Patterns in Software

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Patterns
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Display 205 Quick-Sort Realization of Sorting Pattern (part 2 of 2)
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39 40 41 42 43 44 template <class T> void join(T a[], int begin, int splitPt, int end) { //Nothing to do }
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For example, the merge sort realization of split divides the array into two roughly equal parts, and a merge sort is indeed very ef cient It can be shown, although we will not do so here, that the merge sort runs in time O(N log N) and that any sorting algorithm (that meets some minimal and reasonable constraints) cannot be faster than O(N log N) So, a merge sort is, in some sense, optimal The quick-sort realization of split divides the array into two portions that might be almost equal or might be very different in size depending on the choice of a splitting value Since in extremely unfortunate cases the split might be very uneven, the worstcase running time for a quick sort is O(N 2 ), which is much slower than the O(N log N) we obtained for the merge sort However, for an array that is lled with randomly chosen values, most splitting values will produce a division that is close enough to an even division So, under a suitable de nition of average case, the quick sort has an averagecase running time that is O(N log N) In practice, the quick sort is one of the bestperforming sorting algorithms The selection sort algorithm, which we discussed in 5, divides the array into two pieces, one with a single element and one with the rest of the array interval (see Self-Test Exercise 2) Because of this uneven division, the selection sort has a worst-case, and even average-case, running time that is O(N 2 ) In practice, the selection sort is not a very fast sorting algorithm, although it does have the virtue of simplicity We do not have room to provide proof of these running times in this book, but you can nd such results in almost any data structures or analysis of algorithms text
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PRAGMATICS
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PATTERNS
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You should not feel compelled to follow all the fine details of a pattern Patterns are guides, not requirements For example, we did the quick-sort implementation by exactly following the pattern in order to have a clean example In practice, we would have taken some liberties Notice that, with a quick sort, the join function does nothing In practice, we would simply eliminate the calls to join These calls incur overhead and accomplish nothing
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Patterns and UML
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The Divide-and-Conquer Sorting pattern must divide the interval being sorted into two smaller intervals If there were cases in which the split function divided the interval into an empty interval and the rest, then one subinterval would be the same as the original interval being divided, and infinite recursion would result In the quick-sort realization we avoided this infinite recursion with the following lines from the definition of split:
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if (up > 0) return (begin + up); else return (begin + 1); //Ensures that both pieces are nonempty
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Without this extra adjustment, the function split could compute 0 as the value of up and so divide the interval into two intervals with one of the two being empty That would produce infinite recursion The way this is usually avoided with a quick-sort realization (and the way that produces the nicest code) is to separate out the split point and divide only the remaining element, so that the array interval is divided into three pieces: the split point, the subinterval before the split point, and the subinterval after the split point This guarantees that even if one subinterval is empty, the other is shorter than the interval being divided Thus, infinite recursion is avoided When these points are taken into consideration, you are likely to change the Sorting pattern to the following when you are designing the quick-sort realization:
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if ((end - begin) > 1) { int splitPt = split(a, begin, end); sort(a, begin, splitPt - 1); sort(a, splitPt + 1, end); }//else sorting one (or fewer) elements, so do nothing
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