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4.7: Adding Recursion

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  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    Once your merge method is working correctly, you can try out a simple version of merge sort:

    public Deck almostMergeSort() {
        // divide the deck into two subdecks
        // sort the subdecks using selectionSort
        // merge the two halves and return the result

    An exercise at the end of the chapter asks you to implement this algorithm. Once you get it working, the real fun begins! The magical thing about merge sort is that it is inherently recursive.

    At the point where you sort the subdecks, why should you invoke the slower algorithm, selectionSort? Why not invoke the spiffy new mergeSort you are in the process of writing? Not only is that a good idea, it is necessary to achieve the log2 performance advantage.

    To make mergeSort work recursively, you have to add a base case; otherwise it repeats forever. A simple base case is a subdeck with 0 or 1 cards. If mergeSort receives such a small subdeck, it can return it unmodified since it would already be sorted.

    The recursive version of mergeSort should look something like this:

    public Deck mergeSort() {
        // if the deck is 0 or 1 cards, return it
        // divide the deck into two subdecks
        // sort the subdecks using mergeSort
        // merge the two halves and return the result

    As usual, there are two ways to think about recursive programs: you can think through the entire flow of execution, or you can make the “leap of faith” (see Section 6.8). This example should encourage you to make the leap of faith.

    When you used selectionSort to sort the subdecks, you didn’t feel compelled to follow the flow of execution. You just assumed it works because you had already debugged it. And all you did to make mergeSort recursive was replace one sorting algorithm with another. There is no reason to read the program any differently.

    Well, almost. You might have to give some thought to getting the base case right and making sure that you reach it eventually. But other than that, writing the recursive version should be no problem.

    This page titled 4.7: Adding Recursion is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Allen B. Downey (Green Tea Press) .

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