Distances in and Layering of a DAG (1711.03256v2)

Abstract: The diameter of an undirected unweighted graph $G=(V,E)$ is the maximum value of the distance from any vertex $u$ to another vertex $v$ for $u,v \in V$ where distance i.e. $d(u,v)$ is the length of the shortest path from $u$ to $v$ in $G$. DAG, is a directed graph without a cycle. We denote the diameter of an unweighted DAG $G=(V,E)$ by $\delta (G)$. The stretch of a DAG $G$ is the length of longest path from $u$ to $v$ in $G$, for all choices of $(u, v) \in V$ denoted by $\Delta (G)$. The diameter of an undirected graph can be computed in $O(|V|(|V|+|E|))$ time by executing breadth first search $|V|$ times. We show that stretch and diameter of a DAG can be computed in $O(|V|+|E|)$ time and $O(|V||E|)$ time respectively. A DAG is balanced if and only if a consistent assignment of level numbers to all vertices is possible. Layering refers to such an assignment. A balanced DAG is defined. An efficient algorithm that either detects whether a given DAG is unbalanced or layers it otherwise is designed with a running time of $O(|V|+|E|)$. \ Key words: Diameter, directed acyclic graph, longest directed path, graph algorithms, complexity.