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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 $\{1,\ldots,k\}$. 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 $O(n+m)$.