Distances in and Layering of a DAG

TL;DR

This paper introduces efficient algorithms for computing the diameter and stretch of DAGs, and for detecting and layering balanced DAGs, significantly improving computational complexity over previous methods.

Contribution

The paper presents algorithms that compute DAG stretch and diameter in linear and near-linear time, and a method to detect and layer balanced DAGs efficiently.

Findings

DAG stretch can be computed in O(|V|+|E|) time.

DAG diameter can be computed in O(|V||E|) time.

Balanced DAGs can be detected and layered in O(|V|+|E|) time.

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.