Dissections, orientations, and trees, with applications to optimal mesh encoding and to random sampling

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optimal mesh encoding and to random sampling

Eric Fusy, Dominique Poulalhon, Gilles Schaeffer

To cite this version:

Eric Fusy, Dominique Poulalhon, Gilles Schaeffer. Dissections, orientations, and trees, with applica-

tions to optimal mesh encoding and to random sampling. ACM Transactions on Algorithms, Associ-

ation for Computing Machinery, 2008, 4 (2), pp.Art.19. �hal-00330580v2�

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and to random sampling

ÉRICFUSY,DOMINIQUEPOULALHONandGILLESSCHAEFFER

É.FandG.S: LIX,ÉolePolytehnique. D.P:Liafa,Univ. Paris 7. Frane

Wepresentabijetionbetweensomequadrangulardissetionsofanhexagonandunrootedbinary

trees,withinterestingonsequenesforenumeration,meshompressionandgraphsampling.

Ourbijetionyieldsaneientuniformrandomsamplerfor3-onnetedplanargraphs,whih

turns out to bedeterminant for the quadrati omplexityof the urrent best known uniform

randomsamplerforlabelledplanargraphs[Fusy,AnalysisofAlgorithms2005℄.

ItalsoprovidesanenodingforthesetP(n)^ofn^-edge^3-onneted^planar^graphs^that^mathes

theentropybound

1

nlog₂|P(n)|= 2 +o(1)^bits^per^edge^(bpe). ^This^solves^a^theoretial^problem

reentlyraisedinmeshompression,asthesegraphsabstrattheombinatorialpartofmesheswith

spherialtopology.Wealsoahievetheoptimalparametrirate

1

nlog₂|P(n, i, j)|^bpe^for^graphs

ofP(n)^withi^verties^andj^faes,^mathingⁱⁿ^partiular^the^optimal^rate^fortriangulations.

Ourenoding relies on alinear timealgorithmto ompute an orientation assoiatedto the

minimalShnyderwoodofa3-onnetedplanarmap.Thisalgorithmisofindependentinterest,

anditisforinstaneakeyingredientinareentstraightlinedrawingalgorithmfor3-onneted

planargraphs[Bonihonetal.,GraphDrawing2005℄.

CategoriesandSubjetDesriptors: G.2.1[DisreteMathematis℄:Combinatorialalgorithms

GeneralTerms: Algorithms

AdditionalKeyWordsandPhrases:Bijetion,Counting,Coding,Randomgeneration

1. INTRODUCTION

OneoriginofthisworkanbetraedbaktoanartileofEdBenderintheAmer-

ianMathematial Monthly [Bender1987℄,whereheaskedforasimpleexplanation

oftheremarkableasymptotiformula

|P(n, i, j)| ∼

1 3

⁵

2

⁴

ijn

2i

−

2 j + 2

2j

−

2 i + 2

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fortheardinalityofthesetof 3-onneted(unlabelled) planargraphswith

i

^ver-

ties,

j

^faes^and

n = i + j

−

2

^edges,

n

^going^toînnity. ^Byâ^theoremôf^Whitney

[1933℄,thesegraphshaveessentiallyauniqueembeddingonthesphereuptohome-

omorphisms,sothattheirstudyamountstothatofrooted3-onnetedmaps,where

amapisagraphembeddedintheplaneandrooted meanswithamarkedoriented

edge.

1.1 Graphs,dissetionsandtrees

Anotherknownpropertyof3-onnetedplanargraphswith

n

^edges^is^the^fat^that

theyarein diret one-to-one orrespondenewith dissetions ofthesphereinto

n

quadranglesthathavenonon-faial4-yle. Theheartofourpaperliesinafurther

one-to-oneorrespondene.

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Theorem 1.1. There is a one-to-one orrespondene between unrooted binary

trees with

n

^nodes ^and ^unrooted quadrangular dissetions of an hexagon with

n

interiorvertiesandnonon-faial 4-yle.

