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Graph Coverage (3)Graph Coverage (3)

Reading AssignmentReading AssignmentP. Ammann and J. Offutt

“Introduction to Software Testing”◦Section 2.2◦Section 2.3

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OutlineOutlineData Flow Criteria

◦DU Pairs and DU PathsGraph Coverage for Source Code

◦Basic Block◦Control Flow Graph (CFG) – If statements◦CFG – Loops

CFG – Test requirements and Test paths

Data Flow Graph Coverage for Source Code

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Data Flow CriteriaData Flow Criteria

Definition (def) : A location where a value for a variable is stored into memory

Use : A location where a variable’s value is accessed

Goal: Try to ensure that values are computed and used correctly

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Data Flow CriteriaData Flow Criteria

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4X = 42

Z = X-8

Z = X*2

Defs: def (0) = {X}

def (4) = {Z}

def (5) = {Z}

Uses: use (4) = {X}

use (5) = {X}

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DU Pairs and DU PathsDU Pairs and DU PathsDU pair : A pair of locations (li, lj)

such that a variable v is defined at li and used at lj

Def-clear : A path from li to lj is def-clear with respect to variable v if v is not given another value on any of the nodes or edges in the path

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DU Pairs and DU PathsDU Pairs and DU PathsReach : If there is a def-clear path

from li to lj with respect to v, the def of v at li reaches the use at lj

du-path : A simple subpath that is def-clear with respect to v from a def of v to a use of v

du (ni, nj, v) – the set of du-paths from ni to nj

du (ni, v) – the set of du-paths that start at ni

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Data Flow Test CriteriaData Flow Test Criteria

All-defs coverage (ADC)All-defs coverage (ADC) : For each set of du-paths : For each set of du-paths SS = = dudu ( (nn, , vv), TR contains at least one path ), TR contains at least one path dd in in SS..

All-uses coverage (AUC)All-uses coverage (AUC) : For each set of du-paths to : For each set of du-paths to uses uses SS = = dudu ( (nnii, n, njj, , vv), TR contains at least one path ), TR contains at least one path dd

in in SS..

• Then we make sure that every def reaches all possible uses

• First, we make sure every def reaches a use

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Data Flow Test CriteriaData Flow Test Criteria

All-du-paths coverage (ADUPC)All-du-paths coverage (ADUPC) : For each set : For each set SS = = dudu ( (ni, njni, nj, , vv), TR contains every path ), TR contains every path dd in in SS..

• Finally, we cover all the paths between defs and uses

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Data Flow Testing Data Flow Testing ExampleExample

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4X = 42

Z = X-8

Z = X*2

All-defs for X

[ 0, 1, 3, 4 ]

All-uses for X

[ 0, 1, 3, 4 ]

[ 0, 1, 3, 5 ]

All-du-paths for X

[ 0, 1, 3, 4 ]

[ 0, 2, 3, 4 ]

[ 0, 1, 3, 5 ]

[ 0, 2, 3, 5 ]

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Graph Coverage Criteria Graph Coverage Criteria Subsumption Subsumption

Node Coverage

NC

Edge Coverage

EC

Edge-Pair Coverage

EPC

Prime Path Coverage

PPC

Complete Path Coverage

CPC

All-DU-Paths Coverage

ADUP

All-uses Coverage

AUC

All-defs Coverage

ADC

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Control Flow Graphs (CFG) – Control Flow Graphs (CFG) – Source CodeSource CodeA CFG models all executions of a

method by describing control structures

Nodes : Statements or sequences of statements (basic blocks)

Edges : Transfers of control

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Control Flow Graphs (CFG) – Control Flow Graphs (CFG) – Source CodeSource CodeBasic Block : A sequence of

statements such that if the first statement is executed, all statements will be (no branches)

CFGs are sometimes annotated with extra information◦ branch predicates◦ defs◦ uses

Rules for translating statements into graphs …

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CFG : The if Statement CFG : The if Statement if (x < y){ y = 0; x = x + 1;}else{ x = y;}

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x >= yx < y

x = yy = 0

x = x + 1

if (x < y){ y = 0; x = x + 1;}

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2x >= y

x < y

y = 0x = x + 1

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CFG : The if-Return StatementCFG : The if-Return Statement

if (x < y){ return;}print (x);return;

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2x >= y

x < y

return

print (x)return

No edge from node 2 to 3.The return nodes must be distinct.

