# MPI-I-93-119

## Generalized topological sorting in linear time

### Hagerup, Torben and Maas, Martin

#### May 1993, 10 pages.

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##### Status: available - back from printing

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 $O(n+m)$.

**URL to this document: **https://domino.mpi-inf.mpg.de/internet/reports.nsf/NumberView/1993-119

**BibTeX**
`@TECHREPORT{``HagerupMaas93``,`

` AUTHOR = {Hagerup, Torben and Maas, Martin},`

` TITLE = {Generalized topological sorting in linear time},`

` TYPE = {Research Report},`

` INSTITUTION = {Max-Planck-Institut f{\"u}r Informatik},`

` ADDRESS = {Im Stadtwald, D-66123 Saarbr{\"u}cken, Germany},`

` NUMBER = {MPI-I-93-119},`

` MONTH = {May},`

` YEAR = {1993},`

` ISSN = {0946-011X},`

`}`