**Introduction**

In the field of mathematics, there are things that need understanding by the men and women of that field. This is present in computer science and engineering. It may seem complex but it is theoretical in essence. Today, we’ll know The Best Asymptotic Runtime Complexity Algorithm.

Mathematical analytics of algorithms of asymptotic runtime complexity. This is the method to get data of a model of calculations that may run through infinity. It may seem hard for those that have not developed the taste for math. But there is a way to better clear the cloud from sight.

First of all, to quell the itch with regards to the question at hand. There are two methods to achieve the purpose through asymptotic runtime complexity. Professionals in the field choose **Insertion Sort** and **Heap Sort** would qualify as the best.

There are many other methods of sorting data. Each of which may come in different computer languages. A few of these sort methods, aside from those mentioned above, are Merge, Quick, and Bubble Sort. These methods are under discussion in detail in a later section.

Computer constructs of these sort methods would be C++, C#, Java, Python, and PHP, among the more popular. These languages are also favorites of programmers. Depending on the build, it can also be a cross-platform program that runs on any operating system.

**Definition of Terms**

For those that are new to algorithms, the need to understand the meaning of a few words is essential. Beginners in programming benefits as well.

We will use the closest definition applicable for our purpose. That is, of course, in the field of mathematics and computer science. The reason for this is the second term has a medical definition as well.

1. Algorithm – a set of instructions designed to solve a particular problem either recurring or not. It may also be a sequence of actions or procedures for the same purpose. A computer block of instructions or a mathematical procedure are algorithm. Both is examples aimed at solving a small problem where the output will be the input to the next algorithm.

2. Asymptotic – a line that ever comes nearer to a curve. But that line never intersects or touches the curve; the line and the curve are asymptotic to each other. In simple terms, a function is not allowed to go into infinity else it will loop and will never end.

3. Runtime Complexity – in the analysis of the performance of an algorithm, two things emerge. Time complexity and space complexity are in consideration for analysis. Each has different scopes to cover the whole extent of the sort method.

Time complexity is the amount of time the algorithm finishes its task. Space complexity is the amount of memory space needed to finish the same task.

Time and space complexity will reveal the characteristics of the sort method. But the one with the fastest time to finish may not be the best. Likewise, the one with the least used memory may not also be the better one.

In computer programming, several things come to play. The size of the raw data and the microchip inside the machine are important. Also, the type and size of the virtual and physical memory inside the machine factors.

**Different Sort Methods**

Before we go deeper in this section, asymptotic runtime complexity is an algorithm. It refers to any sort methods in computer programming languages. Be it in C++ or Python or any of the others does not matter. It is a set of programming instructions with the aim to arrive at a result or output.

In the world of computing, the growing internet that spews a lot of products every year, data is every growing. Cyber security is now a profession. Server professionals are sought out, and Big Data is an increasing concern.

Sort methods have been around since the first-ever integrated circuit came about. It is not novel but the industry is in need of new ones to augment the existing ones. The following list will not expound on any specific data but its use or method unless expressly so indicated.

**1. **Bubble Sort

*This sorting algorithm is one of the simplest form of sorting, if not the simplest one among the many. Its previous name was “sorting by exchange” prior to 1962. And currently someone gave an alternative name, “sinking sort”. In 1962, a man by the name of Iverson first used the phrase “bubble sort” in a publication.

*Having an unsorted array, this sorting algorithm works by comparing the first and second elements. Whether the choice of the sort is ascending or not, the comparison will exchange or swap the elements. If ascending, the greater value element is then compared with the third element in the array. This process goes on until the last element of the array. A second iteration is necessary on the semi-sorted array. Same comparison from the first and second element down to the last element.

*The number of iterations depends on the number of elements in the array. In the first instance when there is no more exchange or swap, the iteration stops with the array in full sort.

**2**. Bucket Sort

*The name always precludes what the method adapts. Another name for this is bin sort. For this sorting method, the use of buckets (plural), are important to sort a data set of arrays. These buckets are virtual, of course. This is one of the more advanced sorting methods as compared to the others. Hence, analysis of a more complex set of data is possible with this sort. This uses what is termed as “the scatter and gather” technique with due explanation later.

*This sort method breaks down a set of data into categories called “buckets”. The size and quantity of these buckets will depend on the size of the data. If the array has two elements only then a straight sort will get the job done. If there are many elements and the values are far between, using more buckets will help. Each bucket is with different ranges of values.

*Put all elements into the corresponding bucket. If the first bucket has a range value of “0 to 9” and an element in the unsorted array is “8” then “8” goes to the first bucket. This is the scattering stage. After that, sort each bucket on its own. Then gather all elements from each sorted bucket. The bucket with the lowest range of values will be the first to put back into the original array. The rest will follow in sequence.