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LoopsLoops

Loops require “extra” nodes to be added

Nodes that do not represent statements or basic blocks

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CFG : while and for LoopsCFG : while and for Loops

x = 0;while (x < y){ y = f (x, y); x = x + 1;}

1x = 0

43y =f(x,y)x = x + 1

x >= yx < y

for (x = 0; x < y; x++){ y = f (x, y);}

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x = x + 1

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x >= yx < y

y = f (x, y)

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dummy node

x = 0implicitly

initializes loop

implicitly increments loop

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CFG : The case (switch) CFG : The case (switch) StructureStructure

read ( c) ;switch ( c ){ case ‘N’: y = 25; break; case ‘Y’: y = 50; break; default: y = 0; break;}print (y);

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1 read ( c );

c == ‘N’

y = 0;break;

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c == ‘Y’ default

y = 50;break;

y = 25;break;

print (y);

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Example Control Flow – Example Control Flow – computeStatscomputeStatspublic static void computeStats (int [ ] numbers){ int length = numbers.length; double med, var, sd, mean, sum, varsum;

sum = 0; for (int i = 0; i < length; i++) { sum += numbers [ i ]; } med = numbers [ length / 2 ]; mean = sum / (double) length;

varsum = 0; for (int i = 0; i < length; i++) { varsum = varsum + ((numbers [ I ] - mean) * (numbers [ I ] - mean)); } var = varsum / ( length - 1.0 ); sd = Math.sqrt ( var );

System.out.println ("length: " + length); System.out.println ("mean: " + mean); System.out.println ("median: " + med); System.out.println ("variance: " + var); System.out.println ("standard deviation: " + sd);}

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Control Flow Graph for Control Flow Graph for computeStatscomputeStatspublic static void computeStats (int [ ] numbers){ int length = numbers.length; double med, var, sd, mean, sum, varsum;




i = 0

i >= length

i < lengthi++

i >= lengthi < length

i = 0

i++

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Control Flow TRs and Test Control Flow TRs and Test Paths – ECPaths – EC

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TRA. [ 1, 2 ]B. [ 2, 3 ]C. [ 3, 4 ]D. [ 3, 5 ]E. [ 4, 3 ]F. [ 5, 6 ]G. [ 6, 7 ]H. [ 6, 8 ]I. [ 7, 6 ]

Test Path[ 1, 2, 3, 4, 3, 5, 6, 7, 6, 8 ]

Edge Coverage

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Control Flow TRs and Test Paths Control Flow TRs and Test Paths – EPC– EPC

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TRA. [ 1, 2, 3 ]B. [ 2, 3, 4 ]C. [ 2, 3, 5 ]D. [ 3, 4, 3 ]E. [ 3, 5, 6 ]F. [ 4, 3, 5 ]G. [ 5, 6, 7 ]H. [ 5, 6, 8 ]I. [ 6, 7, 6 ]J. [ 7, 6, 8 ]K. [ 4, 3, 4 ]L. [ 7, 6, 7 ]

Test Pathsi. [ 1, 2, 3, 4, 3, 5, 6, 7, 6, 8 ]ii. [ 1, 2, 3, 5, 6, 8 ]iii. [ 1, 2, 3, 4, 3, 4, 3, 5, 6, 7, 6, 7, 6, 8 ]

Edge-Pair Coverage

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Control Flow TRs and Test Paths Control Flow TRs and Test Paths – PPC– PPC

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TRA. [ 3, 4, 3 ]B. [ 4, 3, 4 ]C. [ 7, 6, 7 ]D. [ 7, 6, 8 ]E. [ 6, 7, 6 ]F. [ 1, 2, 3, 4 ]G. [ 4, 3, 5, 6, 7 ]H. [ 4, 3, 5, 6, 8 ]I. [ 1, 2, 3, 5, 6, 7 ]J. [ 1, 2, 3, 5, 6, 8 ]

Test Pathsi. [ 1, 2, 3, 4, 3, 5, 6, 7, 6, 8 ]ii. [ 1, 2, 3, 4, 3, 4, 3, 5, 6, 7, 6, 7, 6, 8 ]iii. [ 1, 2, 3, 4, 3, 5, 6, 8 ]iv. [ 1, 2, 3, 5, 6, 7, 6, 8 ]v. [ 1, 2, 3, 5, 6, 8 ]

Prime Path Coverage

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Data Flow Graph Coverage Data Flow Graph Coverage for Source Codefor Source Codedef : a location where a value is

stored into memory◦x appears on the left side of an

assignment (x = 44;)◦x is an actual parameter in a call

and the method changes its value◦x is an input to a program

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Data Flow Graph Coverage for Data Flow Graph Coverage for Source CodeSource Codeuse : a location where variable’s

value is accessed◦x appears on the right side of an

assignment◦x appears in a conditional test◦x is an actual parameter to a method◦x is an output of the program◦x is an output of a method in a

return statement

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Example Data Flow – Example Data Flow – computeStatscomputeStatspublic static void computeStats (int [ ] numbers)

{ int length = numbers.length; double med, var, sd, mean, sum, varsum;




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Control Flow Graph for Control Flow Graph for computeStats computeStats

( numbers )sum = 0length = numbers.length

i = 0

i >= length

i < length

sum += numbers [ i ]i++

med = numbers [ length / 2 ]mean = sum / (double) length;varsum = 0i = 0

i >= length

i < length

varsum = …i++

var = varsum / ( length - 1.0 )sd = Math.sqrt ( var )print (length, mean, med, var, sd)

