Bounding overcount to prove O(g^4) running time of the distance algorithm on generalized Bolza surfaces

Establish a rigorous upper bound on the overcount—the average number of times the same polygon is revisited—during execution of the depth-first search distance algorithm (Algorithm 1) on generalized Bolza surfaces S_g, and use this bound to prove the empirically observed O(g^4) running time.

Background

The paper proposes a depth-first search algorithm (Algorithm 1) to compute distances by exploring images of the fundamental polygon under the Fuchsian group. Empirical results for generalized Bolza surfaces S_g suggest that the algorithm runs in O(g^4) time. However, a rigorous proof of this complexity is lacking because the authors cannot bound the ‘overcount,’ i.e., the average number of times the same polygon is searched.

Proving an upper bound on the overcount would provide a formal complexity guarantee for the algorithm’s performance on S_g, aligning theory with the observed behavior.