A  good sample is a point set such that any ball of radius $\epsilon$ contains a constant number of points. The Delaunay triangulation of a good sample is proved to have linear size, unfortunately this is not enough to ensure a good time complexity of the randomized incremental construction of the Delaunay triangulation. In this paper we prove that a random Bernoulli sample of a good sample has a triangulation of linear size. This result allows to prove that the randomized incremental construction needs an expected linear size and an expected $O(n\log n)$ time.Un bon échantillon est un ensemble de points tel que toute boule de rayon $\epsilon$ contienne un nombre constant de points.Il est démontré que la triangulation de Delaunay d'un bon échantillon a une taille linéaire, malheureusement cela ne suffit pas à assurerune bonne complexité à la construction incrémentale randomisée de latriangulation de Delaunay.Dans ce rapport, nous démontrons que la triangulation d'un échantillon aléatoire de Bernoullid'un bon échantillon a une taille linéaire. Nous en déduisonsque la construction incrémentale randomisée peut être faite en temps$O(n \log n)$ et espace $O(n)$

Devillers, Olivier

Glisse, Marc

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HAL Id: hal-01568030https://hal.inria.fr/hal-01568030Submitted on 24 Jul 2017HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Delaunay triangulation of a random sample of a goodsample has linear sizeOlivier Devillers, Marc GlisseTo cite this version:Olivier Devillers, Marc Glisse. Delaunay triangulation of a random sample of a good sample has linearsize. [Research Report] RR-9082, Inria Saclay Ile de France; Inria Nancy - Grand Est. 2017, pp.6.￿hal-01568030￿ISSN0249-6399ISRNINRIA/RR--9082--FR+ENGRESEARCHREPORTN° 9082July 2017Project-Teams Datashape &GambleDelaunay triangulation ofa random sample of agood sample has linearsizeOlivier Devillers Marc GlisseRESEARCH CENTRESACLAY – ÎLE-DE-FRANCEParc Orsay Université4 rue Jacques Monod91893 Orsay CedexDelaunay triangulation of a random sample ofa good sample has linear sizeOlivier Devillers∗†‡ Marc Glisse§Project-Teams Datashape & GambleResearch Report n° 9082 — July 2017 — 8 pagesAbstract: A good sample is a point set such that any ball of radius ε contains a constantnumber of points. The Delaunay triangulation of a good sample is proved to have linear size,unfortunately this is not enough to ensure a good time complexity of the randomized incrementalconstruction of the Delaunay triangulation. In this paper we prove that a random Bernoulli sampleof a good sample has a triangulation of linear size. This result allows to prove that the randomizedincremental construction needs an expected linear size and an expected O(n log n) time.Key-words: Probabilistic analysis – Worst-case analysis – Randomized incremental constructionsPart of this work is supported by the Advanced Grant of the European Research Council GUDHI (GeometricUnderstanding in Higher Dimensions).∗ Inria, Centre de recherche Nancy - Grand Est, France.† CNRS, Loria, France.