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Graph Coverage (1)Graph Coverage (1)

Reading AssignmentReading AssignmentP. Ammann and J. Offutt

“Introduction to Software Testing”◦Section 2.1

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OutlineOutlineCovering Graphs

◦Definition of a Graph◦Paths in Graph

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Covering GraphsCovering GraphsGraphs are the most commonly

used structure for testingGraphs can come from many

sources◦ Control flow graphs◦ Design structure◦ FSMs and statecharts◦ Use cases

Tests usually are intended to ‘cover’ the graph in some way.

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Definition of a GraphDefinition of a GraphA set N of nodes, N is not empty

A set N0 of initial nodes, N0 is not empty

A set Nf of final nodes, Nf is not empty

A set E of edges, each edge from one node to another◦ ( ni , nj ), ni is predecessor, nj is successor◦ Initial nodes appear in edges with no predecessor◦ Final nodes are denoted with heavy borders

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Three Example GraphsThree Example Graphs0

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N0 = { 0 }

Nf = { 3 }

0

21

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N0 = { }

Nf = { 3 }

9

0

43

7

1

5

8

2

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N0 = { 0, 1, 2 }

Nf = { 7, 8, 9 }

Not aNot avalidvalidgraphgraph

Sometimes graphs are restricted to those having oneinitial node and one final node. (how?)

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Paths in GraphsPaths in Graphs

Path : A sequence of nodes – [n1, n2, …, nM]◦ Each pair of nodes is an edge

Length : The number of edges◦ A single node is a path of length 0

Subpath : A subsequence of nodes in p is a subpath of pReach (n) : Subgraph that can be reached from n

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Paths in GraphsPaths in Graphs

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0 1 2

43 5 6

Paths

[ 0, 3, 7 ]

[ 1, 4, 8, 5, 1 ]

[ 2, 6, 9 ]

Reach (0) = { 0, 3, 4, 7, 8, 5, 1, 9 }

Reach ({0, 2}) = G

Reach([2,6]) = {2, 6, 9}

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Test Paths and SESEsTest Paths and SESEs

Test Path : A path that starts at an initial node and ends at a final node

Test paths represent execution of test cases◦ Some test paths can be executed by many

tests

◦Some test paths cannot be executed by any tests

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Test Paths and SESEsTest Paths and SESEsSESE graphs : All test paths start

at a single node and end at another node◦ Single-entry, single-exit

◦ N0 and Nf have exactly one node

0

2

1

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Double-diamond graphFour test paths[ 0, 1, 3, 4, 6 ][ 0, 1, 3, 5, 6 ][ 0, 2, 3, 4, 6 ][ 0, 2, 3, 5, 6 ]

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Visiting and TouringVisiting and Touring

Visit : A test path p visits node n if n is in p A test path p visits edge e if e is in pTour : A test path p tours subpath q if q is a subpath of p

Path [ 0, 1, 3, 4, 6 ]

Visits nodes 0, 1, 3, 4, 6

Visits edges (0, 1), (1, 3), (3, 4), (4, 6)

Tours subpaths (0, 1, 3), (1, 3, 4), (3, 4, 6), (0, 1, 3, 4), (1, 3, 4, 6)

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Tests and Test PathsTests and Test Paths

path (t) : The test path executed by test tpath (T) : The set of test paths executed by the set

of tests TEach test executes one and only one test pathA location in a graph (node or edge) can be reached

from another location if there is a sequence of edges from the first location to the second◦ Syntactic reach : A subpath exists in the graph◦ Semantic reach : A test exists that can execute

that subpath

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Tests and Test PathsTests and Test Pathstest 1

test 2

test 3

many-to-one

test 1

test 2

test 3

many-to-manyTest Path 1

Test Path 2

Test Path 3

Non-deterministic software – a test can execute different test paths

Test Path

Deterministic software – a test always executes the same test path

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Key PointsKey PointsGraphs are the most commonly

used structure for testingGraphs are used in Testing

◦Developing a model of the software as a graph

◦Requiring test cases to visit or tour specific sets of nodes, edges or sub-paths.