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918 Continuing from Exercise 917, after part (a), a perform a test so that the smaller subarray is processed by the first recursive call and the larger subarray is processed by the second recursive call b remove the tail recursion by writing a while loop and altering low or high,as necessary c prove that the number of recursive calls is logarithmic in the worst-case 919 Suppose that the recursive quicksort receives an int parameter, depth,from the driver that is initially approximately 2 log N a Modify the recursive quicksort to call mergesort on its current subarray if the level of recursion has reached depth(Hint: decrement depth as you make recursive calls; when it is 0, switch to mergesort
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b Prove that the worst-case running time of this algorithm is O(N log N) c Conduct experiments to determine how often mergesort gets called d Implement this technique in conjunction with tail recursion removal in Exercise 917 e Explain why the technique in Exercise 918 would no longer be needed
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920 An array contains N numbers, and you want to determine whether two of the numbers sum to a given number K For instance, if the input is 8 , 4 , 1 , 6 and K is 10, the answer is yes (4 and 6) A number may be used twice Do the following a Give an O(N2) algorithm to solve this problem b Give an O(N log N ) algorithm to solve this problem (Hint: Sort the items first After doing so, you can solve the problem in linear time) c Code both solutions and compare the running times of your algorithms 921 Repeat Exercise 920 for four numbers Try to design an O(N210g N) algorithm (Hint: Compute all possible sums of two elements, sort these possible sums, and then proceed as in Exercise 920) 922 Repeat Exercise 920 for three numbers Try to design an O(N2) algorjthm 923 In Exercise 628 you were asked to find the single integral solution ~oA%B~+C~+D~+E~=F~W~~~O<AIBICIDIEIFIN, where N is 75 Use the ideas explored in Exercise 921 to obtain a solution relatively quickly by sorting all possible values of A5 + B5 + C5 and F5 - (D5 + E5), and then seeing if a number in the first group is equal to a number in the second group In terms of N, how much space and time does the algorithm require
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924 Compare the performance of Shellsort with various increment sequences, as follows Obtain an average time for some input size N by generating several random sequences of N items Use the same input for all increment sequences In a separate test obtain the average number of comparable comparisons and Comparable copies Set the number of repeated trials to be large but doable within I hour of CPU time The increment sequences are
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a Shell's original sequence (repeatedly divide by 2) b Shell's origiilal sequence, adding 1 if the result is nonzero but even c Gonnet's sequence shown in the text, with repeated division by 22 d Hibbard's increments: 1 , 3 , 7 , , 2k - 1 e Knuth's increments: 1 , 4 , 13, , ( 3 k - 1 ) / 2 f Sedgewick's increments: 1, 5, 19, 41, 109, , with each term having the form of either 9 4 k - 9 2k + 1 or 4k - 3 2k + 1
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925 Code both Shellsort and quicksort and compare their running times Use the best implementations in the text and run them on a integers b real numbers of type d o u b l e c strings 926 Many implementations of quicksort use primitive arrays and pointer hopping (see Appendix D) as an alternative to the normal arrayindexing mechanism In Section D34 we argue that pointer hopping is not always a good idea Implement quicksort by using both techniques on primitive arrays (you will need to pass the number of items to sort to the driver routine) and determine which is faster on average for sorting 100,000 integers 927 Write a template indirect sort, using the techniques in Section 992 In the template, use a function object to pass the underlying sorting algorithm to your indirect sort function 928 Write a function that removes all duplicates in an array A of N items Return the number of items that remain in A Your function must run in O(N log N ) average time (use quicksort as a preprocessing step) 929 Exercise 92 addressed stable sorting Write a function template that performs a stable quicksort To do so, create an array of records; each record is to contain a data item and its initial position in the array (you can use a p a i r object; see Section 55) Then sort the array; if two records have identical data items, use the initial position to break the tie After the array of records has been sorted, rearrange the original array 930 Redo Exercise 929 by using pointers to avoid excessive data movement
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