Algorithms and Data Structures

N.Wirth. Algorithms and Data Structures. Oberon version 143
4.3.3 An Application: Partial Ordering (Topological Sorting)
An appropriate example of the use of a flexible, dynamic data structure is the process of topological
sorting. This is a sorting process of items over which a partial ordering is defined, i.e., where an ordering is
given over some pairs of items but not between all of them. The following are examples of partial orderings:
1. In a dictionary or glossary, words are defined in terms of other words. If a word v is defined in terms
of a word w, we denote this by v 〈 w. Topological sorting of the words in a dictionary means
arranging them in an order such that there will be no forward references.
2. A task (e.g., an engineering project) is broken up into subtasks. Completion of certain subtasks
must usually precede the execution of other subtasks. If a subtask v must precede a subtask w, we
write v 〈 w. Topological sorting means their arrangement in an order such that upon initiation of each
subtask all its prerequisite subtasks have been completed.
3. In a university curriculum, certain courses must be taken before others since they rely on the material
presented in their prerequisites. If a course v is a prerequisite for course w, we write v 〈 w.
Topological sorting means arranging the courses in such an order that no course lists a later course as
prerequisite.
4. In a program, some procedures may contain calls of other procedures. If a procedure v is called by a
procedure w, we write v 〈 w. Topological sorting implies the arrangement of procedure declarations in
such a way that there are no forward references.
In general, a partial ordering of a set S is a relation between the elements of S. It is denoted by the
symbol "〈", verbalized by precedes, and satisfies the following three properties (axioms) for any distinct
elements x, y, z of S:
1. if x 〈 y and y 〈 z, then x 〈 z (transitivity)
2. if x 〈 y, then not y 〈 x (asymmetry)
3. not z 〈 z (irreflexivity)
For evident reasons, we will assume that the sets S to be topologically sorted by an algorithm are finite.
Hence, a partial ordering can be illustrated by drawing a diagram or graph in which the vertices denote the
elements of S and the directed edges represent ordering relationships. An example is shown in Fig. 4.13.
2
1
10
6
4
8 9
3
5
7
Fig. 4.13. Partially ordered set
The problem of topological sorting is to embed the partial order in a linear order. Graphically, this implies
the arrangement of the vertices of the graph in a row, such that all arrows point to the right, as shown in
Fig. 4.14. Properties (1) and (2) of partial orderings ensure that the graph contains no loops. This is exactly
the prerequisite condition under which such an embedding in a linear order is possible.