HAL Id: inria-00372234
https://hal.inria.fr/inria-00372234
Submitted on 31 Mar 2009
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of
sci-entific research documents, whether they are
pub-lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
Univariate Algebraic Kernel and Application to
Arrangements
Sylvain Lazard, Luis Peñaranda, Elias Tsigaridas
To cite this version:
Sylvain Lazard, Luis Peñaranda, Elias Tsigaridas. Univariate Algebraic Kernel and Application to
Arrangements. [Research Report] RR-6893, INRIA. 2009, pp.17. �inria-00372234�
a p p o r t
d e r e c h e r c h e
N
0
2
4
9
-6
3
9
9
IS
R
N
IN
R
IA
/R
R
--6
8
9
3
--F
R
+
E
N
G
Thème SYM
Univariate Algebraic Kernel and Application to
Arrangements
Sylvain Lazard — Luis Peñaranda — Elias Tsigaridas
N° 6893
Centre de recherche INRIA Nancy – Grand Est
LORIA, Technopôle de Nancy-Brabois, Campus scientifique,
SylvainLazard ,Luis Peñaranda , EliasTsigaridas y
ThèmeSYMSystèmessymboliques Équipe-ProjetVEGAS
Rapportdere her he n° 6893Mars200918pages
Abstra t: Solvingpolynomialsandperformingoperationswithrealalgebrai numbersare riti alissuesin geometri omputing, in parti ularwhendealing with urvedobje ts. Moreover,therealrootsneedtobe omputedina erti ed wayinordertoavoidpossiblein onsisten yingeometri algorithms. Developing e ientsolutions forthis problem is thus animportant issuefor the develop-ment oflibrariesof omputationalgeometry algorithms and, in parti ular, for thestate-of-the-art(opensour e) gal library. Wepresenta gal-based uni-variatealgebrai kernel,whi hprovides erti edreal-rootisolationofunivariate polynomialswith integer oe ientsandstandardfun tionalitiessu hasbasi arithmeti operations,greatest ommondivisor(g d)andsquare-free fa toriza-tion,aswellas omparisonandsignevaluationsofrealalgebrai numbers.
We ompareour kernelwith other omparablekernels, demonstrating the e ien yofourapproa h. Ourexperimentsareperformedonlargedatasets in- ludingpolynomialsofhighdegree(upto2000)andwithverylarge oe ients (up to25000bitsper oe ient).
We also address the problem of omputing arrangements of x-monotone polynomial urves. We apply our kernel to this problem and demonstrate its e ien y ompared to previoussolutions available in gal. We also present the rst bit- omplexity analysis of the standard sweep-line algorithm for this problem.
Key-words: CGAL, algebrai kernel, RS, root isolation,algebrai number omparison
INRIANan y-GrandEst,LORIA,Fran e. FirstName.Nameloria.fr y
INRIA Sophia-Antipolis - Méditerranée, Fran e. FirstName.Namesophia.inr ia. fr . MostpartofthisworkwasdonewhiletheauthorwasatLORIA-INRIANan y-GrandEst.
Résumé : Résoudre des systèmes polynomiaux et ee tuer des opérations ave des nombres algébriques réels ont une importan e ritique en géométrie algorithmique, notamment dans la gestion d'objets ourbes. De plus, les ra- inesréelles né essitentd'être isolées d'unemanière erti ée pour évitertoute in ohéren edanslesalgorithmesgéométriques. La on eptiondesolutions e - a espour eproblèmeestdon unetâ heimportantepourledéveloppementde bibliothèquesd'algorithmesdegéométriealgorithmiqueet,enparti ulier,pour labibliothèque(opensour e)deréféren e gal. Nousprésentonsi iunnoyau algébriqueunivariérépondantauxspé i ations galetpermettantd'isoleret de omparer, de façon erti é, lesra ines réellesde polyn mes univariés à o-e ientsentiers. Cenoyaupermetégalementle al ul depg det l'élimination de fa teurs arrés (square-free fa torization), et il possède les fon tionnalités standards, ommelesopérationsarithmétiques.
Nous omparonsnotrenoyauave lesautresnoyauxsimilaires,démontrant ainsi l'e a ité de notre appro he. Nous avons testé les noyaux sur de gros ensemblesdedonnées omprenantdespolyn mesdehautdegré(jusqu'à2000) etave detrèsgrands oe ients(jusqu'à25000bitspar oe ient).
Nous onsidéronségalementleproblèmedu al uld'arrangementsde ourbes polynomialesmonotonesenx. Noustestonsnotrenoyausur eproblèmeetnous prouvons son e a ité par rapport aux solutions déjà existantes dans gal. Nous présentons en n la première analyse de bit omplexité de l'algorithme standarddebalayage.
Mots- lés: CGAL,noyaualgébrique,RS,isolationdesra ines, omparaison desnombresalgébriques
1 Introdu tion
Implementinggeometri algorithmsrobustlyis knowntobeadi ulttaskfor twomainreasons. First,alldegeneratesituationshaveto behandled and se -ond, algorithms often assume a real-RAM model (a random-a ess ma hine where ea h register anhold areal numberandea harithmeti operationhas unit ost)whi hisnotrealisti inpra ti e. Inre entyears,theparadigmofexa t geometri omputinghasarisenasastandardforrobustimplementations[27℄. In thisparadigm, geometri queries,also alled predi ates,su h asis apoint inside, outside oron a ir le?, are made exa tlyusing, usually, either (i) ex-a t arithmeti ombined, for e ien y, with interval arithmeti ondoubles or (ii)intervalarithmeti onarbitrary- xed-pre ision oating-pointnumbers om-binedwithseparationbounds;ontheotherhand,geometri onstru tions,su h asthe ir le throughthree pointsorpointsofinterse tion betweentwo urves, maybeapproximated.
We address here one re urrent di ulty arising when implementing algo-rithms dealing, in parti ular, with urved obje ts. Su h algorithms usually requireevaluating, manipulatingandsolvingsystemsofpolynomialsequations and omparingtheirroots. Oneofthemost riti alpartsofdealingwith poly-nomialsorpolynomialsystemsistheisolationofthe realrootsandtheir om-parison.
