Bubble sort, sometimes referred to as sinking sort, is a simple sorting algorithm that repeatedly steps through the list to be sorted, compares each pair of adjacent items and swaps them if they are in the wrong order (ascending or descending arrangement). The pass through the list is repeated until no swaps are needed, which indicates that the list is sorted.

In computer science, counting sort is an algorithm for sorting a collection of objects according to keys that are small integers; that is, it is an integer sorting algorithm. It operates by counting the number of objects that have each distinct key value, and using arithmetic on those counts to determine the positions of each key value in the output sequence. Its running time is linear in the number of items and the difference between the maximum and minimum key values, so it is only suitable for direct use in situations where the variation in keys is not significantly greater than the number of items. However, it is often used as a subroutine in another sorting algorithm, radix sort, that can handle larger keys more efficiently. Because counting sort uses key values as indexes into an array, it is not a comparison sort, and the Ω(n log n) lower bound for comparison sorting does not apply to it. Bucket sort may be used for many of the same tasks as counting sort, with a similar time analysis; however, compared to counting sort, bucket sort requires linked lists, dynamic arrays or a large amount of preallocated memory to hold the sets of items within each bucket, whereas counting sort instead stores a single number (the count of items) per bucket. Counting sorting works best when the range of numbers for each array element is very small. Step I In first step we calculate the count of all the elements of the input array A. Then Store the result in the count array C. The way we count is depicted below. Step II In second step we calculate how many elements exist in the input array A which are less than or equals for the given index. Step III In this step we place the input array A element at sorted position by taking help of constructed count array C ,i.e what we constructed in step two. We used the result array B to store the sorted elements. Here we handled the index of B start from zero.

Heapsort is a comparison-based sorting algorithm. Heapsort can be thought of as an improved selection sort: like that algorithm, it divides its input into a sorted and an unsorted region, and it iteratively shrinks the unsorted region by extracting the largest element and moving that to the sorted region. The improvement consists of the use of a heap data structure rather than a linear-time search to find the maximum.

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.

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 implementation preserves the input order of equal elements in the sorted output. Mergesort is a divide and conquer algorithm that was invented by John von Neumann in 1945. An example of merge sort. First divide the list into the smallest unit (1 element), then compare each element with the adjacent list to sort and merge the two adjacent lists. Finally all the elements are sorted and merged. A recursive merge sort algorithm used to sort an array of 7 integer values. These are the steps a human would take to emulate merge sort (top-down).

Quicksort is a divide and conquer algorithm. Quicksort first divides a large array into two smaller sub-arrays: the low elements and the high elements. Quicksort can then recursively sort the sub-arrays The steps are: 1. Pick an element, called a pivot, from the array. 2. 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. 3. 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. Animated visualization of the quicksort algorithm. The horizontal lines are pivot values.

In computer science, radix sort is a non-comparative integer sorting algorithm that sorts data with integer keys by grouping keys by the individual digits which share the same significant position and value. A positional notation is required, but because integers can represent strings of characters (e.g., names or dates) and specially formatted floating point numbers, radix sort is not limited to integers. Where does the name come from? In mathematical numeral systems, the radix or base is the number of unique digits, including the digit zero, used to represent numbers in a positional numeral system. For example, a binary system (using numbers 0 and 1) has a radix of 2 and a decimal system (using numbers 0 to 9) has a radix of 10. The topic of the efficiency of radix sort compared to other sorting algorithms is somewhat tricky and subject to quite a lot of misunderstandings. Whether radix sort is equally efficient, less efficient or more efficient than the best comparison-based algorithms depends on the details of the assumptions made. Radix sort complexity is O(wn) for n keys which are integers of word size w. Sometimes w is presented as a constant, which would make radix sort better (for sufficiently large n) than the best comparison-based sorting algorithms, which all perform O(n log n) comparisons to sort n keys. However, in general w cannot be considered a constant: if all n keys are distinct, then w has to be at least log n for a random-access machine to be able to store them in memory, which gives at best a time complexity O(n log n). That would seem to make radix sort at most equally efficient as the best comparison-based sorts (and worse if keys are much longer than log n).

Selection sort is a sorting algorithm, specifically an in-place comparison sort. It has O(n2) time complexity, making it inefficient on large lists, and generally performs worse than the similar insertion sort. Selection sort is noted for its simplicity, and it has performance advantages over more complicated algorithms in certain situations, particularly where auxiliary memory is limited.

Shellsort, also known as Shell sort or Shell's method, is an in-place comparison sort. It can be seen as either a generalization of sorting by exchange (bubble sort) or sorting by insertion (insertion sort). The method starts by sorting pairs of elements far apart from each other, then progressively reducing the gap between elements to be compared. Starting with far apart elements, it can move some out-of-place elements into position faster than a simple nearest neighbor exchange For our example and ease of understanding, we take the interval of 4. Make a virtual sub-list of all values located at the interval of 4 positions. Here these values are We compare values in each sub-list and swap them (if necessary) in the original array. After this step, the new array should look like this Then, we take interval of 2 and this gap generates two sub-lists We compare and swap the values, if required, in the original array. After this step, the array should look like this > UPD: On the picture below there is a typo and result array is supposed to be [14, 10, 27, 19, 35, 33, 42, 44]. Finally, we sort the rest of the array using interval of value 1. Shell sort uses insertion sort to sort the array.