Generalized topological sorting in linear time

Abstract

The generalized topological sorting problem takes as input a positive integer k and a directed, acyclic graph with some vertices labeled by positive integers, and the goal is to label the remaining vertices by positive integers in such a way that each edge leads from a lower-labeled vertex to a higher-labeled vertex, and such that the set of labels used is exactly {1,...,k}. Given a generalized topological sorting problem, we want to compute a solution, if one exists, and also to test the uniqueness of a given solution. The best previous algorithm for the generalized topological sorting problem computes a solution, if one exists, and tests its uniqueness in O(n log log n+m) time on input graphs with n vertices and m edges. We describe improved algorithms that solve both problems in linear time O(n+m).