How to understand the Uniswap V3 swap fee calculation

Question

i read the whitepaper and the code about uniswapv3. but i am still confused about uni-v3 swap fee calculation.

i can‘t understand 3 formula

qustion1 cant understand Tick struct field feeGrowthOutside(0/1)X128

   
    struct Info {

        uint256 feeGrowthOutside0X128;
        uint256 feeGrowthOutside1X128;

    }

which at whitepaper is

if swap cross it will be

question2 cant understand above tick i and below tick i formula

which at whitepaper is

i know at uniswapv3 code. when add/remove liquidity will calculate fee owned because liqudity is discrete。 but i really dont konw what 3 formula mean although i tried the fee distribution result is right

Answer 1

1. the thought of recording fee is 'trade space for time'
2. record the outside fee sum to save time further, that is, calculate tick's left/right fee sum and the global fee sum, that is useful for calculating the fee sum while a swap cross many tick

eg. calculate the fee sum when the price range is [tick_-1000,tick_1000]

just do subtraction for 2 time globalSum - tick_-1000.oustsideSum - tick_1000.outsideSum, instead of add every tick interval fee [-1000,-999).fee + [-999,-998).fee + ... + [999,1000).fee

3. left sum & right sum & global sum

a tick splits the whole tick space into two part, (-∞,Price_t) (Price_t,+∞)

for every tick, every swap's price belong to either the left or the right space, so there is a left sum and a right sum of historical fee correspond to that two part

left sum  = global sum - right sum
right sum = global sum - left sum

when calculate [-1000,1000], we need -1000.leftSum and also 1000.rightSum.

4. what is outside sum

we use only one variable to record that sum because we only use one of them for a particular interval（aka position). In fact, outsideSum is the leftSum or the rightSum depending on which one will not change in the future (after this cross, before the next time we cross it). And it is easy to calculate another(see 3).

Don't worry if you get confused about "cross", you'll understand what happened the moment you crossed the tick at part 5

5. When to change the outside value of a tick

when the price cross a tick from left to right, this tick.leftSum is definite, but rightSum still changing when the price is belong to the right space of that tick and new swaps occur, so we will renew tick.outsideSum == tick.leftSum. What's the value of tick.outsideSum now? it is the tick.rightSum because last time we crossed that tick from right to left (or never crossed it before thus it's zero, still holds) do the crossing renew see 4

   tick.outsideSum = left sum

on the other hand, renew the outsideSum to the latest rightSum when cross a tick from right to left.

6. an example

a swap makes the price's tick change from -998 to -1000, what's the fee sum on the interval [-1000,-998]?

ic : -998
fb : tick_-1000.leftSum 
     tick_-1000.outsideSum is leftSum 
     see formula 6.18.i
fa : tick_-998.rightSum 
     tick_-998.outsideSum is leftSum because last time we crossed -998 from left to right
     calculate the rightSum easily
     see formula 6.17.i

answer = global - fa - fb

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