## 1. Introduction

Sorting is one of the most fundamental and common tasks in computer science. Sorting is the process of arranging data in a specific order, such as ascending or descending, based on some criteria. Sorting can help us to organize, search, and analyze data more efficiently and effectively.

In this blog, you will learn about **sorting algorithms**, which are methods of implementing sorting in a programming language. You will learn how to use **Java** to implement different types of sorting algorithms, such as selection sort, bubble sort, insertion sort, merge sort, quick sort, counting sort, and radix sort. You will also learn how to compare sorting algorithms based on their performance, complexity, stability, and adaptability.

By the end of this blog, you will have a solid understanding of sorting algorithms and how to apply them in Java. You will also be able to choose the best sorting algorithm for your specific problem and data set. You will also gain some valuable skills and knowledge in Java programming and data structures.

Are you ready to dive into the world of sorting algorithms? Let’s get started!

## 2. What are Sorting Algorithms?

A **sorting algorithm** is a method of arranging data in a specific order, such as ascending or descending, based on some criteria. For example, you can sort a list of numbers from smallest to largest, or a list of names alphabetically.

Sorting algorithms are important for many reasons. They can help us to:

- Organize data in a meaningful and logical way.
- Search data faster and more efficiently.
- Analyze data and find patterns or trends.
- Optimize the performance and memory usage of programs.

There are many different types of sorting algorithms, each with its own advantages and disadvantages. Some sorting algorithms are faster than others, some are more memory-efficient, some are more stable, and some are more adaptable to different types of data.

How do sorting algorithms work? The basic idea is to compare two or more elements of the data and decide which one should come first in the sorted order. Depending on the algorithm, this comparison can be done in different ways, such as:

**Swap**: Exchange the positions of two elements if they are not in the correct order.**Partition**: Divide the data into two or more subsets based on some pivot element, and sort each subset recursively.**Count**: Count the number of occurrences of each element in the data, and use the counts to determine the positions of the elements in the sorted order.**Radix**: Sort the data based on the individual digits or characters of each element, starting from the least significant digit or character.

In the next section, we will learn more about the different types of sorting algorithms and how they are classified.

## 3. Types of Sorting Algorithms

Sorting algorithms can be classified into two main categories: **comparison-based** and **non-comparison-based**. The difference between these two categories is the way they compare the elements of the data to determine their order.

**Comparison-based sorting algorithms** use a **comparison** function to compare two or more elements of the data and decide which one should come first in the sorted order. For example, if we want to sort a list of numbers in ascending order, we can use a comparison function that returns true if the first number is smaller than the second number, and false otherwise. Some examples of comparison-based sorting algorithms are selection sort, bubble sort, insertion sort, merge sort, and quick sort.

**Non-comparison-based sorting algorithms** do not use a comparison function to compare the elements of the data. Instead, they use some other information or property of the elements, such as their digits, characters, or counts, to determine their order. For example, if we want to sort a list of numbers in ascending order, we can use a non-comparison-based sorting algorithm that counts the number of occurrences of each number in the list, and then uses the counts to place the numbers in the sorted order. Some examples of non-comparison-based sorting algorithms are counting sort and radix sort.

Both comparison-based and non-comparison-based sorting algorithms have their own advantages and disadvantages. In general, comparison-based sorting algorithms are more versatile and adaptable, as they can sort any type of data that has a well-defined comparison function. However, they also have a lower bound on their time complexity, which means that they cannot be faster than a certain limit. On the other hand, non-comparison-based sorting algorithms can be faster and more efficient than comparison-based sorting algorithms, as they do not need to compare the elements of the data. However, they also have some limitations and restrictions, such as the range and size of the data, and the amount of extra space they need.

In the next section, we will learn how to implement some of the most common and popular sorting algorithms in Java, and see how they work in action.

### 3.1. Comparison-based Sorting Algorithms

In this section, you will learn about some of the most common and popular **comparison-based sorting algorithms**. These are sorting algorithms that use a **comparison** function to compare two or more elements of the data and decide which one should come first in the sorted order.

Some of the comparison-based sorting algorithms that you will learn are:

**Selection sort**: A simple and intuitive algorithm that repeatedly selects the smallest or largest element from the unsorted part of the data and places it at the end of the sorted part.**Bubble sort**: A simple and slow algorithm that repeatedly swaps adjacent elements that are out of order until the data is sorted.**Insertion sort**: A simple and efficient algorithm that builds the sorted part of the data one element at a time by inserting each element into its correct position.**Merge sort**: A fast and stable algorithm that divides the data into two or more subsets, sorts each subset recursively, and then merges the sorted subsets into one.**Quick sort**: A fast and widely used algorithm that partitions the data into two or more subsets based on some pivot element, and then sorts each subset recursively.

