Distances in and Layering of a DAG

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 ∈ 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 δ(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) ∈ V denoted by ∆(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|).