On the Convergence Rate of the One-Hop Transfer Algorithm

TL;DR

This paper establishes the convergence rate of the transfer algorithm for on-chain swap routing, proving it terminates within a specific number of rounds based on network parameters.

Contribution

It provides the first explicit convergence rate bound for the transfer algorithm, improving understanding of its efficiency in on-chain swap routing.

Findings

The algorithm terminates in at most O(Nκ log(1/ε)) rounds.

The convergence rate depends on the number of AMMs, liquidity heterogeneity, and tolerance.

The result clarifies the efficiency of the transfer algorithm in practical settings.

Abstract

The transfer algorithm~\cite{jiang} solves the on-chain one-hop swap routing problem. In \cite{jiang}, the convergence is proved but the convergence rate is left open. We prove that the algorithm terminates in at most $\mathcal{O}(N\kappa\log\frac{1}{\varepsilon})$ rounds, where $N$ is the number of AMMs, $\kappa$ a liquidity heterogeneity parameter and $\varepsilon$ a tolerance parameter.