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14.2 GraphFunctionsAndProcedures
167
Inthetwoargumentversion,onlyedges ofthetypegivenassecondargumentarecon-
tained. Thethreeargumentversionbehavesasthetwoargumentversion,butadditionally
onlyedgesincidenttoanoppositenodeofthetypegivenasthirdargumentarecontained.
EXAMPLE(83)
1
rulefoo {
2
src:Node -e:Edge->;
3
if(!isIncoming(src, e));
4
5
modify{
6
eval {
7
if(incident(src,NiftyEdge,NiftyNode).size()>2)
8
{
9
for(ne:NiftyEdgein outgoing(src, NiftyEdge))
10
{
11
ne.attr= 42;
12
}
13
}
14
}
15
}
16
}
Some fabricated example showing how to use the incoming, outgoing, and incident func-
tions in n their r boolean n predicate, set t function, , andfor iteration n versions, , withand without
constrainingtheedgeandnodetypes.
Availablearequeriesfortheneighbouringnodes:
adjacent(.)
returnsthesetofthenodesthatareadjacenttothenodegivenasargumentvalue.
adjacentIncoming(.)
sameastheadjacentabove,butrestrictedtonodesreachableviaincomingedges.
adjacentOutgoing(.)
sameastheadjacentabove,butrestrictedtonodesreachableviaoutgoingedges.
In the e two o argument t version, , nodes s incident to an edge e of the type given n as s second
argumentarecontained. Thethreeargumentversionbehavesasthetwoargumentversion,
butadditionallyonlynodesofthenodetypegivenasthirdargumentarecontained.
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168
GraphTypeandComputations
EXAMPLE(84)
Anexampleshowinghowtoprogramaruleby handwiththe graphqueryoperationsfor
matchingandthegraphupdateoperationsforrewriting.ThisiswhatGrGendoesunderthe
covers,andasyoucanseefromthevolumeandstyleofcodeveryhelpful{usethepattern
language,don’tfallbacktothecomputationslanguageunlessreallyneeded.
1
ruleexample
2
{
3
x:N -e:E->y:N<--> z:M;
4
if {e.a==42; }
5
6
modify{
7
delete(y);
8
xn:NN<x>;
9
xn--> yn:N-->z;
10
}
11
}
12
// thisprocedurebehavessimilarilyto the ruleabove
13
procedureexample
14
{
15
// matchLHSpattern
16
defvarleave:boolean= false;
17
for(x:Ninnodes(N)) // lookupnoftype Ninthegraph
18
{
19
x.visited =true; // (see14.5 VisitedFlagsbelow)
20
for(e:Einoutgoing(x,E)) //fromxonfindoutgoingedgeeoftypeE
21
{
22
if(!(e.a==42)) {//with e.a ==42
23
continue;
24
}
25
def y:Node =target(e); //andtarget nodey oftypeN
26
if(!(typeof(y)<=N)){
27
continue;
28
}
29
if(y.visited) {//that isnot the sameasx
30
continue;
31
}
32
for(z:Node inadjacent(y,Edge, M))//fromy on findadjacentnodez of typeM
33
{//N and Maredisjoint,can’tmatcheachother,otherwisevisited wouldbeneeded
34
35
//rewriteaccording toRHSpattern
36
rem(y);
37
(defxn:NN)=retype(x,NN);
38
(defyn:N)=add(N);
39
add(Edge,xn, yn);
40
add(Edge,yn, z);
41
42
leave= true;break;
43
}
44
if(leave) {break;}
45
}
46
x.visited =false;
47
if(leave) {break;}
48
}
49
return;
50
}
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14.2 GraphFunctionsAndProcedures
169
TransitiveNodeNeighbourhood
Besidesdirectneighbourhood,transitiveneighbourhoodcanbequeriedwiththereachability
functions.
Availablearequeriesforthereachableedges:
reachableEdges(.)
returnsthesetoftheedgesthatarereachablefromthenodegivenasargumentvalue.
reachableEdgesIncoming(.)
sameasthereachableEdgesabove,butrestrictedtoincomingedges.
reachableEdgesOutgoing(.)
sameasthereachableEdgesabove,butrestrictedtooutgoingedges.
