Radix Sort Algorithm Explained Simply

Analysis of Radix Sort

Radix Sort by taking each of the digits (least significant first, on the right side) and sorting it (in place). e.g., with the decimal system (base 10), it would be by the units column (that is, the 1 s column), then by the tens column, etc. The algorithm employs the use of the counting sort as a subroutine to take care of sorting of characters at each place position. It means that in case of a 3-digit number in the base 10, counting sort will be invoked to sort 1 place, 10 place and 100 place successively, essentially performing a complete sort in this case. Radix Sort is a stable sorting algorithm, i.e. it maintains relative sequence of elements with equal key. Such stability is also essential since with the sorting of a based on the following digits, the sequence which has been formulated by the preceding digits should remain in place to bring the end outcome accurately.

The Workings of Radix Sort

Radix, or base, refers to the number of unique symbols in a number system. Decimal has 10 different digits (0-9). Radix Sort rests on this principle by putting decimal numbers into 10 various boxes depending on which number is under focus. Once placed in buckets, values are backfilled in the array after which the algorithm proceeds to the next journey digit. The process repeats itself until all the digits are taken into account. Radix Sort operates on non-negative integers only. 

The phases of Radix Sort Algorithm may be outlined as follows:

° Start with the least important digit: You should start a sorting process with the right-most digit of the numbers.

° Organize by the digit under consideration: according to the front digit, put the place values into the corresponding buckets and then restore them into the abstract in the correct sequence.

° Advance to the next digit: Repeat the process of sorting each of the digit positions, proceeding with the least significant to the most significant bit, till all the figures have been sorted.

Suppose, you have an unsorted array such as \(\):

° Starting State: represents the array and a radixArray having 10 empty buckets (0-9).

° Unit Place (LS greatest Digit):

The elements are shifted into radixArray based on unit digit. To give an example, 40 enters bucket 0, 33 enters bucket 3, 24 enters bucket 4, 45 and 25 enter bucket 5, and 17 enters bucket 7.

The radixArray turns to \([,,,,,,,,, ]\). It is worth recalling that 45 and 25 keep their relative position as 45 was originally prior to 25 in the initial arrangement and they showed stability.

The elements are then transferred back to the initial array using radixArray and that leaves \(\ ca nut in

° Tens Place(Next Significant Digit) Sort:

We concentrated on ten places.

Are stored in the radixArray according to their tens digit: 17 in bucket 1, 24 and 25 in bucket 2, 33 in bucket 3 and 40 and 45 in bucket 4.

The radixArray turns to \([,,,,,,,,, ]\).

The radixArray is used to move elements back to the original array thus bringing the results to the sorted array \(\).

This process is repeated for all the other significant digits until the array is sorted completely. The passes are equivalent to the quantity of maximum elements in every integer of the array. The radixArray must be of at least two dimensions to allow a multiplicity of values to follow each other in the same digit at a particular radix or index.

Code examples and implementation Details

There are a few components that would be required so as to use the Radix Sort algorithm:

° A list of non-negative integers that should be ordered.

° A two-dimensional array of index 0 to 9 to temporarily store the values according to their current radix.

° A loop that will transfer the values on the unsorted array to the right places on the two-dimensional radix array.

° A loop to push the values out of the radix array and push it back into the original array.

° An external loop which is repeated by bookkeeping the quantity of digits in the biggest worth in the array.

The individual digit sorting step of Radix Sort may be any stable sorting algorithm e.g. counting sort or bubble sort. Counting sort is frequently called explicitly as a subroutine in many others.

The following is the code snippet on implementing Radix Sort in different programming languages and these should normally contain a helper function to determine the largest value in the array and a countSort that would sort the array based on the value of the particular digit.

An Example of Python Implementation

The primary elements of this Python code snippet can be explained as follows: countingSort the function that sorts the array according to a specific power (exp1) that represents the value of the powers of 10 (current digits place value), and the radixSort that includes the repeated use of countingSort, with the less significant and the most significant digits.