For each of these algorithms, you will learn how they work, how to implement them in **Java**, and what are their advantages and disadvantages. You will also see some examples and code snippets to help you understand them better.

Are you ready to explore the world of comparison-based sorting algorithms? Let’s begin with selection sort!

### 3.2. Non-comparison-based Sorting Algorithms

In this section, you will learn about some of the **non-comparison-based sorting algorithms**. These are sorting algorithms that do not use a **comparison** function to compare the elements of the data. Instead, they use some other information or property of the elements, such as their digits, characters, or counts, to determine their order.

Some of the non-comparison-based sorting algorithms that you will learn are:

**Counting sort**: A fast and simple algorithm that counts the number of occurrences of each element in the data, and then uses the counts to place the elements in the sorted order.**Radix sort**: A fast and efficient algorithm that sorts the data based on the individual digits or characters of each element, starting from the least significant digit or character.

For each of these algorithms, you will learn how they work, how to implement them in **Java**, and what are their advantages and disadvantages. You will also see some examples and code snippets to help you understand them better.

Are you ready to explore the world of non-comparison-based sorting algorithms? Let’s begin with counting sort!

## 4. How to Implement Sorting Algorithms in Java

In this section, you will learn how to implement some of the sorting algorithms that you learned in the previous section in **Java**. You will see how to write the code for each algorithm, and how to test and run it on some sample data.

Before you start coding, you need to set up your Java development environment. You will need the following tools:

**Java Development Kit (JDK)**: This is a software package that contains the tools and libraries that you need to compile and run Java programs. You can download the latest version of JDK from here.**Integrated Development Environment (IDE)**: This is a software application that provides a user-friendly interface for writing, editing, debugging, and running Java code. You can use any IDE that supports Java, such as Eclipse, IntelliJ IDEA, or NetBeans.

Once you have installed the JDK and the IDE, you are ready to create your Java project. You can follow the steps below to create a new Java project in Eclipse:

- Open Eclipse and select File -> New -> Java Project.
- Enter a name for your project, such as SortingAlgorithms, and click Next.
- Select the Java version that matches your JDK, such as Java SE 16, and click Finish.
- Right-click on the src folder in the Package Explorer and select New -> Class.
- Enter a name for your class, such as SortingAlgorithms, and click Finish.
- You should see a new Java file with the following code:

public class SortingAlgorithms { public static void main(String[] args) { // TODO Auto-generated method stub } }

This is the main class of your project, where you will write the code for the sorting algorithms and test them on some sample data. You can use the main method as the entry point of your program, where you can create and initialize an array of integers, and then call the sorting methods that you will define later.

For example, you can write the following code in the main method to create an array of 10 random integers between 1 and 100, and print it before and after sorting:

public class SortingAlgorithms { public static void main(String[] args) { // Create an array of 10 random integers between 1 and 100 int[] array = new int[10]; for (int i = 0; i < array.length; i++) { array[i] = (int) (Math.random() * 100) + 1; } // Print the array before sorting System.out.println("Array before sorting:"); printArray(array); // Call the sorting method of your choice, such as selectionSort selectionSort(array); // Print the array after sorting System.out.println("Array after sorting:"); printArray(array); } // A helper method to print an array public static void printArray(int[] array) { for (int element : array) { System.out.print(element + " "); } System.out.println(); } // TODO: Define the sorting methods here }

Now, you can define the sorting methods below the main method, using the algorithms that you learned in the previous section. You can use the following template for each sorting method:

// A method to sort an array using the algorithm of your choice, such as selectionSort public static void selectionSort(int[] array) { // Write the code for the algorithm here }

In the next subsections, you will see how to implement some of the sorting algorithms in Java, and what are their code snippets.

### 4.1. Selection Sort

**Selection sort** is a simple and intuitive algorithm that repeatedly selects the smallest or largest element from the unsorted part of the data and places it at the end of the sorted part. The algorithm works as follows:

- Find the smallest element in the unsorted part of the data and swap it with the first element of the unsorted part.
- Move the boundary of the sorted part by one position to the right.
- Repeat steps 1 and 2 until the entire data is sorted.

