Automated Market Makers (AMM) in Blockchain / Solidity - Time & Space Complexity

Time Complexity: Automated Market Makers (AMM)

O(1)

Understanding Time Complexity

When using Automated Market Makers (AMM), it is important to understand how the time to process trades grows as more users interact with the system.

We want to know how the number of operations changes when the number of trades or liquidity providers increases.

Scenario Under Consideration

Analyze the time complexity of the following simplified AMM swap function.

function swap(inputAmount, inputReserve, outputReserve) {
  const inputAmountWithFee = inputAmount * 997;
  const numerator = inputAmountWithFee * outputReserve;
  const denominator = inputReserve * 1000 + inputAmountWithFee;
  const outputAmount = numerator / denominator;
  return outputAmount;
}
    

This function calculates how many output tokens a user receives when swapping input tokens, based on current reserves.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: The function performs a fixed number of arithmetic calculations.
• How many times: These calculations happen once per swap call, no loops or recursion inside.

How Execution Grows With Input

The number of operations stays the same no matter how large the input amount or reserves are.

Input Size (n)	Approx. Operations
10	5 arithmetic operations
100	5 arithmetic operations
1000	5 arithmetic operations

Pattern observation: The work done does not increase with input size; it stays constant.

Final Time Complexity

Time Complexity: O(1)

This means the time to calculate a swap does not grow as the input or reserves grow; it takes the same amount of time every call.

Common Mistake

[X] Wrong: "The swap function takes longer if the input amount is bigger because it has to do more work."

[OK] Correct: The function only does a fixed set of calculations regardless of input size, so bigger numbers don't mean more steps.

Interview Connect

Understanding that AMM swap calculations run in constant time helps you explain how decentralized exchanges handle many trades efficiently.

Self-Check

"What if the swap function had to loop through a list of liquidity pools to find the best rate? How would the time complexity change?"