Generation of Union Closed Sets and Moore families

0.07 MB·

by  Gunnar Brinkmann

DiscreteMathematicsandTheoreticalComputerScience DMTCSvol. (subm.):1,bytheauthors,#rev
Generation of Union Closed Sets and
Moore families
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2 Gunnar Brinkmann∗ Robin Deklerck†

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a 1 UniversiteitGent,Toegepastewiskunde,informaticaenstatistiek,Krijgslaan281S9,
J 2 B9000Gent,Belgium

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1

In this article we will describe an algorithm to constructively enumerate non-isomorphic Union Closed Sets and
]
O Mooresets.WeconfirmthenumberofisomorphismclassesofUnionClosedSetsandMooresetsonn≤6elements
presented by other authors and give the number of isomorphism classes of Union Closed Sets and Moore sets on
C
7elements. Duetotheenormousgrowthofthenumberofisomorphismclassesitseemsunlikelythatconstructive
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h enumerationfor8ormoreelementswillbepossibleintheforeseeablefuture.
t
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m Keywords:set,generation,orderly,homomorphism,Moorefamily
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1
Introduction
v
1
5 All sets in this article are finite. A Union Closed Set is a set U of sets with the property that for all
7 A,B ∈ U we have that A∪B ∈ U. We call ΩU = SA∈UA the universe of U. Two Union Closed
3 SetswithuniverseΩU,resp. ΩU′ aredefinedtobeisomorphicifthereisabijectivemappingΩU →ΩU′
0
inducingabijectionbetweentheUnionClosedSets.
.
1 Asweareonlyinterestedinisomorphismclasses, wemayassumeΩ = Ω = {1,...,n}forsome
U n
0 n. WhilethewholeuniverseΩ isbydefinitionanelementofaUnionClosedSetU,thisisnotthecase
U
7
fortheemptyset. AstheemptysethasnoimpactonthestructureofaUnionClosedSet,oneofteneither
1
requirestheemptysettobeanelementofeachUnionClosedSetorforbidsittobeanelement.Wechoose
:
v forthefirstconvention,soourUnionClosedSetscontainΩ aswellastheemptyset. We denotea set
n
i
X containingonerepresentativeofeachisomorphismclassofUnionClosedSetswithuniverseΩnasRn.
AlthoughthefamousUnionClosedSetsconjecture(orFrankl’sconjecture)isexactlyaboutthestruc-
r
a tureswegeneratehere,ourapproachisnotsuitablefortestingthisconjecture. Alotisknownaboutthe
structureofpossiblecounterexamplestotheUnionClosedSetsconjecture(see[BruhnandSchaudt(2015)]
forasurvey),soanyapproachtoextendtheknowledgeonthesmallestsizeofapossiblecounterexample
byconstructiveenumerationmustfocusonthesubsetofUnionClosedSetswiththoseadditionalstruc-
turalproperties(e.g. withsmallaveragesizeofthesets,withoutsomesubconfigurationslikesingletons,
etc.).
∗[email protected]
†[email protected]
ISSNsubm.toDMTCS(cid:13)c bytheauthorsbytheauthor(s)DistributedunderaCreativeCommonsAttribution4.0InternationalLicense
2 GunnarBrinkmann,RobinDeklerck
UnionClosedSetsarecloselyrelatedtoMooreFamilies. AMoorefamilyforauniverseΩ isasetof
n
subsetsofΩ thatisclosedunderintersectionandcontainsΩ . ItiseasytoseethatforaUnionClosed
n n
SetU thesetUc ={Ω \A|A∈U}isaMoorefamily.ForaMoorefamilyMthesetMc ={Ω \A|A∈
n n
M}isclosedunderunion,butastheemptysetisnotnecessarilycontainedinM, itisaUnionClosed
SetforΩn\TA∈MA,whichisisomorphictoaUnionClosedSetforsomeΩn′ withn′ ≤n.
AsetM ofrepresentativesofMoorefamilies(withthecanonicaldefinitionofisomorphism)forthe
n
universeΩn can be obtainedfrom sets R0,...,Rn of representativesof Union Closed Sets containing
theemptysetas
M =Sn {Uc|U ∈R }
n i=0 i
ifthecomplementisineachcasetakenintheuniverseΩ .
n
In[Higuchi(1998)],[HabibandNourine(2005)]and[Colombetal.(2010)Colomb,Irlande,andRaynaud]
Moorefamiliesareenumerated.Inthemostadvancedofthesearticles–[Colombetal.(2010)Colomb,Irlande,andRaynaud]
–allMoorefamiliesforn≤6werecountedandrepresentativesofisomorphismclassesweregenerated.