For example, suppose you want to sort the following array of 10 random integers between 1 and 100 in ascending order using selection sort:

int[] array = {34, 67, 12, 89, 45, 23, 78, 90, 11, 56};

The algorithm will perform the following steps:

Step | Sorted part | Unsorted part | Smallest element | Swap |

0 | [] | [34, 67, 12, 89, 45, 23, 78, 90, 11, 56] | 11 | 11 and 34 |

1 | [11] | [34, 67, 12, 89, 45, 23, 78, 90, 56] | 12 | 12 and 34 |

2 | [11, 12] | [34, 67, 89, 45, 23, 78, 90, 56] | 23 | 23 and 34 |

3 | [11, 12, 23] | [34, 67, 89, 45, 78, 90, 56] | 34 | No swap |

4 | [11, 12, 23, 34] | [67, 89, 45, 78, 90, 56] | 45 | 45 and 67 |

5 | [11, 12, 23, 34, 45] | [67, 89, 78, 90, 56] | 56 | 56 and 67 |

6 | [11, 12, 23, 34, 45, 56] | [67, 89, 78, 90] | 67 | No swap |

7 | [11, 12, 23, 34, 45, 56, 67] | [89, 78, 90] | 78 | 78 and 89 |

8 | [11, 12, 23, 34, 45, 56, 67, 78] | [89, 90] | 89 | No swap |

9 | [11, 12, 23, 34, 45, 56, 67, 78, 89] | [90] | 90 | No swap |

10 | [11, 12, 23, 34, 45, 56, 67, 78, 89, 90] | [] | - | - |

As you can see, the algorithm sorts the array in 10 steps, by selecting the smallest element in each step and swapping it with the first element of the unsorted part.

To implement selection sort in Java, you can use the following code:

// A method to sort an array using selection sort public static void selectionSort(int[] array) { // Loop through the array from left to right for (int i = 0; i < array.length - 1; i++) { // Assume the current element is the smallest in the unsorted part int minIndex = i; // Loop through the unsorted part of the array from right to left for (int j = i + 1; j < array.length; j++) { // If an element is smaller than the current minimum, update the minimum index if (array[j] < array[minIndex]) { minIndex = j; } } // Swap the smallest element with the first element of the unsorted part int temp = array[i]; array[i] = array[minIndex]; array[minIndex] = temp; } }

The code uses two nested loops to iterate through the array and find the smallest element in each iteration. It then swaps the smallest element with the first element of the unsorted part, and moves the boundary of the sorted part by one position to the right. The code also uses a temporary variable to store the value of the element that is being swapped, to avoid losing any data.

Some of the advantages of selection sort are:

- It is simple and easy to implement.
- It performs well on small data sets.
- It has a fixed number of swaps, which is at most n - 1, where n is the size of the data.
- It is stable, which means that it preserves the relative order of equal elements.

Some of the disadvantages of selection sort are:

- It is slow and inefficient on large data sets.
- It has a high time complexity, which is O(n^2), where n is the size of the data.
- It is not adaptive, which means that it does not take advantage of the existing order in the data.

Now that you have learned how to implement selection sort in Java, you can try it on some sample data and see how it works. You can also compare it with other sorting algorithms that you will learn in the next subsections.

### 4.2. Bubble Sort

**Bubble sort** is a simple and slow algorithm that repeatedly swaps adjacent elements that are out of order until the data is sorted. The algorithm works as follows:

- Compare the first and second elements of the data and swap them if they are in the wrong order.
- Move to the next pair of elements and repeat step 1 until you reach the end of the data.
- At this point, the largest element of the data is at the last position. This is the sorted part of the data.
- Repeat steps 1 to 3 for the remaining unsorted part of the data, until the entire data is sorted.

For example, suppose you want to sort the following array of 10 random integers between 1 and 100 in ascending order using bubble sort:

int[] array = {34, 67, 12, 89, 45, 23, 78, 90, 11, 56};

The algorithm will perform the following steps:

### 4.3. Insertion Sort

**Insertion sort** is a simple and intuitive sorting algorithm that works by inserting each element of the data into its correct position in the sorted order. It is similar to how you would sort a deck of cards by picking one card at a time and placing it in the right place.

The algorithm works as follows:

- Start from the second element of the data and assume that the first element is already sorted.
- Compare the current element with the previous elements in the sorted order and find the position where it belongs.
- Shift all the elements that are larger than the current element to the right by one position.
- Insert the current element into the found position.
- Repeat steps 2 to 4 for each element until the entire data is sorted.

Here is an example of how insertion sort works on an array of numbers:

// The array to be sorted int[] arr = {5, 2, 4, 6, 1, 3}; // Loop from the second element to the last element for (int i = 1; i < arr.length; i++) { // The current element to be inserted int key = arr[i]; // The index of the previous element int j = i - 1; // Compare the current element with the previous elements // and shift them to the right if they are larger while (j >= 0 && arr[j] > key) { arr[j + 1] = arr[j]; j--; } // Insert the current element into the found position arr[j + 1] = key; } // The sorted array // [1, 2, 3, 4, 5, 6]

Insertion sort is an **in-place** and **stable** sorting algorithm, which means that it does not require extra space to sort the data and it preserves the relative order of equal elements. However, it is also a **comparison-based** and **adaptive** sorting algorithm, which means that its performance depends on the number of comparisons and swaps it makes and the initial order of the data.

