CS447/ECE453/SE465
Software Testing Tutorial
Winter 2008
Based on the tutorials by Prof. Kontogiannis, Winter 2006
Presented by Jiang Wu
j32wu@cs.uwaterloo.ca
CS447/ECE453/SE465
Prof. Alencar
University of Waterloo
1
Outline of the 12 tutorials

Graphs
Tutorial 1

Example Problems
Tutorial 2

Project Kickoff
Tutorial 3

Functional Testing
Tutorial 4, 5

Structural Testing
Tutorial 6, 7, 8

Integration Testing
Tutorial 9, 10

System testing
Tutorial 11

Software reliability
Tutorial 12
CS447/ECE453/SE465
Prof. Alencar
Graphs

Undirected Graphs








Directed Graphs








Degree of a Node
Incidence Matrices
Adjacency Matrices
Paths
Connectedness
Condensation Graphs
Cyclomatic Number
Indegrees and Outdegrees
Types of Nodes
Adjacency Matrix of a Directed Graph
Paths and Semi – Paths
Reachability Matrix
N – Connectedness
Strong Components
Graphs for Testing



ECE 453
Program Graphs
Finite State Machines
Petri Nets
University of Waterloo
3
Agenda

Undirected Graphs








Directed Graphs








Degree of a Node
Incidence Matrices
Adjacency Matrices
Paths
Connectedness
Condensation Graphs
Cyclomatic Number
Indegrees and Outdegrees
Types of Nodes
Adjacency Matrix of a Directed Graph
Paths and Semi – Paths
Reachability Matrix
N – Connectedness
Strong Components
Graphs for Testing



ECE 453
Program Graphs
Finite State Machines
Petri Nets
University of Waterloo
4
Undirected Graphs




A graph G = (V, E) is composed of a finite
(and non-empty) set V = {n1, n2,…,nm} of
nodes, and a set E = {e1, e2,…,ep} of
edges, where each edge ek = {ni, nj} for
some nodes ni, nj V
Nodes also called vertices
Edges also called arcs
Nodes also called endpoints of an arc
ECE 453
University of Waterloo
5
Undirected Graphs
n1
e1
e3
n5
n2
e5
e4
n4
e2
V = {n1, n2, n3, n4, n5, n6, n7}
n3
e6
E = {e1, e2, e3, e4, e5, e6, e7}
= {(n1, n2), (n2, n3), (n1, n5), (n4, n5), (n2, n4), (n3, n4), (n4, n6)}
edge {n3, n4}
e7
nodes


n6
n7
Nodes are usually program statements
Edges for flow of control, define/use
relationships etc.
ECE 453
University of Waterloo
6
Agenda

Undirected Graphs








Directed Graphs








Degree of a Node
Incidence Matrices
Adjacency Matrices
Paths
Connectedness
Condensation Graphs
Cyclomatic Number
Indegrees and Outdegrees
Types of Nodes
Adjacency Matrix of a Directed Graph
Paths and Semi – Paths
Reachability Matrix
N – Connectedness
Strong Components
Graphs for Testing



ECE 453
Program Graphs
Finite State Machines
Petri Nets
University of Waterloo
7
Degree of a Node

The degree of a node in a graph is the
number of edges that have that node as
an endpoint. We write deg(n) for the
degree of node n.
n1
e1
e3
n5
n2
e5
e4
e2
n3
e6
n4
e7
n6
ECE 453
n7
deg(n1) = 2
deg(n2) = 3
deg(n3) = 2
deg(n4) = 4
deg(n5) = 2
deg(n6) = 1
deg(n7) = 0
University of Waterloo
8
Agenda

Undirected Graphs








Directed Graphs








Degree of a Node
Incidence Matrices
Adjacency Matrices
Paths
Connectedness
Condensation Graphs
Cyclomatic Number
Indegrees and Outdegrees
Types of Nodes
Adjacency Matrix of a Directed Graph
Paths and Semi – Paths
Reachability Matrix
N – Connectedness
Strong Components
Graphs for Testing



ECE 453
Program Graphs
Finite State Machines
Petri Nets
University of Waterloo
9
Incidence Matrices


