 14.8 14.9
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14.10 Under what circumstances would the merge sort be preferred over the other two O(n lg n) sorting algorithms 14.11 Under what circumstances is the quick sort like the selection sort 14.12 Under what circumstances would the quick sort be preferred over the other two O(n lg n) sorting algorithms 14.13 How is the heap sort similar to the selection sort and the insertion sort 14.14 Which algorithm does the Java API use to implement its java.util.Arrays.sort() methods 14.15 A sorting algorithm is said to be stable if it preserves the order of equal elements. Which of the sorting algorithms are not stable 14.16 Which of the nine sorting algorithms outlined in this chapter require extra array space 14.17 Which of the nine sorting algorithms outlined in this chapter would work best on an external file of records 14.18 The merge sort is parallelizable. This means that parts of it can be performed simultaneously, independent of each other, provided that the computer has multiple processors that can run in parallel. This works for the merge sort because several different parts of the array can be subdivided or merged independently of other parts. Which of the other sorting algorithms described in this chapter are parallelizable
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14.19 Imagine a Web site that has a Java applet for each sorting algorithm that shows how the algorithm works by displaying an animation of a test run on an array a[] of 256 random numbers in the range 0.0 to 1.0. The animation shows on each iteration of the algorithm s main loop a two-dimensional plot of 256 points (x, y), one for each element in the array, where x = i+1 and y = a[i]. Each plot in Figure 14.7 shows the progress halfway through the sort for one of the following six algorithms: selection sort insertion sort merge sort quick sort heap sort radix sort Match each plot with the sorting algorithm that produced it:
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Figure 14.7 Sorting algorithms in motion
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14.1 If an O(n2) algorithm (e.g., the bubble sort, the selection sort, or the insertion sort) takes 3.1 milliseconds to run on an array of 200 elements, how long would you expect it to take to run on a similar array of: a. 400 elements b. 40,000 elements If an O(n lg n) algorithm (e.g., the merge sort, the quick sort, or the heap sort) takes 3.1 milliseconds to run on an array of 200 elements, how long would you expect it to take to run on a similar array of 40,000 elements The insertion sort runs in linear time on an array that is already sorted. How does it do on an array that is sorted in reverse order