Dependency graph
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In mathematics and computer science, a dependency graph is a directed graph representing dependencies of several objects towards each other. It is possible to derive an evaluation order or the absence of an evaluation order that respects the given dependencies from the dependency graph.
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[edit] Definition
Given a set of objects S and a transitive relation R = 
with 
modeling a dependency "a needs b evaluated first", the dependency graph is a graph G = (S, T) with 
and R being the transitive closure of T.
For example, assume a simple calculator. This calculator supports assignment of constant values to variables and assigning the sum of exactly 2 variables to a third variable. Given several equations like "A = B+C; B = 5+D; C=4; D=2;", then S = A,B,C,D and R = (A,B),(A,C),(B,D). You can derive this relation directly - A depends on B and C, because you can add two variables only if and only if you know the values of both variables, thus, B and C must be calculated before A can be calculated. However, Ds value is known immediately, because it is a number literal. (TODO: add image for first graph.)
[edit] Recognizing Impossible Evaluations
In a dependency graph, impossible calculations form cycles. Thus, such situations are called circular dependencies. There exist efficient algorithms for detecting those cycles, like Tortoise and Hare. For further information, refer to cycle detection. If one strives for efficiency, this cycle detection can be done during the derivation of an evaluation order. However, this increases the difficulty of outputting the objects in the cycle, for example for special treatment or usage in an error message.
Assume the simple calculator from before. The equation system "A=B; B=D+C; C=D+A; D=12;" contains a circular dependency formed by A, B and C, as B must be evaluated before A, C must be evaluated before B and A must be evaluated before C. (TODO: add little image)
[edit] Deriving an evaluation order
A correct evaluation order is a numbering 
of the objects that form the nodes of the dependency graph so that the following equation holds: 
with 
. This means, if the numbering orders two elements a and b so that a will be evaluated before b, then b must not depend on a. Furthermore, there can be more than a single correct evaluation order. In fact, a correct numbering is a topological order, and any topological order is a correct numbering. Thus, any algorithm that derives a correct topological order derives a correct evaluation order.
Assume the simple calculator from above once more. Given the equation system "A = B+C; B = 5+D; C=4; D=2;", a correct evaluation order would be (D, C, B, A). However, (C, D, B, A) is a correct evaluation order as well.
[edit] Examples
Dependency graphs are used in:
- Automated software installers. They walk the graph looking for uninstalled but needed software packages. The dependency is given by the coupling of the packages.
- Dead code elimination. If no side effected operation depends on a variable, this variable is considered dead and can be removed.
- Spreadsheet calculators. They need to derive a correct calculation order similar to that one in the example used in this article.
- Cooking. One needs to wash the vegetables before putting them in the soup.
[edit] See also
[edit] References
- Balmas, Francoise (2001) Displaying dependence graphs: a hierarchical approach, [1] wcre, p. 261, Eighth Working Conference on Reverse Engineering (WCRE'01)