‡ Université de Lorraine, France§ Inria, Université Paris-Saclay, France.La triangulation de Delaunay d’un échantillon aléatoired’un bon échantillon a une taille linéaireRésumé : Un bon échantillon est un ensemble de points tel que toute boule de rayon εcontienne un nombre constant de points. Il est démontré que la triangulation de Delaunay d’unbon échantillon a une taille linéaire, malheureusement cela ne suffit pas à assurer une bonnecomplexité à la construction incrémentale randomisée de la triangulation de Delaunay. Dans cerapport, nous démontrons que la triangulation d’un échantillon aléatoire de Bernoulli d’un bonéchantillon a une taille linéaire. Nous en déduisons que la construction incrémentale randomiséepeut être faite en temps O(n log n) et espace O(n).Mots-clés : Analyse probabiliste – Analyse dans le cas le pire – construction incrémentalerandomiséeDelaunay triangulation of a random sample of a good sample has linear size 31 IntroductionA good sample of some domain D ⊂ Rd is a point set of size n in Rd such that any ball centeredin D of radius ε contains a constant number of points. When we enforce a hypothesis of gooddistribution of the points in space, a volume counting argument ensures that the local complexityof the Delaunay triangulation around a vertex is bounded by a constant (dependent on ε andd but not on the number of points). Unfortunately, to be able to control the complexity of theusual randomized incremental algorithms [1–4], it is not enough to control the final complexity ofthe triangulation, but also the complexity of the triangulation of a random subset.One would expect that a random sample of size k of a good sample is also a good samplewith high probability. Actually this is not quite true, it may happen with reasonable probabilitythat balls of volume O(1k)may contain log n points or that a ball of volume ΩÄlog kkädoes notcontain any point. Thus this approach can transfer the complexity of a good sample to the oneof a random sample of a good sample but losing log factors.In this paper, we study directly the Delaunay triangulation of a random sample of a goodsample and deduce results about the complexity of randomized incremental constructions.2 Results, Definitions, and NotationsWe define several sampling notions, our point set is in Td the flat torus of dimension d to avoidboundary conditions (T = R/Z):Definition 1. A set X of n points in Td is an (ε, κ)-sample if any ball of radius ε contains atleast one point and at most κ points.Definition 2. A subset Y of set X is a Bernoulli sample of X of parameter α if each point of Xbelongs to Y with probability α independently.Definition 3. A subset Y of set X is a uniform sample of X of size k if Y is any possible subsetof X of size k with equal probability.This short note proves the two following theorems:Theorem 4. Given an (ε, κ)-sample X in Td, a Bernoulli sample Y of probability α of X , and apoint p, the expected number of faces of dimension i of the Delaunay triangulation of Y ∪ {p}that contain p is at most 72di+O(d+i(log iκ)). In particular this number is a constant with respectto n and α.Theorem 5. Given an (ε, κ)-sample X in Td, the randomized incremental construction of theDelaunay triangulation needs O(n log n) expected time and O(n) expected space.An analogous of Theorem 4 for uniform samples instead of Bernoulli samples is probably truebut more difficult to prove. Since Theorem 4 suffices to deduce Theorem 5, we leave the resultfor uniform sample as a conjecture.We denote by Σ(p, r) and B(p, r) the sphere and the ball of center p and radius r respectively.] (Z) denotes the cardinality of the set Z.The volume of the unit ball of dimension d is denoted Vd, the area of its boundary is denotedSd−1, the formulas are Vd = πd2Γ( d2 +1)and Sd = 2πVd−1.RR n° 90824 O. Devillers & M. Glisse3 