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What is Hamming Distance? Applications and Operations

Before going through the blog, let’s take a quick look at the prerequisites. You should have a foundational knowledge of binary representation and basic mathematical operations. It will be beneficial if you have a firm grasp on the concept of strings and sets.

While Hamming distance coding is possible in any programming language, we will utilize C++ 23 in this context. This decision stems from the 2022 survey conducted by HackerEarth, where C++ was identified as the foremost general programming language. Out of the respondents, 53.3% favored C++, followed by Java (26.7%) and Python (10%).

Table of Contents

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What is Hamming Distance?

Alt text – What is Hamming Distance?

By definition, Hamming distance is a metric in information theory that is used for comparing two binary data strings of equal length at corresponding positions. The crux of Hamming distance is to find the minimum number of substitutions needed to change one string into another.

Note: In the blog, we will consider two strings, 'strOne' and 'strTwo' of equal length, 'len', and represented by d(strOne, strTwo)'.

Hamming distance follows multiple conditions. A few are mentioned below:

Here are a few of the important terminologies that you might encounter during your learning journey:

Brief History of Hamming Distance

The American mathematician Richard Hamming first coined the concept of Hamming distance. While working at Bell Labs, he introduced Hamming distance in his “Hamming codes, Error detection, and error correction codes” paper in 1950. 

He was working on developing a method for the identification and correction of bits in data transmission. He concluded that by counting the number of bits that differed between two binary sequences, he could determine how many errors had occurred.

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How to Calculate Hamming Distance?

Calculating the Hamming Distance involves measuring the dissimilarity between two strings of equal length. Here are some processes:

Mathematically, Hamming distance is expressed as:

Hamming Distance = Σ (strOne[i] ≠ strTwo[i])

Hamming Distance denotes the disparity between the strings.
Σ symbolizes the summation of all positions.
strOne[i] represents the element at index ‘i’ in the first string.
strTwo[i] represents the element at index ‘i’ in the second string.
≠ signifies “not equal.”

Hamming Distance Formula

The Hamming distance formula is as follows:

hamming_distance(strOne, strTwo) = number of bits in strOne that are different from bits in strTwo


For example, the Hamming distance between the strings 101010 and 111010 is 1. There is one bit position where the strings are different: the first bit.

Hamming Distance Flow Chart

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Application of Hamming Distance

Alt text – Application of Hamming Distance

Below mentioned are the various fields in which hamming distance is applied: 

It is also seen that Hamming distance is also used for digital forensics to check for alterations and barcoding identifiers for enhanced readability, even when the barcode is distorted. 

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Hamming Distance Vs. Euclidean Distance

Alt text – Hamming Distance Vs. Euclidean Distance

It would be best to have a glimpse of Euclidean distance to have better clarity about Hamming distance. Below, we have mentioned the major differences between Hamming distance and Euclidean distance.

Parameters Hamming Distance  Euclidean Distance
Definition Measures dissimilarity in binary data Measures geometric distance in vectors
Data Type Used for binary or categorical data Applicable to numerical data
Calculation Counts positions with differing values Calculates vector magnitude
Dimensionality Works well with equal-length strings Suits multi-dimensional data
Scale Sensitivity Insensitive to the scale of attributes Sensitive to attribute scales
Zero Values Considers zero values as meaningful Treats zero values as origin
Applicability Suitable for text analysis, error checks Applied in spatial and numeric contexts
Interpretability Provides insights into symbolic data Offers geometric interpretation
Distance Range Range is [0, string length] Range varies based on data
Weighted Features Weighting is not applicable Weighted Euclidean Distance can be used

Let’s Code: Hamming Distance

Here is the implementation of Hamming distance:

Pseudo Code:


Hamming Code C++:

int hammingDistance(const std::string& strOne, const std::string& strTwo) {
    if (strOne.length() != strTwo.length()) {
        throw std::invalid_argument("Strings must have equal lengths");
    int distance = 0;
    for (size_t i = 0; i < strOne.length(); ++i) {
        if (strOne[i] != strTwo[i]) {
    return distance;
int main() {
    std::string strOne, strTwo;
    std::cout << "Enter the first string: ";
    std::cin >> strOne;
    std::cout << "Enter the second string: ";
    std::cin >> strTwo;
    try {
        int distance = hammingDistance(strOne, strTwo);
        std::cout << "Hamming Distance: " << distance << std::endl;
    } catch (const std::invalid_argument& e) {
        std::cerr << "Error: " << e.what() << std::endl;
    return 0;


Enter the first string: 101010
Enter the second string: 111000
Hamming Distance: 3

Hamming Code Python:

def hamming_distance(strOne, strTwo):
    if len(strOne) != len(strTwo):
        raise ValueError("Strings must have equal length for Hamming Distance calculation")
    distance = 0
    for char1, char2 in zip(strOne, strTwo):
        if char1 != char2:
            distance += 1
    return distance
# Test Cases
    strOne = "101010"
    strTwo = "100110"
    distance = hamming_distance(strOne, strTwo)
    print(f"Hamming Distance between '{strOne}' and '{strTwo}' is: {distance}")
except ValueError as ve:
    strOne = "101010"
    strTwo = "10011"
    distance = hamming_distance(strOne, strTwo)
    print(f"Hamming Distance between '{strOne}' and '{strTwo}' is: {distance}"
except ValueError as ve:


