Analysis of Crosscutting in Early Software Development Phases Based on Traceability - Transactions on Aspect-Oriented Software Development - page 78

the case studies (Sect. 5.1). We will now describe the representation of trace
dependencies in traceability matrices.
3 Matrix Representation of Trace Relations
In this section we show how crosscutting can be represented and identified by means
of an extension to traceability matrices. Trace relations are captured in a dependency
matrix, representing the mapping between source and target. As an extension, we
derive the crosscutting matrix from the dependency matrix. We describe how the
crosscutting matrix can be constructed from the dependency matrix with some
auxiliary matrices. This is illustrated with some examples.
3.1 Tracing from Source to Target
Traceability matrices have usually been used to show the relationships between
requirements elicitation and the representation of these requirements in a particular
engineering approach (such as use cases [31] or viewpoints [16]).
In terms of linear algebra, traceability matrices show the mappings between source
and target. We show these mappings in a special kind of traceability matrix that we
called a dependency matrix. A dependency matrix (source x target) represents the
dependency relation between source elements and target elements (inter-level
relationship) . In the rows we have the source elements, and in the columns we have
the target elements. In this matrix a cell with 1 denotes that the source element (in the
row) is mapped to the target element (in the column). Reciprocally this means that the
target element depends on the source element. Scattering and tangling can easily be
visualized in this matrix (see the examples below).
We define a new auxiliary concept crosscutpoint used in the context of dependency
matrices to denote a matrix cell involved in both tangling and scattering (see dark
grey cell in Table 3 ). If there are one or more crosscutpoints then we say we have
crosscutting.
Crosscutting between source elements for a given mapping to target elements, as
shown in a dependency matrix, can be represented in a crosscutting matrix. A
crosscutting matrix (source x source) represents the crosscutting relation between
source elements for a given source-to-target mapping (represented in a dependency
matrix). In the crosscutting matrix, a cell with 1 denotes that the source element in the
row is crosscutting the source element in the column. In the next Sect. 3.2, we explain
how this crosscutting matrix can be derived from the dependency matrix.
A crosscutting matrix should not be confused with a coupling matrix. A coupling
matrix shows coupling relations between elements at the same level of abstraction
(intra-level dependencies). In some sense, the coupling matrix is related to the design
structure matrix [3]. On the other hand, a crosscutting matrix shows crosscutting
relations between elements at one level with respect to a mapping onto elements at
some other level (inter-level dependencies).
We now give an example and use the dependency matrix and crosscutting matrix
to visualize the definitions (S denotes a scattered source element — a grey row; NS
denotes a non-scattered source element; T denotes a tangled target element — a grey
column; NT denotes a non-tangled target element). The example is shown in Table 3 .