Worst and Average Case Time Complexity: O(n*n). Worst case occurs when array is reverse sorted. Best Case Time Complexity: O(n). Best case occurs when array is already sorted. Auxiliary Space: O(1) Boundary Cases: Bubble sort takes minimum time (Order of n) when elements are already sorted. Sorting In Place: Yes Stable: Yes Due to its simplicity, bubble sort is often used to introduce the concept of a sorting algorithm. In computer graphics it is popular for its capability to detect a very small error (like swap of just two elements) in almost-sorted arrays and fix it with just linear complexity (2n). For example, it is used in a polygon filling algorithm, where bounding lines are sorted by their x coordinate at a specific scan line (a line parallel to x axis) and with incrementing y their order changes (two elements are swapped) only at intersections of two lines (Source: Wikipedia)

In computer science, selection sort is an in-place comparison sorting algorithm. It has an O(n2) time complexity, which makes it inefficient on large lists, and generally performs worse than the similar insertion sort. Selection sort is noted for its simplicity and has performance advantages over more complicated algorithms in certain situations, particularly where auxiliary memory is limited. The algorithm divides the input list into two parts: a sorted sublist of items which is built up from left to right at the front (left) of the list and a sublist of the remaining unsorted items that occupy the rest of the list. Initially, the sorted sublist is empty and the unsorted sublist is the entire input list. The algorithm proceeds by finding the smallest (or largest, depending on sorting order) element in the unsorted sublist, exchanging (swapping) it with the leftmost unsorted element (putting it in sorted order), and moving the sublist boundaries one element to the right. The time efficiency of selection sort is quadratic, so there are a number of sorting techniques which have better time complexity than selection sort. One thing which distinguishes selection sort from other sorting algorithms is that it makes the minimum possible number of swaps, n − 1 in the worst case. (Source: Wikipedia)

Insertion sort is a simple sorting algorithm that builds the final sorted array (or list) one item at a time. It is much less efficient on large lists than more advanced algorithms such as quicksort, heapsort, or merge sort. However, insertion sort provides several advantages: Simple implementation: Jon Bentley shows a three-line C version, and a five-line optimized version[1] Efficient for (quite) small data sets, much like other quadratic sorting algorithms More efficient in practice than most other simple quadratic (i.e., O(n2)) algorithms such as selection sort or bubble sort Adaptive, i.e., efficient for data sets that are already substantially sorted: the time complexity is O(kn) when each element in the input is no more than k places away from its sorted position 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 Insertion sort iterates, consuming one input element each repetition, and growing a sorted output list. At each iteration, insertion sort removes one element from the input data, finds the location it belongs within the sorted list, and inserts it there. It repeats until no input elements remain. Sorting is typically done in-place, by iterating up the array, growing the sorted list behind it. At each array-position, it checks the value there against the largest value in the sorted list (which happens to be next to it, in the previous array-position checked). If larger, it leaves the element in place and moves to the next. If smaller, it finds the correct position within the sorted list, shifts all the larger values up to make a space, and inserts into that correct position. The resulting array after k iterations has the property where the first k + 1 entries are sorted ("+1" because the first entry is skipped). In each iteration the first remaining entry of the input is removed, and inserted into the result at the correct position, thus extending the result(Source: Wikipedia)

In computer science, merge sort (also commonly spelled mergesort) is an efficient, general-purpose, comparison-based sorting algorithm. Most implementations produce a stable sort, which means that the order of equal elements is the same in the input and output. Merge sort is a divide and conquer algorithm that was invented by John von Neumann in 1945.[2] A detailed description and analysis of bottom-up mergesort appeared in a report by Goldstine and von Neumann as early as 1948.[3] Conceptually, a merge sort works as follows: Divide the unsorted list into n sublists, each containing one element (a list of one element is considered sorted). Repeatedly merge sublists to produce new sorted sublists until there is only one sublist remaining. This will be the sorted list. Top-down implementation Example C-like code using indices for top-down merge sort algorithm that recursively splits the list (called runs in this example) into sublists until sublist size is 1, then merges those sublists to produce a sorted list. The copy back step is avoided with alternating the direction of the merge with each level of recursion (except for an initial one time copy). To help understand this, consider an array with 2 elements. The elements are copied to B[], then merged back to A[]. If there are 4 elements, when the bottom of recursion level is reached, single element runs from A[] are merged to B[], and then at the next higher level of recursion, those 2 element runs are merged to A[]. This pattern continues with each level of recursion (Source: Wikipedia)

Quicksort (sometimes called partition-exchange sort) is an efficient sorting algorithm. Developed by British computer scientist Tony Hoare in 1959[1] and published in 1961,[2] it is still a commonly used algorithm for sorting. When implemented well, it can be about two or three times faster than its main competitors, merge sort and heapsort.[3][contradictory] Quicksort is a divide-and-conquer algorithm. It works by selecting a 'pivot' element from the array and partitioning the other elements into two sub-arrays, according to whether they are less than or greater than the pivot. The sub-arrays are then sorted recursively. This can be done in-place, requiring small additional amounts of memory to perform the sorting. Quicksort is a comparison sort, meaning that it can sort items of any type for which a "less-than" relation (formally, a total order) is defined. Efficient implementations of Quicksort are not a stable sort, meaning that the relative order of equal sort items is not preserved. Mathematical analysis of quicksort shows that, on average, the algorithm takes O(n log n) comparisons to sort n items. In the worst case, it makes O(n2) comparisons, though this behavior is rare. Algorithm Full example of quicksort on a random set of numbers. The shaded element is the pivot. It is always chosen as the last element of the partition. However, always choosing the last element in the partition as the pivot in this way results in poor performance (O(n²)) on already sorted arrays, or arrays of identical elements. Since sub-arrays of sorted / identical elements crop up a lot towards the end of a sorting procedure on a large set, versions of the quicksort algorithm that choose the pivot as the middle element run much more quickly than the algorithm described in this diagram on large sets of numbers. Quicksort is a divide and conquer algorithm. It first divides the input array into two smaller sub-arrays: the low elements and the high elements. It then recursively sorts the sub-arrays. The steps for in-place Quicksort are: Pick an element, called a pivot, from the array. Partitioning: reorder the array so that all elements with values less than the pivot come before the pivot, while all elements with values greater than the pivot come after it (equal values can go either way). After this partitioning, the pivot is in its final position. This is called the partition operation. Recursively apply the above steps to the sub-array of elements with smaller values and separately to the sub-array of elements with greater values. The base case of the recursion is arrays of size zero or one, which are in order by definition, so they never need to be sorted. The pivot selection and partitioning steps can be done in several different ways; the choice of specific implementation schemes greatly affects the algorithm's performance. (Source: Wikipedia)

Heap sort is one of the sorting algorithms used to arrange a list of elements in order. Heapsort algorithm uses one of the tree concepts called Heap Tree. In this sorting algorithm, we use Max Heap to arrange list of elements in Descending order and Min Heap to arrange list elements in Ascending order. Step by Step Process The Heap sort algorithm to arrange a list of elements in ascending order is performed using following steps... Step 1 - Construct a Binary Tree with given list of Elements. Step 2 - Transform the Binary Tree into Min Heap. Step 3 - Delete the root element from Min Heap using Heapify method. Step 4 - Put the deleted element into the Sorted list. Step 5 - Repeat the same until Min Heap becomes empty. Step 6 - Display the sorted list. (Source: Wikipedia)