Data Structures and Algorithms with Object-Oriented Design Patterns in Python
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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_inline70549. A topological sort   of the vertices of G is a sequence tex2html_wrap_inline71251 in which each element of tex2html_wrap_inline70551 appears exactly once. For every pair of distinct vertices tex2html_wrap_inline70715 and tex2html_wrap_inline71257 in the sequence S, if tex2html_wrap_inline70863 is an edge in G, i.e., tex2html_wrap_inline71265, 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_inline71269 shown in Figure gif. The sequence tex2html_wrap_inline71271 is a topological sort of the vertices of tex2html_wrap_inline71269. To see that this is so, consider the set of vertices:

displaymath71245

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

   figure50454
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_inline71269:

eqnarray50683

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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