Infinite Reduction of Divisors on Metric Graphs

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 $\omega^{\Theta({\rm deg}(D))}$.