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).

Supported by the ESPRIT Basic Research Actions Program of the EC under contract No. 7141 (project ALCOM II). Part of the research was carried out while both authors were with the Universität des Saarlandes.