Email/password signup in Firebase - Time & Space Complexity

Time Complexity: Email/password signup

O(n)

Understanding Time Complexity

When a user signs up with email and password, the system must create a new account securely.

We want to know how the time to complete signup changes as more users sign up.

Scenario Under Consideration

Analyze the time complexity of the signup operation using Firebase Authentication.

const auth = getAuth();
createUserWithEmailAndPassword(auth, email, password)
  .then((userCredential) => {
    const user = userCredential.user;
    // User account created
  })
  .catch((error) => {
    // Handle signup errors
  });
    

This code creates a new user account with email and password in Firebase Authentication.

Identify Repeating Operations

In this signup process, the main repeated operation is the call to Firebase's createUserWithEmailAndPassword API.

• Primary operation: createUserWithEmailAndPassword API call to Firebase Authentication service.
• How many times: Once per user signup attempt.

How Execution Grows With Input

Each signup is a separate call that does not depend on the number of existing users.

Input Size (n)	Approx. Api Calls/Operations
10	10 calls (one per signup)
100	100 calls
1000	1000 calls

Pattern observation: The number of API calls grows directly with the number of signups, one call per user.

Final Time Complexity

Time Complexity: O(n)

This means the total time grows linearly with the number of users signing up.

Common Mistake

[X] Wrong: "Signup time gets slower as more users exist in the system."

[OK] Correct: Each signup is handled independently by Firebase, so existing users do not slow down new signups.

Interview Connect

Understanding how signup operations scale helps you design systems that handle many users smoothly.

Self-Check

"What if we added email verification steps after signup? How would the time complexity change?"