1. Path Analysis Why path analysis for test case design?
Provides a systematic methodology.
Reproducible
Traceable
Countable
What is path analysis?
Analyzes the number of paths that exist in the system
Facilitates the decision process of how many paths to include in the test

2. Linearly Independent Path
A path through the system is Linearly Independent** from other paths only if it includes some segment that is not covered in the other path. S1
- The statements are represented
by the rectangular and diamond blocks.
- The segments between the blocks are
labeled with numbered circles. 1 2 C1
Path1 : S1 – C1 – S3
Path2 : S1 – C1 – S2 – S3 OR
Path1: segments (1,4)
Path2: segments (1,2,3) 4 S2 3 S3
Path1 and Path2 are linearly independent because each includes some
segment that is not included in the other.
** This definition will require more explanation later.

3. Another Example of Linearly Independent Paths1 S2 8 C1 2
Path1: segments (1,2,8)
Path2: segments (1,5,3,9)
Path3: segments (1,5,6,4,10)
Path4: segments (1,5,6,7)
Note that these are all linearly independent 5 C2 3 S3 9 6 C3 4 10 S4 7 S5

4. Statement Coverage Method
Count all the linearly independent paths
Pick the minimum number of linearly independent paths that will include all the statements (S’s and C’s in the diagram) S1
Path1 : S1 – C1 – S3
Path2 : S1 – C1 – S2 – S3 1 2 C1 4 S2 3 S3
Path1 and Path2 are needed to cover all the statements: (S1,C1,S2,S3) ?

5. Another Example of Statement Coverage1 S2 8 C1 2
The 4 Linearly Independent Paths Covers:
Path1: includes S1-C1-S2-S5
Path2: includes S1-C1-C2-S3-S5
Path3: includes S1-C1-C2-C3-S4-S5
Path4: includes S1-C1-C2-C3-S5 5 C2 3 S3 9 6 C3 4 10 S4 7 S5
For 100% Statement Coverage, all we need are 3 paths : Path1, Path2, and Path3 to cover all the statements (S1,C1,S2,C2,S3,C3,S4,S5)
no need for Path

6. Branch Coverage Method
Identify all the decisions
Count all the branches from the each of the decisions
Pick the minimum number of paths that will cover all the branches from the decisions.

7. Branch Coverage MethodS1
Decision C1 :
B1 : Path1 : C1 – S3
B2 : Path2 : C1 – S2 – S3 1
Branch 1 C1
Branch 2 2 4 S2 3 S3
Path1 and Path2 are needed to cover both branches from C1?

8. Another Example of Branch Coverage
The 3 Decisions:
C1:
- B1 : C1- S2
- B2 : C1- C2
C2:
- B3 : C2 – S3
- B4 : C2 – C3
C3:
- B5 : C3 – S4
- B6 ; C3 – S5 S1 1 S2 8 C1 2 5 C2 3 S3 9 6 C3 4 10 S4
The 4 Linearly Independent Paths Covers:
Path1: includes S1-C1-S2-S5
Path2: includes S1-C1-C2-S3-S5
Path3: includes S1-C1-C2-C3-S4-S5
Path4: includes S1-C1-C2-C3-S5 7 S5
We need:
Path1 to cover B1,
Path2 to cover B2 and B3,
Path3 to cover B4 and B5,
Path4 to cover B6

9. McCabe’s Cyclomatic Number
Is there a way to know how many linearly independent paths exist?
McCabe’s Cyclomatic number used to study program complexity may be applied. There are 3 ways to get the Cyclomatic Complexity number from a flow diagram.
# of binary decisions + 1
# of edges - # of nodes + 2
# of closed regions + 1
Reference: T.J. McCabe, “A complexity Measure,” IEEE Transactions on Software Engineering, Dec. 1976

10. McCabe’s Cyclomatic Complexity Number Earlier Example
We know there are 2 linearly independent
paths from before:
Path1 : C1 – S3
Path2 : C1 – S2 – S3 S1 1
McCabe’s Cyclomatic Number:
a) # of binary decisions +1 = 1 +1 = 2
b) # of edges - # of nodes +2 = = 2
c) # of closed regions + 1 = = 2 C1 2 4
Closed region S2 3 S3

