Hierarchical metrics and the prime generation problem

Abstract

The field of software metrication can be divided between those program measurement techniques that are largely textual (e.g. the metrics of Halstead and his followers) and those that are more structural, i.e. based on the hierarchical decomposition structure of the underlying flow graph. We are concerned with the latter, but only from the ‘control flow’ perspective. (Note that structural metrics may also focus on the ‘data flow’ model, especially when attempting to measure module cohesion and coupling, but these are not of concern here.) The decomposition is of two varieties; sequence and nesting, each extended recursively to the bottom-most level consisting of ‘prime flow graphs’, those that cannot be constructed from the sequence and nesting operations. These then play a role in the software metric theory not unlike that of prime numbers in arithmetic. We initiate a new method for generating the prime flow graphs, based more firmly on graph-theoretic techniques and classical enumeration theory than in previous work. The technique is rooted in a fundamental reformulation of the flow graph model itself, making use of the notation of a cubic graph. This approach offers new insights into the decompositional methodology and broad implications for the design of effective structural metrics based on that methodology. In particular, we are able to obtain wide-ranging software engineering applications, towards the comparison of individual hierarchical metrics, the generation and enumeration of the prime flow graphs, and the design of a test for flow graph primality.