Graph Theory Techniques in Model-Based Testing

Harry Robinson Semantic Platform Test Group Microsoft

Graphs and testing go well together Hmm … if I am here and I do this I should end up there

here

this

there

Executing a test can be viewed as traversing a state graph ...

1

Königsberg, 1736 ?

Is it possible to cross each bridge exactly once?

The Birth of Graph Theory node

link

Königsberg’s problem: too many odd vertices!

2

So … what does this have to do with the cost of mail delivery in China ?

China, 1962 3

a

?

b

c g

3

5 f

e

d 3

How to visit all the mailboxes in the fewest steps?

3

Chinese Postman algorithm 4

a

!

b b

c g

4

After “Eulerizing”

6 f

e

d 4

d

A solution: a b c b e f g d d

New York, 1977 +1

a

?

c g

-1

b

-1 f

e

d +1

How to visit all the garbage cans with the least driving?

4

New York Street Sweeper algorithm 0

a

!

c g

0

b

b

e

e

0

g f d 0

A solution: a b c b f e g d e g

So … what does this have to do with the cost of software delivery in Redmond?

5

A Testing Problem B

a

?

c g

A

b

C f

e

d D

How can I test all these actions in the fewest steps?

The Street Sweeper strikes again B

a

!

c g

A

b

b

e

e

C

g f d D

A solution: a b c b f e g d e g

6

A ‘Safecracking’ Problem B

a

??

c g

A

b

C f

e

d D

How can I test all the pairs of actions in the fewest steps?

De Bruijn sequences a

b

a c g d

b g

f

c

f

e d

e

A solution: a b c b f e c b g d e f e g

7

Testing the Most Likely Paths P(a)=.9 a

???

B b

P(c)=.02 c

A

g P(g)=.9 P(f)=.08

C e

f

d

P(d)=.1

D

How can I test the likeliest sequences first?

Markov Chains + Depth First Traversal g b a

d

e

0.81

f

e

g

c

b

g

g f

0.0648 0.0162 0.09

e

g

0.0072

abg deg abfeg abcbg defeg

8

Capacitated Vehicle problem c d

????

e a

b

i

f g h

How can I test all these actions in the shortest time?

The Street Sweeper solution c d e a

b

i

f

abcbdbebfbgbhbi

g h

The street sweeper approach takes 15 time intervals.

9

But distributing the work ... a

b

c

b

i

a

b

d

b

i

a

b

e

b

i

a

b

f

b

i

a

b

g

b

i

a

b

h

b

i

… gets the job done in 1/3 the time!

Future Directions ?

Random walks

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Levy Flights

Probability of step’s length is in inverse proportion to the square of that length.

Anti-Random Walks

Rather than taking choosing each step independently of what went before, choose each next step that is most unlike its predecessors.

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Models + Algorithms = Testing • Models are good at representing systems • Graph theory can use models to generate tests • Different algorithms can provide tests to suit your needs: – – – – – – –

Street sweepers Safecracking Markov chains Capacitated vehicles Random walks Levy flights Anti-random walks

Thank you!

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