CS 300 Assignment 3 Due Monday Nov. 20 by 11:59 pm Name: Email: Instructions: Fill in your name and email. Use webct to submit your worksheet. There are four questions, each worth 25 points. Assignments must be done individually; however, if you have questions or do not understand how to do something, you should ask or send your question to the class mailing list. Overview: The purpose of this assignment is to utilize Maple and generating functions to investigate the expected number of comparisons used by a modification of Quicksort called Median of Three Quicksort.

Derive a functional equation for the generating function for the harmonic numbers using the recurrence NiMvJiUiSEc2IyUibkcsJiZGJTYjLCZGJyIiIkYsISIiRiwqJkYsRixGJ0YtRiw=, with NiMvJiUiSEc2IyIiIkYn. First integrate the generating function 1/(1-z) and its series expansion to find a generating function for the sequence 1, 1/2, 1/3, 1/4,... restart; Multiplying the recurrence relation for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiSEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUYvNiVRIm5GJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw== by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUYvNiVRIm5GJ0YyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYn and sum to derive a functional equation for the generating function, 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 for the harmonic numbers. Solve for 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and use series to verify that it generates the harmonic numbers. Use rsolve to solve the recurrence NiMvLSUiQUc2IyUibkcsJiomIiImIiIiLUYlNiMsJkYnRitGKyEiIkYrRisqJiIiJ0YrLUYlNiMsJkYnRisiIiNGL0YrRi8=, A(1) = 1, A(0) = 0. Show how to solve this recurrence using generating functions. I.E. use the recurrence to find the generating function, and then use a partial fraction (see convert/parfrac ) decomposition to find the nth term of the sequence given by the recurrence.

Implement and test the Median of Three modification to quicksort. The Median of Three Quicksort selects the pivot element to be the median of the first three elements of the input array. The base case of this variant is when the input array has fewer than 3 elements. In this case, you may simply use Maple's sort to return the sorted list. The code for quicksort from class is included below. Just like the version of Quicksort, you should return the number of comparisons used using a pass by reference parameter count. When counting comparisons, you should return 0 in the base case and should not count the comparisons used to determine the median. with(combinat): # S := Quicksort(L,count) # Inputs: # L : a list of integers L = [l1,...,ln] # Outputs: # S : a list of integers [s1,...,sn] with s1 <= ... <= sn. # count : positive integer passed by reference. Upon completion count is # set to the number of comparisions used to sort L. Quicksort := proc(L,count::name) local pivot, n, i, L1, L2, S, S1, S2, comps, count1, count2; n := nops(L); comps := 0; if n = 0 or n = 1 then S := L; else pivot := L[1]; L1 := []; L2 := []; for i from 2 to n do if L[i] < pivot then L1 := [L[i],op(L1)]; else L2 := [L[i],op(L2)]; fi; comps := comps + 1; od; S1 := Quicksort(L1,count1); S2 := Quicksort(L2,count2); S := [op(S1),pivot,op(S2)]; comps := count1 + count2 + comps; fi; count := comps; return S; end;

Zio2JEkiTEc2IidJJmNvdW50R0YlSSVuYW1lRyUqcHJvdGVjdGVkRzYtSSZwaXZvdEdGJUkibkdGJUkiaUdGJUkjTDFHRiVJI0wyR0YlSSJTR0YlSSNTMUdGJUkjUzJHRiVJJmNvbXBzR0YlSSdjb3VudDFHRiVJJ2NvdW50MkdGJUYlRiVDJz5GLC1JJW5vcHNHRik2I0YkPkYzIiIhQCU1L0YsRjwvRiwiIiI+RjBGJEMqPkYrJkYkNiNGQT5GLjciPkYvRkg/KEYtIiIjRkFGLEkldHJ1ZUdGKUMkQCUyJkYkNiNGLUYrPkYuNyRGUC1JI29wR0YpNiNGLj5GLzckRlAtRlU2I0YvPkYzLCZGM0ZBRkFGQT5GMS1JKlF1aWNrc29ydEdGJTYkRi5GND5GMi1GaW42JEYvRjU+RjA3JS1GVTYjRjFGKy1GVTYjRjI+RjMsKEY0RkFGNUZBRjNGQT5GJ0YzT0YwRiVGJUYl

Compare the expected number of comparisons used by Quicksort and and the Median of Three variant on inputs of size LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYn3..8 (compute the expected number of comparisons by exhaustively running through all permutations of size 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For the comparison use should compute the ratio (use evalf).

Recall the following recurrence for the expected number of comparisons used by Quicksort. 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 Assume that each of the 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permutations are equally likely to occur as input to the Median of Three quicksort algorithm. Then LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRIUYnLyUnbHNwYWNlR1EkMGVtRicvJSdyc3BhY2VHRk8vJShtaW5zaXplR1EiMUYnLyUobWF4c2l6ZUdRKWluZmluaXR5Ric= is selected as the pivot element when the first three elements in the input array contain 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a number less than LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUSQwZW1GJy8lJ3JzcGFjZUdRM3Zlcnl0aGlja21hdGhzcGFjZUYnLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnand a number greater than LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIi5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUSQwZW1GJy8lJ3JzcGFjZUdGTy8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJy1GNjYwUSJ+RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSFGJ0ZNRlBGUkZVRlg=The probability that this happens is equal to 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 Therefore, following the derivation of the recurrence for the expected number of comparisons used by quicksort, the expected number of comparisons for the Median of Three quicksort algorithm satisfies the following recurrence. 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 For this analysis, we do not count the comparisons required to compute the median and assume that no comparisons are used when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIjxGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkjbW5HRiQ2JFEjMy5GJ0Y5 Write a recursive Maple procedure to compute the expected number of comparisons used by the Median of Three quicksort algorithm. Compare the expected values computed from the recurrence to those obtained empirically in the previous question.Using this function and the one from the lecture on quicksort for the expected number of comparisons used by quicksort, compute a sequence of ratios of the expected number of comparisons of the two algorithms. Does this appear to be approaching a limit?

Multiplying the recurrence by 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 the recurrence in the previous section becomes 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 Use this equation to derive a functional equation for the generating function, 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for the expected number of comparisons used by the Median of Three quicksort algorithm. Hint: The techniques used in the quicksort lecture to derive the functional equation can be used for this problem, and you will get a similar, but higher order, differential equation. You will need to find a generating function for 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For this it is convenient to write this series in terms of series for 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 Use dsolve to solve the equation. Note that the boundary conditions for the differential equation are obtained from the base cases of the recurrence: 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(these conditions must be stated in terms of the function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInpGJ0YvRjIvRjNRJ25vcm1hbEYn and its derivatives - hint, how do you obtain the coefficients of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiTUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInpGJ0YvRjIvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2MFEifkYnRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUlZm9ybUdRIUYnLyUnbHNwYWNlR1EkMGVtRicvJSdyc3BhY2VHRlcvJShtaW5zaXplR1EiMUYnLyUobWF4c2l6ZUdRKWluZmluaXR5Ric=from the Taylor series). Show that the resulting solution to the differential equation generates the sequence of expected values for the number of comparisons for the Median of 3 quicksort, and determine a formula for the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==-th coefficient in the series (this gives a closed form for the expected number of comparisons). By computing the limiting ratio of 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you can theoretically compute the limiting ratio for the expected number of comparisons of the original quicksort to the improved version (Median of Three). How does this compare to your empirical data in Questions 2 and 3? restart;