Radix Sort Algorithm in DSA C# - Time & Space Complexity

Time Complexity: Radix Sort Algorithm

O(d * n)

Understanding Time Complexity

We want to understand how the time taken by Radix Sort changes as the input size grows.

Specifically, how the number of digits and elements affect the work done.

Scenario Under Consideration

Analyze the time complexity of the following Radix Sort code snippet.

public void RadixSort(int[] arr) {
    int max = arr.Max();
    for (int exp = 1; max / exp > 0; exp *= 10) {
        CountingSortByDigit(arr, exp);
    }
}

private void CountingSortByDigit(int[] arr, int exp) {
    int n = arr.Length;
    int[] output = new int[n];
    int[] count = new int[10];
    for (int i = 0; i < n; i++)
        count[(arr[i] / exp) % 10]++;
    for (int i = 1; i < 10; i++)
        count[i] += count[i - 1];
    for (int i = n - 1; i >= 0; i--) {
        output[count[(arr[i] / exp) % 10] - 1] = arr[i];
        count[(arr[i] / exp) % 10]--;
    }
    for (int i = 0; i < n; i++)
        arr[i] = output[i];
}
    

This code sorts an array by processing digits from least significant to most significant using counting sort.

Identify Repeating Operations

Look at the loops and repeated steps:

• Primary operation: Counting sort runs once for each digit place.
• How many times: Number of digits in the largest number (d times).
• Each counting sort loops over all n elements multiple times.

How Execution Grows With Input

For n elements and d digits per number, work grows roughly as n times d.

Input Size (n)	Digits (d)	Approx. Operations
10	3	About 30
100	3	About 300
1000	4	About 4000

Pattern observation: Operations grow linearly with number of elements and digits.

Final Time Complexity

Time Complexity: O(d * n)

This means the time grows linearly with the number of elements and the number of digits in the largest number.

Common Mistake

[X] Wrong: "Radix Sort always runs in linear time O(n)."

[OK] Correct: The number of digits d affects the time, so if numbers have many digits, time grows with d too.

Interview Connect

Understanding Radix Sort's time complexity helps you explain sorting choices clearly and shows you can analyze multi-step algorithms.

Self-Check

"What if the numbers were fixed-length strings instead of integers? How would that affect the time complexity?"

Practice

1. What is the main idea behind the Radix Sort Algorithm?

easy

A. Sort numbers digit by digit starting from the least significant digit

B. Sort numbers by comparing each pair directly

C. Sort numbers by dividing them into two halves recursively

D. Sort numbers by swapping adjacent elements repeatedly

Solution

1. Step 1: Understand Radix Sort approach

   Radix Sort sorts numbers by processing each digit separately, starting from the least significant digit (rightmost).

2. Step 2: Compare with other sorting methods

   Unlike comparison-based sorts, Radix Sort uses digit positions to sort without direct comparisons between whole numbers.

3. Final Answer:

   Sort numbers digit by digit starting from the least significant digit -> Option A

4. Quick Check:

   Radix Sort = Digit by digit sorting [OK]

Hint: Radix Sort processes digits from right to left [OK]

Common Mistakes:

• Thinking Radix Sort compares whole numbers directly
• Confusing Radix Sort with quicksort or mergesort
• Assuming it sorts from most significant digit first

2. Which of the following is the correct way to initialize a queue in C# for use in Radix Sort?

easy

A. Queue<int> buckets = new Queue<int>[10];

B. Queue<int> buckets = new Queue<int>(10);

C. Queue buckets = new Queue<int>(10);

D. Queue<int>[] buckets = new Queue<int>[10];

Solution

1. Step 1: Understand array of queues declaration

   We need an array of 10 queues, so the correct syntax is an array of Queue<int>: Queue<int>[] buckets = new Queue<int>[10];

2. Step 2: Check other options

   Queue<int> buckets = new Queue<int>[10]; tries to assign an array to a single Queue<int> variable, which is invalid. Queue buckets = new Queue<int>(10); uses non-generic Queue incorrectly. Queue<int> buckets = new Queue<int>(10); creates a single Queue with capacity 10, not an array.