Complexity of Radix Sort

Radix Sort works on \(n\) numbers, each of which has no more than \(d\) digits, and the value of each digit can be from \(b\) different numbers (where \(b\) is the base the number is written in). For instance, in base 10, a digit can be from 0 to 9.

Time Complexity

The time complexity of Radix Sort is O(d(n+b)). The time for handling \(n\) numbers in one pass is $O(n+b))\ and the number of passes is \(d\).

° Best Case: If the dataset contains significantly large values but each value has only a few digits, for instance, one million values where the largest value is 999 (three digits), the time complexity can be reduced to \(O(n))\.

° Worst Case: If the maximum number of digits in the highest value is the same as the number of values to sort, the time complexity might be \(O(n^2))\. However, this is rarely the case.

° Average Case: Provided that the number of digits \(k\) is approximately, then the complexity of Radix Sort will be \(O(n \cdot \log n)).\) As an example, sorting 1,000,000 values with 6 digits would fit here.

Generally, a Radix Sort is time complexity is $O(n \cdot k))\, where \(n) is the quantity of the input and \(k\) is the number of digits in the biggest number.

Space Complexity

Radix Sort is based on counting sort and this algorithm needs additional arrays of size \(n\) (the size of the output) and \(k\) (how many elements are needed to count the possible key values), where n is the number of elements in the original array and k is the largest array element to be sorted on among the digits. Thus, Radix Sort has a space complexity of O(n+k)). It cannot be an in place sorting algorithm since it needs this extra space. The space complexity is O(n +b ) where n is the element and b is the base.

The Pros and Cons of Radix Sort

Advantages

Short keys: Radix Sort is fast on short keys (numbers or strings), i.e. when the range of the array elements is small.

Stability: it is a stable type, that is, it maintains the relative order of elements with the equal value. This is a very important property because it has a multi-pass implementation.

Linear time complexity: In a few circumstances, Radix Sort is capable of attaining linear time complexity; in such cases, it is inferior to comparison-based sorts, such as QuickSort, with respect to particular input distributions.

Application in suffix array construction: It finds use in algorithms used in constructing suffix arrays, in particular the Manber algorithm and the DC3 algorithm.

Finding positions in big ranges: Radix Sort can be successfully used to find positions of the numbers in a wide range of numbers.

Disadvantages

Less flexible: Radix Sort is not so flexible as compared to other sorting algorithms, in the sense that it depends on digits or letters so it will have to be re-written when there is a shift in type of data.

Greater constant factor: It can be greater by constant factor than other sorting algorithms.

Greater space requirement: Radix Sort usually needs additional space than in-place sorting algorithms such as QuickSort.

Badly designed operations: Should the operations (like sub-inset lists, delete functions and procedure of isolating the required figures) be inefficient, Radix Sort may be slower in comparison to other algorithms (such as Merge Sort and QuickSort).

Radix Sort in the classroom and Uncodemy Courses

Sorting algorithms remain an essential subject within the field of computer programming, and they were discussed in classes such as the programming classes, data structures classes as well as classes on algorithms. Radix Sort has been taught with the rest of the common sorting algorithms, such as selection sort, insertion sort, bubble sort, merge sort, quick sort and heap sort. In teaching and learning purposes, educational tools commonly use the visual representation of animation to explain how these algorithms better work. Uncodemy offers a course in data structure and algorithm course in Noida through practical exposure to tools and development technologies. Although the curriculum at Uncodemy does not describe Radix Sort in specific detail, chances are high that a course in Data Structure and Algorithms is the most likely course to incorporate sorting algorithms, which would by extension include Radix Sort in the Data Structure and Algorithm course. Another online learning site, Udemy also provides other step-by-step procedures of solving problems with algorithms, including the basic algorithm sorting and searching entirely in Python.