Path Analysis

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)
S2
4
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 Paths
S1
1
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
S3
C2
3
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 (Ss
  and Cs in the diagram)

S1
Path1 S1 C1 S3 Path2 S1 C1 S2 S3
1
2
C1
S2
4
3
S3
Path1 and Path2 are needed to cover all the
statements (S1,C1,S2,S3) ?
5
Another Example of Statement Coverage
S1
1
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
S3
C2
3
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 Path4 - - - -
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 Method
S1
Decision C1 B1 Path1 C1 S3
B2 Path2 C1 S2 S3
1
C1
Branch 1
Branch 2
2
S2
4
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
S3
C2
3
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
McCabes Cyclomatic Number

• Is there a way to know how many linearly
  independent paths exist?
• McCabes 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
McCabes Cyclomatic Complexity NumberEarlier
Example
We know there are 2 linearly independent paths
from before Path1 C1 S3
Path2 C1 S2 S3
S1
1

• McCabes Cyclomatic Number
• a) of binary decisions 1 1 1 2
• b) of edges - of nodes 2 4-42 2
• c) of closed regions 1 1 1 2

C1
2
S2
4
Closed region
3
S3
11
McCabes Cyclomatic Complexity NumberAnother
Example

• McCabes Cyclomatic Number
• a) of binary decisions 1 2 1 3
• b) of edges - of nodes 2 7-62 3
• c) of closed regions 1 2 1 3

S1
1
4
C1
2
C2
5
S2
There are 3 Linearly Independent Paths
Closed Region
Closed Region
S4
7
3
6
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
S4
S5
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 21,0 40,1
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 224
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
1
path1
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 224
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
1
path1
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.?