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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.
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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 $O(n+m)$.
