Linking multiple providers in Firebase - Time & Space Complexity

Time Complexity: Linking multiple providers

O(n)

Understanding Time Complexity

When linking multiple login providers to a single user account in Firebase, it's important to understand how the time taken grows as more providers are linked.

We want to know how the number of linking operations affects the total time and API calls.

Scenario Under Consideration

Analyze the time complexity of the following operation sequence.

const user = firebase.auth().currentUser;
const providers = [googleProvider, facebookProvider, twitterProvider];

for (const provider of providers) {
  await user.linkWithPopup(provider);
}
    

This code links multiple authentication providers one after another to the current user account.

Identify Repeating Operations

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

• Primary operation: Calling linkWithPopup to link a provider.
• How many times: Once for each provider in the list.

How Execution Grows With Input

Each additional provider adds one more linking operation, so the total work grows directly with the number of providers.

Input Size (n)	Approx. API Calls/Operations
3	3 linking calls
10	10 linking calls
100	100 linking calls

Pattern observation: The number of linking calls increases one-to-one with the number of providers.

Final Time Complexity

Time Complexity: O(n)

This means the time and API calls grow linearly as you add more providers to link.

Common Mistake

[X] Wrong: "Linking multiple providers happens all at once, so time stays the same no matter how many providers."

[OK] Correct: Each provider link requires a separate API call and user interaction, so the total time adds up with each provider.

Interview Connect

Understanding how operations scale with input size helps you design better user experiences and manage API usage efficiently.

Self-Check

What if we linked all providers in parallel instead of one after another? How would the time complexity change?