AMM Routing & Aggregation Algorithms

ROLE: You are a DeFi infrastructure developer who builds swap routing and aggregation algorithms. You understand the graph theory and optimization techniques behind finding the best price across dozens of liquidity sources, and you help traders get the best execution by splitting orders and routing through optimal paths.

CONTEXT: Modern DEX trading does not happen on a single AMM — aggregators like 1inch, Jupiter, and CowSwap route trades across multiple liquidity sources to minimize slippage. I want to understand how routing algorithms work, how to build or improve them, and how to evaluate aggregator performance for my trades.

TASK:
1. Routing Problem Definition — Explain the swap routing optimization problem. Cover formulating DEX aggregation as a graph optimization problem (tokens are nodes, pools are edges with price functions), the objective function (maximize output tokens for a given input), constraints (gas costs, pool reserves, minimum output amounts), why this is a hard problem (non-linear pricing functions, combinatorial explosion of paths), the difference between finding the best single path vs splitting across multiple paths, and how real-time reserve changes make this a dynamic optimization problem.
2. Single-Path Routing Algorithms — Detail algorithms for finding the best single swap route. Cover Dijkstra-style shortest path adaptation for AMM routing, handling different pool types (constant product, stable pools, concentrated liquidity) within the same graph, gas cost incorporation into the path cost function, multi-hop routing (trading through intermediate tokens for a better price), the challenge of accurate price quotes that account for slippage at each hop, and optimizing the algorithm for speed (sub-second response times are required).
3. Split-Route Optimization — Walk through the more complex problem of splitting orders across multiple routes. Cover why splitting often gives better prices than single-path routing (spreading slippage across pools), the mathematical formulation of the split optimization (maximize output subject to total input constraint), iterative approaches vs convex optimization for finding the optimal split, handling discrete pool types that do not fit standard optimization frameworks, balancing the benefit of more splits against increased gas costs, and the diminishing returns of adding more route segments.
4. Cross-Chain & Cross-Protocol Routing — Explain how aggregators route across different chains and protocols. Cover integrating bridge routes into the routing graph (chain-to-chain hops with bridge fees and delays), combining CEX and DEX liquidity in hybrid aggregators (like UniswapX or 1inch Fusion), intent-based routing where solvers compete to fill orders, batch auction systems that match trades against each other before hitting AMMs, and the latency challenges of cross-chain price discovery.
5. Aggregator Performance Evaluation — Describe how to evaluate and compare DEX aggregator performance. Cover defining price execution quality (actual output vs quoted output), comparing quotes across aggregators for the same trade, measuring positive and negative slippage distributions, evaluating gas efficiency (some aggregators use more gas for marginal price improvement), analyzing performance across different trade sizes (aggregators may vary at different scales), and meta-aggregator tools that compare multiple aggregators simultaneously.
6. Building a Custom Routing Engine — Guide through building a basic routing engine. Cover data collection (pool discovery, reserve monitoring, fee structures), building the liquidity graph data structure, implementing a basic routing algorithm in Python or TypeScript, quote caching and invalidation strategies for performance, testing against live aggregator quotes for accuracy validation, and scaling considerations for production deployment (concurrency, latency, data freshness requirements).