Complexity metrics

1
Complexity metrics

• measure certain aspects of the software (lines of
  code, of if-statements, depth of nesting, )
• use these numbers as a criterion to assess a
  design, or to guide the design
• interpretation higher value ? higher complexity
  ? more effort required ( worse design)
• two kinds
• intra-modular inside one module
• inter-modular between modules

2
Sized-based complexity measures

• counting lines of code (intra-modular)
• differences in scale for different programming
  languages
• Halsteads metrics counting operators and
  operands

3
Halsteads metrics

• n1 number of unique operators
• n2 number of unique operands
• N1 total number of operators
• N2 total number of operands

4
Example program

• public static void sort(int x )
• for (int i0 i lt x.length-1 i)
• for (int ji1 j lt x.length j)
• if (xi gt xj)
• int savexi
• xixj xjsave

operator, 1 occurrence
operator, 2 occurrences
5
operator of occurrences

• public
• sort()
• int
• for
• if ()
• lt
• n1 17

• 1
• 1
• 4
• 7
• 4
• 2
• 1
• 5
• 2
• N1 39

6
Example program

• public static void sort(int x )
• for (int i0 i lt x.length-1 i)
• for (int ji1 j lt x.length j)
• if (xi gt xj)
• int savexi
• xixj xjsave

operand, 2 occurrences
operand, 2 occurrences
7
operand of occurrences

• x
• length
• i
• j
• save
• 0
• 1
• n2 7

• 9
• 2
• 7
• 6
• 2
• 1
• 2
• N2 29

8
Metrics

• size of vocabulary n n1 n2
• program length N N1 N2
• volume V N log2n
• level of abstraction L V/ V, Vvolume of fct
  prototype. For sort(x), V 2 log 2. For main(),
  L is high.
• approximation L (2/n1)(n2/N2)
• programming effort E V/L
• estimated programming time T E/18
• estimate of N N n1log2n1 n2log2n2
• for this example N 68, N 89, L .015, L
  .028

9
Remember

• Metrics are only estimates, give some idea on
  complexity
• critique
• different definitions of operand and operator
• explanations are not convincing (empirical, may
  not apply to other projects)
• Source code must exist

10
Other metrics (structure-based)

• use
• control structures
• data structures
• or both
• example complexity measure based on data
  structures average number of instructions
  between successive references to a variable
• best known measure is based on the control
  structure McCabes cyclomatic complexity

11
Example program
1
2
3

• public static void sort(int x )
• for (int i0 i lt x.length-1 i)
• for (int ji1 j lt x.length j)
• if (xi gt xj)
• int savexi
• xixj xjsave

4
5
11
6
7
8
9
10
12
Cyclomatic complexity
1
2
3

• e number of edges (13)
• n number of nodes (11)
• p number of connected components (1)
• CV e - n p 1 (4)

4
5
11
6
7
8
9
10
13
Intra-modular complexity measures, summary

• for small programs, the various measures
  correlate well with programming time
• however, a simple length measure such as LOC does
  equally well
• complexity measures are not very context
  sensitive
• complexity measures take into account few aspects
• it might help to look at the complexity density
  instead

14
System structure inter-module complexity

• measures dependencies between modules
• draw graph
• modules nodes
• edges connecting modules may denote several
  relations, most often A uses B (ex procedure
  calls)

15
The uses relation

• the call-graph
• chaos (general directed graph)
• hierarchy (acyclic graph)
• strict hierarchy (layers)
• tree

16
In a picture
hierarchy
chaos
strict hierarchy
tree
17
Measurements

size nodes edges
width
height
18
Deviation from a tree
hierarchy
strict hierarchy
tree
19
Tree impurity metric

• complete graph with n nodes has e n(n-1)/2
  edges
• a tree with n nodes has e (n-1) edges
• tree impurity for a graph with n nodes and e
  edges
• m(G) 2(e-n1)/(n-1)(n-2)
• (0 for tree, 1 for complete graph)

20
Any tree impurity metric

• m(G) 0 if and only if G is a tree
• m(G1) gt m(G2) if G1 G2 an extra edge
• if G1 and G2 have the same of extra edges wrt
  their spanning tree, and G1 has more nodes than
  G2, then m(G1) lt m(G2)
• m(G) ? m(Kn) 1, where G has n nodes, and Kn is
  the (undirected) complete graph with n nodes

21
Information flow metric

• tree impurity metrics only consider the number of
  edges, not their thickness
• Shepperds metric
• there is a local flow from A to B if
• A invokes B and passes it a parameter
• B invokes A and A returns a value
• there is a global flow from A to B if A updates
  some global structure and B reads that structure

22
Shepperds metric

• fan-in(M) (local and global) flows whose sink
  is M
• fan-out(M) (local and global) flows whose
  source is M
• complexity(M) (fan-in(M) fan-out(M))2

23
Point to ponderWhat does this program do?

• procedure X(A array 1..n of int)
• var i, k, small int
• begin
• for i 1 to n do
• small Ai
• for k i to n-1 do
• if small lt Ak
• then swap (Ak, Ak1)
• end
• end
• end
• end

24
Object-oriented metrics

• WMC Weighted Methods per Class
• DIT Depth of Inheritance Tree
• NOC Number Of Children
• CBO Coupling Between Object Classes
• RFC Response For a Class
• LCOM Lack of COhesion of a Method

25
Weighted Methods per Class

• measure for size of class
• WMC ? c(i), i 1, , n (number of methods)
• c(i) complexity of method i
• mostly, c(i) 1

26
Depth of Class in Inheritance Tree

• DIT distance of class to root of its
  inheritance tree
• Good forest of classes of medium height

27
Number Of Children

• NOC counts immediate descendants in inheritance
  tree
• higher values NOC are considered bad
• possibly improper abstraction of the parent class
• also suggests that class is to be used in a
  variety of settings

28
Coupling Between Object Classes

• two classes are coupled if a method of one class
  uses a method or state variable of another class
• CBO count of all classes a given class is
  coupled with
• high values something is wrong
• all couplings are counted alike refinements are
  possible

29
Response For a Class

• RFC size of the response set of a class
• response set set of methods M ? set methods
  called by M1 R1 ? R2 ?

R1
M2
M1
M3
30
Lack of Cohesion of a Method

• cohesion glue that keeps the module (class)
  together, eg. all methods use the same set of
  state variables
• if some methods use a subset of the state
  variables, and others use another subset, the
  class lacks cohesion
• LCOM number of disjoint sets of methods in a
  class
• two methods in the same set share at least one
  state variable

31
OO metrics

• WMC, CBO, RFC, LCOM most useful
• Predict fault proneness during design
• Strong relationship to maintenance effort
• Many OO metrics correlate strongly with size