1. BASIS PATH TESTING

2. Path Testing Program graph is a directed graph in which
nodes are either entire statements or fragments of a statement,
edges represent flow of control.
The importance of the program graph is that program executions correspond to paths from the source to the sink nodes.

3. DD-Paths
The best known form of structural testing is based on a construct known as a decision-to-decision path (DD-Path) [Miller 77].
The name refers to a sequence of statements that, in Miller’s words, begins with the “outway” of a decision statement and ends with the “inway” of the next decision statement.
There are no internal branches in such a sequence, so the corresponding code is like a row of dominoes lined up so that when the first falls, all the rest in the sequence fall.
DD-Paths in terms of paths of nodes in a directed graph.
We might call these paths chains, where a chain is a path in which the initial and terminal nodes are distinct, and every interior node has indegree = 1 and outdegree = 1.

4. A DD-Path is a chain in a program graph such that
Case 1: it consists of a single node with indeg = 0,
Case 2: it consists of a single node with outdeg = 0,
Case 3: it consists of a single node with indeg ≥ 2 or outdeg ≥ 2,
Case 4: it consists of a single node with indeg = 1 and outdeg = 1,
Case 5: it is a maximal chain of length ≥ 1.
DD-Path graph is the directed graph in which
nodes are DD-Paths of its program graph, and
edges represent control flow between successor DD-Paths.

5. Cases 1 and 2 establish the unique source and sink nodes of the program graph of a structured program as initial and final DD-Paths.
Case 3 deals with complex nodes; it assures that no node is contained in more than one DD-Path.
Case 4 is needed for “short branches”; it also preserves the one fragment, one DD-Path principle.
Case 5 is the “normal case”, in which a DD-Path is a single entry, single exit sequence of nodes (a chain).
The “maximal” part of the case 5 definition is used to determine the final node of a normal (non-trivial) chain.

6. DD-Path Graph for the Triangle Program
Program Graph of the Triangle Program

8. Basis Path Testing
Notion of a “basis” has attractive possibilities for structural testing.
A basis in terms of a structure called a “vector space”, which is a set of elements (called vectors) and which has operations that correspond to multiplication and addition defined for the vectors.
Basis path testing is a white-box testing technique first proposed by Tom McCabe [MCC76].
The basis path method enables the test case designer to derive a logical complexity measure of a procedural design and use this measure as a guide for defining a basis set of execution paths.
Test cases derived to exercise the basis set are guaranteed to execute every statement in the program at least one time during testing.

9. Cyclomatic complexity is a software metric that provides a quantitative measure of the logical complexity of a program.
CC defines the number of independent paths in the basis set of a program and
Provides us with an upper bound for the number of tests that must be conducted to ensure that all statements have been executed at least once.
An independent path is any path through the program that introduces at least one new set of processing statements or a new condition.

10. Complexity is computed in one of three ways:
1. The number of regions of the flow graph correspond to the cyclomatic complexity.
2. Cyclomatic complexity, V(G), for a flow graph, G, is defined as
V(G) = E - N + 2
where E is the number of flow graph edges, N is the number of flow graph nodes.
3. Cyclomatic complexity, V(G), for a flow graph, G, is also defined as
V(G) = P + 1
where P is the number of predicate nodes contained in the flow graph G.

11. 1. The flow graph has four regions.
2. V(G) = 11 edges nodes + 2 = 4.
3. V(G) = 3 predicate nodes + 1 = 4.

12. DD-Path Graph for the Triangle Program
The following are basis paths for Triangle pbm.
DD-Path Graph for the Triangle Program

13. The problem here is that there are several inherent dependencies in the triangle problem.
One is that if three integers constitute sides of a triangle, they must be one of the three possibilities: equilateral, isosceles, or scalene.
A second dependency is that the three possibilities are mutually exclusive: if one is true, the other two must be false.

14. We can identify several rules for the pbm:
If node B is traversed, then we must traverse nodes D and E.
If node C is traversed, then we must traverse nodes D and L.
If node E is traversed, then we must traverse one of nodes F, H, and J.
If node F is traversed, then we cannot traverse nodes H and J.
If node H is traversed, then we cannot traverse nodes F and J.
If node J is traversed, then we cannot traverse nodes F and I.

15. The following feasible basis path set:
The triangle problem is atypical in that there are no loops.
The program has only 18 topologically possible paths, and of these, only the four basis paths listed above are feasible.