Themappingfrombinarytreestodissetions,whihweallthelosure,iseasily

desribedandresemblesonstrutionsthatwerereentlyproposedforsimplerkinds

ofmaps [Shaeer1997;Bouttieret al.2002; Poulalhonand Shaeer2006℄. The

proof that the mapping is a bijetion is instead rather sophistiated, relying on

new properties of onstrained orientations [Ossona de Mendez 1994℄, related to

Shnyder woods of triangulations and 3-onneted planar maps [Shnyder 1990;

di Battistaet al.1999;Felsner2001℄.

Conversely, thereonstrution of the treefrom thedissetion relies onalinear

time algorithm to ompute the minimal Shnyder woods of a 3-onneted map

(or equivalently, the minimal

α

0-orientation of the assoiated derived map, see Setion 9). This problem is of independant interest and our algorithm has for

example appliations in the graph drawing ontext [Bonihon et al. 2007℄. It is

akin to Kant's anonial ordering [Kant 1996; Chuang et al. 1998; Bonihon et

al.2003;Castelli-AleardiandDevillers2004℄,but againtheproofoforretnessis

quiteinvolved.

Theorem 1.1 leads diretly to the impliit representation of the numbers |Pn^′|

ountingrooted 3-onnetedmaps with

n

êdges^due ^to^Tutte ^[1963℄), ândîts

renement asdisussed in Setion 5yields that of |P_ij^′ | ^the ^number ^of ^rooted ^3-

onnetedmapswith

i

^verties^and

j

^faes^(due^to^Mullin^andShellenberg[1968℄) from whih Formula(1)follows. It partiallyexplains theombinatorisoftheo-

urrene of the ross produt of binomials, sine these are typial of binary tree

enumerations. Letus mentionthattheone-to-one orrespondenespeializespar-

tiularlynielytoountplanetriangulations(i.e.,3-onnetedmapswithallfaes

ofdegree 3),leadingtotherstbijetivederivationoftheountingformulaforun-

rootedplane triangulations with

i

^verties,^originally^found^by^Brown^[1964℄^using

algebraimethods.

1.2 Randomsampling

A seond byprodut of Theorem 1.1 is an eient uniform random sampler for

rooted3-onnetedmaps,i.e.,analgorithmthat,given

n

^,ôutputsâ^randomêlement

in theset P_n^′ ^of ^rooted^3-onneted ^maps ^with

n

êdges^with êqual^hanes ^forâll

elements. Thesamepriniplesyield auniformsamplerforP_ij^′ ^.

The uniform random generation of lasses of maps like triangulations or 3-

onneted graphs was rst onsidered in mathematial physis(see referenes in

[Ambjørn et al. 1994; Poulalhon and Shaeer 2006℄), and various types of ran-

dom planargraphs areommonly used fortesting graph drawingalgorithms (see

[de Fraysseixetal.℄).

Thebest previouslyknown algorithm[Shaeer1999℄ hadexpetedomplexity

O(n

^5/3

)

^forP_n^′^, ând^was^muh^lessêient^forP_ij^′ ^,^havingêvenexponentialom- plexityfor

i/j

^or

j/i

^tending^to ²^(due^to ^Euler's^formula^these^ratio^are^bounded

aboveby2for3-onnetedmaps). InSetion6,weshowthatourgeneratorforP_n^′

orP_ij^′ ^performsⁱⁿ ^linear^time ^exept^if

i/j

^or

j/i

^tends ^to ²^where^it ^beomes ^at

mostubi.

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Fromthetheoretialpointofview,itisalsodesirabletoworkwiththeuniform

distributiononplanargraphs. However,random(labelled)planargraphsappearto

behallengingmathematial objets[Osthus et al.2003; MDiarmidet al. 2005℄.

