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On the Convergence Rate of the One-Hop Transfer Algorithm

Ruichao Jiang, Long Wen

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

On the Convergence Rate of the One-Hop Transfer Algorithm

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 rounds, where is the number of AMMs, a liquidity heterogeneity parameter and a tolerance parameter.

Paper Structure

This paper contains 5 sections, 5 theorems, 32 equations, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Main result
  4. Experiment
  5. Discussion

Key Result

lemma 1

$\varphi(x_D)<0$.

Theorems & Definitions (13)

  • definition 1: Liquidity heterogeneity
  • lemma 1
  • proof
  • remark 1
  • lemma 2: Sandwich lemma
  • proof
  • lemma 3: Improvement lower bound
  • proof
  • lemma 4: Gradient estimate
  • proof
  • ...and 3 more