We restri t here our attention to the ase of univariate polynomials and addressthisprobleminthe ontextof gal, aC++ComputationalGeometry AlgorithmsLibrary,whi hisanopensour eproje tandbe ameastandardfor theimplementationofgeometri algorithms[4℄.
Cgal is designed in a modular fashion following the paradigm of generi programming. Algorithmsaretypi allyparameterizedbyatraits lasswhi h en- apsulatesthegeometri obje ts,predi atesand onstru tionsusedbythe algo-rithm. Algorithms anthustypi allybeimplementedindependentlyofthetype ofinputobje ts. Forinstan e,the oreofaline-sweepalgorithmfor omputing arrangementsofplane urves[7℄ anbeimplementedindependentlyofwhether the urvesare lines, line segments, orgeneral urves; on the other hand, the elementaryoperationsthatdependonthetypeoftheobje ts(su has, ompar-ingx- oordinatesofpointsofinterse tion)areimplementedseparatelyintraits lasses. Similarly, the model of omputation, su h as exa t arbitrary-length integerarithmeti orapproximate xed-pre ision oating-pointarithmeti , are en apsulatedinthe on eptofkernel. Animplementationisthustypi ally sep-arated in three or four layers,(i) the geometri algorithm whi h relies on (ii) a traits lass, whi h itself relies on (iii) a kernel for elementary (typi ally ge-ometri ) operations. Cgal provides several prede ned Cartesian kernels, for instan e allowing standard Cartesian geometri operations on inputs de ned with doublesand providingapproximate onstru tions(i.e., de ned with dou-ble)butexa tpredi ates. However,akernel analsorelyon(iv)anumbertype whi h essentiallyen apsulatesthetypeof number(su h as,double, arbitrary-length integers, intervals) and the asso iated arithmeti operations. A hoi e of traits lasses,kernelsand numbertypesis usefulasit givesfreedom to the usersanditmakesiteasierto ompareandimprovethevariousbuildingblo ks ofanimplementation.
OurContributions. Wepresentinthispapera gal- ompliantalgebrai ker-nelthatprovidesreal-rootisolationofunivariateintegerpolynomialsandbasi operations, i.e. omparisons and sign evaluations, of real algebrai numbers. Thisopen-sour ekernelfollowsthe galspe i ationsforalgebrai kernels[3℄. TherootisolationisbasedontheintervalDes artesalgorithm[5℄andusesthe libraryrs[22℄. Moreover,ourkernelprovidesvariousoperationsfor polynomi-als,su hasg d,whi hare ru ialformanipulatingalgebrai numbers.
We ompareourkernelwithother omparablekernelsanddemonstratethe e ien yofourapproa h. Weperformexperimentsonlargedatasetsin luding polynomialsofhighdegree(upto2000)andwithverylarge oe ients(upto 25000bitsper oe ient).
Finally, we apply our kernel to the problem of omputing arrangements of x-monotone polynomial urvesand demonstrate its e ien y ompared to previoussolutionsavailable in gal. We alsopresentanoutput-sensitive bit- omplexityanalysisofthestandardsweep-linealgorithmfor thisproblem. We establishaboundof e O B (n+k)d 3 ( 2 +s 2 )
,wherenisthenumberof urves,k isthenumberofinterse tions,dboundsthedegreeofthepolynomials, bounds thebitsizeoftheir oe ients,andsisthelogarithmoftheminimumdistan e betweenthe ( omplex) rootsof the dieren eof anytwopolynomials(that is
roughlyspeaking the bitsizeof this distan e); the e O B
() notation ignoresthe
logarithmi fa tors. If N =maxfn;k;d; ;sg, this bound isin e O B (N 6 ) whi h is,asexpe ted, quadrati in thesize oftheinputintheworst ase;indeed,the input onsistsofnpolynomialsofdegreedandwith oe ientsofbitsize and is thus in (N
3
) in the worst ase. Tothe best of ourknowledge, this isthe rstbit- omplexityanalysisofthisalgorithm.
Related work. Combiningalgebraandgeometryformanipulating non-linear obje ts has been a long-standing hallenge. Previouswork in ludes, but it is not limited to, map [17℄ a library for manipulating points that are de ned algebrai allyand handling urvesin theplane. Morere ently,the library ex-a us [2℄, whi h handles urves and surfa es in omputational geometry and supports various algebrai operations, wasdeveloped and partially integrated into gal. Thenotionof algebrai kernelfor gal wasproposedin 2004[12℄; in this work, the underlying algebrai operations were based on the synaps library[18℄. Severalmethods and algebrai kernelshavebeen developed sin e then.
OnekernelwasdevelopedbyHemmerandLimba h[16℄followingthegeneri programming paradigm using the C++ template me hanism. This kernel is templatedbytherepresentationofalgebrai numbersandbythereal root iso-lationmethod, for whi h two lasseshavebeendeveloped;oneisbasedonthe Des artes methodand theother ontheBitstreamDes artes method [9℄. This approa hhas theadvantagetoallow, in prin iple,usingthebest instan esfor bothtemplatearguments.
Anotherkerneldevelopedatinriareliesonthesynapslibrary[18℄. Inthis kernelthereareseveralapproa hes on erningrealrootisolation,i.e., methods basedonSturmsubdivisions, sleevesapproximations, ontinuedfra tions, and asymboli -numeri ombinationofthesleeveand ontinuedfra tionsmethods (see [11℄). Moreover,there are spe ialized methods for polynomials of degree lessorequalthanfour[24℄.
Emirisetal.[11℄presentedsomeben hmarksofthesevariousapproa hesin thesetwokernelsaswellassometestsonthekernelwepresenthere. Theauthors mention that ourkernelbased onintervalDes artes performs similarlyto one approa h(refertoasn f2)basedon ontinuedfra tions[23℄for oe ientswith (very)largebitsizebut n f2is moree ientforsmall bitsize. They on lude that, rst, dedi ated algorithms for polynomials of degree lessthan (or equal to) four is alwaysthe most e ient approa h and, se ond, that n f2 always performthe best ex ept for low-degreeandhigh-bitsize polynomials,in whi h ase the kernel based on the Bitstream Des artes method performs the best. Wemoderateherethese on lusions.
