Test configuration generation is an important part of Combinatorial test design process. Combinatorial design process is shown in the figure below :

As can be seen from the above figure, combinatorial test design process has 3 main steps :

- Modeling of the input space and configuration space.
- Generation of a combinatorial object with the help of a combinatorial design procedure. Combinatorial object obtained is an array of factors and levels.
- Generation of a test set or test configuration using the combinatorial object.

**Note :**

The input variables are called **factors **and values that are assignable to a factor is called a **level**.

**Test Design Algorithm :**

To design test configurations, we make use of a Test Design algorithm. This algorithm takes as input n number of factors (input variables) and outputs a set of factor combinations such that all level pairs are covered. Test Design algorithm has the following steps :

**Relabel the Factors :**

The given factors must first be relabelled as F1, F2, F3, ….. Fn such that :|F1| >= |F2| >= |F3| >= .........|Fn-1| >= |Fn| Let b = |F1| and k = |F2| Here, |Fi| is the number of levels in Factor Fi

**Prepare a Table :**

Create a table containing n number of columns labeled as F1, F2, F3, …..Fn, and (b x k) number of rows divided into b number of blocks. Here, each block contains k number of rows. A sample table is shown below for n = b = k = 3 :**Fill the columns F1 and F2 :**

We fill the column F1 with 1s in Block 1, 2s in Block 2, and similarly for other blocks. For column F2, we fill the block 1 with the sequence 1, 2, 3 …. k in rows 1 through k and repeat this for the rest of the blocks. A sample table is shown below :**Find MOLS of order k and fill the rest of the columns :**

Before filling the columns, we must first understand MOLS and how to find them.MOLS (Mutually Orthogonal Latin Squares) are used for selecting a subset of factor combinations from a complete set. A

**Latin square**of order n is an n x n matrix where no element appears more than once in a row and column.**Example-1 :**

If S = {X, Y}, then Latin squares of order 2 will be :

X Y Y X and Y X X Y

**Example-2 :**

If S = {1, 2, 3}, then Latin squares of order 3 will be :1 2 3 2 3 1 3 1 2 and 2 3 1 1 2 3 3 1 2 and 2 1 3 3 2 1 1 3 2

To construct a Latin square, create the first row with n distinct elements and fill the additional rows by permuting the first row. For example,

If S = {1, 2, 3, 4}, then a Latin square could be :

1 2 3 4 2 3 4 1 3 4 1 2 4 1 2 3

To create**MOLS**, let M1 and M2 be two Latin squares, each of order n.Let M1 ( i, j ) and M2 ( i, j ) denote element at the ith row and jth column in Latin squares M1 and M2 respectively. Now, we create an n x n matrix L from M1 and M2 such that L ( i, j ) is M1 ( i, j ) M2 ( i, j ), i.e. we juxtapose corresponding elements of M1 and M2.

If each element in L appears exactly once, i.e. it is unique, then M1 and M2 are said to be mutually orthogonal Latin squares of order n.

For example,

1 2 3 2 3 1 Consider M1 = 2 3 1 and, M2 = 1 2 3 3 1 2 3 1 2

So, L will be constructed as follows:

12 23 31 L = 21 32 13 33 11 22

Since elements in L are unique, so, M1 and M2 are MOLS of order 3.

**Note :**

When n is a prime number or a power of a prime number, then MOLS(n) contains n -1 MOLS.

Also, MOLS do not exist for n = 2 and n = 6, however, they do exist for all other values greater than 2.

Now, we can find MOLS of order k in order to fill remaining columns. Number these MOLS as M1, M2, and so on.

Here, s < k for k > 1, where s = number of MOLS of order k. Fill Block 1 of column F3 with elements from column 1 of M1, Block 2 with elements from column 2 of M1, and so on.If b > k, then reuse columns of M1 to fill rows in remaining (b-k) blocks. This process is repeated for columns F4 through Fn using MOLS M2 through Ms. If s < n – 2, then we can fill remaining columns by randomly selecting values of the factors.

For example,

If n = k = 3

Then MOLS of order 3 are :

1 2 3 1 2 3 M1 = 2 3 1 and, M2 = 3 1 2 3 1 2 2 3 1

We can fill remaining columns of the table using these MOLS. Refer to table shown below for a better understanding.

**Check if constraints are satisfied :**

If there are no constraints given, then Step 5 and 6 should be skipped. Otherwise, entries in rows should be marked with a box if they do not satisfy constraints given. Constraints could be given like :- Safari browser supports only Mac OS.
- Firefox browser works on Windows, Linux.
- Windows OS supports only LAN and PPP protocols.

**Remove configurations that do not satisfy given constraints :**

The configurations that are highlighted with a box in the table have to be removed. This is done by following a two-step procedure to remove them and retain pairwise coverage :- Modify the highlighted rows so that the constraints are maintained.
- Add new configurations that cover the pairs that were left uncovered while replacing the highlighted rows.

**Replace numbers in columns with given factor values :**

In this step, we finally obtain test configurations by replacing values in columns of table with the actual values for factors. For example,If F1 is a factor called Operating Systems and levels of F1 includes {MacOs, Windows, Linux}

Then if column F1 has values like:

2 3 1

Then, they should be replaced as follows :

Windows Linux MacOS

where Windows, Linux and MacOS represent 2, 3 and 1 respectively.

After performing this step for all columns, the table obtained will contain the final test configurations.

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