For n = 7 the approach is not suitable for generating a set of representatives and only the number of
labeledMoorefamilies–thatiswithoutconsideringisomorphisms–wasdeterminedbycleveruseofthe
structureofrepresentativesofMoorefamiliesforn = 6. Inouralgorithmwedeterminethenumberof
labeledUnionClosed Sets resp. labeled Moorefamiliesforn = 7 fromrepresentativesandtheirauto-
morphismgroupsoftheUnionClosedSetsforn = 7,resp. n ≤ 7. Thisgivesaverygoodindependent
test fortheimplemetationin [Colombetal.(2010)Colomb,Irlande,andRaynaud] aswellas forourim-
plementation.WhencomputingthenumberoflabeledMoorefamiliesforΩ7fromthenumberoflabeled
UnionClosedSetswithn ≤ 7,forthoseUnionClosedSetswithn < 7,thepossiblewaystochoosen
verticesfrom{1,...,7}arealsotakenintoaccount.
The algorithm
AsubsetA⊆Ωnisrepresentedasanumberb(A)givenasthebinarynumberbn−1...b0–possiblywith
leadingzeros–withb =1if(i+1)∈Aandb =0otherwise.
i i
WeuseanorderingofthesubsetsofΩ . ForA,B ⊆ Ω wedefineA > B if|A| < |B|(sosetswith
n n
moreelementsareconsideredsmallerinthisorder)or|A| = |B|andb(A) > b(B). Wheneverwerefer
tolargerorsmallersets,werefertothisordering.
TheconstructionalgorithmgeneratesUnionClosedSetsrecursivelybasedonthefollowingeasylemma:
Lemma1. IfU 6={Ω }isaUnionClosedSetandAisthelargestnon-emptyelementinU,thenU\{A}
n
isalsoaUnionClosedSet.
This implies that Union Closed Sets for universe Ω can be recursively constructed from the Union
n
ClosedSet{Ω ,∅}ofsmallestsizebysuccessivelyaddingsubsetsofΩ thatarelargerthanthelargest
n n
non-emptysetalreadypresent.OfcourseitisnotassuredthataddingasmallersettoaUnionClosedSet
doesnotviolatetheconditionthatthesetmustbeclosedunderunions.
Inordertoturnthisintoanefficientalgorithm,twoteststhatare(inprinciple)appliedtoeachstructure
generatedmustbeveryfast:
(i) Thetestwhetherthesetthathasbeenconstructedbyaddinganewsetisclosedunderunion.
(ii) Thetestforisomorphisms.
GenerationofUnionClosedSetsandMoorefamilies 3
We will first discuss (i). A straightforwardway to test (i) for a Union Closed Set U to which a new
set A is addedwould be to form all unionsA∪B with B ∈ U and test whether they are in U ∪{A}.
Althoughallthese steps canbe implementedasefficientbit-operations,theirnumberwouldslow down
theprogram.Wedefine:
Definition1. ForaUnionClosedSetU wedefinethereducedsetred(U)as
red(U)={A∈U|A6=∅andthereisnoA1 6=∅inU,A1 (A}
Lemma 2. Let U be a Union Closed Set for a universe Ω and let A ⊂ Ω , that is larger than any
n n
non-emptysetinU. ThenU ∪{A}isclosedunderunionifandonlyifA∪B ∈U forallB ∈red(U).
Proof:
AssumefirstthatU ∪{A}isclosedunderunionandletB ∈red(U). ThenA∪B ∈(U∪{A})andas
AislargerthanB,wehaveA∪B 6=A,soA∪B ∈U.
FortheotherdirectionassumethatA∪B ∈U forallB ∈red(U)andletD ∈U.
ChooseanyD′ ⊂DsothatD′ ∈red(U). ThenA′ =A∪D′ ∈U andthereforealsoA′∪D ∈U asU
isclosedunderunion,butA′∪D =A∪D–soA∪D ∈U ∪{A}andU ∪{A}isclosedunderunion.
It wouldbe inefficientto computered(U) each time a new Union Closed Set is constructed, butas a
new Union Closed Set U′ is constructedby addinga new smallest elementA to U, the set red(U′) can
easilybeconstructedfromred(U)byaddingAandremovingelementsthatcontainA. Neverthelessthe
few lines of code testing whether the potentialUnion Closed Set is closed under union take more than
50%ofthetotalrunningtimewhencomputingUnionClosedSetsforΩ6, whichisthelargestcasethat
canbeprofiled.
Inordertosolveproblem(ii)efficiently–thatisavoidthegenerationofisomorphiccopies,weusea
combinationofRead/Faradzˇevtypeorderlygeneration[Faradzˇev(1976)][Read(1978)]andthehomomor-
phismprinciple(seee.g. [Brinkmann(2000)]).