Hamming Distance between ‘101010’ and ‘100110’ is: 2
Strings must have equal length for Hamming Distance calculation

Hamming Code Java:

import java.util.Scanner;
public class IntellipaatHammingDistance {
    public static void main(String[] args) {
        Scanner scanner = new Scanner(;
        System.out.print("Enter the first string: ");
        String string1 = scanner.nextLine();
        System.out.print("Enter the second string: ");
        String string2 = scanner.nextLine();
        if (string1.length() != string2.length()) {
            System.out.println("Error: The two strings must have equal length.");
        } else {
            int hammingDistance = calculateHammingDistance(string1, string2);
            System.out.println("Hamming Distance: " + hammingDistance);
    public static int calculateHammingDistance(String str1, String str2) {
        int distance = 0;
        for (int i = 0; i < str1.length(); i++) {
            if (str1.charAt(i) != str2.charAt(i)) {
        return distance;


Enter the first string: 101010
Enter the second string: 111110
Hamming Distance: 1

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Operations on Hamming Distance

Here are a few of the operations that you can perform using Hamming distance:

Hamming Distance Between Two Integers

Problem Statement: Write a program to find the hamming distance between two integers. Considering the inputs as intOne = 10 (1010), intTwo = 14 (1110), and output: 3. 


using namespace std;
int hammingDistance(int intOne, int intTwo)
int d = intOne ^ intTwo;
int setBits = 0;
while (d > 0) {
setBits += d & 1;
d >>= 1;
return setBits;
// Driver code
int main()
int intOne = 10, intTwo = 14;
cout << hammingDistance(10, 14) << "/n";
return 0;

Time Complexity: O(log2x)

Auxiliary Space: O(1)

We can also solve this problem using two more approaches:

Reduce Hamming Distance by Swapping Two Characters

Problem Statement: Write a program to find the positions of two characters to be swapped in the strings strOne and strTwo using Hamming distance in such a way that the distance between the two strings is the smallest. Taking inputs as strOne = “Intellipaat”, and strTwo = “Inllitepata”, and output: 6 3.


using namespace std;#define MAX 26
void Swap(string one, string two, int n)
int dp[MAX][MAX];
memset(dp, -1, sizeof dp);
int t = 0;
for (int i = 0; i < n; i++)
if (one[i] != two[i])
for (int i = 0; i < n; i++) {
int a = one[i] - 'a';
int b = two[i] - 'a';
if (a == b)
if (dp[a][b] != -1) {
cout << i + 1 << " " << dp[a][b] + 1 << endl;
dp[b][a] = i;
int A[MAX], B[MAX];
memset(A, -1, sizeof A);
memset(B, -1, sizeof B);
for (int i = 0; i < n; i++) {
int a = one[i] - 'a';
int b = two[i] - 'a';
if (a == b)
if (A[b] != -1) {
cout << i + 1 << " " << A[b] + 1 << endl;
if (B[a] != -1) {
cout << i + 1 << " " << B[a] + 1 << endl;
A[a] = i;
B[b] = i;
cout << -1 << endl;
// Driver code
int main()
string strOne = "Intellipaat";
string strTwo = "Inllitepata";
int n = strOne.length();
if (strOne == "" || strTwo == "")
cout << "Required string is empty.";
Swap(strOne, strTwo, n);
return 0;

Time Complexity: O(n)

Auxiliary Space: O(MAX * MAX)

Finding Rotation with Maximum Hamming Distance

Problem Statement: Write a program to find a rotation with the maximum Hamming distance. Input an array of size ‘N’ and another array that contains the rotation of the first array and print the maximum distance between them. Take N = 3, arr = {1,4,1}, and output: 2.


using namespace std;
int maxHamming(int arrOne[], int N)
int arrTwo[2 * N + 1];
for (int i = 0; i < N; i++) {
arrTwo[i] = arrOne[i];
arrTwo[N + i] = arrOne[i];
int maxHam = 0;
for (int i = 1; i < N; i++) {
int Ham = 0;
for (int j = i, k = 0; j < (i + N); j++, k++)
if (arrTwo[j] != arrOne[k])
if (Ham == N)
return N;
maxHam = max(maxHam, Ham);
return maxHam;
// Driver program
int main()
int arrOne[] = { 1, 4, 1 };
int N = sizeof(arrOne) / sizeof(arrOne[0]);
cout << maxHamming(arrOne, N);
return 0;

Time Complexity: O(n2)

Auxiliary Space: O(n)

We have used a naive approach to solve the problem. But this problem can also be solved using two other approaches:

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In the blog, ‘What is Hamming Distance? Operations and Applications’, we briefly discussed Hamming distance, its applications, and various operations you can perform using it. Hamming distance is helpful for error correction, computer programming, and computer networks. 

To expand your knowledge, we recommend exploring related topics like error-correcting codes and information theory. To solidify your understanding, practicing more problems involving Hamming Distance will help you master its applications. Computer science is like an ocean, and nobody has seen it all. Hence, consistent efforts are required to explore various untouched concepts. 

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