Token bucket algorithm in Rest API - Time & Space Complexity

Time Complexity: Token bucket algorithm

O(1)

Understanding Time Complexity

We want to understand how the time it takes to check and update tokens grows as requests come in.

How does the algorithm handle more requests over time?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

def allow_request(bucket, tokens_needed=1):
    now = current_time()
    elapsed = now - bucket['last_checked']
    bucket['tokens'] = min(bucket['capacity'], bucket['tokens'] + elapsed * bucket['rate'])
    bucket['last_checked'] = now
    if bucket['tokens'] >= tokens_needed:
        bucket['tokens'] -= tokens_needed
        return True
    return False

This code checks if enough tokens are available to allow a request and updates the token count based on elapsed time.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Simple arithmetic and comparisons done once per request.
• How many times: Once each time a request is checked.

How Execution Grows With Input

The work done for each request stays about the same no matter how many requests come in.

Input Size (n)	Approx. Operations
10	10 simple checks and updates
100	100 simple checks and updates
1000	1000 simple checks and updates

Pattern observation: The time grows linearly with the number of requests, but each request takes a fixed small amount of work.

Final Time Complexity

Time Complexity: O(1)

This means each request is handled in constant time, no matter how many requests happen.

Common Mistake

[X] Wrong: "The token bucket algorithm must loop over all tokens or requests, so it takes longer as requests increase."

[OK] Correct: The algorithm only updates counts using simple math each time a request comes in, without looping over past requests or tokens.

Interview Connect

Understanding this helps you explain how rate limiting works efficiently in real systems, showing you can reason about performance in practical code.

Self-Check

"What if we changed the algorithm to store each token as an individual object and check them one by one? How would the time complexity change?"

Practice

1.

What is the main purpose of the token bucket algorithm in REST APIs?

easy

A. To encrypt API responses

B. To store user data securely

C. To control the rate of incoming requests by using tokens

D. To manage database connections

Solution

1. Step 1: Understand the token bucket algorithm concept

   The token bucket algorithm limits how many requests can be processed by controlling tokens that refill over time.

2. Step 2: Identify the purpose in REST APIs

   It helps prevent too many requests at once, protecting the server from overload.

3. Final Answer:

   To control the rate of incoming requests by using tokens -> Option C

4. Quick Check:

   Token bucket controls request rate = C [OK]

Hint: Token bucket limits request rate using tokens [OK]

Common Mistakes:

• Confusing token bucket with data storage
• Thinking it encrypts data
• Assuming it manages database connections

2.

Which of the following is the correct way to represent a token bucket refill rate in pseudocode?

1. refill_rate = tokens_per_second
2. refill_rate = seconds_per_token
3. refill_rate = max_tokens * time
4. refill_rate = tokens / max_tokens

easy

A. refill_rate = seconds_per_token

B. refill_rate = tokens_per_second

C. refill_rate = max_tokens * time

D. refill_rate = tokens / max_tokens

Solution

1. Step 1: Understand refill rate meaning

   The refill rate is how many tokens are added per second to the bucket.

2. Step 2: Match with options

   refill_rate = tokens_per_second correctly shows tokens added per second, which is the refill rate.

3. Final Answer:

   refill_rate = tokens_per_second -> Option B

4. Quick Check:

   Refill rate = tokens per second [OK]

Hint: Refill rate means tokens added each second [OK]

Common Mistakes:

• Confusing refill rate with time per token
• Multiplying max tokens by time incorrectly
• Using ratios instead of rates

3.

Given a token bucket with max_tokens = 5, refill_rate = 1 token/second, and an empty bucket at time 0, what is the number of tokens available at time 3 seconds?

medium

A. 3 tokens

B. 5 tokens

C. 0 tokens

D. 1 token

Solution

1. Step 1: Calculate tokens refilled after 3 seconds

   Since refill rate is 1 token per second, after 3 seconds, 3 tokens are added.

2. Step 2: Check max tokens limit

   The bucket max is 5 tokens, so 3 tokens fit without exceeding the max.

3. Final Answer:

   3 tokens -> Option A

4. Quick Check:

   3 seconds * 1 token/sec = 3 tokens [OK]

Hint: Multiply seconds by refill rate, cap at max tokens [OK]

Common Mistakes:

• Assuming bucket fills instantly to max
• Ignoring max token limit
• Using refill rate incorrectly

4.

Consider this pseudocode snippet for token bucket check:
if tokens <= 0:
  reject_request()
else:
  tokens -= 1
  allow_request()
What is the bug in this logic?

medium

A. It should check if tokens > 0 before allowing

B. It should increase tokens instead of decreasing

C. It should reject when tokens > 0

D. It should check if tokens < 1, not <= 0

Solution

1. Step 1: Recall proper token bucket logic

   To consume 1 token, check if tokens >= 1 before decrementing (equivalent to reject if tokens < 1).

2. Step 2: Identify the bug

   The code rejects only if tokens <= 0. For fractional tokens (common in real implementations), if 0 < tokens < 1, it allows the request, decrementing to negative, which is incorrect.

3. Final Answer:

   It should check if tokens < 1, not <= 0 -> Option D

4. Quick Check:

   Reject if tokens < 1 [OK]

Hint: Allow only if tokens >= 1 [OK]

Common Mistakes:

• Using <= 0 instead of < 1 causes off-by-one errors
• Increasing tokens on request instead of decreasing
• Rejecting requests when tokens are available

5.

You want to implement a token bucket that allows bursts of up to 10 requests and refills tokens at 2 tokens per second. If a client sends 15 requests instantly after being idle for 3 seconds, how many requests will be allowed immediately?

hard

A. 6 requests

B. 5 requests

C. 15 requests

D. 10 requests

Solution

1. Step 1: Calculate tokens available after 3 seconds idle

   Refill rate is 2 tokens/second, so after 3 seconds: 2 * 3 = 6 tokens. Max tokens allowed is 10, so bucket fills to 6 tokens.

2. Step 2: Consider burst capacity

   Since the bucket max is 10, if it was full before idle, it would have 10 tokens. But starting empty, after 3 seconds it has 6 tokens.

3. Step 3: Determine allowed requests

   The client sends 15 requests instantly, but only 6 tokens are available, so only 6 requests allowed immediately.

4. Final Answer:

   6 requests -> Option A

5. Quick Check:

   3 sec * 2 tokens/sec = 6 tokens available [OK]

Hint: Tokens = min(max_tokens, refill_rate * idle_time) [OK]

Common Mistakes:

• Assuming bucket always full at max tokens
• Allowing more requests than tokens available
• Ignoring refill rate and idle time