Data Structures and Algorithms with Object-Oriented Design Patterns in C#
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Topological Sort

 

A topological sort is an ordering of the vertices of a directed acyclic graph given by the following definition:

Definition (Topological Sort) Consider a directed acyclic graph tex2html_wrap_inline70252. A topological sort   of the vertices of G is a sequence tex2html_wrap_inline70954 in which each element of tex2html_wrap_inline70254 appears exactly once. For every pair of distinct vertices tex2html_wrap_inline70418 and tex2html_wrap_inline70960 in the sequence S, if tex2html_wrap_inline70566 is an edge in G, i.e., tex2html_wrap_inline70968, then i<j.

Informally, a topological sort is a list of the vertices of a DAG in which all the successors of any given vertex appear in the sequence after that vertex. Consider the directed acyclic graph tex2html_wrap_inline70972 shown in Figure gif. The sequence tex2html_wrap_inline70974 is a topological sort of the vertices of tex2html_wrap_inline70972. To see that this is so, consider the set of vertices:

displaymath70948

The vertices in each edge are in alphabetical order, and so is the sequence S.

   figure50106
Figure: A directed acyclic graph.

It should also be evident from Figure gif that a topological sort is not unique. For example, the following are also valid topological sorts of the graph tex2html_wrap_inline70972:

eqnarray50335

One way to find a topological sort is to consider the in-degrees  of the vertices. (The number above a vertex in Figure gif is the in-degree of that vertex). Clearly the first vertex in a topological sort must have in-degree zero and every DAG must contain at least one vertex with in-degree zero. A simple algorithm to create the sort goes like this:

Repeat the following steps until the graph is empty:

  1. Select a vertex that has in-degree zero.
  2. Add the vertex to the sort.
  3. Delete the vertex and all the edges emanating from it from the graph.



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Bruno Copyright © 2001 by Bruno R. Preiss, P.Eng. All rights reserved.