Inthetwoargumentversion,onlyedges ofthetypegivenassecondargumentarecon-
tainedandfollowed. Thethreeargumentversionbehavesasthetwoargumentversion,but
additionallyonlyedgesincidenttoanoppositenodeofthetypegivenasthirdargumentare
containedandfollowed.
Availablearequeriesforthereachablenodes:
reachable(.)
returnsthesetofthenodesthatarereachablefromthenodegivenasargumentvalue.
reachableIncoming(.)
sameasany of the reachables above, , butrestrictedtonodes s reachableviaincoming
edges.
reachableOutgoing(.)
same as anyofthe reachables above, but restrictedtonodes reachable via outgoing
edges.
Inthetwoargumentversion,nodesincidenttoanedgeofthetypegivenassecondargu-
mentarecontainedandfollowed. Thethreeargumentversionbehavesasthetwoargument
version,butadditionallyonlynodesofthenodetypegivenasthirdargumentarecontained
andfollowed.
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170
GraphTypeandComputations
EXAMPLE(85)
1
rulebar {
2
src:Node;
3
tgt:Node;
4
if(isReachableOutgoing(src,tgt, NiftyEdge));
5
6
modify{
7
eval {
8
if(!(reachableIncoming(src) &reachableOutgoing(src)).empty())
9
{
10
for(ne:NiftyEdge inreachable(src))
11
{
12
ne.attr= 42;
13
}
14
}
15
}
16
}
17
}
SomefabricatedexampleshowinghowtousetheisReachableOutgoingfunctiontocheckfor
aniteratedpathbetweenthesrcandtargetnodes,howtocheckforloopsbyintersecting
thesetsofnodesreachablebyoutgoingedgesfromanodeandreachablebyincomingedges
toanode,andhowtoiteratewithoneloopoveralledgesreachableineither wayfroma
node.
TheisReachablefunctionsgivethemost ecientandmostconvenientwaytocheckforan
iteratedpathinGrGen,ifyouneedmoreelaboratecheckingthanconstrainingtheedgetype
toonetypeandthetargetnodetypetoonetypeyouneedtoprogramtheiteratedpathwith
subpatternrecursion.
Thereachableiterationisthemostconcisewaytonotedownadepthrstwalkoveragraph,
visitingallelementsreachablefromasourcenodeon.
BoundedTransitiveNodeNeighbourhood
Transitive neighbourhood d can be e queried also o with a a path h of f bounded d length, , with the
boundedreachabilityfunctions(queryingforaboundediteratedpath).
Availablearequeriesforthereachable-within-boundsedges:
boundedReachableEdges(.,.)
returns thesetoftheedgesthatarereachablefromthenodegivenasrst argument
value,withinatmostasmuchstepsasspeciedbythesecondargument.
boundedReachableEdgesIncoming(.,.)
sameasthereachableEdgesabove,butrestrictedtoincomingedges.
boundedReachableEdgesOutgoing(.,.)
sameasthereachableEdgesabove,butrestrictedtooutgoingedges.
Inthethreeargumentversion,onlyedges ofthetypegivenasthirdargumentarecon-
tainedandfollowed.Thefourargumentversionbehavesasthethreeargumentversion,but
additionally only edgesincident toanoppositenodeofthetype givenasfourthargument
arecontainedandfollowed.
Availablearequeriesforthereachable-within-boundsnodes:
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14.2 GraphFunctionsAndProcedures
171
boundedReachable(.,.)
returnsthesetofthenodesthatarereachablefromthenodegivenasrst argument
value,withinatmostasmuchstepsasspeciedbythesecondargument.
boundedReachableIncoming(.,.)
sameasany of the reachables above, , butrestrictedtonodes s reachableviaincoming
edges.
boundedReachableOutgoing(.,.)
same as anyofthe reachables above, but restrictedtonodes reachable via outgoing
edges.
Inthethreeargumentversion,nodesincidenttoanedgeofthetypegivenasthirdargu-
mentarecontainedandfollowed.Thefourargumentversionbehavesasthethreeargument
version,butadditionallyonlynodesofthenodetypegivenasfourthargumentarecontained
andfollowed.
