Question on uniswap v3 Fee growth

[yathindrak]

From https://updraft.cyfrin.io/courses/uniswap-v3/fee-algorithm/fee-growth-below

Why does the algorithm subtract from fg instead of simply adding the fee growth values in this area for example?
fg₀ + fg₁.., fg6 is past isnt it?
i mean the timeline would be fg₀ + fg₁ + fg₂ + ... + fg(current)?

btw the fee growth chart for ref.

Additionally, great if someone could explain this whole table values a bit more (both fb ad f0,i). been struggling to understand the logic under the hood.

[EngrPips]

I honestly don't think I can do a better job explaining this than @t4sk but just to try my best to make it clear, The table reflects the fee growth below the tick and the below the tick is where we have those red boxes, The height of those red boxes intrinsically contains the height of the above tick which is the other side of the tick where we have no boxes. We are trying to calculate the fee growth below the tick and not the total fee growth, so whenever we do our calculation, the formula subtracts the fee growth above from the fee growth below(total fee to be precise to get the actual fee growth below).

To cut the long story short, we only have a single line which we use to represent both fee growth below and fee growth above which mean realistically we only have fee growth so to calculate the fee growth below we consider the height of the boxes below the tick and subtract the height of the above tick it contains to ensure we are calculating the accurate fee growth below. If we want to calculate fee growth above as well, we will need to subtract the fee growth below from it to ensure that we are getting the accurate fee growth above

I hope I have not confused you more. Please ask questions if you have any.

[t4sk]

Why does the algorithm subtract from fg instead of simply adding the fee growth values in this area for example?
fg₀ + fg₁.., fg6 is past isnt it?

fg is always increasing (It's the cumulative of fees).
To calculate the fee growth below for the second row, second column (highlighted in pink), you would need to calculate the sum of the heights of the red boxes up to the current height of fg. While fg <= fg2, the sum of the heights are

fg - (fg1 - fg0) = fg - fg1 + fg0

fg1 - fg0 is the fee growth above tick i that accumulated between time t0 and t1.

For more info, check here
https://github.com/Cyfrin/advanced-defi-2024/blob/main/excalidraw/amm/uniswap-v3/uniswap-v3-fee-growth-below.png

[yathindrak]

Fully understood now. Thank you so much @EngrPips and @t4sk for the support. I just completed the course. 🔥

[EngrPips]

I am glad you figured it out. Keep crushing it!