How efficient is insertion sort? How does it compare with other sorting algorithms? We will answer these questions in the next section.

### 4.4. Merge Sort

**Merge sort** is a powerful and efficient sorting algorithm that works by dividing the data into smaller and smaller subarrays, sorting each subarray, and then merging them back together in the correct order. It is based on the principle of **divide and conquer**, which is a common technique for solving complex problems by breaking them down into simpler subproblems.

The algorithm works as follows:

- Divide the data into two equal or nearly equal halves.
- Recursively sort each half using merge sort.
- Merge the two sorted halves into one sorted array.

Here is an example of how merge sort works on an array of numbers:

// The array to be sorted int[] arr = {5, 2, 4, 6, 1, 3}; // A helper method to merge two sorted subarrays public static void merge(int[] arr, int left, int mid, int right) { // The size of the left subarray int n1 = mid - left + 1; // The size of the right subarray int n2 = right - mid; // Create temporary arrays to store the subarrays int[] L = new int[n1]; int[] R = new int[n2]; // Copy the data from the original array to the subarrays for (int i = 0; i < n1; i++) { L[i] = arr[left + i]; } for (int j = 0; j < n2; j++) { R[j] = arr[mid + 1 + j]; } // Initialize the indices of the subarrays int i = 0; int j = 0; // Initialize the index of the merged array int k = left; // Merge the subarrays by comparing their elements while (i < n1 && j < n2) { if (L[i] <= R[j]) { arr[k] = L[i]; i++; } else { arr[k] = R[j]; j++; } k++; } // Copy the remaining elements of the left subarray if any while (i < n1) { arr[k] = L[i]; i++; k++; } // Copy the remaining elements of the right subarray if any while (j < n2) { arr[k] = R[j]; j++; k++; } } // The main method to implement merge sort public static void mergeSort(int[] arr, int left, int right) { // Check if the array has more than one element if (left < right) { // Find the middle point of the array int mid = (left + right) / 2; // Recursively sort the left half mergeSort(arr, left, mid); // Recursively sort the right half mergeSort(arr, mid + 1, right); // Merge the two sorted halves merge(arr, left, mid, right); } } // Call the merge sort method on the array mergeSort(arr, 0, arr.length - 1); // The sorted array // [1, 2, 3, 4, 5, 6]

Merge sort is an **out-of-place** and **stable** sorting algorithm, which means that it requires extra space to sort the data and it preserves the relative order of equal elements. However, it is also a **comparison-based** and **non-adaptive** sorting algorithm, which means that its performance depends on the number of comparisons it makes and it does not benefit from the initial order of the data.

How efficient is merge sort? How does it compare with other sorting algorithms? We will answer these questions in the next section.

### 4.5. Quick Sort

**Quick sort** is a fast and elegant sorting algorithm that works by choosing a pivot element from the data and partitioning the data into two subarrays, one with elements smaller than the pivot and one with elements larger than the pivot. It then recursively sorts the subarrays using the same technique.

The algorithm works as follows:

- Choose a pivot element from the data, such as the first, last, or middle element, or a random element.
- Partition the data into two subarrays, such that all the elements in the left subarray are smaller than or equal to the pivot, and all the elements in the right subarray are larger than or equal to the pivot.
- Recursively sort the left and right subarrays using quick sort.

Here is an example of how quick sort works on an array of numbers:

// The array to be sorted int[] arr = {5, 2, 4, 6, 1, 3}; // A helper method to swap two elements in the array public static void swap(int[] arr, int i, int j) { int temp = arr[i]; arr[i] = arr[j]; arr[j] = temp; } // A helper method to partition the array around a pivot element public static int partition(int[] arr, int low, int high) { // Choose the last element as the pivot int pivot = arr[high]; // Initialize the index of the smaller element int i = low - 1; // Loop from the low to the high element for (int j = low; j < high; j++) { // If the current element is smaller than or equal to the pivot if (arr[j] <= pivot) { // Increment the index of the smaller element i++; // Swap the current element with the smaller element swap(arr, i, j); } } // Swap the pivot element with the element at the index of the smaller element + 1 swap(arr, i + 1, high); // Return the index of the pivot element return i + 1; } // The main method to implement quick sort public static void quickSort(int[] arr, int low, int high) { // Check if the array has more than one element if (low < high) { // Partition the array around a pivot element and get its index int pi = partition(arr, low, high); // Recursively sort the left subarray quickSort(arr, low, pi - 1); // Recursively sort the right subarray quickSort(arr, pi + 1, high); } } // Call the quick sort method on the array quickSort(arr, 0, arr.length - 1); // The sorted array // [1, 2, 3, 4, 5, 6]

Quick sort is an **in-place** and **unstable** sorting algorithm, which means that it does not require extra space to sort the data and it does not preserve the relative order of equal elements. However, it is also a **comparison-based** and **adaptive** sorting algorithm, which means that its performance depends on the number of comparisons and swaps it makes and the initial order of the data.

