Radix Sort: A Comprehensive Guide to Efficient Non-Comparison Sorting

9.3.2 Radix Sort

Sorting is a fundamental operation in computer science, and understanding different sorting algorithms is crucial for efficient data manipulation. In this section, we delve into Radix Sort, a non-comparison-based sorting algorithm that stands out for its efficiency in sorting integers. Unlike traditional comparison-based sorting algorithms like Quick Sort or Merge Sort, Radix Sort processes individual digits of numbers, making it particularly effective for large datasets with fixed digit lengths.

Introduction to Radix Sort

Radix Sort is a unique sorting algorithm that sorts numbers by processing each digit individually, from the least significant digit (LSD) to the most significant digit (MSD). This approach is known as LSD Radix Sort. It leverages a stable sorting algorithm, such as Counting Sort, as a subroutine to sort the numbers based on each digit.

Key Characteristics of Radix Sort

• Non-Comparison-Based: Radix Sort does not compare elements directly. Instead, it categorizes numbers based on their digits.
• Stable Sorting: The algorithm maintains the relative order of elements with equal keys, which is crucial for sorting complex data structures.
• Efficient for Fixed-Length Numbers: Radix Sort excels in scenarios where the number of digits (d) is fixed, making it suitable for sorting large datasets of integers.

How Radix Sort Works

Radix Sort processes numbers digit by digit, starting from the least significant digit. For each digit, it uses a stable sorting algorithm to sort the numbers. This process is repeated for each digit position until the entire number is sorted.

Step-by-Step Process

1. Identify the Maximum Number: Determine the number with the maximum digits in the array. This will dictate the number of passes required.
2. Sort by Each Digit: Starting with the least significant digit, sort the array using a stable sorting algorithm like Counting Sort.
3. Repeat for Each Digit: Move to the next significant digit and repeat the sorting process.
4. Complete the Sorting: Continue until all digit positions have been processed.

Implementing Radix Sort in JavaScript

Let’s implement Radix Sort for integers in JavaScript. We’ll use Counting Sort as the stable sorting algorithm for sorting based on individual digits.

function radixSort(arr) {
  const maxNum = Math.max(...arr);
  let digit = 1;
  while (parseInt(maxNum / digit) > 0) {
    countingSortByDigit(arr, digit);
    digit *= 10;
  }
  return arr;
}

function countingSortByDigit(arr, digit) {
  const count = new Array(10).fill(0);
  const output = new Array(arr.length);
  
  // Count occurrences of each digit
  for (let i = 0; i < arr.length; i++) {
    const digitValue = Math.floor(arr[i] / digit) % 10;
    count[digitValue]++;
  }
  
  // Accumulate counts
  for (let i = 1; i < 10; i++) {
    count[i] += count[i - 1];
  }
  
  // Build the output array
  for (let i = arr.length - 1; i >= 0; i--) {
    const digitValue = Math.floor(arr[i] / digit) % 10;
    output[count[digitValue] - 1] = arr[i];
    count[digitValue]--;
  }
  
  // Copy the sorted numbers back to the original array
  for (let i = 0; i < arr.length; i++) {
    arr[i] = output[i];
  }
}

Understanding the Code

• radixSort Function: This function initializes the sorting process by determining the maximum number in the array and iteratively sorting the array by each digit.
• countingSortByDigit Function: This helper function performs Counting Sort on the array based on the current digit. It counts the occurrences of each digit, accumulates the counts, and constructs the sorted output array.

Time Complexity of Radix Sort

The time complexity of Radix Sort is O(d * (n + b)), where:

• d is the number of digits in the largest number.
• n is the number of elements in the array.
• b is the base of the number system (e.g., 10 for decimal numbers).

Radix Sort is efficient when the number of digits (d) is significantly smaller than the number of elements (n), making it suitable for sorting large datasets of integers with a fixed number of digits.

When to Use Radix Sort

Radix Sort is particularly useful in scenarios where:

• You need to sort large datasets of integers.
• The number of digits in the numbers is fixed or relatively small.
• Stability is a requirement, as Radix Sort maintains the relative order of elements with equal keys.

Experimenting with Radix Sort

To fully grasp the power of Radix Sort, experiment with arrays containing numbers of varying digit lengths. Observe how the algorithm efficiently handles sorting without direct comparisons between elements.

Visualization of Radix Sort

Visualizing the sorting process can enhance understanding. Below is a flowchart illustrating the Radix Sort process:

    flowchart TD
	    A[Start] --> B[Identify Maximum Number]
	    B --> C[Initialize Digit to 1]
	    C --> D{Is MaxNum / Digit > 0?}
	    D -->|Yes| E[Perform Counting Sort by Digit]
	    E --> F[Multiply Digit by 10]
	    F --> D
	    D -->|No| G[End]

Best Practices and Common Pitfalls

• Ensure Stability: Use a stable sorting algorithm like Counting Sort for sorting by digits to maintain stability.
• Handle Negative Numbers: Radix Sort, as implemented here, is designed for non-negative integers. Additional logic is required to handle negative numbers.
• Optimize for Base: While the base is typically 10 for decimal numbers, experimenting with different bases can yield performance improvements in specific scenarios.

Conclusion

Radix Sort is a powerful non-comparison-based sorting algorithm that excels in sorting large datasets of integers with fixed digit lengths. By processing each digit individually and leveraging a stable sorting algorithm, Radix Sort offers efficient and stable sorting capabilities. Understanding and implementing Radix Sort in JavaScript equips you with a valuable tool for tackling sorting challenges in various applications.

Quiz Time!