Practical Software Engineering

Complexity -- Software Metrics

Complexity is a measure of the resources which must be expanded in developing, maintaining, or using a software product.

A large share of the resources are used to find errors, debug, and retest; thus, an associated measure of complexity is the number of software errors.

Items to consider under resources are:

Consideration of storage, complexity, and processing time may or may not be considered in the conceptualization stage. For example, storage of a large database might be considered, whereas if one already existed, it could be left until system-specification.

Measures of complexity are useful to:

  1. Rank competitive designs and give a "distance" between rankings.
  2. Rank difficulty of various modules in order to assign personnel.
  3. Judge whether subdivision of a module is necessary.
  4. Measure progress and quality during development.

Complexity Measures

Complexity theory does not yet allow us to measure the separate aspects, e.g. problem, algorithm, design, language, functions, modules and combine them to measure and compare overall complexity. Most complexity measures concentrate on the program. Instruction counts are used to predict man-hours and development costs. A high level language may have between 4:1 and 10:1 ratio to machine language.

Statistical approach, used by Halstead, looked at total number of operators and operands, and related this to man-hours. This measure was rooted in information theory -- Zipf's laws of natural languages, Shannon's information theory.

Zipf's Laws

Zipf studied the relationship between frequency of occurrence nr and rank r for words from English, Chinese and Latin, and found that

n(r) /n . r = constant (c)

or n(r) = c n / r

n is sample size

Graph Theoretic Complexity Measures

Consideration of the length and difficulty of individual program statements does not account for the interrelationships among instructions, i.e. structural intricacy. Recent work on structural complexity relates to graph topological measures. Graph theory is not able to represent storage, or operands. It concentrates on transfer of control.


Similar to set theory: collection X, mapping M

e.g. X-set elements, M-set mapping rules

Each element belonging to X is represented by a point called a vertex or node. Mapping rules define connections between the vertices. If there is an order in the connection, or arc then it is a directed graph or digraph. If there is no order the connecting line is an edge.

A directed graph and a "regular" graph

Directed Graphs

A strongly connected graph

In linear algebra, a set of vectors x(1), x(2), ..., x(n) is linearly dependent if there exists a set of numbers not all zero s.t. a(1)x(1) + a(2)x(2) + ... + a(n)x(n ) = 0. If no such set of numbers exists then the set of vectors are linearly independent.

Any system of n linearly independent vectors e(1), e(2), ..., e(n) forms a basis in n-dimensional vector space.

In graph theory, if the concept of a vector is associated with each circuit or cycle, then there are analogous concepts of independent cycles and a cycle basis.

Cyclomatic Number of a Strongly Connected Graph

The cyclomatic number v(G) of graph G is equal to the maximum number of linearly independent cycles. The cyclomatic number v(G) of a strongly connected digraph is equal to the maximum number of linearly independent circuits. v(G) = m - n + p

where there are m arcs

n vertices

and p separate components

e.g. m=9, n=5, p=1, v(G)=5

There are various other methods for calculating the cyclomatic number of a graph.

Graph Theoretic Complexity Measures

A program graph is obtained from shrinking each process and decision symbol in a flowchart to a point (vertex) and lines joining them (arcs). Direction in a program is important, hence a directed graph (digraph) is obtained.

e.g. digraph of program flowchart given:

nodes a and k correspond to start and stop nodes

node c corresponds to GET A

node i corresponds to L = L + 1

The cycle (loop) is composed of arcs 4, 5, 6, 7, 8

IF THEN ELSE appears as 5, 6 and 9, 10.

The cyclomatic number of the digraph gives a complexity metric. Comparisons show that a high cyclomatic complexity is associated with error-prone programs.

A graph is constructed with a single start node and a single stop node. If there is an unreachable node, that represents unreachable code or coding error which must be corrected.

In general the graph is not strongly connected, so form a loop round the entire program (phantom arc).

e.g. v(G)= 13 - 11 + 1 = 3

number of arcs m = 13

number of nodes (vertices) n = 11

separate parts p = 1


McCabe says that v(G) is a good measure of complexity and all modules should have v(G)<10. This replaces the general rule of thumb that a module should be one page of code (50 to 60 lines). He has shown that programs with a large v(G) are error-prone.

Much theory exists in this area, but as yet has not been applied on a wide scale.

Practical Software Engineering, Department of Computer Science 12-Jan-96