How efficient is quick sort? How does it compare with other sorting algorithms? We will answer these questions in the next section.

### 4.6. Counting Sort

**Counting sort** is a simple and fast sorting algorithm that works by counting the number of occurrences of each element in the data and using the counts to determine the positions of the elements in the sorted order. It is based on the assumption that the data consists of integers in a fixed range.

The algorithm works as follows:

- Create an array of size equal to the range of the data and initialize it with zeros.
- Loop through the data and increment the count of each element in the corresponding index of the array.
- Modify the array by adding the count of the previous element to the current element, so that each element represents the cumulative count of the elements up to that index.
- Create a new array of the same size as the data and loop through the data from the end.
- For each element, find its position in the sorted order by looking up its count in the array, and place it in the new array.
- Decrement the count of the element in the array by one.

Here is an example of how counting sort works on an array of numbers:

// The array to be sorted int[] arr = {5, 2, 4, 6, 1, 3}; // The range of the data int k = 6; // Create an array of size k and initialize it with zeros int[] count = new int[k]; for (int i = 0; i < k; i++) { count[i] = 0; } // Loop through the data and increment the count of each element for (int i = 0; i < arr.length; i++) { count[arr[i] - 1]++; } // Modify the array by adding the count of the previous element for (int i = 1; i < k; i++) { count[i] += count[i - 1]; } // Create a new array of the same size as the data int[] output = new int[arr.length]; // Loop through the data from the end for (int i = arr.length - 1; i >= 0; i--) { // Find the position of the element in the sorted order int index = count[arr[i] - 1] - 1; // Place the element in the new array output[index] = arr[i]; // Decrement the count of the element count[arr[i] - 1]--; } // The sorted array // [1, 2, 3, 4, 5, 6]

Counting sort is an **out-of-place** and **stable** sorting algorithm, which means that it requires extra space to sort the data and it preserves the relative order of equal elements. However, it is also a **non-comparison-based** and **non-adaptive** sorting algorithm, which means that its performance does not depend on the number of comparisons and swaps it makes and it does not benefit from the initial order of the data.

How efficient is counting sort? How does it compare with other sorting algorithms? We will answer these questions in the next section.

### 4.7. Radix Sort

**Radix sort** is a unique and fast sorting algorithm that works by sorting the data based on the individual digits or characters of each element, starting from the least significant digit or character and moving to the most significant one. It is based on the assumption that the data consists of integers or strings of fixed length.

The algorithm works as follows:

- Find the maximum number of digits or characters in the data.
- Loop from the least significant digit or character to the most significant one.
- For each digit or character, use a stable sorting algorithm, such as counting sort, to sort the data according to that digit or character.

Here is an example of how radix sort works on an array of numbers:

// The array to be sorted int[] arr = {5, 2, 4, 6, 1, 3}; // The maximum number of digits in the data int d = 1; // A helper method to get the digit at a specific position of a number public static int getDigit(int num, int pos) { // Divide the number by 10^pos and get the remainder return (num / (int) Math.pow(10, pos)) % 10; } // A helper method to implement counting sort on a specific digit public static void countingSort(int[] arr, int pos) { // The range of the digits int k = 10; // Create an array of size k and initialize it with zeros int[] count = new int[k]; for (int i = 0; i < k; i++) { count[i] = 0; } // Loop through the data and increment the count of each digit for (int i = 0; i < arr.length; i++) { count[getDigit(arr[i], pos)]++; } // Modify the array by adding the count of the previous digit for (int i = 1; i < k; i++) { count[i] += count[i - 1]; } // Create a new array of the same size as the data int[] output = new int[arr.length]; // Loop through the data from the end for (int i = arr.length - 1; i >= 0; i--) { // Find the position of the digit in the sorted order int index = count[getDigit(arr[i], pos)] - 1; // Place the element in the new array output[index] = arr[i]; // Decrement the count of the digit count[getDigit(arr[i], pos)]--; } // Copy the new array to the original array for (int i = 0; i < arr.length; i++) { arr[i] = output[i]; } } // The main method to implement radix sort public static void radixSort(int[] arr) { // Loop from the least significant digit to the most significant digit for (int i = 0; i < d; i++) { // Use counting sort to sort the data according to the current digit countingSort(arr, i); } } // Call the radix sort method on the array radixSort(arr); // The sorted array // [1, 2, 3, 4, 5, 6]

Radix sort is an **out-of-place** and **stable** sorting algorithm, which means that it requires extra space to sort the data and it preserves the relative order of equal elements. However, it is also a **non-comparison-based** and **non-adaptive** sorting algorithm, which means that its performance does not depend on the number of comparisons and swaps it makes and it does not benefit from the initial order of the data.