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CFG for computeStats – With Defs CFG for computeStats – With Defs & Uses& Uses

def (1) = { numbers, sum, length }

def (2) = { i }

def (5) = { med, mean, varsum, i }use (5) = { numbers, length, sum }

def (8) = { var, sd }use (8) = { varsum, length, mean, med, var, sd }

use (3, 5) = { i, length }

use (3, 4) = { i, length }

def (4) = { sum, i }use (4) = { sum, numbers, i }

use (6, 8) = { i, length }

use (6, 7) = { i, length }

def (7) = { varsum, i }use (7) = { varsum, numbers, i, mean }

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Defs and Uses Tables for Defs and Uses Tables for computeStatscomputeStats

Node Def Use

1 { numbers, sum, length }

2 { i }

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4 { sum, i } { numbers, i, sum }

5 { med, mean, varsum, i }

{ numbers, length, sum }

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7 { varsum, i } { varsum, numbers, i, mean }

8 { var, sd } { varsum, length, var, mean, med, var, sd }

Edge Use

(1, 2)

(2, 3)

(3, 4) { i, length }

(4, 3)

(3, 5) { i, length }

(5, 6)

(6, 7) { i, length }

(7, 6)

(6, 8) { i, length }

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DU Paths for computeStatsDU Paths for computeStatsvariable DU Pairs DU Paths

numbers (1, 4)(1, 5)(1, 7)

[ 1, 2, 3, 4 ][ 1, 2, 3, 5 ][ 1, 2, 3, 5, 6, 7 ]

length (1, 5)(1, 8)(1, (3,4))(1, (3,5))(1, (6,7))(1, (6,8))

[ 1, 2, 3, 5 ][ 1, 2, 3, 5, 6, 8 ][ 1, 2, 3, 4 ][ 1, 2, 3, 5 ][ 1, 2, 3, 5, 6, 7 ][ 1, 2, 3, 5, 6, 8 ]

med (5, 8) [ 5, 6, 8 ]var (8, 8) No path neededsd (8, 8) No path neededsum (1, 4)

(1, 5)(4, 4)(4, 5)

[ 1, 2, 3, 4 ][ 1, 2, 3, 5 ][ 4, 3, 4 ][ 4, 3, 5 ]

variable DU Pairs DU Pathsmean (5, 7)

(5, 8)[ 5, 6, 7 ][ 5, 6, 8 ]

varsum (5, 7)(5, 8)(7, 7)(7, 8)

[ 5, 6, 7 ][ 5, 6, 8 ][ 7, 6, 7 ][ 7, 6, 8 ]

i (2, 4)(2, (3,4))(2, (3,5))(4, 4)(4, (3,4))(4, (3,5))(5, 7)(5, (6,7))(5, (6,8))(7, 7)(7, (6,7))(7, (6,8))

[ 2, 3, 4 ][ 2, 3, 4 ][ 2, 3, 5 ][ 4, 3, 4 ][ 4, 3, 4 ][ 4, 3, 5 ][ 5, 6, 7 ][ 5, 6, 7 ][ 5, 6, 8 ][ 7, 6, 7 ][ 7, 6, 7 ][ 7, 6, 8 ]

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Test Cases and Test PathsTest Cases and Test Paths

There are 12 unique DU paths[ 1, 2, 3, 4 ][ 1, 2, 3, 5 ][ 1, 2, 3, 5, 6, 7 ][ 1, 2, 3, 5, 6, 8 ][ 2, 3, 4 ][ 2, 3, 5 ]

[ 4, 3, 4 ][ 4, 3, 5 ][ 5, 6, 7 ][ 5, 6, 8 ][ 7, 6, 7 ][ 7, 6, 8 ]

5 expect a loop not to be “entered”

5 require at least one iteration of a loop

2 require at least two iteration of a loop

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Test Cases and Test PathsTest Cases and Test Paths

Test Case : numbers = (44) ; length = 1Test Path : [ 1, 2, 3, 4, 3, 5, 6, 7, 6, 8 ]Note: At least one iteration of a loop

Test Case : numbers = (2, 10, 15) ; length = 3Test Path : [ 1, 2, 3, 4, 3, 4, 3, 4, 3, 5, 6, 7, 6, 7, 6, 7, 6, 8 ]Note: At least two iterations of a loop

Other DU paths require arrays with length 0 to skip loopsBut the method fails with divide by zero on the statement … mean = sum / (double) length;

A fault wasfound

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Key Points (1/2)Key Points (1/2)Data Flow Criteria tries to ensure that

values are computed and used correctly.Graph coverage criteria sub-sumptionControl Flow Graph – Source Code

◦ If statement◦ Loops◦ Case statements

Applying the graph test criteria to control flow graphs is relatively straightforward◦ Most of the developmental research work

was done with CFGs◦ A few subtle decisions must be made to

translate control structures into the graph

Key Points (2/2)Key Points (2/2)DU Pairs

◦ def : a location where a value is stored into memory

◦ use : a location where variable’s value is accessed

◦ Test cases and DU Paths

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