Our first aim is to define a unique representative for each isomorphism class – called the canonical
representative– andthenonlyconstructUnionClosed Sets thatare thethe canonicalrepresentativesof
theirclass.WerepresentaUnionClosedSetU withk+1elementsA1 <A2 <···<Ak <∅asthestring
s(U)=b(A1),...,b(Ak)ofnumbers.ForagivenisomorphismclassofUnionClosedSetsforauniverse
Ω wechoosetheUnionClosedSetwiththelexicographicallysmalleststringastherepresentative.
n
It is in principle easy to test whether a given Union Closed Set U is the representative of its class
by applying all n! possible permutations to U and comparing the strings. As n ≤ 7 this would not
be extremely expensive, but due to the large number of times that this test has to be applied, still too
expensivetoconstructtheUnionClosedSetsforΩ7. Inthesequelwewilldescribeawayhowthiscan
beoptimized.
In orderto increase the efficiencyby makingit an orderlyalgorithmof Read/Faradzˇevtype, we will
usethecanonicitytestnotonlyforstructuresweoutput,butalsoduringtheconstruction: non-canonical
structuresareneitheroutputnorusedintheconstruction. Thiswillonlyleadtoacorrectalgorithmifwe
canprovethatcanonicalrepresentativesareconstructedfromcanonicalrepresentatives:
Theorem1. LetU 6={Ω ,∅}beaUnionClosedSetfortheuniverseΩ thatisthecanonicalrepresen-
n n
tativeforitsisomorphismclass. IfU ={A1,A2,...,Ak,∅}withA1 <A2 <···<Ak and1≤m≤k,
then{A1,A2,...,Am,∅}isalsothecanonicalrepresentativeofitsclass.
4 GunnarBrinkmann,RobinDeklerck
Proof: We provetheresultform = k−1. Fork = m itistheassumptionandform < k−1itthen
followsbyinduction.
Lets(U)=(s1,...,sk). ForapermutationΠofΩnandaUnionClosedSetU wewriteΠ(U)forthe
UnionClosed Set obtainedby replacingeach elementofa set in U by its imageunderΠ. Assume that
U′ = {A1,A2,...,Ak−1,∅}isnotthecanonicalrepresentativeofitsclass. SothereisapermutationΠ
of Ωn with s(Π(U′)) < s(U′). Let s(Π(U′)) = (p1,...,pk−1) and we have s(U′) = (s1,...,sk−1).
Letj bethefirstpositionsothatp < s . Letusnowlookats(Π(U)) = (p′,...,p′)andletr bethe
j j 1 k
positionofΠ(A )inthisstring. Ifr > j thenp′ = p = s for1 ≤ i < j andp′ = p < s –sothere
k i i i j j j
isasmallerrepresentativefortheisomorphismclassofU. Ifr ≤ j thenp′ = p = s for1 ≤i < rand
i i i
p′ <p ≤s andagainwehavefoundasmallerrepresentativecontradictingtheminimalityofs(U).
r r r
This theorem proves that the algorithm can reject non-canonicalUnion Closed Sets and is a correct
orderlyalgorithm,butthecostofthecanonicitytestwouldmakeitimpossibletodeterminethenumber
ofUnionClosedSetsforΩ7.
ForagivenUnionClosedSetUwithuniverseΩ and1≤m≤nwewriteU forthesubsetcontaining
n m
allsetsofsizek ≥mandΠ (U)forallpermutationsΠofΩ withthepropertythatΠ(U )=U .
m n m m
Lemma 3. LetU 6= {Ωn,∅}beanon-canonicalUnionClosedSetfortheuniverseΩn with setsA1 <
A2 < ··· < Ak < ∅,sothat{A1,A2,...,Ak−1,∅}iscanonicaland|Ak| = m. Thenallpermutations
Πwiths(Π(U))<s(U)areinΠm+1(U).
Proof:
AnypermutationΠnotinΠm+1(U)wouldbydefinitiongiveanisomorphicbutdifferentUnionClosed
SetΠ(Um+1). AsduetoTheorem1s(Um+1)isminimal,s(Π(Um+1))wouldbelargerandthereforealso
thepartofthestringofs(Π(U))describingsetsofsizeatleastm+1wouldimplys(Π(U))>s(U).
Lemma 3 speeds up the canonicity test dramatically: We start with a list of all n! permutations as
Π (U). WhentestingcanonicityofaUnionClosedSetwiththesmallestsetofsizek <n,weonlyapply
n
permutationsfromΠk+1(U). Duringthisapplication,wecanalreadycomputeΠk(U)bysimplyadding
exactlythosepermutationstotheinitiallyemptysetΠ (U)thatfixU. Asweworkwithsmallsets, itis
k
noproblemtostoreandusethesetofallgroupelementsinsteadofjustasetofgenerators.