Alternative representation of graphs
The incidence matrix of a graph G = (V, E)
with m nodes and n edges is an m x n
matrix, where the element in row i, column
j is 1 if and only if node i is an endpoint of
edge j, otherwise the element is 0
ECE 453
University of Waterloo
10
Incidence Matrices
n1
e1
e3
n5
n2
e5
e4
e2
n3
e6
n4
e7
n6
n7
Sum of a row:
degree
Sum of a column: 2
ECE 453
University of Waterloo
11
Incidence Matrices

The sum of the entries in any column is 2


It is a form of integrity checking
The row sum is the degree of the node

ECE 453
When zero, the node is isolated (unreachable
code, objects never used)
University of Waterloo
12
Agenda

Undirected Graphs








Directed Graphs








Degree of a Node
Incidence Matrices
Adjacency Matrices
Paths
Connectedness
Condensation Graphs
Cyclomatic Number
Indegrees and Outdegrees
Types of Nodes
Adjacency Matrix of a Directed Graph
Paths and Semi – Paths
Reachability Matrix
N – Connectedness
Strong Components
Graphs for Testing



ECE 453
Program Graphs
Finite State Machines
Petri Nets
University of Waterloo
13
Adjacency Matrices




Useful supplement to the incidence matrix
The adjacency matrix of a graph G=(V, E)
with m nodes is an m x m matrix, where
the element in row i, column j is a 1 if and
only if there is an edge between node i
and node j, otherwise the element is 0
Symmetric: element (i, j) = element (j, i)
The row sum is the degree of the node
ECE 453
University of Waterloo
14
Adjacency Matrices
n1
e1
e3
n5
n2
e5
e4
e2
n3
e6
n4
e7
n6
n7
Symmetric
ECE 453
University of Waterloo
15
Agenda

Undirected Graphs








Directed Graphs








Degree of a Node
Incidence Matrices
Adjacency Matrices
Paths
Connectedness
Condensation Graphs
Cyclomatic Number
Indegrees and Outdegrees
Types of Nodes
Adjacency Matrix of a Directed Graph
Paths and Semi – Paths
Reachability Matrix
N – Connectedness
Strong Components
Graphs for Testing



ECE 453
Program Graphs
Finite State Machines
Petri Nets
University of Waterloo
16
Paths


A path is a sequence of edges such that,
for any adjacent pair of edges ei, ej in the
sequence, the edges share a common
(node) endpoint
Sequences of nodes or sequences of
edges
ECE 453
University of Waterloo
17
Paths
n1
e1
e3
n5
n2
e5
e4
e2
n3
e6
n4
e7
n6
ECE 453
n7
University of Waterloo
18
Paths

If we multiply an adjacency matrix with itself the product will show
the number of all two edge paths of the graph. If we further multiply
the product with the original adjacency matrix the new product will
show the number of all three edge paths of the graph
2
=
ECE 453
University of Waterloo
19
Agenda

Undirected Graphs








Directed Graphs








Degree of a Node
Incidence Matrices
Adjacency Matrices
Paths
Connectedness
Condensation Graphs
Cyclomatic Number
Indegrees and Outdegrees
Types of Nodes
Adjacency Matrix of a Directed Graph
Paths and Semi – Paths
Reachability Matrix
N – Connectedness
Strong Components
Graphs for Testing



ECE 453
Program Graphs
Finite State Machines
Petri Nets
University of Waterloo
20
Connectedness


Nodes ni and nj are connected if and only
if they are in the same path
Connectedness is an equivalence relation
on the node set of a graph
Reflexive: every node is in a path of length 0
with itself
 Symmetric: if (ni, nj) 
path, then (nj, ni) 
path
 Transitive: if (ni, nj)
path and (nj, nw) 
path,
then (ni, nw) 
path

ECE 453
University of Waterloo
21
Connectedness

Equivalence relations induce a partition


Connectedness defines a partition on the
node set of a graph
A component of a graph is a maximal set
of connected nodes
n1
e1
e3
n5
n2
e5
e4
e2
n3
e6
{n1, n2, n3, n4, n5, n6}, {n7}
n4
e7
n6
ECE 453
n7
{n1, n2} is not a component
University of Waterloo
22
Agenda