**3. **Counting Sort

*Counting sort algorithm is better than comparison-based sort algorithms in particular situations only. But if the elements in the array are large and have multiple counts, its performance will suffer.

*First to do is to know the greatest element (max) before the start of the sort algorithm. An allocation of an auxiliary array with slots to cover the max value plus 1. If the max is “8” then it will be a breeze. If it is like “387”, a large array with 388 slots is vital to complete the task.

*The algorithm will then count the number of times each element occurs in the unsorted array. For instance, if “5” occurs 17 times in the unsorted array, the algorithm will put “17” on the sixth slot. The first slot of the auxiliary array is 0, the next is 1, and so on. The sort will then go to a cumulative count and indexing.

**4. **Heap Sort

*For this sort, the learning curve for a considerable number of students is high. Despite that, this is one of the more popular sorts. It is also one of the more efficient algorithms to use. Understanding two areas of this sort in a thorough manner is important. So that one can be able to come up, write, or understand the workings of the heap sort algorithm.

*These two are necessary to understand: data structure of heap and binary tree. The algorithm treats all the elements in the array as a thorough and special type of binary tree. A function called heapify iterates over and over until the array is in full sort. In essence, this sorting algorithm is an exchange sort much like bubble sort but far more complex.

*J.W.J. Williams is the creator of heap sort in 1964 with further enhancement in the same year by R.W. Floyd.

**5. **Insertion Sort

*As the name simply states, the algorithm will check each element in the array. Starting with the first one then the second and compare which one is lower. The lower one will go ahead of the higher one as an insert. The algorithm then will proceed to the third element in the array. It compares that to the previous two and decide if it is higher or lower than the previous two.

*A Binary Insertion Sort is available to reduce the number of comparisons. The time in swapping elements is still a concern.

*This sort is old, 1959 or earlier, even before the existence of any home or personal computer. The algorithm is simple but nobody knows who made it first.

**6. **Merge Sort

*Two methods or approaches are available for opt under the merge sort. These are the top-down approach, and the bottom-up merge sort. Whichever you choose of the two approaches, the same principle applies. They break down the unsorted elements in the array into smaller pieces. Then the handling of the sort algorithm will begin.

*The Top-Down approach is to break down all elements in an unsorted array into a one-element array. A one-element array becomes a sorted array in programming. Each mini-array is then compared with another then combined in the order selected.

*The Bottom-Up approach is to identify the size of elements to fit. Usually two elements into the smaller units for the breakdown. Sort and combine each unit, then sort unit to unit until done.

*This was a creation by John von Neumann in 1945. A detailed description of which came out three years after with a colleague. This is also one of those categorized as the divide and conquer sorting algorithm.

**7. **Quick Sort

*Another divide and conquer sorting algorithm. Quick sort is the baby of a visiting student to a university in Moscow, Tony Hoare. This happened a little over a decade after merge sort.

*Unlike merge sort, quick sort does the heavy work at the dividing stage of the algorithm. Merge sort has the heavy work done in the combining stage, after the dividing stage.

*The division of the unsorted array into sub-arrays happens first. In each sub-array is a pivot element. Another division occurs into that sub-array. Then comparison of elements where the lesser value elements are on the left of the pivot element. The greater value elements are to the right of the pivot element. This is what is partitioning. Recursive sorting and repeated pivot selection will result in a sorted array. This is the combining stage where, in practicality, there was no activity made.

*Performance taken into account, this sorting algorithm is not stable. It means it will move all elements in the array even if two are of the same value.

*This sorting algorithm will have its worst-case scenario when a certain event occurs. The pivot element it chooses is the largest or the smallest in the unsorted array. The resulting is a long time. Space complexity is not that much of a concern for this.

**8. **Selection Sort

*In this sort algorithm, it is much simpler. That is why the performance of this sort is best when the unsorted array is small. Element checking is still required. If the unsorted array is a jumble of elements in no order, the performance of this sort is on average. If the unsorted array comes in a sorted order, the sort will have its worst performance. How much more if the unsorted array is large?

*Like the shell sort, it has a temp variable. There will be no need of those sequences of increments procedures to get to the output. For it to work, the selection sort algorithm will compare the first and second elements in the array. Whichever is smallest goes to the temp which is then compared to the third element. If the third element is smaller, it goes into the temp, if not, then the fourth element comparison. This happens until the last element. The smallest element will go into the first slot of the unsorted array. A second iteration will proceed but skipping the first element will happen.

