Tag Archives: insertionsort

Sorting Algorithms (Take Two)

Sorting is an essential part of our everyday lives. Because of this importance, many people have devoted much time towards improving the performance of different sorting procedures. Previously, I wrote a script comparing many of these algorithms. I decided to break that single script up into several scripts (one for each algorithm) so that the individual algorithms are easier to access and learn more about.

When I first learned about BubbleSort, I thought of being in elementary school and how my teacher would arrange students in line by increasing height. The way she did this was to have us begin by standing in line (in arbitrary order). Then she would start at the beginning of the line and compare the height of each student to that of the student behind him. If the student in front was taller, then she’d have the two students change places. She continued doing this until she reached the back of the line, taking note of whether or not two students had moved or not. If a swap had been made, she would go back to the beginning of the line and start again.

What my teacher was doing (knowingly or unknowingly) was executing the BubbleSort algorithm. A more formal pseudocode version of this algorithm is listed below:

BubbleSort(A, start, end):
swapped <- false while (swapped is false) for (i = start to end) if (A[i] > A[i+1]) then
temp <- A[i] A[i] <- A[i+1] A[i+1] <- temp swapped <- true end if end for end while end BubbleSortHere is an execution of the BubbleSort algorithm.

One of the more common ways that we do sorting in real life is through something similar to selection sort. Suppose we have a list of graded exams and want to have them in descending order before we pass them back to the students. The idea behind selection sort is to search for the largest element in the list, or in this case the highest test score, move that element to the beginning of the list, and then repeat the same procedure on the remaining portion of the list.

Here is a more formal pseudocode for the SelectionSort algorithm:

SelectionSort(A, start, end):
for (i = start to end)
min <- i for (j = i+1 to end) if (A[j] < A[min]) min <- j end if end fortemp <- A[i] A[i] <- A[min] A[min] <- temp end for end SelectionSortHere is an execution of the SelectionSort algorithm

If we think of the same set of (unsorted) graded exams as in the SelectionSort description that we wish to have them in descending order before we pass them back to the students, an alternative way to sort the exams is to have more of a “sort-as-you-go” type of a procedure where each test is only compared to an already sorted subset of the exams. The way this is carried out is to divide the exams into two piles. Originally, one pile (call it Pile 1) is empty and the other pile (Pile 2) contains the entire set of exams. We take the exam from the top of Pile 2 and compare it to the exams in Pile 1 and place it in the proper location in this pile (removing it from Pile 1). We continue this until Pile 2 is empty, in which case Pile 1 will have all the exams and the set of exams will be sorted.

Here is the pseudocode for the InsertionSort algorithm:
InsertionSort(A, start, end):
for (i = start to end)
val <- A[i] j <- iwhile (A[j - 1] > val)
A[j] <- A[j-1] j <- j - 1 end whileA[j] <- val end for end InsertionSortHere is an execution of the InsertionSort algorithm

MergeSort is based on a simple “Divide and Conquer” approach. The principle behind it is simple: We want to sort an entire list, but if this entire list is sorted, then each half of this list must also be sorted. It is an easier problem to sort a smaller list than a larger list, so we sort each of the smaller lists. Once these two lists are sorted, we run a merge algorithm which combines the two smaller lists into one larger list. These smaller lists are then sorted using the same principle until our smaller lists are of size 1 or 2, in which case we can simply swap the two items or return the item itself.

The algorithm itself is similar to the hierarchy at work.
The boss is given a job – “sort these 8 documents in decreasing order”.
Boss divides the list into two sets of size 8 and gets two subordinates and says “hey you two, each of you take one of these lists and sort it and get it back to me”.
Each of subordinates then divide their lists into two distinct lists of size 4 and gets two sub-subordinates to sort these smaller lists.
The sub-subordinates then divide their lists into two distinct lists of size 2 and gets two sub-sub-subordinates to sort these smaller lists.
The sub-sub-subordinates now have lists of size 2 which are easy to sort, if they’re ordered, then they say “its already sorted”; if they’re not ordered, they swap the elements and say “here you go”.
Now that the lists of size 2 have been sorted, the sub-subordinates need to combine them into a list of size 4. This can be done by simply reading both lists and taking the maximum element from each list.
Similarly, the subordinates can just combine the lists of size 4 into lists of size 8 using that same principle.
Finally the boss can take the two lists of size 8 and combine it into a sorted list of size 16 using that same principle.

Here is the pseudocode for the MergeSort Algorithm:
MergeSort(A, start, end)
if ((end – start + 1) = 1)
return A
end if

mid <- (end - start) / 2left <- {A[start], A[start+1], ... A[mid]} right <- {A[mid+1], A[mid+2], ... A[end]left <- MergeSort(left, 0, mid) right <- MergeSort(right, 0, end-mid)return Merge(left, right) end MergeSortHere is an execution of the MergeSort algorithm

QuickSort is considered the best sorting algorithm among the ones listed here. Like MergeSort, it operates under a divide and conquer principle. Unlike MergeSort, however, the question of where to divide the list (known as the pivot element) and the recursive calls are a bit more complex decisions.

The algorithm works by choosing a pivot element in the list (by default we can let this be the middle element in the list, but there are more complex variations that help decide what this should be), and then reordering the list so that the elements to the left (call this sub-list “small”) of the pivot element are all less than the pivot element, and the elements to the right (call this sub-list “big”) of the pivot element are all greater than the pivot element. So after the swap part of quicksort, the largest element of “small” will be less than the smallest element of “big”. We then need to check on whether the sub-lists “small” and “big” need to be sorted, which is true if we have not considered those regions yet. In such a case, we call the procedure Quicksort on each of the sub-lists.

Here is the pseudocode for the QuickSort Algorithm:
QuickSort(A, left, right)
i <- left j <- right pivot <- A[floor((left + right) / 2)] while (i <= j) while (A[i] < pivot) i <- i+1 end while while (A[j] > pivot)
j <- j-1 end while if (i <= j) tmp <- A[i] A[i] <- A[j] A[j] <- tmp i <- i+1 j <- j-1 end if end while if (left < j) A <- QuickSort(A, left, j) end if if (i < right) A <- QuickSort(A, i, right) end ifreturn A; end QuickSortHere is an execution of the QuickSort algorithm

Sorting Algorithms

I just added a script which executes and gives examples of some basic sorting algorithms. Its accessible at Sorting Algorithms

Right now, the algorithms consist of BubbleSort, InsertionSort, MergeSort and SelectionSort.

BubbleSort works by comparing items that are next to one another. If the items are not in proper order, they are swapped. The process continues until we pass through the list without making a swap.

InsertionSort divides an array into two parts, a sorted part and an unsorted part. In each iteration, a new element is compared to all the elements of the sorted part of the array to find where it belongs in this subarray. The algorithm terminates when all elements have been inserted into the sorted part of the array.

MergeSort is based on the divide and conquer algorithm. It works by calling itself (the function mergesort) on two smaller arrays: the elements in the first half of the array, and the elements in the second half of the array.

QuickSort is another divide and conquer algorithm. It is based on first choosing a pivot element (we choose the middle element, but it can be any element) and ensuring that all elements that are less than the pivot element are to the left of it in the array, and all elements greater than the pivot element are to the right in the array. Then the quicksort algorithm is recursively called on each part of the array.

SelectionSort repeatedly finds the minimal value in the list and places it in the first (remaining) position in the (unsorted) array.