Therestofthepaperisstru turedasfollows. Inthenextse tionwedes ribe our univariate algebrai kernel. In Se tion 3 we present various experiments on erningrealrootisolationand omparisonofrealalgebrai numbers. Finally, inSe tion4,wesket hourtraits lassforarrangements,wepresentexperiments againstthetraits lassthatis urrentlyavailablein gal,andwepresentthebit omplexity analysisof thealgorithm for omputing thearrangementof urves de ned byunivariatepolynomials.
2 Univariate algebrai kernel
We des ribe here ourimplementation of our univariatealgebrai kernel. The twomainrequirementsofthe gal spe i ations,whi hwedes ribehere,are theisolationofrealrootsandtheir omparison. Wealsodes ribeour implemen-tation of twooperations, theg d omputation and the re nementof isolating intervals,that arebothneededfor omparingalgebrai numbers.
Preliminaries. Thekernelhandlesunivariatepolynomialsandalgebrai num-bers. Thepolynomialshaveinteger oe ientsand are representedby arrays of gmparbitrary-lengthintegers[15℄. Weimplementedin thekernelthebasi fun tions forpolynomials. An algebrai numberthatisarootof apolynomial F isrepresentedby F and an isolatinginterval,that is an interval ontaining this root but no other root of F. We implemented intervals using the mpfi library[19℄,whi hrepresentsintervalswithtwompfr arbitrary- xed-pre ision oating-point numbers [20℄; note that mpfr is developed on top of the gmp libraryformulti-pre isionarithmeti [15℄.
Root isolation. Forisolating the real roots of univariatepolynomials with integer oe ients, we developed an interfa e with the library rs [22℄. This libraryiswritteninCandisbasedonDes artes'ruleforisolatingtherealroots ofunivariatepolynomialswith integer oe ients.
Webrie y detail herethe generaldesign of thers library; see [21℄ for de-tails. rsisbasedonanalgorithmknownasinterval Des artes [5℄;namely,the oe ientsofthepolynomialsobtainedby hangesofvariable,sendingintervals [a;b℄onto[0;+1℄,areonlyapproximatedusingintervalarithmeti whenthisis su ientfordeterminingtheirsigns. Notethattheorderin whi hthese trans-formations are performed in rs is important for memory onsumption. The intervalsandoperationsonthemarehandledbythempfilibrary.
Algebrai number omparison. As mentionedabove,oneofthemain re-quirementsofthe galalgebrai kernelspe i ationsis to ompare two alge-brai numbersr
1 andr
2
. Ifwearelu ky,theirisolatingintervalsdonotoverlap andthe omparisonis straightforward. Thisis, of ourse,notalwaysthe ase. If we knew that they were not equal, we ould re ne both isolating intervals untiltheyaredisjoint;seebelowfordetailsonhowweperformthere nements. Hen e,theproblemredu esto determiningwhether thealgebrai numbersare equalornot.
Todoso,we omputethesquare-freefa torizationoftheg dofthe polyno-mialsasso iatedto thealgebrai numbers(seebelowfor details). Therootsof thisg darethe ommonrootsofbothpolynomials. We al ulatethe interse -tion,I,oftheisolatingintervalsofr
1 andr
2
. Theg dhasarootinthisinterval ifandonlyifr
1 =r
2 .
Todetermine whether theg d hasaroot in intervalI, it su es to he k thesignoftheg dattheendpointsofI: iftheyaredierentoroneofthemis zero,theg dhasaroot inI andr
1 =r 2 ;otherwise,r 1 6=r 2
andwe anre ne bothintervalsuntiltheyaredisjoint.
G d omputations. Computinggreatest ommondivisorsbetweentwo poly-nomialsis nota di ulttask, however, itis nottrivialto doso e iently. A naiveimplementationoftheEu lideanalgorithmworks neforsmall polynomi-alsbuttheintermediate oe ientssueranexponentialgrowinsize,whi his notmanageableformediumtolargesizepolynomials. Wethusimplementeda modular g dfun tion. Wedidnotusesomeexistingimplementationsmainlyfor e ien ybe ause onvertingpolynomialsfromonerepresentationtoanotheris substantially ostly assoon asthe degreeand bitsize are large. Our fun tion al ulatestheg dofpolynomialsmodulosomeprimenumbersandre onstru ts latertheresultwiththehelpoftheChineseremaindertheorem. Detailsonthese algorithms an be found in, e.g. [26℄. Note that modular g d is always more e ientthanregularg danditismu hmoree ientwhenthetwopolynomials haveno ommonroots.
Re ningisolating intervals. Aswementionedbefore,re ningtheinterval representingan algebrai numberis riti al for omparing su h numbers. We providetwoapproa hesforre nement.
Both approa hes require that the polynomial asso iated to the algebrai numberis square free. The rst step thus onsists of omputing the square-free partof the polynomial (by omputing the g d of the polynomial and its derivative).
Our rstapproa hisasimplebise tionalgorithm. It onsistsin al ulating thesignofthepolynomialasso iatedtothealgebrai numberattheendpoints andmidpointoftheinterval. Dependingonthesesigns, were netheisolating intervaltoitsleftofrighthalf.
Ourse ondapproa hisaquadrati intervalre nement[1℄. Roughly speak-ing,thismethodsplitstheintervalinmanypartsand,basedonalinear interpo-lation,guessesin whi h onetherootlies. If theguessis orre t,thealgorithm dividesinthenextre nementsteptheintervalinmorepartsand,ifnot,inless. Unfortunately, even with our areful implementation this approa h turns outto be, onaverage,onlyjust abit faster thanthe bise tion approa h. Our
0 400 800 1200 1600 2000 o e ientbitsize 0 1 2 3 4 isolation time [ms℄ Ourkernel MPI I'skernel SYNAPS'kernel (a) 0 10000 20000 30000 40000 50000 o e ientbitsize 0 10 20 30 40 isolation time [ms℄ Ourkernel MPI I'skernel SYNAPS'kernel
(b)
Figure1: Runningtimeforisolatingalltherealrootsofdegree12polynomials with12realrootsin termsofthemaximumbitsizeoftheir oe ients.
experiments showed that the bottlene k of there nement is theevaluation of polynomials.
