Simple sorting algorithm

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a) Bubble Sort :

Bubble sort is a simple sorting algorithm. It works by repeatedly stepping through the list to be sorted, comparing each pair of adjacent items and swapping them if they are in the wrong order. The pass through the list is repeated until no swaps are needed, which indicates that the list is sorted. The algorithm gets its name from the way smaller elements “bubble” to the top of the list. Because it only uses comparisons to operate on elements, it is a comparison sort.

Bubble sort has worst-case and average complexity both О(n²), where n is the number of items being sorted.

Step-by-step example

Let us take the array of numbers “5 1 4 2 8”, and sort the array from lowest number to greatest number using bubble sort algorithm. In each step, elements written in bold are being compared.

b) Heap Sort:

Heapsort (method) is a comparison-based sorting algorithm, and is part of the selection sort family. Although somewhat slower in practice on most machines than a good implementation of quick sort, it has the advantage of a worst-case Θ(n log n) runtime. Heapsort is an in-place algorithm, but is not a stable sort.

The heap sort works as its name suggests. It begins by building a heap out of the data set, and then removing the largest item and placing it at the end of the sorted array. After removing the largest item, it reconstructs the heap and removes the largest remaining item and places it in the next open position from the end of the sorted array. This is repeated until there are no items left in the heap and the sorted array is full. Elementary implementations require two arrays - one to hold the heap and the other to hold the sorted elements.

Heapsort inserts the input list elements into a heap data structure. The largest value (in a max-heap) or the smallest value (in a min-heap) are extracted until none remain, the values having been extracted in sorted order. The heap's invariant is preserved after each extraction, so the only cost is that of extraction.

During extraction, the only space required is that needed to store the heap. In order to achieve constant space overhead, the heap is stored in the part of the input array that has not yet been sorted.(The structure of this heap is described at Binary heap: Heap implementation.)

Heapsort uses two heap operations: insertion and root deletion. Each extraction places an element in the last empty location of the array. The remaining prefix of the array stores the unsorted elements.

Comparison with other sorts:

Heapsort primarily competes with quick sort, another very efficient general purpose nearly-in-place comparison-based sort algorithm.

Quicksort is typically somewhat faster, due to better cache behavior and other factors, but the worst-case running time for quicksort is O(n2), which is unacceptable for large data sets and can be deliberately triggered given enough knowledge of the implementation, creating a security risk. See quick sort for a detailed discussion of this problem, and possible solutions.

Thus, because of the O(n log n) upper bound on heapsort's running time and constant upper bound on its auxiliary storage, embedded systems with real-time constraints or systems concerned with security often use heapsort.

c) Shell Sort:

Shell sort is a sorting algorithm that is a generalization of insertion sort, with two observations:

  • insertion sort is efficient if the input is “almost sorted”, and
  • insertion sort is typically inefficient because it moves values just one position at a time.

Implementation:

The original implementation performs O (n2) comparisons and exchanges in the worst case. A minor change given in V. Pratt's book improved the bound to O(n log2 n). This is worse than the optimal comparison sorts, which are O(n log n).

Shell sort improves insertion sort by comparing elements separated by a gap of several positions. This lets an element take “bigger steps” toward its expected position. Multiple passes over the data are taken with smaller and smaller gap sizes. The last step of Shell sort is a plain insertion sort, but by then, the array of data is guaranteed to be almost sorted.

Consider a small value that is initially stored in the wrong end of the array. Using an O(n2) sort such as bubble sort or insertion sort, it will take roughly n comparisons and exchanges to move this value all the way to the other end of the array. Shell sort first moves values using giant step sizes, so a small value will move a long way towards its final position, with just a few comparisons and exchanges.

One can visualize Shell sort in the following way: arrange the list into a table and sort the columns (using an insertion sort). Repeat this process, each time with smaller number of longer columns. At the end, the table has only one column. While transforming the list into a table makes it easier to visualize, the algorithm itself does its sorting in-place (by incrementing the index by the step size, i.e. using i += step_size instead of i++).