Undirected Graphs








Directed Graphs








Degree of a Node
Incidence Matrices
Adjacency Matrices
Paths
Connectedness
Condensation Graphs
Cyclomatic Number
Indegrees and Outdegrees
Types of Nodes
Adjacency Matrix of a Directed Graph
Paths and Semi – Paths
Reachability Matrix
N – Connectedness
Strong Components
Graphs for Testing



ECE 453
Program Graphs
Finite State Machines
Petri Nets
University of Waterloo
23
Condensation Graphs

Given a graph G = (V, E), its condensation
graph is formed by replacing each
component by a condensing node
n1
e1
e3
n5
n2
e5
e4
e2
n3
e6
→
n4
e7
n6
n7
S1 = {n1, n2, n3, n4, n5, n6}
S2 = {n7}
ECE 453
University of Waterloo
24
Agenda

Undirected Graphs








Directed Graphs








Degree of a Node
Incidence Matrices
Adjacency Matrices
Paths
Connectedness
Condensation Graphs
Cyclomatic Number
Indegrees and Outdegrees
Types of Nodes
Adjacency Matrix of a Directed Graph
Paths and Semi – Paths
Reachability Matrix
N – Connectedness
Strong Components
Graphs for Testing



ECE 453
Program Graphs
Finite State Machines
Petri Nets
University of Waterloo
25
Cyclomatic Number


The cyclomatic number of a graph G is given by
V(G) = e – n + p, e: number of edges, n: number
of nodes, p: number of components
Number of distinct regions in a graph
n1
e1
e3
n5
n2
e5
e4
e2
n3
e6
Cyclomatic Number: 7 – 7 + 2 = 2
n4
e7
n6
ECE 453
n7
University of Waterloo
26
Cyclomatic Number
2
ECE 453
0
0
Agenda

Undirected Graphs








Directed Graphs








Degree of a Node
Incidence Matrices
Adjacency Matrices
Paths
Connectedness
Condensation Graphs
Cyclomatic Number
Indegrees and Outdegrees
Types of Nodes
Adjacency Matrix of a Directed Graph
Paths and Semi – Paths
Reachability Matrix
N – Connectedness
Strong Components
Graphs for Testing



ECE 453
Program Graphs
Finite State Machines
Petri Nets
University of Waterloo
28
Directed Graphs

A directed graph (digraph) D = (V, E)
consists of a finite set V = {n1, n2,…,nm} of
nodes, and a set E = {e1, e2,…,ep} of
edges, where each edge ek = <ni, nj> is an
ordered pair of nodes ni, nj V


ni: initial (start) node, nj terminal (finish) node
Software concepts represented with
directed graphs: sequential behavior, timeordered events, messages etc.
ECE 453
University of Waterloo
29
Directed Graphs
e1
n1
e3
n5
n2
e5
e4
e2
n3
e6
V = {n1, n2, n3, n4, n5, n6, n7}
E = {e1, e2, e3, e4, e5, e6, e7}
= {<n1, n2>, <n2, n3>, <n5, n2>, <n4, n5>, <n2, n4>, <n3, n4>, <n4, n6>}
n4
e7
n6
ECE 453
n7
University of Waterloo
30
Agenda

Undirected Graphs








Directed Graphs








Degree of a Node
Incidence Matrices
Adjacency Matrices
Paths
Connectedness
Condensation Graphs
Cyclomatic Number
Indegrees and Outdegrees
Types of Nodes
Adjacency Matrix of a Directed Graph
Paths and Semi – Paths
Reachability Matrix
N – Connectedness
Strong Components
Graphs for Testing



ECE 453
Program Graphs
Finite State Machines
Petri Nets
University of Waterloo
31
Indegrees and Outdegrees



