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## 4.3.5.1 Topological Sorting

Topological sorting proceeds by finding a class C in SC such that no other class precedes that element according to the elements in R. The class C is placed first in the result. Remove C from SC, and remove all pairs of the form (C,D), D<ELEMENT-OF>SC, from R. Repeat the process, adding classes with no predecessors to the end of the result. Stop when no element can be found that has no predecessor.

If SC is not empty and the process has stopped, the set R is inconsistent. If every class in the finite set of classes is preceded by another, then R contains a loop. That is, there is a chain of classes C1,...,Cn such that Ci precedes Ci+1, 1<=i<n, and Cn precedes C1.

Sometimes there are several classes from SC with no predecessors. In this case select the one that has a direct subclass rightmost in the class precedence list computed so far. (If there is no such candidate class, R does not generate a partial ordering---the Rc, c<ELEMENT-OF>SC, are inconsistent.)

In more precise terms, let {N1,...,Nm}, m>=2, be the classes from SC with no predecessors. Let (C1...Cn), n>=1, be the class precedence list constructed so far. C1 is the most specific class, and Cn is the least specific. Let 1<=j<=n be the largest number such that there exists an i where 1<=i<=m and Ni is a direct superclass of Cj; Ni is placed next.

The effect of this rule for selecting from a set of classes with no predecessors is that the classes in a simple superclass chain are adjacent in the class precedence list and that classes in each relatively separated subgraph are adjacent in the class precedence list. For example, let T1 and T2 be subgraphs whose only element in common is the class J. Suppose that no superclass of J appears in either T1 or T2, and that J is in the superclass chain of every class in both T1 and T2. Let C1 be the bottom of T1; and let C2 be the bottom of T2. Suppose C is a class whose direct superclasses are C1 and C2 in that order, then the class precedence list for C starts with C and is followed by all classes in T1 except J. All the classes of T2 are next. The class J and its superclasses appear last.

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