For example, consider a list of numbers like [ 13 14 94 33 82 25 59 94 65 23 45 27 73 25 39 10 ]. If we started with a step-size of 5, we could visualize this as breaking the list of numbers into a table with 5 columns. This would look like this:

13 14 94 33 82

25 59 94 65 23

45 27 73 25 39

10

We then sort each column, which gives us

10 14 73 25 23

13 27 94 33 39

25 59 94 65 82

45

When read back as a single list of numbers, we get [ 10 14 73 25 23 13 27 94 33 39 25 59 94 65 82 45 ]. Here, the 10 which was all the way at the end, has moved all the way to the beginning. This list is then again sorted using a 3-gap sort, and then 1-gap sort (simple insertion sort). then we can split the list as 3-step sequence

10 14 73

25 23 13

27 94 33

39 25 59

94 65 82

45

d) Insertion Sort :

Data structure   Array

Worst case performance    Ðž(n2)

Best case performance   O(n)

Average case performance   Ðž(n2)

Worst case space complexity   Ðž(n) total, O(1) auxiliary

Optimal   Inserting a few values into a sorted list,

     or substantially sorted sets.

Insertion sort is a simple sorting algorithm, a comparison sort in which the sorted array (or list) is built one entry at a time. It is much less efficient on large lists than more advanced algorithms such as quick sort, heap sort, or merge sort. However, insertion sort provides several advantages:

  • simple implementation
  • efficient for (quite) small data sets
  • adaptive, i.e. efficient for data sets that are already substantially sorted: the time complexity is O(n + d), where d is the number of inversions
  • more efficient in practice than most other simple quadratic (i.e. O(n2)) algorithms such as selection sort or bubble sort: the average running time is n2/4, and the running time is linear in the best case
  • stable, i.e. does not change the relative order of elements with equal keys
  • in-place, i.e. only requires a constant amount O(1) of additional memory space
  • online, i.e. can sort a list as it receives it

Most humans when sorting-ordering a deck of cards, for example-use a method that is similar to insertion sort.

Algorithm:

Every iteration of insertion sort removes an element from the input data, inserting it into the correct position in the already-sorted list, until no input elements remain. The choice of which element to remove from the input is arbitrary, and can be made using almost any choice algorithm.

Sorting is typically done in-place. The resulting array after k iterations has the property where the first k + 1 entries are sorted. In each iteration the first remaining entry of the input is removed, inserted into the result at the correct position, thus extending the result:

Array prior to insertion of x

Becomes

Array after the insertion of x

with each element greater than x copied to the right as it is compared against x.

The most common variant of insertion sort, which operates on arrays, can be described as follows:

  1. Suppose there exists a function called Insert designed to insert a value into a sorted sequence at the beginning of an array. It operates by beginning at the end of the sequence and shifting each element one place to the right until a suitable position is found for the new element. The function has the side effect of overwriting the value stored immediately after the sorted sequence in the array.
  2. To perform an insertion sort, begin at the left-most element of the array and invoke Insert to insert each element encountered into its correct position. The ordered sequence into which the element is inserted is stored at the beginning of the array in the set of indices already examined. Each insertion overwrites a single value: the value being inserted.

Best, worst, and average cases

The best case input is an array that is already sorted. In this case insertion sort has a linear running time (i.e., O (n)). During each iteration, the first remaining element of the input is only compared with the right-most element of the sorted subsection of the array.

The worst case input is an array sorted in reverse order. In this case every iteration of the inner loop will scan and shift the entire sorted subsection of the array before inserting the next element. For this case insertion sort has a quadratic running time (i.e., O(n2)).

The average case is also quadratic, which makes insertion sort impractical for sorting large arrays. However, insertion sort is one of the fastest algorithms for sorting arrays containing fewer than ten elements.

e) Quick Sort :

Algorithm

Quicksort sorts by employing a divide and conquer strategy to divide a list into two sub-lists.

Quicksort is a very efficient sorting algorithm.

It has two phases:

  • Partition phase- Most of the work is done in the partition phase
  • Sort phase - simply sorts the two smaller problems that are generated in the partition phase

f) Merge Sort :

Merge-sort is based on the divide-and-conquer paradigm. The Merge-sort algorithm can be described in general terms as consisting of the following three steps:

  • Divide Step

If given array A has zero or one element, return S; it is already sorted. Otherwise, divide A into two arrays, A1 and A2, each containing about half of the elements of A.

  • Recursion Step

Recursively sort array A1 and A2.

  • Conquer Step

Combine the elements back in A by merging the sorted arrays A1 and A2 into a sorted sequence.

We can visualize Merge-sort by means of binary tree where each node of the tree represents a recursive call and each external nodes represent individual elements of given array A. Such a tree is called Merge-sort tree. The heart of the Merge-sort algorithm is conquer step, which merge two sorted sequences into a single sorted sequence.

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