**9. **Shell Sort

*The shell sort algorithm is much akin to the insertion sort algorithm. Shell sort is a generalized edition of insertion sort. In a way, it is another type of exchange sort only with an addition. The addition is a set of procedures. The aim is to achieve the desired sort and to get the best performance rating. This is what is the sequence of increments. There are many that have evolved through the years from several individuals since. Donald Lewis Shell is responsible for the creation of shell sort in 1959.

*The use of a variable called temp will happen after selecting the sequence of increments. Storage of the element that is lesser in value goes into temp. The greater value goes to the place vacated by the lesser value element. The one in temp then occupies the vacated place of the greater value element. This goes on until all elements are done.

*The second iteration on the array will start by using the second sequence of increments. This will be for all the elements in the array.

**10. **Radix Sort

*This is a sorting algorithm that is more complex than others in the sense that it uses integers to sort the data. The integer keys will then separate the elements into separate groups like a bucket sort. Each of the separated groups is under a subroutine. Then the application of the counting sort algorithm. Application of the radix sort on a data array that is larger than usual is possible. The data may even contain non-integers or non-base 10 digits.

*There are sources that claim the radix sort was born at MIT in 1954 by Mr. Seward. He is responsible for integrating this sorting algorithm into computers. The aim of having memory efficiency was the driving force.

*There are sources also that say Herman Hollerith, invented the machine, the Hollerith machine. The machine was for the tally of the population in 1890. Radix sort algorithm is at its core. The company Mr. Hollerith started became the International Business Machine (IBM) in 1924.

**Sort Complexity / Performance Rating**

In performance analysis, the professionals settled to measure in agreed units. Through the years, they have standardized the measures down to the appropriate indicators.

Journals and such would go much deeper than the list below. The items inside the brackets [] are a few of the units of measure used to compare one sort algorithm from another.

• Time Complexity: Best, Average, Worst Scenarios [O(n+k); O(n2)O(n), O(n(log(n))]

• Space Complexity:[O(max); O(1); O(n+k)]

• Stability: Yes/No

**Frequently Asked Questions**

**1**. Which among the sort algorithms have good or favorable stability?

Considering all possible situations, there are a few. Good and favorable stability would be: Bubble, Counting, Insertion and Merge Sorts. Unstable sort algorithms, with tweaks to their code, make it stable.

**2**. How do we know if a sorting algorithm is ideal to use?

First, there is no particular sorting algorithm that is perfect. Not one is ideal at all times, anywhere or everywhere. There are different uses and applications in the real world. Choosing the sort algorithm that is best suited for that application is the best way to go.

If we proceed, though, the following properties would be needful:

a) high speed when raw data has only very few unique elements or when the array is almost sorted right from the start;

b) requiring least extra space in memory;

c) stable property;

d) its worst-case scenario is O(n log n) when comparing elements.

Not even one sorting algorithm has all these ideal properties!

**3**) What computer languages are used to construct a sorting algorithm?

Any computer language will do. The purpose is to make an implementation of any of the sorting algorithms. The most basic BASIC had its own sort. There used to exist old languages for use for sort algorithms such as COBOL, FORTRAN, and Pascal. These may not be in use today but these were the immediate predecessors of the modern languages.

The new and current languages that are also options include C, C++, C#, Java, JavaScript, and Python. There are other languages but these are the popular ones at the moment.

**4**) Differentiate External sorting from Internal sorting?

The use of the computer’s memory is a big factor. Choosing which sort algorithm to use for which practical application is what matters. The bigger the memory, the better.

Internal sorting is when the storage of the data set is into memory in its entirety. This is while the sort algorithm is in progress. External sorting keeps the raw data outside of memory. It brings only little chunks of data into memory at a time. The data when stored on the machine’s disk is a drawback since accessing data on disk is very slow.

**5**) What are the fastest and the slowest sort algorithms?

Basing on the ten sort algorithms we have tabulated above, the fastest among them is Quick Sort. This has the same performance level in the best and average scenarios. It is linearithmic. Based again on the list above, the slowest of the lot is Bubble Sort with quadratic time complexity.

**6**) Are there other sorting algorithms available aside from the ones listed above?

There are other sorting algorithms available. These are not as popular or appropriate to most common applications. These are Stooge Sort, Slow Sort, Sleep Sort, Smooth Sort, Tim Sort, and two variations of Bogo Sort. Other sort algorithms not mentioned is not, by any chance, intentional.

**7)** Is there a name to each of those Big O notations?

Yes, there is a name to each of those Big O notations. The name is according to the mathematical formula they would fall into. These are applicable to both time and space complexity and we start from left to right.

Constant [O(1)]; Double Logarithmic [O(log log n)]; Logarithmic [O(log n)]; Sq. Root [O(√n)]; Linear [O(n)]; Linearithmic [O(n log n)]; Quadratic [O(n2)]; Exponential [O(2n)]; Factorial [O(n!)].