3 Kernel ben hmarks
In this se tion, weanalyze therunning time ofthe twomain fun tions of our algebrai kernel, that (i) isolate the roots of a polynomial and (ii) ompare twoalgebrai numbersthatis, omparetherootsof twopolynomials. Wealso ompare the performan e of our kernel with the one based on the Bistream Des artes method [9℄ and developed by Hemmer and Limba h [16℄ (referred to asmpii's kernel)
1
and with a kernel based on ontinued fra tions [23℄ and developedontopofthesynapslibrary[18℄(referredtoassynaps'kernel).
All tests were ran on asingle- ore3.2 GHzIntelPentium4 with 2 Gb of RAMand2048kbof a hememory,using64-bitLinux.
Root isolation. We onsider two suites of experiments in whi h we either x thedegreeofthepolynomialsand varythebitsizeofthe oe ientsorthe
1
To omparebothalgebrai kernelswiththesameinputs,weparameterizedmpii'skernel to use BitstreamDes artes as root isolator, algebrai _real_bfi_rep as algebrai number representationandCoreintegersandrationalstorepresentthe oe ientsofthepolynomials andthe isolationboundsofalgebrai numbers,respe tively. The hoi eof Core(vs. leda) wasindu edbytheneedoftestingthekernelsinthesame onditions,thatis,relyingongmp.
0 5000 10000 15000 20000 25000 o e ientbitsize 0 50 100 150 200 250 isolation time [ms℄ Ourkernel MPI I'skernel SYNAPS'kernel (a) 0 10 20 30 40 50
d
0 1 2 3 4 5 6 isolation time [s℄ Ourkernel MPI I'skernel SYNAPS'kernel(b)
Figure 2: Running time forisolating allthe realroots of (a) degree100 poly-nomialsintermsofthemaximumbitsizeoftheir oe ientsand(b)Mignotte polynomialsoftheform f =x
d
2(kx 1) 2
intermsofthedegreed.
onverse; see Figs. 1 and 2. In ea h experiment, we report the running time forisolatingalltherootsperpolynomial,averagedoverdierenttrials,forour kernel,mpii'sandsynaps'kernel.
Varying bitsize. Westudyherepolynomialswithratherlowdegree(12)but withno omplexrootandpolynomialswithreasonablylargedegree(100)with random oe ients(andthuswithfewrealroots).
The rst test sets omes from [16℄. See Fig.1. It onsists of polynomials of degree 12, ea h one being the produ t of six degree-two polynomials with tworoots,at leastoneofthem intheinterval[0;1℄;everypolynomialthus has 12realroots. Wevary themaximumbitsizeofallthe oe ientsoftheinput polynomialfrom100to50000andaverageea htestover250trials.
Se ondly, we onsider random polynomials with onstant degree 100 and oe ientswith varying bitsize. SeeFig. 2(a). Note that su h random poly-nomialshave fewroots: the expe ted numberof real rootsof apolynomialof degreedwith oe ientsindependently hosenfrom thestandardnormal dis-tribution is 2 ln (d)+C+ 2 d +O(1=d 2
) where C 0:625735 [8℄; this gives, for degree 100 an average of about 3.6 roots (note that this bound mat hes extremelywellexperimentalobservations). Wevarythemaximumbitsizeofall the oe ientsfrom2000to25000andaverageea htestover100trials.
Varyingdegree. We onsidertwosetsofexperimentsinwhi hwestudy ran-dompolynomialsandMignottepolynomials(whi hhavetwovery loseroots). We rst onsider polynomialswith random oe ients of xed bitsize for variousvaluesbetween32and1000.Wethenvarythedegreeofthepolynomials from100to2000andaverageourexperimentsover100trials(seeFig.3). Note that theaboveformula givesanexpe ted numberof rootsvarying from 3.6to 5.5. Weobservethat therunningtime is almostindependent ofthe bitsizein the onsidered range.
Finally,wetestMignottepolynomials,thatispolynomialsoftheformx d
2(kx 1) 2
. Su h polynomials are known to be hallenging for Des artes al-gorithmsbe ause twooftheir roots arevery loseto ea h other; theisolating
0 500 1000 1500 2000 p olynomialdegree 0 5 10 15 20 25 isolation time [s℄ Ourkernel MPI I'skernel SYNAPS'kernel (a) 0 500 1000 1500 2000 p olynomialdegree 0 10 20 30 40 50 60 70 isolation time [s℄ Ourkernel MPI I'skernel SYNAPS'kernel
(b)
Figure 3: Runningtime for isolatingallthe real rootsof random polynomials with oe ientsofbitsize(a)32and(b)1000,anddependingonthedegree.
intervals for these two roots are thus very small. For these tests, we used Mignottepolynomialswith oe ientsofbitsize50,withvaryingdegreedfrom 5 to 50. See Fig. 2(b). We averagedthe running time over5 trials for ea h degree. We observed essentially nodieren e betweenour kerneland MPII's one;theytakeroughly0.2 and5.5se onds forMignotte polynomialsofdegree 20and 50,respe tively. However,synaps'kernelismu hmoree ientasthe ontinuedfra tionsalgorithmis notsoae ted bythe losenessoftheroots.
Dis ussion. We observe (Fig. 1(a)) that synaps' kernel is more e ient thanbothourandmpii'skernelinthe aseofpolynomialsofsmalldegree(e.g., twelve) andsmall to moderately large oe ients(up to 2000bits per oe - ient). However,for extremely large oe ients mpii's kernelis substantially moree ient (by afa torofupto 3for oe ientsof upto50000bits)than bothourandsynaps'kernels,whi h performsimilarly.