EXAMPLE(86)
1
// iterativedeepening istrivialduetoboundedReachable(encapsulatinga depth-bounded
depthfirstsearch)
2
procedureIterativeDeepening(root:Node, var maxdepth:int) : (boolean)
3
{
4
for(depth:intin[0:maxdepth])
5
{
6
for(n:NodeinboundedReachableOutgoing(root,depth))
7
{
8
if(foundCondition()) {
9
return(true);
10
}
11
}
12
}
13
return(false);
14
}
An example showing how to use boundedReachable e inimplementing iterative deepening.
You ndit t and d anexample for r depth-rst search, , as s well l as s an n example forbreadth-rst
searchimplementedwithadequeintest/should
pass,in/DfsBfsSearch.grg.
Available are queries for thereachable-within-bounds nodes s that t return n the minimum
distancetothestartnodeinaddition:
boundedReachableWithRemainingDepth(.,.)
returns the e mapof the nodes that are reachable from the node givenas rst t argu-
mentvalue,withinatmostasmuchstepsasspeciedbythesecondargument,tothe
remainingdepth.
boundedReachableWithRemainingDepthIncoming(.,.)
sameasany of the reachables above, , butrestrictedtonodes s reachableviaincoming
edges,totheremainingdepth.
boundedReachableWithRemainingDepthOutgoing(.,.)
same as anyofthe reachables above, but restrictedtonodes reachable via outgoing
edges,totheremainingdepth.
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172
GraphTypeandComputations
Inthethreeargumentversion,nodesincidenttoanedgeofthetypegivenasthirdargu-
mentarecontainedandfollowed.Thefourargumentversionbehavesasthethreeargument
version,butadditionallyonlynodesofthenodetypegivenasfourthargumentarecontained
andfollowed.Thedepthvaluefortheminimumdistanceofthenodetotherootnodeinthe
resultmapiscounteddownwardsfromthedepthrequestedoriginally,youmustsubtractit
fromthedepthrequestedtogettherealdistance.
14.3 SubgraphOperations
Severalfunctionsandproceduresreturningandacceptingsubgraphsareavailable;theyare
especiallyusefulinstatespaceenumeration,cf.19.7,butalsoingraph-orientedprogramming
withthestructuringandinformationhidingsupportedby hierarchicallynestedgraphs,cf.
16.3.
Inadditiontothosecomputationsexplainedbelow,youcanaccessthecurrentlyprocessed
graphviathethisvariablethatisavailableintheexpressionsofthesequencesandtherules.
Bydefaultitisboundtothehostgraph,butifprocessingwasrelocatedinthesequencesto
asubgraph,itisboundtothecurrentlyprocessedsubgraph.
Theglobalfunctionsallowtocompute(node-or-edge)inducedsubgraphsandclonesub-
graphs:
inducedSubgraph(.)
returnstheinducedsubgraph(type:graph)ofthehostgraphforthesetofnodesgiven
asargumentvalue.
definedSubgraph(.)
returns thedened(edge-induced)subgraph(type: : graph)ofthehostgraphfor r the
setofedgesgivenasargumentvalue.
copy(.)
returnsacloneoftheoriginalsubgraphgivenasargument.
Youmayfurtheroncheckforsubgraphequalityagainstasetofsubgraphsatonce:
equalsAny(.,.)
returnswhetherthe(sub)graphgivenasrstargument(type: graph)equalsanyofthe
(sub)graphs available intheset of subgraphs givenwiththesecondargument(type:
set<graph>).
equalsAnyStructurally(.,.)
returns whether the (sub)graphgivenas rst argument (type: : graph) ) isstructurally
(neglecting attribute values) ) equal to o any of the (sub)graphs available in the set of
subgraphsgivenwiththesecondargument(type: set<graph>).
Therstfunctionemploys ==inaloopuntiladecisionisreached,thesecondfunction
doessowith~~. Theyareabitmoreconvenientinusagecomparedtowritingtheloopby
hand,butaboveallaretheyamenabletoparallelization. Youmustspecifythe(maximum)
numberofworkerthreadstoenableparallelexecution,seesection22.4.
Theglobalproceduresallowtoinsertclonesofsubgraphscomputedwiththepreviously
introducedfunction,ortoinsertclonesofinducedsubgraphsdirectly.
insert(.)
inserts a givensubgraph h to thecurrent t host graph(disjoint union n of the nodes s and
edges);theoriginalgraphisdestroyedbythis(movesemantics).
