Loop unrolling of UCA models: distance labeling

A proper circular-arc (PCA) model is a pair $M = (C, A)$ where $C$ is a circle and $A$ is a family of inclusion-free arcs on $C$ whose extremes are pairwise different. The model $M$ represents a digraph $D$ that has one vertex $v(a)$ for each $a \in A$ and one edge $v(a) \to v(b)$ for each pair of arcs $a,b \in A(M)$ such that the beginning point of $b$ belongs to $a$. For $k \geq 0$, the $k$-th power $D^k$ of $D$ has the same vertices as $D$ and $v(a) \to v(b)$ is an edge of $D^k$ when $a\neq b$ and the distance from $v(a)$ to $v(b)$ in $D$ is at most $k$. A unit circular-arc (UCA) model is a PCA model $U = (C,A)$ in which all the arcs have the same length $\ell+1$. If $\ell$, the length $c$ of $C$, and the extremes of the arcs of $A$ are integer, then $U$ is a $(c,\ell)$-CA model. For $i \geq 0$, the model $i \times U$ of $U$ is obtained by replacing each arc $(s,s+\ell+1)$ with the arc $(s,s+i\ell+1)$. If $U$ represents a digraph $D$, then $U$ is $k$-multiplicative when $i \times U$ represents $D^i$ for every $0 \leq i \leq k$. In this article we design a linear time algorithm to decide if a PCA model $M$ is equivalent to a $k$-multiplicative UCA model when $k$ is given as input. The algorithm either outputs a $k$-multiplicative UCA model $U$ equivalent to $M$ or a negative certificate that can be authenticated in linear time. Our main technical tool is a new characterization of those PCA models that are equivalent to $k$-multiplicative UCA models. For $k=1$, this characterization yields a new algorithm for the classical representation problem that is simpler than the previously known algorithms.