The indegree of a node in a directed graph
is the number of distinct edges that have
the node as a terminal node (indeg(n))
The outdegree of a node in a directed
graph is the number of distinct edges that
have the node as an start point (outdeg(n))
deg(n) = indeg(n) + outdeg(n)
ECE 453
University of Waterloo
32
Indegrees and Outdegrees
e1
n1
e3
n5
n2
e5
e4
e2
n3
e6
n4
e7
n6
ECE 453
n7
indeg(n1) = 0
indeg(n2) = 2
indeg(n3) = 1
indeg(n4) = 2
indeg(n5) = 1
indeg(n6) = 1
indeg(n7) = 0
University of Waterloo
outdeg(n1) = 1
outdeg(n2) = 2
outdeg(n3) = 1
outdeg(n4) = 2
outdeg(n5) = 1
outdeg(n6) = 0
outdeg(n7) = 0
33
Agenda

Undirected Graphs








Directed Graphs








Degree of a Node
Incidence Matrices
Adjacency Matrices
Paths
Connectedness
Condensation Graphs
Cyclomatic Number
Indegrees and Outdegrees
Types of Nodes
Adjacency Matrix of a Directed Graph
Paths and Semi – Paths
Reachability Matrix
N – Connectedness
Strong Components
Graphs for Testing



ECE 453
Program Graphs
Finite State Machines
Petri Nets
University of Waterloo
34
Types of Nodes





Source Node: indegree = 0
Sink Node: outdegree = 0
Transfer Node: indegree ≠ 0 &&
outdegree ≠ 0
Isolated Node: indegree = 0 &&
outdegree = 0
External Entities: Source and Sink Nodes
ECE 453
University of Waterloo
35
Types of Nodes
e1
n1
e3
n5
n2
e5
e4
e2
n3
e6
Sink Nodes: n6
Transfer Nodes: n2, n3, n4, n5
n4
e7
n6
ECE 453
Source Nodes: n1
Isolated Nodes: n7
n7
University of Waterloo
36
Agenda

Undirected Graphs








Directed Graphs








Degree of a Node
Incidence Matrices
Adjacency Matrices
Paths
Connectedness
Condensation Graphs
Cyclomatic Number
Indegrees and Outdegrees
Types of Nodes
Adjacency Matrix of a Directed Graph
Paths and Semi – Paths
Reachability Matrix
N – Connectedness
Strong Components
Graphs for Testing



ECE 453
Program Graphs
Finite State Machines
Petri Nets
University of Waterloo
37
Adjacency Matrix of a Directed Graph




The adjacency matrix of a directed graph
D = (V, E) with m nodes is a m x m matrix
A=(a(i, j)), where a(i, j) is a 1 if and only if
there is an edge from node i and node j,
otherwise the element is 0
Not necessarily symmetric
Row sum = outdegree of node
Column sum = indegree of node
ECE 453
University of Waterloo
38
Adjacency Matrix of a Directed Graph
e1
n1
e3
n5
n2
e5
e4
e2
n3
e6
n4
e7
n6
ECE 453
n7
University of Waterloo
39
Agenda

Undirected Graphs








Directed Graphs








Degree of a Node
Incidence Matrices
Adjacency Matrices
Paths
Connectedness
Condensation Graphs
Cyclomatic Number
Indegrees and Outdegrees
Types of Nodes
Adjacency Matrix of a Directed Graph
Paths and Semi – Paths
Reachability Matrix
N – Connectedness
Strong Components
Graphs for Testing



ECE 453
Program Graphs
Finite State Machines
Petri Nets
University of Waterloo
40
Paths and Semi – Paths



A (directed) path is a sequence of edges such
that, for any adjacent pair of edges ei, ej, in the
sequence, the terminal node of the first edge is
the initial node of the second edge
A cycle is a directed path that begins and ends
at the same node
A (directed) semi – path is a sequence of edges
such that, for at least one adjacent pair of edges
ei, ej in the sequence, the initial node of the first
edge is the initial node of the second edge or the
terminal node of the first edge is the terminal
node of the second edge
ECE 453
University of Waterloo
41
Paths and Semi – Paths

Directed paths also called chains
e1
n1
e3
n5
n2
e5
e4
e2
n3
e6
Cycle: n2 → n4 → n5 → n2
n4
Semi – Path: e1 , e3
e7
n6
ECE 453
Path: n1 → n2 → n3
n7
Semi – Path: e2 , e5
University of Waterloo
42
Agenda