For polynomials of reasonablelarge degree, both our and synaps' kernels aremu hmoree ientthatmpii'skernel;furthermorethesetwokernelsbehave similarlyfordegreesupto1500andourkernelbe omesmoree ientforhigher degrees(byafa tor2fordegree2000).
We also observe that the running time is highly dependent of the various settings. Forinstan e,ourkernelisupto5timesslowerwhenusingapproximate evaluationforhigh-degreeandhigh-bitsizepolynomials. Also,mpii'skernelisin some asesabout10timesslowerwhen hangingthearithmeti kerneltoleda, the representationof algebrai numbers andsome internal algorithms su h as there nementfun tion. Thisexplainswhyourben hmarksonbothmpii'sand synaps'kernelsaresubstantiallybetterthanin Emiriset al. experiments[11℄. We also observethat the running time of mpii's kernel is unstable in our experiments (Figs. 1and 2(a)); surprisingly, this instability o urs when the experimentsareperformedona64-bitsar hite ture,butitis stableon32-bits ar hite tureasshowninpreviousexperiments[11℄.
Comparisonofalgebrai numbers. We onsiderthreesuitesofexperiments for omparingalgebrai numbers;see Fig.4. Re allthat an algebrai number is here represented by a polynomial F that vanishes at and an isolating interval ontainingbutnootherrootofF. Re allalsothatthe omparisonof
0 500 1000 1500 2000 2500 3000 o e ientbitsize 0 250 500 750 1000 omparison time [ms℄ Ourbise tion MPI I'squadrati MPI I'sbise tion
(a) 0 500 1000 1500 2000 2500 3000 o e ientbitsize 0 250 500 750 1000 omparison time [ms℄ Ourbise tion MPI I'squadrati MPI I'sbise tion
(b)
Figure 4: Runningtime for omparing twodistin t lose roots of two almost identi alpolynomialsofdegree20with(a)no ommonrootsand(b)a ommon fa torof degree10.
twoalgebrai numbersisdoneby(i)testingwhether theintervalsaredisjoint; ifso,reporttheordering,otherwise(ii) omputetheg dofthetwopolynomials andtestwhethertheg dvanishesin theinterse tionofthetwointervals;ifso, reporttheequalityofthenumbers,otherwise(iii)re netheintervalsuntilthey aredisjoint.
First, we analyzethe ostof trivial omparisonsthat is, when the two in-tervalsrepresentingthenumbersaredisjoint. Forthatwe omparetherootsof tworandompolynomials. Weobservethat,asexpe ted,the omparisontimeis negligibleandindependentofboththedegreeofthepolynomialsandthebitsize oftheir oe ients.
Se ond,weanalyzethe ostof omparingrootsthat are very loseto ea h other but whose asso iate polynomials have no ommon root. This ase is expensivebe auseweneedtore netheintervalsuntiltheydonotoverlap;this is,however,nottheworstsituationbe ausetheg dofthetwopolynomialsis1 whi histestede ientlywithamodularg d. Weperformtheseexperimentsas follows. Wegeneratepairsofpolynomials,onewithrandom oe ientsandthe otherby onlyadding 1to oneofthe oe ientsofthe rstpolynomial. Su h polynomialsare su h that thei-th rootsof bothpolynomialsarevery loseto ea hother. Wegeneratesu hpairsofpolynomialswith onstantdegree(equal to20)andvarythemaximumbitsizeofthe oe ients. Asthebitsizein reases, thepairs of roots that are lose be ome even loserand thus the omparison timein reases. TheresultsoftheseexperimentsarepresentedinFig.4(a),whi h reports the average running time for omparing two loseroots. We show in this gurethree urves,one orrespondingtoourbise tionalgorithm,andtwo orresponding the two re nement methods implemented in the mpii's kernel: theusualbise tion andaquadrati re nementalgorithm.
Third,we onsider the,apriori,mostexpensives enarioin whi hwe om-parerootsthatareeitherequalorvery losetoea hothersandsu hthat their asso iatepolynomialshavesomerootsin ommon. Inthis ase,wea umulate the ostof omputinganon-trivialg dofthetwopolynomialswiththe ostof re ningintervalswhen omparingtwonon-equalroots. Inpra ti e, we gener-atepairsofdegree-20polynomialsea hde nedastheprodu toftwodegree-10 terms;oneofthesefa torsisrandomand ommontothetwopolynomials;the
0 200 400 600 800 1000 o e ientbitsize 0 50 100 150 200 onstru tion time [s℄ Ourkernel CORE (a) 0 40 80 120 160 200 degree 0 50 100 150 200 onstru tion time [s℄ Ourkernel CORE (b)
Figure 5: Arrangements of ve polynomials, shifted four times ea h, (a) of degree20andvaryingbitsizeand(b)ofbitsize32andvaryingdegree.
other fa tor is random in one of thepolynomialsand slightly modi ed in the other polynomial where, slightly modi ed means, asabove, that we add 1to oneofthe oe ients. Wethen varythemaximumbitsizeofthe oe ients.
Dis ussion. Weseein Fig. 4that thempii'squadrati re nementalgorithm largelyoutperformsthetwobise tionmethods. However,ourbise tionmethod isfasterthanmpii'sone,byafa torupto10. Wealsoobservedthattherunning timefor omparingequalrootsisnegligible omparedtothe ostof omparing losebutdistin troots. (TherunningtimereportedinFig.4(b)isa tuallythe totaltimefor omparingallpairsofrootsdividedbythenumberof omparisons of losebut distin t roots.) This explainswhyour kernelbehavessimilarlyin Figs. 4(a)and4(b). Overall,itappearsthat omparingalgebrai numbersthat arevery loseisfairlytime onsumingandthat themosttime- onsumingpart ofthe omparisonistheevaluationofpolynomialsperformedduringtheinterval re nements.