How efficient is radix sort? How does it compare with other sorting algorithms? We will answer these questions in the next section.

## 5. How to Compare Sorting Algorithms

Sorting algorithms are not all equal. Some are faster, some are more memory-efficient, some are more stable, and some are more adaptable than others. How can we measure and compare these properties of sorting algorithms? In this section, we will introduce some common criteria and methods to evaluate and compare sorting algorithms.

The main criteria that we will use to compare sorting algorithms are:

**Time complexity**: How fast does the algorithm run? How does the running time change with the size and order of the data?**Space complexity**: How much extra space does the algorithm need? How does the space usage change with the size and order of the data?**Stability**: Does the algorithm preserve the relative order of equal elements?**Adaptability**: Does the algorithm benefit from the initial order of the data? Can it handle different types of data?

To measure and compare the time and space complexity of sorting algorithms, we will use the **Big O notation**, which is a mathematical notation that describes the asymptotic behavior of a function as the input size grows. The Big O notation gives us an upper bound on the worst-case scenario of the function, ignoring the constant factors and lower-order terms. For example, if a function has a running time of 2n + 3, we say that it has a Big O of O(n), because the linear term n dominates the constant term 3 as n grows large.

To measure and compare the stability and adaptability of sorting algorithms, we will use some examples and scenarios to illustrate how different algorithms behave on different types of data. For example, we will see how insertion sort is stable and adaptive, while quick sort is unstable and non-adaptive.

In the next subsections, we will apply these criteria and methods to the sorting algorithms that we have learned in the previous sections, and see how they compare with each other.

### 5.1. Time Complexity

The **time complexity** of a sorting algorithm is a measure of how fast it can sort a given data set. It is usually expressed in terms of the **Big O notation**, which gives an upper bound on the worst-case scenario of the algorithm's running time as the size of the data grows.

The time complexity of a sorting algorithm depends on two factors: the number of **comparisons** and the number of **swaps** that it makes. A comparison is a basic operation that determines the order of two elements, such as `arr[i] < arr[j]`

. A swap is a basic operation that exchanges the positions of two elements, such as `swap(arr, i, j)`

.

Different sorting algorithms have different time complexities, depending on how they compare and swap the elements. For example, selection sort has a time complexity of O(n^{2}), because it makes n^{2} comparisons and n swaps. Merge sort has a time complexity of O(n log n), because it makes n log n comparisons and n log n swaps.

The time complexity of a sorting algorithm can also vary depending on the initial order of the data. Some sorting algorithms are **adaptive**, which means that they can take advantage of the existing order and perform faster. For example, insertion sort has a time complexity of O(n^{2}) in the worst case, but it can sort an already sorted data in O(n) time, because it makes only n comparisons and no swaps.

In the following table, we summarize the time complexity of the sorting algorithms that we have learned in the previous sections, in terms of the best case, the average case, and the worst case scenarios.

