On the Convergence Rate of the One-Hop Transfer Algorithm

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 $\mathcal{O}(N\kappa \log \frac{1}{\epsilon })$ rounds, where $N$ is the number of AMMs, $\kappa$ a liquidity heterogeneity parameter and $\epsilon$ a tolerance parameter.