Semi-Fungibility
Last updated
Last updated
There are lots of optimization protocols for full-range V2 AMM LPs, but very few for concentrated V3 LPs. The reason is because all V2 positions are the same range, so they can issue a fungible ERC20 token for protocols to build around. V3 positions on the other hand can have unique price ranges which is why those AMMs issue non-fungible ERC721 tokens for each position. This makes things like auto-compounders incredibly hard to build because without any special handling, they'd have to compound each NFT position one by one which is incredibly gas intensive.
And this is why our Semi-Fungibility is so pivotal. By breaking down concentrated V3 positions into smaller pieces, each individual piece becomes fungible although the overall position remains non-fungible. But these pieces are cleverly shared so pieces from multiple positions can be grouped together for a significant reduction in gas cost. In computer science terms, we reduce the operational overhead from to .
The exact way we do this get's rather complicated. It isn't as simple as just picking a few buckets to split your position into. It follows a tree structure where different levels have overlapping ranges and multiple levels are updated on every action. Overall, we call this structure the Liquidity Tree.
The underlying data structure for the liquidity tree is binary-indexed segment tree. We use the liquidity tree to both track our liquidity distribution and the fees accumulated for each range.
The segment tree spans the entire tick range of the AMM. The leaves of the tree cover a single tick and there is one node per tick on the lowest level. Each level above combines a pair of child nodes into their union'd range. For example nodes on the second level up will cover two ticks, the third will cover four ticks, and so forth. Each node represents a contiguous subrange of the overall price range.
When someone deposits a liquidity position into Hyperplex, we split the original range into the minimal set of nodes such that the union of those nodes' ranges will equal the original range. This set can be efficiently computed by walking up from the low end and high end of the desired range. Once the breakdown is found, all of the deposit's earnings and compounding is calculated on a node basis for each of the nodes in this breakdown.
The segment tree is further augmented with subtree information which helps us efficiently identify token balances, ambient vault balances, liquidity availability, fee attribution, and more.
For example, when we need to distribute fees over the range [A,B], instead of depositing those fees individually to every tick between A and B, we can simply find the breakdown set for [A,B] and deposit the fees into those nodes' subtree fee accumulators. Then whenever a node needs to determine their fee accumulation, they walk up the tree through their ancestors and sum the subtree fee accumulators. The depth of the tree is bounded by the logarithm of the total tree range and therefore remains gas efficient.
Overall, it is the combination of the segment tree, tree augmentation, efficient breakdown computation, efficient tree walking, and some approximation techniques that enable Hyperplex's advanced accounting capabilities.