4 Arrangements
Asanexampleofpossiblebene tofhavinge ientalgebrai kernelsin gal, weusedourimplementationto onstru tarrangementsofpolynomialfun tions. Wein and Fogel [25℄ provided a gal pa kage for al ulatingarrangementsof general urves whi h requires as parameter atraits lass ontaining the data stru turestostorethe urvesandvariousprimitiveoperations,su has ompar-ingtherelativepositionsofpointsofinterse tion. Weimplementedatraits lass whi husesthefun tionsof ouralgebrai kerneland ompareditsperforman e withanothertraits lasseswhi h omeswith gal's arrangementpa kageand usestheCorelibrary[6℄.
Inorder to generate hallenging datasets we pro eed asfollows. First we generatenrandompolynomials. Toea hofthemweadd1tothe onstant oef- ient,mtimes,thusprodu ingadatasetofn(m+1)univariatepolynomials. Noti e that thearrangementofthe graphsofthese polynomialsis guaranteed to bedegenerate,i.e., thereareinterse tionswith thesamex- oordinate. The arrangementsgeneratedthiswayhavefourparameters: thenumbernofinitial
polynomials, the number m of shifts that we perform, the degree d of the polynomials,andthebitsize oftheir oe ients. Weranexperimentsvarying thevaluesofthelastthreeofthese parametersandsettingn=5.
Fig. 5(a) shows the running time in terms of the bitsize for a data set whered=20andm=4(giving25polynomials). Fig. 5(b) showstherunning timein termsof thedegreedfor ase ond data set where =32 andm =4. Wesee from these experimentsthat running timeusing Coreis onsiderably higherthanwhenusingourkernel. Wealso makethefollowingobservations.
Fig. 5(a) shows that the running time depends on the bitsize. When we hangethebitsizeofthe oe ientsoftherandompolynomials,thesize ofthe arrangementdoesnot hange;thatmeansthatthenumberof omparisonsand rootisolationsthekernelmustperform isroughlythesamein allthe arrange-ments of the test suite. The isolation time for random polynomials does not depend mu h on thebitsize(as shown in Fig.2(a)), but the omparison time does. Itfollowsthat therunningtimein reaseswiththebitsize.
Fig. 5(b) shows that the running time depends also on the degree of the inputpolynomials. As wesawin Se tion 3, theexpe ted numberof realroots of a random polynomial depends on its degree. The size of the arrangement thusin reaseswiththedegreeoftheinputpolynomials: ea hvertexistheroot of thedieren e between twoinput polynomials,therefore there will be more verti es. Thus, when we in rement the degree of the inputs, the number of omparisonsandisolationsin reases;furthermore,therunningtimeforea hof theseoperationsin reaseswiththedegreeoftheinput.
Weranadditionalteststoseetheimpa toftheinputshiftsinthe al ulation time. Wegenerated verandompolynomialsofbitsize1000anddegree20. We al ulatedarrangements,then,varyingthenumberofshiftsweperformtoea h polynomial. AsFig. 6shows,wewereonlyabletosolve,usingCore,the rst arrangement,generatedwithoutshifts(notethepointontheverti alaxis). We note that the running time in reases fast with the number of shifts. This is reasonablesin e, ea h time wein rease by 1 the number of shifts, we add to thearrangementnpolynomials,hen ein reasingthenumberofverti esofthe arrangement. Sin e theroot isolation and omparison time remainsthe same (be ause the degreeand the bitsizeare onstant), the running time in reases withthenumberoftheseoperations.
4.1 Complexity analysis
Inthisse tionwepresentanoutput-sensitiveanalysis ofthebit omplexityof thestandardline-sweepalgorithmfor omputingarrangementsofgraphsof uni-variatepolynomialfun tions. Thesameanalysis,andthusthesame omplexity bound,appliesalsointhe aseof rationalunivariatefun tions.
In what follows, ombinatorial and bit omplexities will be denoted by O andO B , respe tively. The e O and e O B
notationsreferto omplexities in whi h weignore (poly-)logarithmi fa tors. Wealso referto theseparation bound of aunivariatepolynomialastotheminimumdistan ebetweenanytwo(possibly omplex) roots of the polynomial. The bitsize of theseparationbound isthe numberofbits, s,neededtorepresentthelargestlowerbound oftheform2
s
thatissmallerthantheseparationbound.
First,wenotethattheresultsof[10,13℄ aneasilybegeneralizedtoexpress the omplexities of isolating the real roots of a polynomial in terms of the
separation bounds of the onsidered instan es of polynomials rather than in termsofworst- aseseparationbounds.
Proposition 1. The real roots of a univariate polynomial of degree d with integer oe ientsofbitsize atmost andseparation boundof bitsizes anbe isolated, with their multipli ities, in
e O B (d 3 ( 2 +s 2
)) time. The bitsize of the endpointsof theisolating intervals isin O(s).
Proof. In[10℄it wasproven that theworst ase omplexityis e O B (d 4 +d 4 2 ). The result is based on the fa t that the bitsize of the worst ase separation boundisO(d ). Toderiveanoutputsensitiveresultwerepla ethisbys.
Then, to isolate a root we need to perform at most e
O( +s) step. At ea hstepweperformapolynomialshift,inthe aseofDes artes'solver,oran evaluationofaSturmsequen e,withanumberofbitsize +s,resultinga ost, inboth ases,of e O B (d 2 2 +d 2 s+d 2 s 2
)forea hroot. Theresultfollowsifwe multiplybythenumberofroots,d.
Wealsore allanoutput-sensitiveresultonthe omplexityof omparingthe rootsoftwopolynomials.
Proposition2([13℄). Tworealalgebrai numbersde nedasrootsof polynomi-alsofdegreeatmostdwithinteger oe ientsofbitsizeatmost andseparation bounds ofbitsize atmost s anbe omparedin
e O B (d 2 ( +s))time.
These omplexityresultsyield,almostdire tly,thefollowingoutput-sensitive bit omplexityofthestandardline-sweepalgorithmfor omputingarrangements ofgraphsofunivariatepolynomialfun tions
Theorem 3. The arrangement of n urves,de nedby univariate polynomials ofdegreeatmostd, withinteger oe ientsofbitsizeatmost ,andseparation bound of bitsize at most s an be omputed in time
e O B ((n+k)d 3 ( 2 +s 2 )), wherek isthe numberof interse tion pointsbetweenthe urves.
