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 needed by Radix Sort changes as the list size grows.

Specifically, how does the sorting time increase when we have more numbers or bigger numbers?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

void radixSort(int arr[], int n) {
    int max = arr[0];
    for (int i = 1; i < n; i++) {
        if (arr[i] > max) max = arr[i];
    }
    for (int exp = 1; max / exp > 0; exp *= 10) {
        countingSortByDigit(arr, n, exp);
    }
}
    

This code finds the largest number, then sorts the array digit by digit using counting sort.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: The outer loop runs once for each digit place (ones, tens, hundreds, etc.).
• How many times: Number of digits in the largest number (d times).
• Inside each iteration: Counting sort runs over all n elements.

How Execution Grows With Input

As the number of elements (n) grows, counting sort runs on all elements for each digit.

Input Size (n)	Approx. Operations (d * n)
10	About 30 (if max has 3 digits)
100	About 300
1000	About 3000

Pattern observation: Operations grow roughly in a straight line with n, multiplied by the number of digits d.

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 time also depends on d, the number of digits. If numbers are very large, d grows and so does time.

Interview Connect

Understanding Radix Sort's time helps you explain when it is faster than comparison sorts and why digit count matters.

Self-Check

"What if we used a base other than 10 for digit grouping? How would that affect the time complexity?"

Practice

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

easy

A. Divide the array into halves and sort each half recursively

B. Compare each element with every other element and swap if needed

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

D. Use a pivot to partition the array and sort partitions separately

Solution

1. Step 1: Understand Radix Sort process

   Radix Sort sorts numbers by processing each digit from least significant to most significant.

2. Step 2: Compare with other sorting methods

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

3. Final Answer:

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

4. Quick Check:

   Radix Sort = Digit by digit sorting [OK]

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

Common Mistakes:

• Confusing Radix Sort with comparison-based sorts
• Thinking it sorts from most significant digit first
• Assuming it uses pivot or recursion

2. Which of the following is the correct way to initialize the counting array for Radix Sort in C?

easy

A. int count[10] = {0};

B. int count[10]; // no initialization

C. int count[10] = {1};

D. int count[10] = NULL;

Solution

1. Step 1: Understand counting array role

   The counting array stores frequency of digits 0-9 and must start at zero.

2. Step 2: Check initialization options

   Only int count[10] = {0}; sets all elements to zero correctly in C.

3. Final Answer:

   int count[10] = {0}; -> Option A

4. Quick Check:

   Counting array must start with zeros [OK]

Hint: Initialize counting array with zeros using {0} [OK]

Common Mistakes:

• Leaving counting array uninitialized
• Initializing with all ones
• Assigning NULL to an int array

3. What is the output of the following C code snippet using Radix Sort on array arr = {170, 45, 75, 90, 802, 24, 2, 66}?

void printArray(int arr[], int n) {
    for (int i = 0; i < n; i++)
        printf("%d ", arr[i]);
    printf("\n");
}

int main() {
    int arr[] = {170, 45, 75, 90, 802, 24, 2, 66};
    int n = sizeof(arr)/sizeof(arr[0]);
    radixSort(arr, n); // Assume radixSort is correctly implemented
    printArray(arr, n);
    return 0;
}

medium

A. 24 2 45 66 75 90 170 802

B. 170 45 75 90 802 24 2 66

C. 802 170 90 75 66 45 24 2

D. 2 24 45 66 75 90 170 802

Solution

1. Step 1: Understand Radix Sort output

   Radix Sort sorts the array fully in ascending order after processing all digits.

2. Step 2: Sort given array

   Sorted array is [2, 24, 45, 66, 75, 90, 170, 802].

3. Final Answer:

   2 24 45 66 75 90 170 802 -> Option D

4. Quick Check:

   Sorted array matches 2 24 45 66 75 90 170 802 [OK]

Hint: Radix Sort outputs fully sorted array ascending [OK]

Common Mistakes:

• Assuming output is unchanged
• Thinking output is reversed
• Confusing partial digit sorting with final output

4. Identify the error in this Radix Sort counting step code snippet:

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]] = arr[i];
    count[(arr[i] / exp) % 10]--;
}

medium

A. Counting array size should be 100, not 10

B. Indexing output array should be count[...] - 1

C. Loop for count prefix sum should start from 0

D. Division by exp should be replaced by modulo exp

Solution

1. Step 1: Analyze output array indexing

   Indexing output array must use count[...] - 1 because count stores cumulative counts.

2. Step 2: Check other parts

   Counting array size 10 is correct for digits 0-9; prefix sum loop starts at 1; division by exp is correct.

3. Final Answer:

   Indexing output array should be count[...] - 1 -> Option B

4. Quick Check:

   Output index = count - 1 [OK]

Hint: Subtract 1 from count for output indexing [OK]

Common Mistakes:

• Using count directly as index without subtracting 1
• Wrong counting array size
• Incorrect loop start for prefix sums
• Misusing division and modulo operators

5. Given an array of strings representing numbers with leading zeros, e.g., ["002", "010", "001", "100"], how can Radix Sort be adapted to sort them correctly as numbers?

hard

A. Pad all strings to equal length and sort digit by digit from right to left

B. Sort strings lexicographically without changes

C. Convert strings to integers first, then apply Radix Sort

D. Use Radix Sort only on the first character of each string

Solution

1. Step 1: Understand string number sorting

   Strings with leading zeros must be equal length to compare digits correctly.

2. Step 2: Adapt Radix Sort

   Pad strings to same length, then sort digit by digit from right to left to preserve numeric order.

3. Final Answer:

   Pad all strings to equal length and sort digit by digit from right to left -> Option A

4. Quick Check:

   Equal length padding + digit-wise sort = correct numeric order [OK]

Hint: Pad strings to same length before digit-wise sorting [OK]

Common Mistakes:

• Sorting strings lexicographically without padding
• Converting to int loses leading zeros importance
• Sorting only first character causes wrong order