Problem Description

The problem describes an authentication system that issues authentication tokens, each with an expiration time set in seconds. The expiration time (timeToLive) is a fixed duration from the moment the token is either generated or renewed. The currentTime is a timestamp indicating the current time in seconds. The system should perform three main functions:

1. Generate a new token with a unique tokenId and set its expiry time.
2. Renew an existing unexpired token with the given tokenId. If the token is expired or does not exist, no action is taken.
3. Count the number of unexpired tokens at a given currentTime.

The catch is that the expiration is processed before any other action if they both occur at the same time. The task is to implement the AuthenticationManager class that can handle these operations.

Intuition

The solution is to implement the AuthenticationManager class with the methods that handle tokens and their expiration times.

• When a new token is generated, we just add the tokenId and its expiration time (current time + timeToLive) to a dictionary. In this dictionary, the keys are the tokenIds, and the values are their respective expiration times.

• To renew a token, we check if the token exists and is not expired by comparing its stored expiration time with the currentTime. If it's not expired, we update its expiration time. If the token is expired or doesn't exist, we ignore the renew request.

• To count the unexpired tokens, we iterate through the dictionary's values, which are the expiration times, and count how many of these times are greater than the currentTime.

The choice of a dictionary makes these operations efficient. The main challenge of the problem is managing the expiration logic correctly and ensuring that we're not attempting to renew or count expired tokens.

Solution Approach

The solution utilizes a Python dictionary for storing tokens and their expiration times, which allows for quick lookups, inserts, and updates. Here's how the implementation works:

• Initialization (__init__ method): The AuthenticationManager is initialized with the timeToLive value, which determines the lifespan of each token. We also define a dictionary self.d that will map each tokenId to its expiration time.

• Generating tokens (generate method): When a new token is generated, we calculate its expiration time (currentTime + self.t) and store it in the dictionary self.d with tokenId as the key. This process is instant due to the efficient nature of dictionary operations in Python.

  def generate(self, tokenId: str, currentTime: int) -> None:
      self.d[tokenId] = currentTime + self.t

• Renewing tokens (renew method): To renew a token, we first check whether the token exists and is not expired by comparing the current time with the stored expiration time. If these conditions are met, we update the token's expiration time to the new current time plus timeToLive. If the token is not found or is expired, we do nothing.

  def renew(self, tokenId: str, currentTime: int) -> None:
      if tokenId in self.d and self.d[tokenId] > currentTime:
          self.d[tokenId] = currentTime + self.t

• Counting unexpired tokens (countUnexpiredTokens method): This function iterates over all expiration times in the dictionary and counts the number of times greater than the current time, indicating the tokens still valid.

  def countUnexpiredTokens(self, currentTime: int) -> int:
      return sum(exp > currentTime for exp in self.d.values())

The use of the sum function combined with a generator expression makes counting efficient. The expression exp > currentTime evaluates to True or False, which in Python correspond to 1 and 0. Thus, sum effectively counts the number of unexpired tokens.

With this implementation, the AuthenticationManager accomplishes the requested operations with a time complexity of O(1) for generating or renewing a token and O(n) for counting unexpired tokens, where n is the number of tokens being managed.

Example Walkthrough

Let's walk through an example to understand how the AuthenticationManager is implemented and functions with the given solution approach.

• Assume the timeToLive value for tokens is set to 5 seconds.
• The currentTime when a token is generated is 1.

Now, let's run through the methods:

1. Initialization:

   manager = AuthenticationManager(5)

   We have created an instance of AuthenticationManager named manager with timeToLive set to 5 seconds. This means each token will expire 5 seconds after it has been created or renewed.

2. Generating a token:

   manager.generate("token123", 1)

   After this call, the internal dictionary self.d will contain:

   {"token123": 6}

   This indicates that token123 will expire at time 6 (currentTime + timeToLive).

3. Renewing a token: Suppose the current time now is 3, and we wish to renew token123:

   manager.renew("token123", 3)

   The internal dictionary gets updated:

   {"token123": 8}

   This is because the current time is 3, and adding the timeToLive value of 5 seconds gives us the new expiration time at 8.

4. Attempting to renew an expired token: If we fast-forward the time to 7 and attempt to renew token123:

   manager.renew("token123", 7)

   Nothing changes since token123 was set to expire at time 8, so it's still valid and can be renewed:

   {"token123": 12}

5. Counting unexpired tokens: Let's now count the unexpired tokens at the current time, which is 7:

   count = manager.countUnexpiredTokens(7)

   The count will be 1 because token123 will now expire at 12, which is greater than the current time of 7.

This example demonstrates how the AuthenticationManager efficiently manages tokens, handles renewals, and counts unexpired tokens as required by the problem description. The implementation ensures rapid generation and renewal using dictionary operations and provides an O(n) approach to count unexpired tokens, where n is the total number of tokens.

Solution Implementation

Time and Space Complexity

Time Complexity:

• __init__: O(1) because we are just setting up variables.
• generate: O(1) as it involves a single operation of assigning a new expiration time for a token.
• renew: O(1) for accessing the dictionary and updating a token's expiration time conditionally. The conditional check and dictionary update happen in constant time.
• countUnexpiredTokens: O(n) where n is the number of tokens in the dictionary. This is due to the iteration over all values to sum the number of unexpired tokens.

Space Complexity:

• The space complexity of the entire class is O(n), where n is the number of different tokens stored. The dictionary self.d grows as new tokens are generated or renewed.

Learn more about how to find time and space complexity quickly using problem constraints.