Examples of Recursion: Problems 3

Problem : What is the base case for quicksort? How would combining quicksort with another sorting algorithm like selection sort change the base case?

The base case for quick sort is when the size of the partition we're working on is one element. As it is in order by definition, there is nothing more to do and the recursion can stop. If we were to combine quicksort with another sort like selection sort, the base case for quicksort becomes the size of the partition at which we switch sorts; this is often around five or six elements.

Problem : How does mergesort achieve its O(nlogn) efficiency?

Mergesort continually splits the data set in half, and at each step it works on O(n) elements. Since the data set can be split in half O(logn) times, the total work for mergesort is O(nlogn).

Problem : While mergesort and quicksort are two "smart" and efficient sorts, there are plenty of inefficient sorts out there, none of which you would ever want to use in a program. One such sort is the permutation sort. A permutation of a data set is one configuration, one ordering of the data. If there are n data elements in a data set, then there are n! permatuations (you have n choices for which element goes first, then n - 1 choices for which element goes second, n - 2 choices for which element goes third, etc, so n!). The permutation sort algorithm computes every permutation of the data set, and for each one checks to see if it is in order If it is, the algorithm ends. If not, it continues on to the next permuation. Write permuation sort recursively (the easiest way to do it). Note that a recursive algorithm can still have loops.

int sort(int arr[], int n, int i) { int j, flag, swap; int true = 1, false = 0; /* Check to see if list is sorted */ flag = 1; for (j=0; j<n-1; j) { if (arr[j] >= arr[j+1]) { flag = 0; break; } } if (flag) return 1; /* Compute each permutation recursively */ for(j=i+1; j<n; j) { swap = arr[i]; arr[i] = arr[j]; arr[j] = swap; if (sort(arr, n, i+1)) return true; swap = arr[i]; arr[i] = arr[j]; arr[j] = swap; } return false; } void permutationsort(int arr[], int n) { sort(arr, n, 0); }

Problem : Your friend Jane proposes the following algorithm for a sort:

random_sort(data set) { -randomly swap two elements -check to see if the data is in order -if it is return as we're done -otherwise call random_sort }

Jane claims that although this algorithm is incredibly inefficient, it will work. You claim that even if you lucked out and got good random swaps, in most cases it would cause your computer program to crash. Why?

After every swap, the function will make another recursive call to itself. Due to the incredible number of function calls necessary to get the array into order, the space on the call stack will be exhausted far earlier than a solution could be found.

Problem : Your friend John claims that quicksort has a worst case running time of O(n²). Is he right?

Yes, quicksort does have a worst case running time of O(n²). If the value of the pivot causes the split to create two sets in every recursive step, one with only 1 element and one with the rest of the elements, then there will be O(n) recursive calls, each one doing O(n) work. Thus an O(n²) algorithm. However, with a good implementation of quicksort that uses pivot picking methods such as random selection and tri-median, the chances of this hapenning are minimal. Quicksort is often the best sort to use, and is used in many commercial and academic programs.

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