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An efficient search algorithm for inverting the sweep map on rational Dyck paths

Given a coprime pair $(m,n)$ of positive integers, rational $(m,n)$-Dyck paths are lattice paths in the $m\times n$ rectangle that never go below the diagonal. The sweep map of a rational $(m,n)$-Dyck paths $D$ is the rational Dyck path $Φ(D)$ obtained by sorting the steps of $D$ according to the ranks of their starting points, where the rank of $(a,b)$ is $bm-an$. It is conjectured to be a bijection, but to this date, $Φ$ is only known to be bijective for the Fuss case ($m=kn\pm 1$). In this paper we give an efficient search algorithm for inverting the $Φ$ map. Roughly speaking, given $σ\in \cal D_{m,n}$, by searching through a $d$-array tree of certain depth, we can output all $D$ such that $Φ(D)=σ$, where $d$ is the remainder of $m$ when divided by $n$. In particular, we show that $Φ$ is invertible for the Fuss case by giving a simple recursive construction for $Φ^{-1} (σ)$.