Theimpactisimmediatelyclear: thenumberofpermutationsthathastobecomputedismuchsmaller
andassoonassomeΠ (U)containsonlytheidentity–whichhappensveryfast–nocanonicitytestshave
k
to be performed,so that the total time for isomorphismcheckingis only about7% of the total running
timewhencomputingUnionClosedSetsforΩ6.
The implementation
The algorithm was implemented in C. Next to an efficient algorithm, of course also implementation
details are of crucial importance to be able to compute the results for Ω7. We precomputedthe action
ofallpermutationsonallsets,sothattheycouldbeappliedveryfastanduseddatastructuresthatallow
to check whether a set is contained in a Union Closed Set in constant time. Special functions were
writtenthataddsetswithonlyoneelement. Asnosetsofsmallersize willbeadded,noupdatesofthe
automorphismgroupsarenecessaryanditturnedoutthatatthisstageitisalsonotefficientanymoreto
GenerationofUnionClosedSetsandMoorefamilies 5
n UnionClosedSet labeledUnionClosedSet
1 1 1
2 3 4
3 14 45
4 165 2.271
5 14.480 1.373.701
6 108.281.182 75.965.474.236
7 2.796.163.091.470.050 14.087.647.703.920.103.947
Tab.1: ThenumberofUnionClosedSetsandlabeledUnionClosedSets.
n Moorefamilies labeledMoorefamilies
1 2 2
2 5 7
3 19 61
4 184 2480
5 14.664 1.385.552
6 108.295.846 75.973.751.474
7 2.796.163.199.765.896 14.087.648.235.707.352.472
Tab.2: ThenumberofMoorefamiliesandlabeledMoorefamilies.
removesetsfromred(U)thatareasupersetofanotherelement. Detailsontheimplementationcanbest
beseeninthecodewhichcanbeobtainedfromtheauthors.
Results
Tables1and2givethenumbersofisomorphismclassesofUnionClosedSetsandMoorefamiliesaswell
asthenumbersoflabeledstructures. Upto5elementstherunningtimesarelessthan0.01seconds. For
n=6itis8.2secondsonaXeon(R)CPUE5-26900runningwith2.90GHzandahighload(whichcan
makeadifferencefortheseprocessors).Forn=7thejobswereruninparallelondifferentmachinesand
somepartshadtobedividedfurtherinordertofinish,soitishardtogiveprecisetimes. Estimatingthe
totalrunningtimefromthosepartsthatwererunonthesamemachineusedforn = 6,thetotaltimeon
thistypeofmachineshouldbearound10to12CPU-years.
A Union Closed Set on n elements is called sparse if the average number of elements in a set – not
countingtheemptyset–isatmost n. ForUnionClosedSetsthatarenotsparse,theUnionClosedSets
2
conjectureistriviallytrue.ThefollowingtablegivesthenumberofsparseUnionClosedSets.Thesenum-
berswerecomputedoncebyfilteringallUnionClosedSetsandoncebyanindependentimplementation
usingspecialboundingcriteriadescribedin[Deklerck(2016)].
6 GunnarBrinkmann,RobinDeklerck
n sparseUnionClosedSet
1 0
2 0
3 0
4 2
5 27
6 3.133
7 5.777.931
Tab.3: ThenumberofsparseUnionClosedSets.
Testing
In[Deklerck(2016)]anindependentimplementationofthealgorithmtogetherwithspecialboundingcri-
teriaforsparseUnionClosedSetswasdeveloped. Theimplementationwasusedtogenerateallisomor-
phismclassesofUnionClosedSetsforΩ1,...,Ω6,and–usingspecialboundingcriteria–toconfirmthe
numbersofisomorphismclassesofsparseUnionClosedSetsforΩ7.
A furtherandindependentconfirmationofthe numbersforΩ1,...,Ω6 andalso an independentcon-
firmationforΩ7 wasobtainedbycomputingthenumberoflabeledstructurescorrespondingtoeachrep-
resentativefromthesizeoftheautomorphismgroup. Notethatasthesizeoftheautomorphismgroupis
knowninthealgorithmanyway,theadditionalcostsforthistestcanbeneglected.Fromthiswecomputed
thenumberof(labeled)Moorefamiliesandgotcompleteagreementwith[Colombetal.(2010)Colomb,Irlande,andRaynaud]
forΩ1,...,Ω7. Duetothecompletelydifferentapproachesthismakesimplementationerrorsleadingto
thesameincorrectresultsinbothcasesextremelyunlikely.
Acknowledgements
We wantto thankCraig Larsonforintroducingustothese interestingstructuresandforinterestingdis-
cussions. FurthermorewewanttothankAnastasiiaZharkovaandMikhailAbrosimov. Thefirstversion
ofthisalgorithmwasintendedasahands-onexampletoillustrateisomorphismrejectiontechniquesina
coursetheyattendedat Ghentuniversity. Withouttheir visit, thisalgorithmmightnothave beendevel-
oped.
References
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