Undirected Graphs








Directed Graphs








Degree of a Node
Incidence Matrices
Adjacency Matrices
Paths
Connectedness
Condensation Graphs
Cyclomatic Number
Indegrees and Outdegrees
Types of Nodes
Adjacency Matrix of a Directed Graph
Paths and Semi – Paths
Reachability Matrix
N – Connectedness
Strong Components
Graphs for Testing



ECE 453
Program Graphs
Finite State Machines
Petri Nets
University of Waterloo
43
Reachability Matrix

The reachability matrix of a directed graph
D = (V, E) with m nodes is a m x m matrix
R = (r(i, j)), where r(i, j) is a 1 if and only if
there is a path from node i and node j,
otherwise the element is 0
ECE 453
University of Waterloo
44
Reachability Matrix
e1
n1
e3
n5
n2
e5
e4
e2
n3
e6
n4
e7
n6
ECE 453
n7
University of Waterloo
45
Agenda

Undirected Graphs








Directed Graphs








Degree of a Node
Incidence Matrices
Adjacency Matrices
Paths
Connectedness
Condensation Graphs
Cyclomatic Number
Indegrees and Outdegrees
Types of Nodes
Adjacency Matrix of a Directed Graph
Paths and Semi – Paths
Reachability Matrix
N – Connectedness
Strong Components
Graphs for Testing



ECE 453
Program Graphs
Finite State Machines
Petri Nets
University of Waterloo
46
N – Connectedness

Two nodes ni and nj in a directed graph
are:
0-connected iff there is no path between ni
and nj
 1-connected iff there is a semi-path but NO
path between ni and nj
 2-connected iff there is a path between ni and
nj
 3-connected iff there is a path from nj to ni and
a path from ni to nj

ECE 453
University of Waterloo
47
N – Connectedness
e1
n1
e3
n5
n2
e5
e4
e2
n3
e6
ECE 453
n1 and n8 are 1-connected
n2 and n6 are 2-connected
n4
n3 and n5 are 3-connected
e7
n8
n1 and n7 are 0-connected
n6
n7
University of Waterloo
48
Agenda

Undirected Graphs








Directed Graphs








Degree of a Node
Incidence Matrices
Adjacency Matrices
Paths
Connectedness
Condensation Graphs
Cyclomatic Number
Indegrees and Outdegrees
Types of Nodes
Adjacency Matrix of a Directed Graph
Paths and Semi – Paths
Reachability Matrix
N – Connectedness
Strong Components
Graphs for Testing



ECE 453
Program Graphs
Finite State Machines
Petri Nets
University of Waterloo
49
Strong Components


A strong component of a directed graph is
a maximal set of 3-connected nodes
Practically loops and isolated nodes are
removed and replaced by strong
components (directed acyclic graphs)
ECE 453
University of Waterloo
50
Condensation Graphs
e1
n1
e3
n5
n2
e5
e4
e2
n3
n1
e6
→
n4
e1
S1
e7
e7
n6
n6
n7
S2
S1 = {n2, n3, n4, n5}
S2 = {n7}
ECE 453
University of Waterloo
51
Agenda

Undirected Graphs








Directed Graphs








Degree of a Node
Incidence Matrices
Adjacency Matrices
Paths
Connectedness
Condensation Graphs
Cyclomatic Number
Indegrees and Outdegrees
Types of Nodes
Adjacency Matrix of a Directed Graph
Paths and Semi – Paths
Reachability Matrix
N – Connectedness
Strong Components
Graphs for Testing



ECE 453
Program Graphs
Finite State Machines
Petri Nets
University of Waterloo
52
Program Graphs

A program graph is a directed graph in which:



Traditional Definition: Nodes are program statements,
and edges represent flow of control (there is an edge
from node i to node j iff the statement corresponding
to node j can be executed immediately after the
statement corresponding to node i)
Improved Definition: Nodes are either entire
statements or fragments of a statement, and edges
represent flow of control (there is an edge from node i
to node j iff the statement or statement fragment
corresponding to node j can be executed immediately
after the statement or statement fragment
corresponding to node i)
Primarily used at the unit testing level
ECE 453
University of Waterloo
53
Program Graphs
ECE 453
University of Waterloo
54
Agenda