A Markovhain onverging to the uniform distribution on planar graphswith

i

vertieswasgivenbyDeniseetal.[1996℄,butitresistsknownapproahesforper-

fetsampling[Wilson2004℄,andhasunknownmixing time. Asopposed tothis,a

reursiveshemeto sampleplanargraphswasproposed byBodirsky etal. [2003℄,

with amortizedomplexity

O(n

^6.5

)

^. ^This ^resultîs ^basedônâ^reursive^deompo-

sition of planar graphs: a planar graphan be deomposed into a tree-struture

whosenodesareoupiedbyrooted3-onnetedmaps. Generatingaplanargraph

redues toomputing branhingprobabilitiessoastogenerate thedeomposition

treewithsuitableprobability;thenarandomrooted3-onnetedmapisgenerated

for eah node of the deomposition tree. Bodirsky et al.[2003℄ usethe so-alled

reursive method [Nijenhuis and Wilf 1978; Flajoletet al. 1994; Wilson 1997℄ to

takeadvantageof thereursivedeomposition ofplanargraphs. Ournewrandom

generatorforrooted3-onneted mapsredues theiramortizedostto

O(n

³

)

^. ^Fi-

nallyanew uniformrandomgeneratorfor planargraphswasreentlydevelopped

byoneoftheauthors[Fusy2005℄,thatavoidstheexpensivepreproessingompu-

tations of[Bodirsky et al.2003℄. The reursivesheme issimilar to the oneused

in [Bodirsky et al. 2003℄, but the method to translate it to a random generator

reliesonBoltzmannsamplers,anewgeneralframeworkfortherandomgeneration

reently developed in [Duhon et al. 2004℄. Thanks to our random generatorfor

rooted 3-onneted maps, the algorithm of [Fusy 2005℄ has a time-omplexity of

O(n²

)

^forêxat^sizeûniform^samplingândêven^performsⁱⁿ^linear^time^forâpprox-

imatesizeuniformsampling.

1.3 Suintenoding

A third byprodut ofTheorem 1.1 is thepossibility to enodein linear timea 3-

onneted planargraphwith

n

^edges ^by ^a^binary ^tree^with

n

^nodes. ^In^turn ^the

treeanbeenodedbyabalanedparenthesiswordof

2n

^bits. ^Thisôdeîsôptimal

intheinformationtheoretisense: theentropyperedgeofthislassofgraphs,i.e.,

the quantity

1

n

log

₂|P(n)|^, ^tends ^to ²^when

n

^goes^to înnity, ^so ^that â ôde ^for P

(n)

ânnot^giveâ^better^guaranteeôn^theômpression^rate.

Appliationsallingforompatstorageandfasttransmissionof3Dgeometrial

mesheshavereentlymotivatedahugeliteratureonompression,inpartiularfor

theombinatorialpartofthemeshes. Therstompression algorithmsdealtonly

withtriangularfaes[Rossigna1999;ToumaandGotsman1998℄,butmanymeshes

inlude largerfaes,sothatpolygonalmesheshavebeome prominent(see[Alliez

andGotsman2003℄forareentsurvey).

Thequestionofoptimalityofoderswasraisedinrelationwithexeptionodes

produed by several heuristiswhen dealingwith meshes withspherialtopology

[Gotsman2003;Khodakovskyet al.2002℄. Sinethesemeshesareexatlytriangu-

lations(fortriangularmeshes)and3-onnetedplanargraphs(forpolyhedralones),

theodersin[PoulalhonandShaeer2006℄andin thepresentpaperrespetively

provethat traversalbasedalgorithmsanahieveoptimality.

On the other hand, in the ontext of suint data strutures, almost optimal

algorithmshavebeenproposed[Heetal.2000;Lu2002℄,thatarebasedonseparator

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theorems. Howeverthese algorithmsarenottruly optimal(they get

ε

^lose^to^the

entropybut at theost of an unontrolled inrease of theonstantsin thelinear

omplexity). Moreover,although theyrelyonasophistiated reursivestruture,

theydonotsupporteientadjaenyrequests.

As opposed to that, ouralgorithm shares with [Heet al.1999; Bonihon et al.