Step | Sorted part | Unsorted part | Swap |

0 | [] | [34, 67, 12, 89, 45, 23, 78, 90, 11, 56] | - |

1 | [] | [34, 67, 12, 89, 45, 23, 78, 90, 11, 56] | 12 and 67 |

2 | [] | [34, 12, 67, 89, 45, 23, 78, 90, 11, 56] | 12 and 34 |

3 | [] | [12, 34, 67, 89, 45, 23, 78, 90, 11, 56] | No swap |

4 | [] | [12, 34, 67, 89, 45, 23, 78, 90, 11, 56] | No swap |

5 | [] | [12, 34, 67, 89, 45, 23, 78, 90, 11, 56] | 23 and 45 |

6 | [] | [12, 34, 67, 89, 23, 45, 78, 90, 11, 56] | 23 and 89 |

7 | [] | [12, 34, 67, 23, 89, 45, 78, 90, 11, 56] | 23 and 67 |

8 | [] | [12, 34, 23, 67, 89, 45, 78, 90, 11, 56] | 23 and 34 |

9 | [] | [12, 23, 34, 67, 89, 45, 78, 90, 11, 56] | 11 and 56 |

10 | [] | [12, 23, 34, 67, 89, 45, 78, 90, 56, 11] | 11 and 90 |

11 | [] | [12, 23, 34, 67, 89, 45, 78, 56, 90, 11] | 11 and 78 |

12 | [] | [12, 23, 34, 67, 89, 45, 56, 78, 90, 11] | 11 and 56 |

13 | [] | [12, 23, 34, 67, 89, 45, 11, 56, 78, 90] | 11 and 45 |

14 | [] | [12, 23, 34, 67, 89, 11, 45, 56, 78, 90] | 11 and 89 |

15 | [] | [12, 23, 34, 67, 11, 89, 45, 56, 78, 90] | 11 and 67 |

16 | [] | [12, 23, 34, 11, 67, 89, 45, 56, 78, 90] | 11 and 34 |

17 | [] | [12, 23, 11, 34, 67, 89, 45, 56, 78, 90] | 11 and 23 |

18 | [] | [12, 11, 23, 34, 67, 89, 45, 56, 78, 90] | 11 and 12 |

19 | [90] | [11, 12, 23, 34, 67, 89, 45, 56, 78] | No swap |

20 | [90] | [11, 12, 23, 34, 67, 89, 45, 56, 78] | No swap |

21 | [90] | [11, 12, 23, 34, 67, 89, 45, 56, 78] | No swap |

22 | [90] | [11, 12, 23, 34, 67, 89, 45, 56, 78] | No swap |

23 | [90] | [11, 12, 23, 34, 67, 89, 45, 56, 78] | No swap |

24 | [90] | [11, 12, 23, 34, 67, 89, 45, 56, 78] | No swap |

25 | [90] | [11, 12, 23, 34, 67, 89, 45, 56, 78] | 45 and 89 |

26 | [90] | [11, 12, 23, 34, 67, 45, 89, 56, 78] | 56 and 89 |

27 | [90] | [11, 12, 23, 34, 67, 45, 56, 89, 78] | 78 and 89 |

28 | [89, 90] | [11, 12, 23, 34, 67, 45, 56, 78] | No swap |

29 | [89, 90] | [11, 12, 23, 34, 67, 45, 56, 78] | No swap |

30 | [89, 90] | [11, 12, 23, 34, 67, 45, 56, 78] | No swap |

31 | [89, 90] | [11, 12, 23, 34, 67, 45, 56, 78] | No swap |

32 | [89, 90] |

Algorithm | Best Case | Average Case | Worst Case |
---|---|---|---|

Selection Sort | O(n^{2}) | O(n^{2}) | O(n^{2}) |

Bubble Sort | O(n) | O(n^{2}) | O(n^{2}) |

Insertion Sort | O(n) | O(n^{2}) | O(n^{2}) |

Merge Sort | O(n log n) | O(n log n) | O(n log n) |

Quick Sort | O(n log n) | O(n log n) | O(n^{2}) |

Counting Sort | O(n + k) | O(n + k) | O(n + k) |

Radix Sort | O(d * (n + k)) | O(d * (n + k)) | O(d * (n + k)) |

Where n is the size of the data, k is the range of the data, and d is the number of digits or characters in the data.

As you can see, the time complexity of sorting algorithms can vary significantly, depending on the algorithm, the data, and the scenario. How can we choose the best sorting algorithm for our problem? How can we measure and compare the other properties of sorting algorithms, such as space complexity, stability, and adaptability? We will answer these questions in the next subsections.

### 5.2. Space Complexity

The **space complexity** of a sorting algorithm is a measure of how much extra space it needs to sort a given data set. It is usually expressed in terms of the **Big O notation**, which gives an upper bound on the worst-case scenario of the algorithm's space usage as the size of the data grows.

The space complexity of a sorting algorithm depends on whether it is **in-place** or **out-of-place**. An in-place sorting algorithm does not require any extra space to sort the data, and it only modifies the original data. An out-of-place sorting algorithm requires extra space to store the sorted data, and it does not modify the original data.

Different sorting algorithms have different space complexities, depending on whether they are in-place or out-of-place. For example, selection sort, bubble sort, insertion sort, and quick sort are in-place sorting algorithms, and they have a space complexity of O(1), because they only need a constant amount of extra space to store some temporary variables. Merge sort, counting sort, and radix sort are out-of-place sorting algorithms, and they have a space complexity of O(n), because they need an extra array of the same size as the data to store the sorted data.

The space complexity of a sorting algorithm can also vary depending on the initial order of the data. Some sorting algorithms are **adaptive**, which means that they can reduce the space usage if the data is already partially sorted. For example, merge sort can have a space complexity of O(log n) if the data is already sorted, because it only needs a stack of size log n to store the recursive calls.

In the following table, we summarize the space complexity of the sorting algorithms that we have learned in the previous sections, in terms of the best case, the average case, and the worst case scenarios.

Algorithm | Best Case | Average Case | Worst Case |
---|---|---|---|

Selection Sort | O(1) | O(1) | O(1) |

Bubble Sort | O(1) | O(1) | O(1) |

Insertion Sort | O(1) | O(1) | O(1) |

Merge Sort | O(log n) | O(n) | O(n) |

Quick Sort | O(log n) | O(log n) | O(n) |

Counting Sort | O(n + k) | O(n + k) | O(n + k) |

Radix Sort | O(n + k) | O(n + k) | O(n + k) |

Where n is the size of the data, k is the range of the data, and d is the number of digits or characters in the data.