Undirected Graphs








Directed Graphs








Degree of a Node
Incidence Matrices
Adjacency Matrices
Paths
Connectedness
Condensation Graphs
Cyclomatic Number
Indegrees and Outdegrees
Types of Nodes
Adjacency Matrix of a Directed Graph
Paths and Semi – Paths
Reachability Matrix
N – Connectedness
Strong Components
Graphs for Testing



ECE 453
Program Graphs
Finite State Machines
Petri Nets
University of Waterloo
55
Finite State Machines

A Finite State Machine (FSM) is a directed
graph (S, T, Ev, Act), in which
S: set of nodes (states)
 T: set of edges (transitions)
 Ev, Act: sets of events and actions associated
with the transitions in T


Primarily used to describe system level
behavior
ECE 453
University of Waterloo
56
Finite State Machines

Example: Lamp
Node
Transition
Event
Action
ECE 453
University of Waterloo
57
Agenda

Undirected Graphs








Directed Graphs








Degree of a Node
Incidence Matrices
Adjacency Matrices
Paths
Connectedness
Condensation Graphs
Cyclomatic Number
Indegrees and Outdegrees
Types of Nodes
Adjacency Matrix of a Directed Graph
Paths and Semi – Paths
Reachability Matrix
N – Connectedness
Strong Components
Graphs for Testing



ECE 453
Program Graphs
Finite State Machines
Petri Nets
University of Waterloo
58
Petri Nets

A Petri net is a bipartite directed graph (P,
T, In, Out), in which
P (places), T (transitions): disjoint sets of
nodes
 In, Out: sets of edges, where In P x T, and
Out T x P


Primarily used to describe system level
behavior
ECE 453
University of Waterloo
59
Petri Nets
P = {p1, p2, p3, p4}
T = {t1, t2, t3}
In = {<p1, t1>, <p1, t2>, <p2, t2>, <p4, t3>}
Out = {<t1, p3>, <t2, p4>, <t3, p3>}
ECE 453
University of Waterloo
60
Petri Nets


A marked Petri net is a 5-tuple (P, T, In,
Out, M) in which (P, T, In, Out) is a Petri
net and M (marking set) is a set of
mappings of places to positive integers
The number associated with a place refers
to the number of tokens that are said to be
“in” the place

ECE 453
e.g. number of times a place has been used
University of Waterloo
61
Petri Nets
P = {p1, p2, p3, p4}
T = {t1, t2, t3}
In = {<p1, t1>, <p1, t2>, <p2, t2>, <p4, t3>}
Out = {<t1, p3>, <t2, p4>, <t3, p3>}
M = {2, 0, 1, 3}
ECE 453
University of Waterloo
62
Petri Nets

A transition in a Petri net is enabled if
there is at least one token in each of its
input places
Enabled Transitions: t1, t3
ECE 453
University of Waterloo
63
Petri Nets

When an enabled Petri net transition fires,
one token is removed from each of its
input places, and one token is added to
each of its output places
ECE 453
University of Waterloo
64
Petri Nets
t3 fired
→
M = {<2, 0, 1, 3>, <2, 0, 2, 2>}
ECE 453
University of Waterloo
65
Petri Nets


Conservative Petri nets: total number of
tokens never changes
Petri net conflicts
Transitions t1 and t2 are in conflict
with respect to place p1
ECE 453
University of Waterloo
66
Petri Net Example

http://www.informatik.unihamburg.de/TGI/PetriNets/introductions/a
alst/elevator1.swf
ECE 453
Agenda

Undirected Graphs








Directed Graphs








Degree of a Node
Incidence Matrices
Adjacency Matrices
Paths
Connectedness
Condensation Graphs
Cyclomatic Number
Indegrees and Outdegrees
Types of Nodes
Adjacency Matrix of a Directed Graph
Paths and Semi – Paths
Reachability Matrix
N – Connectedness
Strong Components
Graphs for Testing



ECE 453
Program Graphs
Finite State Machines
Petri Nets
University of Waterloo
68
References

Software Testing A Craftsman's Approach
2nd edition, Paul C. Jorgensen, CRC
Press
ECE 453
University of Waterloo
69