Infinite Reduction of Divisors on Metric Graphs

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 ${\omega }^{\mathrm{\Theta }(\mathrm{d}\mathrm{e}\mathrm{g}(D))}$.