2003℄ thepropertythat it produesessentiallythe ode ofaspanning tree. More

preiselyitisjustthebalanedparenthesisodeofabinarytree,andadjaeniesof

theinitialdissetionthatarenotpresentinthetreeanbereoveredfromtheode

byasimplevariationontheinterpretation ofthesymbols. Adjaenyqueriesan

thusbe dealt within time proportionalto thedegreeof verties[Castelli-Aleardi

etal.2006℄usingtheapproahof[MunroandRaman1997;Heetal.1999℄.

FinallyweshowthattheodeanbemodiedtobeoptimalonthelassP

(n, i, j)

^.

Sine the entropy of this lass is stritly smaller than that of P

(n)

^as ^soon ^as

|i−

n/2| ≫ n

^1/2^,^the^resulting^parametriôderîs ^moreêientⁱⁿ ^this^range. În

partiularinthease

j = 2i

−

4

ôur^newâlgorithm^speializes^toânôptimalôder

fortriangulations.

1.4 Outlineofthepaper

Thepaperstartswithtwosetionsofpreliminaries: denitionsofthemapsandtrees

involved (Setion 2), and somebasi orrespondenes between them (Setion 3).

Then omes our main result (Setion 4), the mappingbetweenbinary trees and

somedissetionsofthehexagonbyquadrangularfaes. Thefatthatthismapping

isabijetionfollowsfromtheexisteneanduniquenessofaertaintri-orientationof

ourdissetions. Theproofofthisauxiliarytheorem,whihrequirestheintrodution

oftheso-alledderivedmapsandtheir

α

0-orientations,isdelayedtoSetion8,that is, after the three setions dediated to appliations of our main result: in these

setions we suessively disuss ounting (Setion 5), sampling (Setion 6) and

oding (Setion 7) rooted 3-onneted maps. The third appliation leads us to

our seond important result: in Setion 9 we present a linear time algorithm to

ompute the minimal

α

0-orientation of the derived map of a 3-onneted planar map (whih also orresponds to the minimal Shnyder woods alluded to above).

Finally,Setion10isdediatedtotheorretnessproofofthisorientationalgorithm.

Figure 1summarizes theonnetions betweenthe dierent familiesof objetswe

onsider.

2. DEFINITIONS

2.1 Planarmaps

Aplanarmap isaproperembeddingofanunlabelledonnetedgraphintheplane,

where proper means that edges are smooth simple ars that do notmeet but at

their endpoints. A planar map is said to be rooted if oneedge of the outerfae,

alled the root-edge, is marked and oriented suh that the outer fae lays on its

right. Theoriginoftheroot-edgeisalled root-vertex. Vertiesandedgesaresaid

to be outer orinner dependingon whetherthey areinidentto theouter fae or

not.

Aplanarmapis3-onnetedifithasatleast4edgesandannotbedisonneted

bytheremovaloftwoverties. Therst3-onnetedplanarmapisthetetrahedron,

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3-onnetedplanargraphs

3-onnetedmaps derivedmaps irreduible

quadrangulations

derivedmaps

withorientation dissetions

ofthehexagon

dissetionsofthehexagon

withorientation binarytrees

parenthesisode

iterative

algorithm transposition

operations opening

losure

rejetion

Whitney

folklore

Fig.1. Relationsbetweeninvolvedobjets.

whihhas6edges. WedenotebyP_n^′ (respetivelyP_ij^′ ⁾^the^set^of^rooted^3-onneted

planarmapswith

n

^edges^(resp.

i

^verties^and

j

^faes). ^A^3-onneted^planar^map

isouter-triangular ifitsouterfaeistriangular.

2.2 Planetrees,andhalf-edges

Planetrees are planarmaps withasinglefaetheouterone. A vertexis alled

a leaf ifit has degree 1,and node otherwise. Edgesinident to a leaf are alled

stems,andtheotherarealledentireedges. Observethatplanetreesareunrooted

trees.

