Anonymous authentication in Firebase - Time & Space Complexity

Time Complexity: Anonymous authentication

O(n)

Understanding Time Complexity

We want to understand how the time it takes to sign in anonymously changes as more users try to sign in.

Specifically, how does the number of users affect the work done by Firebase during anonymous authentication?

Scenario Under Consideration

Analyze the time complexity of the following operation sequence.

// Sign in a user anonymously
firebase.auth().signInAnonymously()
  .then((userCredential) => {
    const user = userCredential.user;
    console.log('Signed in anonymously with uid:', user.uid);
  })
  .catch((error) => {
    console.error('Error during anonymous sign-in:', error);
  });
    

This code signs in a single user anonymously using Firebase Authentication.

Identify Repeating Operations

Identify the API calls, resource provisioning, data transfers that repeat.

• Primary operation: The call to signInAnonymously() which contacts Firebase servers to create a new anonymous user.
• How many times: Once per user sign-in attempt.

How Execution Grows With Input

Each user sign-in is a separate call to Firebase. The work done for one user does not depend on how many users signed in before.

Input Size (n)	Approx. Api Calls/Operations
10	10 calls to signInAnonymously()
100	100 calls to signInAnonymously()
1000	1000 calls to signInAnonymously()

Pattern observation: The number of calls grows directly with the number of users signing in.

Final Time Complexity

Time Complexity: O(n)

This means the total work grows linearly with the number of users signing in anonymously.

Common Mistake

[X] Wrong: "Signing in one user anonymously takes longer if many users have already signed in."

[OK] Correct: Each anonymous sign-in is independent and handled separately by Firebase, so previous sign-ins do not slow down new ones.

Interview Connect

Understanding how user authentication scales helps you design apps that handle many users smoothly and predict performance as your app grows.

Self-Check

"What if we changed from anonymous authentication to email/password sign-in? How would the time complexity change?"