14.4 FileOperations
173
insertCopy(.,.)
inserts acloneofthegivensubgraphtothecurrenthostgraph(disjointunionofthe
nodesandedges);theoriginalsubgraphstaysuntouched.Returnsthecopyofthenode
givenassecondargumentfromthehostgraph.
insertInduced(.,.)
addsacloneofthesubgraphinducedbythesetofnodesgivenasrstargumenttothe
hostgraph,returnsthecloneoftheanchornodegivenassecondargument.
insertDefined(.,.)
addsacloneofthesubgraphdened(edge-induced)bythesetofedgesgivenas rst
argument to o the host t graph, returns the clone e of the anchor r edge e given n as s second
argument.
14.4 FileOperations
Thefunctions fromthebuilt-inpackageFile allowtoimport subgraphsandcheckforle
existance(formoreonpackagessee16.2.2).
File::import(.)
returnsthe(sub)graphstoredinthe.grs-legivenbyitspath(themaingraphisnot
replaced,thegraphmustbecompatibletothemodelofthecurrenthostgraph).
File::exists(.)
returnswhetherthelegivenbyitspathexists.
Theproceduresfromthebuilt-inpackageFileallowtoexportsubgraphsanddeleteles.
File::export(.)
exportsthecurrenthostgraphtoalewiththegivenpath.
File::export(.,.)
exportsthesubgraphgivenasrstargumenttoalewiththegivenpath.
File::delete(.)
deletesthelewiththegivenpath.
174
GraphTypeandComputations
14.5 Graphcomparison
Hereweextendtherelationalexpressionsalreadyintroducedin6.4(andalreadyextendedin
13toincludecontainertypes)withthe(sub)graphtype.
A == B
True,iAisisomorphictoB.
A != B
True,iAisnotisomorphictoB.
A ~~ ~ B
True, i A is s structurally y the e same as B but maybe dierent regardingthe
attributes.
Table14.1: Compareoperatorsongraphexpressions
Thegraphtypesupportthe==,the!=,andthe~~operators;on(sub)graphtypesthey
tellwhetherthe (sub)graphs areisomorphictoeachother(isomorphychecking/graphiso-
morphychecking)ornot,includingtheattributes,orwhetherthe(sub)graphsareisomorphic
disregardingtheattributes.
Theseoperatorsconsistjustoftwocharacters,butdon’tunderestimatetheirimpacton
performance:theydographisomorphychecking,whichisexpensive. Theyareimplemented
inanearlyoutstyle,i.e. themoredierentthegraphsare,theearlierdoesthecheckreturn
withtheresulttheyarenotisomorphic. Butifthegraphsyouarecheckingareisomorphic
(whichwillhappeneasily ifyouuse automorphic patterns),thenyouhave topay thefull
pricefor isomorphy checking;if thisoccurs often, , your r solutionmay becomeprohibitively
costly(includinganexternalgraphcanonizationlibraryorthecanonizefunctionmaybeof
interestinthatcase).
Somenotesontheearlyoutimplementation:rstthenumberofnodesandedgespertype
arechecked,iftheyaredierentthegraphs can’tbeisomorphic. . Thenumbersaredirectly
suppliedbythelgspBackend,refutingisomorphybasedonthemisextremelycheap. Then
theV-Structures(see26.3)usedincomputingbettermatchersatruntimearerstcomputed
andthencompared;iftheyare dierent the graphscan’tbeisomorphic. . Theyareagood
deallessexpensivetocomputethantryingtomatchtheonegraphintheother;onwelltyped
graphstheV-Structurecountsarehighlydiscriminating.
Ifthesetwopruningmethodsfailed,amatcheriscomputedfromonegraphwithsearch
planning based on n the V-Structure information n just computed, , and d then applied on n the
othergraph.Thematchersarestoredinthegraphsfromwhichtheyoriginated,soifyoudo
repeatedcomparisonsofasubgraphwhichdoesnotchange,takecaretoextractthatsubgraph
onlyonce,storeit,e.g.asanattributeinthegraph,andcontinuetocompareagainstit.This
willsaveyoufromthecostof repeatedsearchplanning;inaddition,often-usedisomorphy
matchersgeteventuallycompiledresultinginafurtherspeed-up.
