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Halstead complexity measures

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Halstead complexity measures are software metrics introduced by Maurice Howard Halstead in 1977 as part of his treatise on establishing an empirical science of software development. Halstead made the observation that metrics of the software should reflect the implementation or expression of algorithms in different languages, but be independent of their execution on a specific platform. These metrics are therefore computed statically from the code.

Contents

Halstead's goal was to identify measurable properties of software, and the relations between them. This is similar to the identification of measurable properties of matter (like the volume, mass, and pressure of a gas) and the relationships between them (analogous to the gas equation). Thus his metrics are actually not just complexity metrics.

Calculation

For a given problem, Let:

  • η 1 = the number of distinct operators
  • η 2 = the number of distinct operands
  • N 1 = the total number of operators
  • N 2 = the total number of operands

    • From these numbers, several measures can be calculated:

  • Program vocabulary: η = η 1 + η 2
  • Program length: N = N 1 + N 2
  • Calculated program length: N ^ = η 1 log 2 ⁡ η 1 + η 2 log 2 ⁡ η 2
  • Volume: V = N × log 2 ⁡ η
  • Difficulty : D = η 1 2 × N 2 η 2
  • Effort: E = D × V

    • The difficulty measure is related to the difficulty of the program to write or understand, e.g. when doing code review.

    The effort measure translates into actual coding time using the following relation,

  • Time required to program: T = E 18 seconds

    • Halstead's delivered bugs (B) is an estimate for the number of errors in the implementation.

  • Number of delivered bugs : B = E 2 3 3000 or, more recently, B = V 3000 is accepted.

    • Example

    Let us consider the following C program:

    main(){ int a, b, c, avg; scanf("%d %d %d", &a, &b, &c); avg = (a + b + c) / 3; printf("avg = %d", avg);}

    The unique operators are: main, (), {}, int, scanf, &, =, +, /, printf

    The unique operands are: a, b, c, avg, "%d %d %d", 3, "avg = %d"

  • η 1 = 10 , η 2 = 7 , η = 17
  • N 1 = 16 , N 2 = 15 , N = 31
  • Calculated Program Length: N ^ = 10 × l o g 2 10 + 7 × l o g 2 7 = 52.9
  • Volume: V = 31 × l o g 2 17 = 126.7
  • Difficulty: D = 10 2 × 15 7 = 10.7
  • Effort: E = 10.7 × 126.7 = 1355.7
  • Time required to program: T = 1 , 355.7 18 = 75.4 seconds
  • Number of delivered bugs: B = 1 , 355.7 2 3 3000 = 0.04

    • References

    Halstead complexity measures Wikipedia
    (Text) CC BY-SA


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