Infinite Reduction of Divisors on Metric Graphs

TL;DR

This paper explores the behavior of the greedy reduction algorithm for divisors on metric graphs, showing it can be infinite, but still well-defined, with its complexity characterized by ordinal numbers.

Contribution

It models the greedy reduction process as a transfinite algorithm and establishes bounds on its worst-case running time using ordinal analysis.

Findings

The greedy algorithm may not terminate on metric graphs.

Infinite reductions have well-defined limits.

Worst-case running time is bounded by ω^{Θ(deg(D))}.

Abstract

We demonstrate that the greedy algorithm for reduction of divisors on metric graphs need not terminate by modeling the Euclidean algorithm in this context. We observe that any infinite reduction has a well defined limit allowing us to treat the greedy reduction algorithm as a transfinite algorithm and to analyze its running time via ordinal numbers. We provide lower and upper bounds which establish a worst case running time of $\omega^{\Theta({\rm deg}(D))}$.