14.5 Graphcomparison
175
EXAMPLE(87)
Anexampleshowinghowtosaveagraphexplodedintoitsconnectedcomponents,andhow
toloaditagainfromthem.
1
proceduresaveConnectedComponents()
2
{
3
defvari:int=0;
4
while(!empty()) {
5
defvarn:Node=fetchNode();
6
defrefconnectedComponent:set<Node> =reachable(n) |set<Node>{n};
7
defvarsub:graph=inducedSubgraph(connectedComponent);
8
File::export(sub, "cc"+i+".grs");
9
deleteSubgraph(connectedComponent);
10
i= i +1;
11
}
12
return;
13
}
14
15
procedureloadConnectedComponents()
16
{
17
defvari:int=0;
18
while(File::exists("cc"+i+".grs")){
19
insert(File::import("cc"+i+".grs"));
20
i= i +1;
21
}
22
return;
23
}
24
25
procedureremoveSavedConnectedComponentsFiles()
26
{
27
defvari:int=0;
28
while(File::exists("cc"+i+".grs")){
29
File::delete("cc"+i+".grs");
30
i= i +1;
31
}
32
return;
33
}
34
35
functionfetchNode(): Node
36
{
37
for(n:Nodeinnodes()) {
38
return(n);
39
}
40
return(null);
41
}
42
43
proceduredeleteSubgraph(refsn:set<Node>)
44
{
45
for(n:Nodeinsn){
46
rem(n);
47
}
48
return;
49
}
176
GraphTypeandComputations
EXAMPLE(88)
Anexampleshowingsomesubgraphextraction,comparison,andinsertionoperations.
1
ruleinit
2
{
3
modify{//creates the hostgraphourexampleruleandfunctionareworkingon
4
start1:SN-:contains->s1x:Node;
5
start1
-:contains->s1y:Node;
6
s1x--> s1y;
7
8
start2:SN-:contains->s2x:Node;
9
start2
-:contains->s2y:Node;
10
start2
-:contains->s2z:Node;
11
s2x--> s2y-->s2z--> s2x;
12
}
13
}
14
15
functionunequalContainedSubgraphs(start1:SN,start2:SN) :boolean
16
{
17
defrefadj:set<Node>= adjacentOutgoing(start1,contains);//adj=={s1x,s2x}
18
defvarsub1:graph= inducedSubgraph(adj);//sub1==graph(s1x’-->’ s1y’)--does not
contain s1x,s1ythemselves!
19
defvarsub2:graph= inducedSubgraph(adjacentOutgoing(start2, contains));
20
defvarres:boolean =sub1 ==sub2;//falseas graph(s1x’-->’s1y’)!=graph(s2x’-->’
s2y’ -->’s2z’ -->’s2x’),answeredquicklybecausenumber ofelementsdifferent
21
defvarsub3:graph= copy(sub1);
22
defvarres2:boolean=sub1== sub3; //trueas graph(s1x’-->’s1y’)isomorphic
graph(s1x’’ -->’’ s1y’’), answeredslowlybecauseisomorphymatching
23
return(res); //remark: theoriginalgraphisuntouched
24
}
25
26
ruleexample
27
{
28
start1:SN;start2:SN;
29
if{ unequalContainedSubgraphs(start1, start2); }
30
31
modify{
32
eval{
33
insert(inducedSubgraph(adjacentOutgoing(start2,contains) |set<Node>{start2}));
34
//1.computestheunion of the setof thenodes outgoingfromstart2withthe
setcontainingstart2
35
//2.creates astructurallyequal cloneoftheinducedgraphofthenodeset
handedin(start2, s2x,s2y,s2z, andalledgesin between)
36
//3.insertsthe cloneintotheoriginal graph (disjointunion)
37
//theclonewasdestroyedbyinsert, can’tbeaccessedfurther
38
(defstart2n:SN)=insertInduced(adjacentOutgoing(start2),start2);
39
//does thesame, justa bitsimplerandmoreefficient,
40
//with start2nyouhave anode thatgivesyouaccesstothesubgraphjust
inserted (by computingadjacentOutgoing(start2n,contains)),
41
//soyoucan dofurther processingof thatnewlycreatedpiece,
42
//e.g. linkittoother nodesintheoriginalgraph
43
}
44
}
45
}
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