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Domain-to-range ratio

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Thedomain-to-range ratio (DRR) is a ratio which describes how the number of outputs corresponds to the number of inputs of a given logicalfunction orsoftware component. The domain-to-range ratio is a mathematical ratio ofcardinality between theset of the function's possible inputs (thedomain) and the set of possible outputs (therange).^[1]^[2] For a function defined on a domain, $D {\displaystyle D}$ , and a range, $R {\displaystyle R}$ , the domain-to-range ratio is given as: $DRR={\frac {|D|}{|R|}}$ It can be used to measure the risk of missing potentialerrors when testing the range of outputs alone.^[3]

Example

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Consider the functionisEven() below, which checks the parity of anunsigned short number $x {\displaystyle x}$ , any value between $0 {\displaystyle 0}$ and $65,536$ , and yields aboolean value which corresponds to whether $x {\displaystyle x}$ is even or odd. This solution takes advantage of the fact that integer division in programming typically rounds towards zero.

boolisEven(unsignedshortx){return(x/2)==((x+3)/2-1);}

Because $x {\displaystyle x}$ can be any value from $0 {\displaystyle 0}$ to $65,535$ , the function's domain has a cardinality of $65,536$ . The function yields $0 {\displaystyle 0}$ , if $x {\displaystyle x}$ is even, or $1 {\displaystyle 1}$ , if $x {\displaystyle x}$ is odd. This is expressed as the range $\{0;1\}$ , which has acardinality of $2 {\displaystyle 2}$ . Therefore, the domain-to-range ratio ofisEven() is given by: $DRR={65,536 \over 2}=32,768$ Here, the domain-to-range ratio indicates that this function would require a comparatively large number of tests to find errors. If a test program attempts every possible value of $x {\displaystyle x}$ in order from $0 {\displaystyle 0}$ to $65,535$ , the program would have to perform $32,768$ tests for each of the two possible outputs in order to find errors oredge cases. Because errors in functions with a high domain-to-range ratio are difficult to identify viamanual testing or methods which reduce the number of tested inputs, such asorthogonal array testing or all-pairs testing, morecomputationally complex techniques may be used, such asfuzzing orstatic program analysis, to find errors.

References

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^Voas, J.M.; Miller, K.W. (March 1993). "Semantic metrics for software testability".Journal of Systems and Software.20 (3):207–216.doi:10.1016/0164-1212(93)90064-5.
^Woodward, Martin R.; Al-Khanjari, Zuhoor A. (5 September 2000).Testability, fault size and the domain-to-range ratio: An eternal triangle. ACM SIGSOFT. pp. 168–172.doi:10.1145/347324.349016.ISBN 978-1-58113-266-3.S2CID 17772461.
^Tarlinder, Alexander (2016).Developer testing : building quality into software. Addison-Wesley.ISBN 978-0-13-429106-2.OCLC 992888632.