Proof. Re all rstthatthe ombinatorial omplexityofthestandardline-sweep algorithmfor omputingarrangementsofnalgebrai urvesofboundeddegreeis O((n+k)logn),wherekisthenumberofinterse tionpoints(see,e.g.,[7,14℄). Toevaluate the bit omplexity of the algorithm, we split the analysis in two parts. Inthe rstpart, we onsider the omplexity ofthe onstru tionof the interse tion points of the urves. In the se ond part, we onsider the ost of omparingthex- oordinatesoftheinterse tionpoints.
Inorderto ompute(thatis,toisolate)theinterse tionpointsoftwo urves y = f 1 (x) and y =f 2 (x), represented by polynomials f 1 ;f 2 2 Z[x℄of degree atmostdwithinteger oe ientsofbitsizeatmost andseparationbound of bitsize at mosts, we an rst isolate thereal roots of thepolynomialf(x) =
f 1 (x) f 2 (x)intime e O B (d 3 ( 2 +s 2
))(byProposition1). We anthen ompute
theimagebyf oftheseintervalsofintime e O B
(d( +ds))(byHorner'srule). Tobeginthealgorithmweneedto omputeaverti allinethat istotheleft all the interse tion points between the urves. The ostof omputingsu h a lineis
e O B
(nd )andisdominatedbytheotherstepsofthealgorithm. Wethen omputetheinterse tionpointsofthislinewithallthe urves,so thattoorder the urvesalongthesweepline. Wethen omputetheinterse tionbetweenthe
n 1pairsofadja ent urvesalongthesweepline. Thus,weinitially perform O(n)interse tionsbetweenpairsof urves.
Then, duringthesweep,everytimeaninterse tionpointisen ounteredby thesweep line, weex hange two urvesis thelist of urvesinterse ted bythe sweeplineandwe omputetheinterse tionbetween(atmost)twonewpairsof adja ent urvesinthislist. Hen e,weperform,intotal,O(n+k)interse tions
betweenpairsof urvesin e O B ((n+k)d 3 ( 2 +s 2 ))time.
Wenow onsiderthe ostof omparingthex- oordinatesoftheinterse tion points when updatingthe eventlist. Everytime two urvesbe omeadja ent alongtheverti allineofsweep,weinserttheir rstinterse tionpointthatisto theright ofthe line. Sin e we only insertone interse tion point (ratherthan d), this requires in total O((n+k)logn) omparisons whi h an be done in
e O B ((n+k)d 2 ( +s))timebyProposition2.
Notethat,asmentionedinse tion1,ifN =maxfn;k;d; ;sg,thisboundis
in e O B (N 6
)whi his,asexpe ted,quadrati inthesizeoftheinputintheworst ase.
5 Con lusion
We presented a new gal- ompliant algebrai kernel that provides erti ed real-root isolationof univariatepolynomialswith integer oe ientsbasedon theintervalDes artes algorithm. This kernelalso provides the omparison of algebrai numbersandotherstandardfun tionalities.
We ompared ourkernelwith other omparable kernelsonlarge data sets in luding,forthe rsttime,polynomialsof highdegree(upto 2000)andwith extremelylarge oe ients(upto25000bitsper oe ient). Wedemonstrated the e ien y of our approa h and showed that it performs similarly, in most ases,withonekernelbasedonthesynapslibrary;morepre isely,ourkernelis moree ientforpolynomialsofverylargedegree(greaterthan1800)andless e ient forpolynomialsof very smalldegree and withsmall to moderate size oe ients. Also, our kernel is a lot more e ient that the kernel developed at mpii for polynomials of large degree (greater than 200); it is however less e ientforpolynomialsofsmalldegreeandwith extremelylarge oe ients.
Ourtestsindi ate thatthekerneldevelopedatmpiiappearstobeless e - ientthantheothertwoforpolynomialsoflargedegree. Howeveritshould be stressedthat this kernelis theonly oneamong thethree that istempletedby thenumbertypeofthe oe ients. Of oursethisdoesnotimplythate ien y isne essarilylostby followingthegeneri programmingparadigm,butit does implythat,fromtheuserpointofview,somesubstantialgainof e ien y an sometimesbemadebyusingakernelthatdoesnotfollowthisparadigm.
Wealso omparedthe performan e ofthe kernelsonthe omparisonof al-gebrai numbers. Weobservedin theseteststhat thebise tionalgorithm runs mu hfasterwhenitisspe ializedonanumbertypesin eitallowsforlowlevel optimizations, on rming thus the assertion in the previous paragraph. On theother hand,it be omes evidentthat thebise tion method is notthe most e ient algorithm when a large number of re nements is needed, and mpii's quadrati re nementisthefastestmethodbyfar.
Afairly large hoi eof algebrai kernelsand, in parti ular, of methods for isolatingthe realroots ofpolynomials,is nowavailable in Cgal. This allows, inparti ular,to ompareandimprovethevariousmethods. Itappearsthat be-tweenthetwobig lassesofmethods,basedon ontinuedfra tionsandDes artes algorithms, neitheris learlymu h betterthan theother. However,some sub-stantialdieren esappearbetweenthevariousimplementations,but,of ourse, itis alwaysverydi ultto ben hmarkimplementations. Forinstan e, we ob-servedherethattherunningtimesarehighly dependentofthevarioussettings andar hite tures.
Finally,wealsoaddresstheproblemof omputingarrangementsofx-monotone polynomial urves. We apply our kernel to this problem and demonstrate its e ien y ompared to previoussolutions available in gal. We also present the rst bit- omplexity analysis of the standard sweep-line algorithm for this problem.
A knowledgments
TheauthorsaregratefultoF.Rouillierforvariousdis ussionsandsuggestions. WethankZ.Zafeirakopoulosfordis overingmanybugsinourimplementation. WealsothankM.Hemmer,E.Berberi h,M.Kerber,andS.Limba hforfruitful dis ussiononthekerneldevelopedatmpiiandontheexperiments.
Referen es
[1℄ J. Abbott. Quadrati intervalre nement forreal roots. InInternational Symposiumon Symboli and Algebrai Computation (ISSAC), poster pre-sentation,2006.