Binary trees areplanetreeswhosenodeshavedegree3. Byonventionweshall

requirethat arootedbinary tree hasaroot-edgethat is astem. The root-edgeof

a rooted binary tree thus onnets a node, alled the root-node, to a leaf, alled

theroot-leaf. Withthisdenitionofrootedbinarytree,upondrawingthetreeina

topdownmannerstartingwiththeroot-leaf,everynode(inludingtheroot-node)

hasafather, aleft sonandarightson. This (veryminor) variationon theusual

denition ofrootedbinarytreeswillbeonvenientlateron. For

n

≥

1

^, ^we^denote

respetivelybyBn ândB^′_n ^the^setsôf^binaryând^rooted^binary^trees^with

n

^nodes

(theyhave

n + 2

^leaves,âs^proved^byîndutionôn

n

^). ^These^rooted^trees^are^well

knownto beountedbytheCatalannumbers: |B_n^′|

=

_n+1¹ ²ⁿ_n

.

Thevertiesofabinarytreeanbegreedilybioloredsayinblakorwhite

sothat adjaentvertieshavedistintolors. Thebiolorationisuniqueuptothe

hoie of the olor of the rst node. As a onsequene, rooted biolored binary

trees are either blak-rooted or white-rooted, depending on the olor of the root

node. Thesetsofblak-rooted(resp. white-rooted)binarytreeswith

i

^blak^nodes

and

j

^white ^nodes îs ^denoted ^by B_ij^• ^(resp. ^by B_ij^◦^); ând ^the ^total ^set ôf ^rooted

bioloredbinarytreeswith

i

^blak^nodes^and

j

^white^nodes^is^denoted^byB^′_ij^.

Itwillbeonvenientto vieweahentireedgeofatreeasapairofoppositehalf-

edges eahoneinidenttooneextremityoftheedgeandtovieweahstemas

asinglehalf-edgeinidenttothenodeholdingthestem. Moregenerallyweshall

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onsidermapsthathaveentireedges (madeoftwohalf-edges)andstems (madeof

onlyonehalf-edge). Itisthenalsonaturaltoassoiateonefaeto eahhalf-edge,

say,thefae onitsright. In theaseof trees,thereis onlytheouterfae,sothat

allhalf-edgesgetthesameassoiatedfae.

2.3 Quadrangulationsanddissetions

A quadrangulation is a planar map whose faes (inluding the outer one) have

degree4. Adissetionof thehexagonbyquadrangularfaes isaplanarmapwhose

outerfaehasdegree6andinnerfaeshavedegree4.

Cylesthatdonotdelimitafaearesaidtobeseparating. Aquadrangulationor

adissetionofthehexagonbyquadrangularfaesissaidtobeirreduible ifithasat

least 4faes andhasno separating4-yle. Therst irreduiblequadrangulation

is the ube, whih has 6 faes. We denote by Q^′_n ^the ^set ^of ^rooted ^irreduible

quadrangulations with

n

^faes, înluding ^the ôuter ône. Êuler's ^relation ênsures

thatthese quadrangulationshave

n + 2

^verties. ^We^denote^byDn ⁽D^′_n⁾^the^set^of

(rooted, respetively)irreduible dissetionsof thehexagonwith

n

^inner ^verties.

These have

n + 2

quadrangular faes, aording to Euler's relation. From now on, irreduible dissetions of the hexagon by quadrangular faes will simply be

alled irreduible dissetions. The lasses of rooted irreduible quadrangulations

and of rooted irreduible dissetions arerespetivelydenoted byQ^′

=

∪nQ^′_n ^and D^′

=

∪nD^′_n^.

As faes of dissetions and quadrangulations haveeven degree, the verties of

these maps an begreedily biolored, say, in blak and white, so that eah edge

onnets a blak vertex to awhite one. Suh abioloration is unique up to the

hoie of the olors. We denote by Q^′_ij ^the ^set ^of ^rooted ^biolored ^irreduible

quadrangulationswith

i

^blak^verties^and

j

^white^verties^and^suh^that^the^root-

vertexisblak;and byD_ij^′ ^the^set ^of^rooted^biolored^irreduible ^dissetions^with

i

^blak^inner ^verties^and

j

^white^inner ^verties^and^suh ^that ^theroot-vertexis blak.