11. McCabe’s Cyclomatic Complexity Number Another Example
McCabe’s Cyclomatic Number:
a) # of binary decisions +1 = 2 +1 = 3
b) # of edges - # of nodes +2 = = 3
c) # of closed regions + 1 = = 3 S1 1 4 C1 2 C2 5 S2
There are 3 Linearly
Independent Paths
Closed Region
Closed Region S4 7 6 3 S3

12. An example of 2n total path
Since for each binary decision, there are 2 paths and
there are 3 in sequence, there are 23 = 8 total “logical” paths
path1 : S1-C1-S2-C2-C3-S4
path2 : S1-C1-S2-C2-C3-S5
path3 : S1-C1-S2-C2-S3-C3-S4
path4 : S1-C1-S2-C2-S3-C3-S5
path5 : S1-C1-C2-C3-S4
path6 : S1-C1-C2-C3-S5
path7 : S1-C1-C2-S3-C3-S4
path8 : S1-C1-C2-S3-C3-S5 S1 1 C1 2 3 S2 4 C2 5 6 S3
How many Linearly Independent paths are there?
Using Cyclomatic number = 3 decisions +1 = 4
One set would be:
path1 : includes segments (1,2,4,6,9)
path2 : includes segments (1,2,4,6,8)
path3 : includes segments (1,2,4,5,7,9)
path5 : includes segments (1,3,6,9) 7 C3 9 8 S5 S4
Note 1: with just 2 paths ( Path1 and Path8) all the statements are covered.
Note2: with just 2 paths ( Path1 and Path8) all the branches are covered.

13. Example with a Loop
Total number of paths may be “ infinite” (very large)
because of the loop S1 1
Linearly Independent Paths = 1 decision +1 = 2
path1 : S1-C1-S3 (segments 1,4)
path2 : S1-C1-S2-C1-S3 (segments 1,2,3,4) C1 4 S3 2
One path will cover all statements: (S1,C1,S2, S3)
path2 : S1-C1-S2-C1-S3 S2 3
One path will cover all branches:
path2 : S1-C1-S2-C1-S3
branch1 (C1-S2) and
branch 2 (C1-S3)

14. More on Linearly Independent Paths
In discussing dimensionality, we talks about orthogonal vectors.
Two dimensional space has two orthogonal vector from which all the other vectors in two dimension can be obtained via “linear combination” of these vectors:
[1,0]
[0,1]
e.g.
[2,4] = 2[1,0] + 4[0,1]
[2,4]
[1,0]
[0,1]

15. More on Linearly Independent Paths
A set of paths is considered to be a Linearly Independent Set if every path may be constructed as a “linear combination” of paths from the linearly independent set. For example:
We already know: a) there are a total of 22=4 logical paths.
b) 2 paths that will cover all statements
and all branches.
c) 2 branches +1 = 3 linearly independent paths. C1 2 1 S1 1 2 3 4 5 6 3
We picked path1,
path2 and path3 as
The Linearly
Independent Set
path1 1 1 1 C2 5
path2 1 1 4
path3 1 1 1 S1
path4 1 1 1 1 6
path 4 = path3 + path1 – path2 = (0,1,1,1,0,0)+(1,0,0,0,1,1)- (1,0,0,1,0,0)
= (1,1,1,1,1,1) - (1,0,0,1,0,0)
= (0,1,1,0,1,1)

16. More on Linearly Independent Paths
We already know: a) there are a total of 22=4 logical paths.
b) 2 paths that will cover all statements
and all branches.
c) 2 branches +1 = 3 linearly independent paths. C1 2 1 S1 1 2 3 4 5 6 3
path1 1 1 1 C2 5
path2 1 1 4
path3 1 1 1 S1
path4 1 1 1 1 6
Although path1 and path3 are linearly independent, they do NOT form a Linearly Independent Set because no linear combination of path1 and path3 can get , say, path4.

17. More on Linearly Independent Paths
Because the Linearly Independent Set of paths display the same characteristics as the mathematical concept of basis in n-dimensional vector space, the testing using the Linearly Independent Set of paths is sometimes called the “basis” testing.

18. Paths Analysis Interested in Total number of “logical” paths
Interested in Linearly Independent paths
Interest in Branch coverage
Interested in Statement coverage
Which one is the largest set, next largest set, , etc.?