[2℄ E.Berberi h,A.Eigenwillig,M.Hemmer,S.Hert,L.Kettner,K.Mehlhorn, J.Rei hel,S.S hmitt,E.S hömer,andN.Wolpert.EXACUS:E ientand Exa tAlgorithmsforCurvesandSurfa es.InPro .13thAnnualEuropean Symposiumon Algorithms (ESA), volume 1669 of LNCS, pages155166. Springer,2005.
[3℄ E. Berberi h, M. Hemmer, M. Karavelas, and M. Teillaud. Revision of the interfa espe i ation of algebai kernel. Te hni al Report ACS-TR-243301-01,ACSEuropean Proje t,2007.
[4℄ Cgal, Computational Geometry Algorithms Library. http://www. gal.org.
[5℄ G.Collins,J.Johnson,andW.Krandi k.IntervalArithmeti inCylindri al Algebrai De omposition. Journal of Symboli Computation, 34(2):145 157,2002.
[6℄ TheCorelibrary. http:// s.nyu.edu/exa t/.
[7℄ M.deBerg,O.Cheong,M.vanKreveld,andM.Overmars.Computational Geometry: AlgorithmsandAppli ations.Springer-Verlag,3rdedition,2008.
[8℄ A. Edelman andE. Kostlan. Howmayzerosof arandompolynomialare real? Bulletinof Ameri anMathemati al So iety,32(1):137,Jan1995.
[9℄ A. Eigenwillig, L. Kettner, W. Krandi k, K. Mehlhorn, S. S hmitt, and N.Wolpert.ADes artesAlgorithmforPolynomialswithBit-Stream Coef- ients. InPro . 8thInt.WorkshoponComputer AlgebrainS ient. Com-put. (CASC),volume3718ofLNCS,pages138149.Springer,2005.
[10℄ A. Eigenwillig, V. Sharma, and C. K. Yap. Almost tight re ursion tree bounds for the Des artes method. In Pro . Int. Symp. on Symboli and Algebrai Computation, pages 7178, New York, NY, USA, 2006. ACM Press.
[11℄ I.Emiris,M.Hemmer,M.Karavelas,S.Limba h,B.Mourrain,E.T sigari-das,andZ.Zafeirakopoulos.Cross-ben hmarksforunivariatealgebrai ker-nels. Te hni alReportACS-TR-363602-02,ACSEuropeanProje t,2008.
[12℄ I. Z. Emiris, A. Kakargias, S. Pion, M. Teillaud, and E. P. Tsigaridas. Towardsand open urved kernel. In Pro . 20th Annual ACM Symp. on Computational Geometry (SoCG),pages438446,NewYork,USA,2004.
[13℄ I.Z. Emiris,B.Mourrain,andE. P.Tsigaridas. RealAlgebrai Numbers: ComplexityAnalysis and Experimentation. In P. Hertling,C. Homann, W. Luther, andN.Revol,editors, Reliable Implementations ofReal Num-berAlgorithms: Theory andPra ti e,volume5045ofLNCS,pages5782. SpringerVerlag,2008.
[14℄ E. Fogel,D. Halperin, L.Kettner, M.Teillaud, R. Wein,and N.Wolpert. Arrangements.InJ.-D.BoissonnatandM.Teillaud,editors,Ee tive Com-putational Geometryfor CurvesandSurfa es,Mathemati sand Visualiza-tion, hapter1.Springer,2006.
[15℄ Gmp, Gnumultiplepre isionarithmeti library. http://gmplib.org/.
[16℄ M.HemmerandS.Limba h.Ben hmarksonageneri univariatealgebrai kernel.Te hni alReportACS-TR-243306-03,ACSEuropeanProje t,2006.
[17℄ J. Keyser, T. Culver, D. Mano ha, and S. Krishnan. E ient and ex-a t manipulationof algebrai pointsand urves. Computer-AidedDesign, 32(11):649662,2000.
[18℄ B. Mourrain, P. Pavone, P. Trébu het, E. P. Tsigaridas, and J. Wintz. Synaps, a library for dedi ated appli ations in symboli numeri om-putations. InM.Stillman,N. Takayama,andJ.Vers helde,editors, IMA VolumesinMathemati sanditsAppli ations,pages81110.Springer,New York,2007. http://synaps.inria.fr.
[19℄ Mpfi, multiple pre ision interval arithmeti library. http://perso.ens-lyon.fr/nathalie.revol/software.html.
[20℄ Mpfr, library for multiple-pre ision oating-point omputations. http://mpfr.org/.
[21℄ F. Rouillier and Z. Zimmermann. E ientisolation of polynomial's real roots. J. ofComputational andApplied Mathemati s,162(1):3350,2004.
[22℄ Rs, a software for real solving of algebrai systems. F. Rouillier. http://fgbrs.lip6.fr.
[23℄ E.P.TsigaridasandI.Z. Emiris. Onthe omplexityofrealrootisolation using Continued Fra tions. Theoreti al Computer S ien e, 392:158173, 2008.
[24℄ E.P. TsigaridasandI.Z.Emiris. Real algebrai numbersand polynomial systemsofsmall degree. Theoreti al Computer S ien e, 409(2):186199, 2008.
[25℄ R. Wein andE. Fogel. Thenewdesign of CGAL's arrangementpa kage. Te hni alreport,Tel-AvivUniversity,2005.
[26℄ C.Yap. FundamentalProblems ofAlgorithmi Algebra. OxfordUniversity Press,Oxford-NewYork,2000.
[27℄ C. Yap. Robust geometri omputation. In J. E. Goodman and J.O'Rourke,editors, Handbook of Dis reteandComputational Geometry, hapter 41, pages927952.Chapmen &Hall/CRC, Bo a Raton, FL, 2nd edition,2004.
0 2 4 6 8 10 verti alshifts 0 100 200 300 onstru tion time [s℄ Ourkernel CORE
Figure6: Arrangementgeneratedfrom verandompolynomialsofbitsize1000 INRIA