3. Final Answer:

   Queue<int>[] buckets = new Queue<int>[10]; -> Option D

4. Quick Check:

   Array of queues = Queue<int>[] [OK]

Hint: Use [] to declare array of queues in C# [OK]

Common Mistakes:

• Declaring a single queue instead of an array
• Using non-generic Queue instead of Queue<int>
• Confusing array initialization syntax

3. What will be the output of the following C# code snippet using Radix Sort on the array [170, 45, 75, 90, 802, 24, 2, 66]?

int[] arr = {170, 45, 75, 90, 802, 24, 2, 66};
RadixSort(arr);
Console.WriteLine(string.Join(", ", arr));

medium

A. [170, 45, 75, 90, 802, 24, 2, 66]

B. [2, 24, 45, 66, 75, 90, 170, 802]

C. [802, 170, 90, 75, 66, 45, 24, 2]

D. [24, 2, 45, 66, 75, 90, 170, 802]

Solution

1. Step 1: Apply Radix Sort on given array

   Radix Sort sorts numbers by digits from least significant to most significant. After sorting, the array becomes sorted in ascending order: 2, 24, 45, 66, 75, 90, 170, 802.

2. Step 2: Verify output matches sorted array

   [2, 24, 45, 66, 75, 90, 170, 802] matches the sorted array. [170, 45, 75, 90, 802, 24, 2, 66] is original unsorted. [802, 170, 90, 75, 66, 45, 24, 2] is descending order. [24, 2, 45, 66, 75, 90, 170, 802] is incorrect order.

3. Final Answer:

   [2, 24, 45, 66, 75, 90, 170, 802] -> Option B

4. Quick Check:

   Radix Sort output = Sorted array ascending [OK]

Hint: Radix Sort outputs sorted array ascending [OK]

Common Mistakes:

• Confusing original and sorted array
• Assuming Radix Sort sorts descending
• Mixing digit order in sorting

4. Identify the error in this Radix Sort code snippet in C#:

for (int i = 0; i < maxDigit; i++)
{
    Queue<int>[] buckets = new Queue<int>[10];
    for (int j = 0; j < 10; j++)
        buckets[j] = new Queue<int>();

    foreach (int num in arr)
    {
        int digit = (num / (int)Math.Pow(10, i)) % 10;
        buckets[digit].Enqueue(num);
    }

    int index = 0;
    foreach (var bucket in buckets)
    {
        while (bucket.Count > 0)
            arr[index++] = bucket.Dequeue();
    }
}

medium

A. The inner foreach loop should use a for loop instead

B. The digit calculation is incorrect; it should use modulus 100

C. There is no error in the code snippet

D. The buckets array is reinitialized inside the outer loop, causing loss of data

Solution

1. Step 1: Analyze the code structure

   The code correctly creates fresh buckets for each digit position, distributes all elements by the current digit using stable queues, and reconstructs the array in place. This is proper LSD Radix Sort implementation.

2. Step 2: Evaluate options

   A: foreach loops are fine for iterating arr and buckets. B: (num / pow(10,i)) % 10 correctly extracts the i-th digit from LSD. C: Reinitializing buckets each pass prevents carry-over from previous passes and does not lose data since all elements are re-enqueued from arr every iteration.

3. Final Answer:

   There is no error in the code snippet -> Option C

4. Quick Check:

   Buckets reinitialized each pass = Correct [OK]

Hint: Buckets reset each digit pass is required, not an error [OK]

Common Mistakes:

• Thinking buckets reinitialization causes data loss
• Misunderstanding digit extraction formula
• Resetting index inside wrong loop

5. You want to adapt Radix Sort to sort an array of strings representing numbers with varying lengths (e.g., ["170", "45", "9", "802", "24"]). Which approach is best to handle different string lengths correctly?

hard

A. Pad shorter strings with leading zeros to match the longest length before sorting

B. Sort strings as they are without padding, digit by digit from left to right

C. Convert strings to integers and apply normal Radix Sort

D. Sort strings by length first, then apply Radix Sort on digits

Solution

1. Step 1: Understand problem with varying string lengths

   Radix Sort requires uniform digit positions. Strings of different lengths cause misalignment when sorting digit by digit.

2. Step 2: Choose padding strategy

   Padding shorter strings with leading zeros makes all strings equal length, allowing digit-by-digit sorting from right to left correctly.

3. Step 3: Evaluate other options

   Sort strings as they are without padding, digit by digit from left to right sorts left to right which breaks Radix Sort logic. Convert strings to integers and apply normal Radix Sort loses string format and may cause overflow. Sort strings by length first, then apply Radix Sort on digits sorting by length first does not solve digit alignment.

4. Final Answer:

   Pad shorter strings with leading zeros to match the longest length before sorting -> Option A

5. Quick Check:

   Pad strings to equal length = Correct approach [OK]

Hint: Pad strings with zeros to equal length before sorting [OK]

Common Mistakes:

• Sorting strings without padding causes wrong order
• Sorting digits left to right breaks Radix Sort
• Converting strings to int may cause errors with large numbers