As you can see, the space complexity of sorting algorithms can vary significantly, depending on the algorithm and the data. How can we choose the best sorting algorithm for our problem? How can we measure and compare the other properties of sorting algorithms, such as time complexity, stability, and adaptability? We will answer these questions in the next subsections.

### 5.3. Stability

The **stability** of a sorting algorithm is a property that determines whether it preserves the relative order of equal elements in the data. A stable sorting algorithm does not change the order of equal elements, while an unstable sorting algorithm may change the order of equal elements.

The stability of a sorting algorithm is important when the data has some additional information or attributes that are not used for sorting, but are still relevant for the problem. For example, suppose you have a list of students with their names and grades, and you want to sort them by their grades. A stable sorting algorithm will keep the order of students with the same grade, while an unstable sorting algorithm may change the order of students with the same grade.

Different sorting algorithms have different stability, depending on how they compare and swap the elements. For example, selection sort and quick sort are unstable sorting algorithms, because they may swap an element with another element that is equal to it. Insertion sort, bubble sort, merge sort, counting sort, and radix sort are stable sorting algorithms, because they do not swap equal elements.

In the following table, we summarize the stability of the sorting algorithms that we have learned in the previous sections.

Algorithm | Stability |
---|---|

Selection Sort | Unstable |

Bubble Sort | Stable |

Insertion Sort | Stable |

Merge Sort | Stable |

Quick Sort | Unstable |

Counting Sort | Stable |

Radix Sort | Stable |

As you can see, the stability of sorting algorithms can vary significantly, depending on the algorithm. How can we choose the best sorting algorithm for our problem? How can we measure and compare the other properties of sorting algorithms, such as time complexity, space complexity, and adaptability? We will answer these questions in the next subsections.

### 5.4. Adaptability

The **adaptability** of a sorting algorithm is a property that determines whether it can benefit from the initial order of the data and perform faster or more efficiently. An adaptive sorting algorithm can adjust its behavior according to the data and optimize its performance. A non-adaptive sorting algorithm does not depend on the initial order of the data and performs the same regardless of the data.

The adaptability of a sorting algorithm is important when the data is already partially sorted or has some patterns or repetitions. An adaptive sorting algorithm can take advantage of these features and reduce the number of comparisons and swaps that it makes. A non-adaptive sorting algorithm will ignore these features and perform the same number of comparisons and swaps as if the data was completely random.

Different sorting algorithms have different adaptability, depending on how they compare and swap the elements. For example, insertion sort and bubble sort are adaptive sorting algorithms, because they can sort an already sorted data in O(n) time, by making only n comparisons and no swaps. Quick sort and radix sort are non-adaptive sorting algorithms, because they do not benefit from the initial order of the data and perform the same number of comparisons and swaps regardless of the data.

In the following table, we summarize the adaptability of the sorting algorithms that we have learned in the previous sections.

Algorithm | Adaptability |
---|---|

Selection Sort | Non-adaptive |

Bubble Sort | Adaptive |

Insertion Sort | Adaptive |

Merge Sort | Non-adaptive |

Quick Sort | Non-adaptive |

Counting Sort | Non-adaptive |

Radix Sort | Non-adaptive |

As you can see, the adaptability of sorting algorithms can vary significantly, depending on the algorithm and the data. How can we choose the best sorting algorithm for our problem? How can we measure and compare the other properties of sorting algorithms, such as time complexity, space complexity, and stability? We will answer these questions in the next section.

## 6. Conclusion

In this blog, you have learned about sorting algorithms, which are methods of arranging data in a specific order, such as ascending or descending, based on some criteria. You have learned how to use Java to implement different types of sorting algorithms, such as selection sort, bubble sort, insertion sort, merge sort, quick sort, counting sort, and radix sort. You have also learned how to compare sorting algorithms based on their properties, such as time complexity, space complexity, stability, and adaptability.

Sorting algorithms are important for many applications, such as organizing, searching, and analyzing data. They can also help you to optimize the performance and memory usage of your programs. However, there is no single best sorting algorithm for every problem and data set. You need to consider the trade-offs and choose the most suitable sorting algorithm for your specific situation.

We hope that this blog has helped you to understand the basics of sorting algorithms and how to apply them in Java. You can use the code snippets and examples that we have provided to practice and experiment with different sorting algorithms. You can also try to implement other sorting algorithms that we have not covered in this blog, such as heap sort, shell sort, or bucket sort. You can also try to modify the existing sorting algorithms to improve their performance or functionality.

Thank you for reading this blog and we hope that you have enjoyed it. If you have any questions, comments, or feedback, please feel free to leave them in the comment section below. We would love to hear from you and learn from your experience. Happy sorting!