All-pairs testing

In computer science, all-pairs testing or pairwise testing is a combinatorial method of software testing that, for each pair of input parameters to a system (typically, a software algorithm), tests all possible discrete combinations of those parameters. Using carefully chosen test vectors, this can be done much faster than an exhaustive search of all combinations of all parameters, by "parallelizing" the tests of parameter pairs.

The most common bugs in a program are generally triggered by either a single input parameter or an interaction between pairs of parameters. Bugs involving interactions between three or more parameters are both progressively less common and also progressively more expensive to find---such testing has as its limit the testing of all possible inputs. Thus, a combinatorial technique for picking test cases like all-pairs testing is a useful cost-benefit compromise that enables a significant reduction in the number of test cases without drastically compromising functional coverage.

More rigorously, if we assume that a test function has ${\displaystyle N}$ parameters given in a set ${\displaystyle \{P_{i}\}=\{P_{1},P_{2},...,P_{N}\}}$. The range of the parameters are given by ${\displaystyle R(P_{i})=R_{i}}$. Let's assume that ${\displaystyle |R_{i}|=n_{i}}$. We note that the all possible conditions that can be used is an exponentiation, while imagining that the code deals with the conditions taking only two pair at a time, might reduce the number of conditions.

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