Abioloredirreduibledissetionisomplete ifthethreeouterwhitevertiesof

thehexagonhavedegreeexatly2. Hene,thesethreevertiesareinidentto two

adjaentedgesonthehexagon.

3. CORRESPONDENCESBETWEENFAMILIESOF PLANARMAPS

This setion reallsa folklore bijetion between irreduiblequadrangulations and

3-onnetedmaps, hereafteralledangularmapping,see [MullinandShellenberg

1968℄,anditsadaptationtoouter-triangular3-onnetedmaps.

3.1 3-onnetedmapsandirreduiblequadrangulations

Letusrstreallhowtheangularmappingworks. Givenarootedquadrangulation

Q

∈ Q^′_n ^endowed ^with^its ^vertex bioloration,let

M

^be ^the^rooted ^map ^obtained

bylinking,foreahfae

f

^of

Q

^(even^the^outer ^fae),^the^two^diagonally^opposed

blak vertiesof

f

^; ^the ^root ^of

M

^is ^hosen^to ^be ^the ^edge orresponding to the outerfaeof

Q

^,^oriented^so^that

M

^and

Q

^have^sameroot-vertex,seeFigure2. The map

M

îs ôftenâlled ^the^primal ^map ôf

Q

^. Â ^similar ônstrution ûsing ^white

vertiesinsteadofblakones wouldgiveitsdual map (i.e., themapwithavertex

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(a)Aquadrangulation (b)withitsblakdiagonals ()givesaplanarmap.

Fig.2.Theangularmapping:fromarootedirreduiblequadrangulationtoarooted3-onneted

planarmap.

in eahfae of

M

^and^edge-setorrespondingto theadjaenies betweenverties andfaesof

M

^).

Theonstrutionof the primalmap is easily invertible. Givenanyrooted map

M

^,^theînverseônstrutionônsistsⁱⁿâddingâ^vertexâlledâ^fae-vertex ⁱⁿêah

fae (eventhe outerone) of

M

^and ^linking ^a^vertex

v

^and^a^fae-vertex

v

f ^by^an

edge if

v

^is ^inident ^to ^the ^fae

f

orresponding to

v

f^. ^Keeping ^only^these ^fae-

vertexinidene edges yields a quadrangulation. The root is hosenas the edge

thatfollowstherootof

M

ⁱⁿ ounter-lokwiseorderarounditsorigin.

Thefollowingtheoremisalassialresultin thetheoryofmaps.

Theorem 3.1 (Angularmapping) . The angular mapping is a bijetion be-

tween P_n^′ ândQ^′_n ând^more^preisely â^bijetion ^betweenP_ij^′ ândQ^′_ij^.

3.2 Outer-triangular3-onnetedmapsandbioloredompleteirreduibledissetions

Thesameprinipleyieldsabijetion,alsoalled angularmapping,betweenouter-

triangular3-onnetedmapsand bioloredompleteirreduibledissetions,whih

will prove very useful in Setions 7 and 8. This mapping is very similar to the

angular mapping: given aomplete dissetion

D

^, ^assoiate^to

D

^the^map

M

^ob-

tainedbylinkingthetwoblakvertiesofeahinnerfae of

D

^by^a^new^edge,^see

Figure3. Themap

M

îsâlled ^the^primal ^map ôf

D

^.

Theorem 3.2 (Angularmappingwith border). Theangularmapping,for-

mulated for omplete dissetions, is a bijetion between biolored omplete irre-

duible dissetions with

i

^blak ^verties ^and

j

^white ^verties ^and outer-triangular 3-onnetedmaps with

i

^verties ^and

j

−

3

^inner^faes.

Proof. Theprooffollowssimilar linesasthatof Theorem3.1,see [Mullinand

Shellenberg1968℄.

3.3 Derivedmaps

Initsversionforompletedissetions,theangularmappinganalsobeformulated

usingtheoneptofderivedmap,whihwillbeveryusefulthroughoutthisartile

(inpartiularwhendealingwithorientations).