Radix Sort

Radix sort is a sorting algorithm that operates by sorting the elements of an array by examining their digits. It is based on the idea of distributing elements into buckets according to the value of a specific digit. This process is repeated for each digit, starting from the least significant to the most significant digit.

The algorithm starts by selecting a digit position and grouping the elements based on their value at that position. For example, if the selected digit position is the ones place, all elements with a 0 in the ones place are grouped together, followed by all elements with a 1 in the ones place, and so on. After grouping the elements, the algorithm concatenates the buckets in the order they were created, resulting in a partially sorted array. This process is repeated for each digit position until the entire array is sorted.

Radix sort has a linear time complexity of O(kn), where k is the maximum number of digits in any element and n is the number of elements in the array. This makes it efficient for large datasets, as it does not depend on the magnitude of the elements being sorted. Additionally, radix sort is stable, meaning that it preserves the relative order of equal elements.

One disadvantage of radix sort is that it requires additional memory to store the buckets. The size of the buckets is determined by the range of possible values for the digit being examined. For example, if the digits are represented in base 10 (0-9), there would be 10 buckets. This can result in high memory usage for large datasets or when working with elements with a large range of possible values.

Another consideration when using radix sort is the choice of digit representation. Different digit representations can affect the speed and memory usage of the algorithm. For example, using binary representation can reduce memory usage but may require additional operations to convert the elements to and from binary.

In conclusion, radix sort is a sorting algorithm that operates by sorting the elements of an array by examining their digits. It has a linear time complexity of O(kn) and is efficient for large datasets. However, it requires additional memory to store the buckets and the choice of digit representation can affect the algorithm's performance.

ALGORITHM:

The steps of radix sort algorithm are as follows:

    1. Determine the maximum number of digits (k) in the input array.

    2. For each digit position (i), starting from the least significant digit to the most significant digit:

        a. Create 10 buckets, numbered 0 to 9.

        b. Iterate through the input array and distribute each element into the appropriate bucket based on the value of its digit at position i.

        c. Concatenate the buckets in the order they were created to form a partially sorted array.

    3. Repeat step 2 for each digit position until the entire array is sorted.

It's important to note that for the bucket distribution step in step 2b, it's common to use a stable sorting algorithm, such as counting sort or insertion sort, to sort the elements within each bucket. This ensures that the relative order of equal elements is preserved in the sorted array.

At the end of the algorithm, the input array will be sorted in non-decreasing order. The time complexity of radix sort is O(kn), where k is the maximum number of digits in any element and n is the number of elements in the array. Radix sort is efficient for large datasets as it does not depend on the magnitude of the elements being sorted, but it does require additional memory to store the buckets.

PSEUDO CODE:

Here's an example of pseudocode for radix sort algorithm:

In this pseudocode, get_max_digits(arr) is a function that returns the maximum number of digits in the input array. get_digit(num, pos) is a function that returns the digit at the specified position pos in the number num. concatenate_buckets(buckets) is a function that concatenates the buckets in the order they were created to form a partially sorted array.

Note that the above pseudocode assumes that all elements in the input array have the same number of digits. If the input array contains elements of varying lengths, some modifications may be necessary to handle this case.

PYTHON CODE:

Here's an implementation of radix sort in Python:

In this Python code, max(arr) returns the maximum value in the input array, and len(str(max(arr))) returns the number of digits in the maximum value. buckets is a list of 10 empty lists to hold the elements in each bucket. The // operator is used for integer division and the ** operator is used for exponentiation. The concatenated buckets are added back into the input array in sorted order.

JAVA CODE:

Here's an implementation of radix sort in Java:

In this Java code, Arrays.stream(arr).max().getAsInt() returns the maximum value in the input array, and Integer.toString() converts it to a string to determine the number of digits. List<Integer>[] buckets is an array of 10 ArrayLists to hold the elements in each bucket. The Math.pow(10, i) expression is used to isolate each digit in the input numbers. The concatenated buckets are added back into the input array in sorted order.

C CODE:

Here's an implementation of radix sort in C:

In this C code, max_digits is determined by iterating over the input array and counting the number of digits in each element. buckets is a 2D array of 10 rows (one for each digit) and n columns (one for each element in the input array) to hold the elements in each bucket. counts is an array of 10 integers to keep track of the number of elements in each bucket. The pow() function from math.h is used to isolate each digit in the input numbers. The concatenated buckets are added back into the input array in sorted order.

C++ CODE:

Here's an implementation of radix sort in C++:

In this C++ code, max_digits is determined by iterating over the input array and computing the number of digits in each element using the log10 function. buckets is a vector of 10 vectors to hold the elements in each bucket. The pow function from cmath is used to isolate each digit in the input numbers. The concatenated buckets are added back into the input vector in sorted order using the insert function.

JAVASCRIPT CODE:

Here's an implementation of radix sort in JavaScript:

In this JavaScript code, maxDigits is determined by computing the maximum number of digits in the input array. buckets is an array of 10 empty arrays to hold the elements in each bucket. The floor and pow functions are used to isolate each digit in the input numbers. The concatenated buckets are added back into the input array in sorted order using the flat function.

Note that in this implementation, the input array is not modified in place, and the sorted array is returned as the result of the function.