Bernoulli SampleLemma 6. For any point p, we can construct a set B ofballs of radius 1 such that any ball of radius bigger than 3having p on its boundary encloses a ball of B, and] (B) ≤ Ad = 2√πΓ(d+12)Γ(d2) Å 18√35ãd−1133Σ(p, 3) p√353Σ(z, r)zxy≤ 3uwΣ(x, 3)≤ 1≤ 1≤ 1Proof. We will build a packing of Σ(p, 3). A ball B of radius 1 centered on Σ(p, 3) verifies:Σ(p, 3)∩B has a (d− 1)-volume bigger that the (d− 1)-volume of a (d− 1) ball of radius√356 (seeFigure above). Now we consider a set of balls B centered on Σ(p, 3) such that the intersections ofthese balls with Σ(p, 3) are disjoint and B is maximal for this property.Let Σ(z, r) be a sphere of radius bigger than 3 having p on its boundary (red boundary of blueball in the figure). Let x be pz ∩ Σ(p, 3), then Σ(x, 3) passes through p and is inside tangent in pto Σ(z, r). Since B is maximal Σ(x, 1) (pink in figure) intersects a unit ball B(y, 1) ∈ B (green onfigure) in some point u. For any point w ∈ B(y, 1) we have ‖xw‖ ≤ ‖xu‖+ ‖uy‖+ ‖yw‖ ≤ 3 andthus Σ(x, 3) encloses B(y, 1).The (d− 1)-volume of Σ(p, 3) is Sd−13d−1, the (d− 1)−volume of Σ(p, 3) ∩B is bigger thanVd−1Ä√356äd−1for all B ∈ B, thus the cardinality of B is bounded by Sd−1Vd−1Ä18√35äd−1. Pluggingthe values of Vd−1 and Sd−1 gives the result.Lemma 7. Let ρ ≤ 1, there exists a covering of the unit ball in dimension d by balls of radius ρwith a number of balls smaller thanÄ3ρäd.Proof. Consider a maximal set of disjoint balls of radius ρ2 with center inside the unit ball,the balls with the same centers and radius ρ cover the unit ball (otherwise it contradicts themaximality). By a volume argument we get that the number of balls is bounded from above byVd(1+ ρ2 )dVd( ρ2 )d ≤Ä3ρädfor ρ < 1.Lemma 8. Any maximal packing of the unit ball in dimension d by balls of radius ρ has a numberof balls greater than (2ρ)−d.Proof. Doubling the radii of the balls gives a covering, then the volume argument gives a lowerbound of VdVd(2ρ)d= (2ρ)−d.Definition 9. Let F (p, r) be the event than the farthest Delaunay neighbor of p is at distancegreater than r from p in the Delaunay triangulation of Y ∪ {p} where Y is a Bernoulli sample ofparameter α of an (ε, κ)-sample X of Td.Lemma 10. P [F (p, 6r)] ≤ Ad expÄ−αrd (2ε)−dä.Proof. We consider Br the set B, defined at Lemma 6, scaled by r around p. If each ball of Brcontains at least one point, we know that all Delaunay neighbors of p are inside Σ(p, 6r).P [F (p, 6r)] ≤ P [∃B ∈ Br, B ∩ Y = ∅] ≤∑B∈BrP [B ∩ Y = ∅] ≤∑B∈Br(1− α)](B∩X )≤ ] (Br) (1− α)(r2ε )d≤ Ad expÅ−α( r2ε)dã,InriaDelaunay triangulation of a random sample of a good sample has linear size 5where the last line uses the lower bound of Lemma 8.Proof of Theorem 4. If the farthest Delaunay neighbor of p is at distance less than 6r, andk = ] (Y ∩B(p, 6r)) then ki is a trivial bound on the number of Delaunay i-faces incident to p.Thus we will compute the following:E [] (Delaunay i-faces incident to p)]≤ Eî] (Y ∩B(p, 6ε))ió+∞∑j=1P[F (p, 6 · 2j−1ε)]Eî](Y ∩B(p, 6 · 2jε))i | F (p, 6 · 2j−1ε)óOn the one hand, we remark thatE [] (Y ∩B(p, 2r)) | F (p, r)] ≤ E [] (Y ∩B(p, 2r))] ,since the knowledge that the farthest neighbor is far implies that there are some points of X thatdo not belong to the sample Y and biases the probability in the above direction. On the otherhand Lemma 2.2 in [5] allows to deduce for any domain D:Eî] (D ∩ Y)ió≤ ii E [] (D ∩ Y)]i .It gives:E [] (Delaunay i-faces incident to p)]≤ ii E [] (Y ∩B(p, 6ε))]i +∞∑j=1P[F (p, 6 · 2j−1ε)]ii E[](Y ∩B(p, 6 · 2jε))]i≤ iiαi] (X ∩B(p, 6ε))i +∞∑j=1P[F (p, 6 · 2j−1ε)]iiαi](X ∩B(p, 6 · 2jε))i≤ iiαiκi18di +∞∑j=0Ad exp(−α(2j−2)d)iiαiκi(18 · 2j)di using Lemmas 7 and 10≤ iiαiκi18di(1 +Ad∞∑j=02dij exp(−α2dj−2d)).The function j  2dij exp(−α2dj−2d)is increas-ing between 0 and j0 and decreasing between j0and ∞, with j0 = 2 + 1d log2(iα). Notice that themaximum of the function is 2dij0 exp(−α2dj0−2d)=22di+i log2(iα ) exp(−α2log2(iα ))≤ 4di(iα)i. A stan-dard sum to integral comparison yields:Sum Integral+ maximumRR n° 90826 O. Devillers & M. GlisseE [] (Delaunay i-faces incident to p)]≤ iiαi18diκiÇ1 +Ad2dij0 exp(−α2dj0−2d)+Ad∫ ∞j=02dij exp(−α2dj−2d)djå≤ iiαi18diκi(1 +Ad4di(iα)i+Ad∫ ∞k=α2−2dÇk22dαåiexp(−k) dkkd ln 2)with k = α2dj−2d≤ iiκi72diÅ1 +Adii +Ad∫ ∞k=0ki−1 exp(−k) 1d ln 2dkã≤ iiκi72diÄ1 +Adii +Ad(i−1)!d ln 2ä≤ 3i2iκi72diAd.4 From Bernoulli Sample to Uniform Sample of Size kProof of Theorem 5. Space complexity. Theorem 4 does not apply directly to the randomizedconstruction of the Delaunay triangulation of X . When the points are inserted in a random order,the kth inserted point pk is a random point in a uniform sample of size k of X , the expected numberof simplices of dimension i created by its insertion is E [] (Delaunay i-faces incident to pk | ] (Y) = k)] .To deduce the expected number of simplices created during the randomized incremental construc-tion we have to sum on i and k.The time complexity can be split in a location part and a construction part. The constructiontime is of the same order as the space complexity. The location time of an insertion in the usualrandomized incremental construction of the Delaunay triangulation of a set X of n points relieson a backward analysis argument. In the classical history graph, the argument is a follows: whenlocating the nth point pn, we trace the conflicts1 within the history of the construction of theDelaunay triangulation; the insertion of pk (2 ≤ k < n) creates m conflicts with pn if and only ifpkpn is an edge of the triangulation of {p1, p2, . . . , pk, pn} incident to m simplices. Since pk is arandom point q in {p1, p2, . . . , pk} we get as expected location time:n−1∑k=2E∑q∈Y;](Y)=kP [pk = q] · 1[qpnedge of DT (Y∪{pn})] · ] (Delaunay d-simplices incident to edge qpn)≤n−1∑k=21k · d · E [] (Delaunay d-simplices incident to pn in DT (Y ∪ {pn}) | ] (Y) = k)]Bernoulli to uniform sampling. Thus, the number of simplices incident to a vertex needsto be controlled in the triangulation of a uniform sample of size k and not of a Bernoulli sampleof probability α. Let fk and gα be the following random variables:fk = ] (Delaunay i-faces of DT (Y ∪ {p}) incident to p) ,where p is any point in the plane and Y is a uniform random sample of X of size k, andgα = ] (Delaunay i-faces of DT (Y ∪ {p}) incident to p)1 A point is in conflict with a simplex if it belongs to its circumscribing ball.InriaDelaunay triangulation of a random sample of a good sample has linear size 7where p is any point in the plane and Y is a Bernoulli sample of X of parameter α. Then theclassical equation isE [gα] =n∑k=0Çnkåαk(1− α)n−k E [fk] =n∑k=1Çnkåαk(1− α)n−k E [fk]since(nk)αk(1− α)n−k (the binomial distribution) is the probability that a Bernoulli sample ofparameter α has size k. Theorem 4 gives E [gα] = O(1).First, we can prove that the expected number of simplices constructed in a randomizedincremental construction is O(n):O(1) =∫ 10E [gα] dα =n∑k=1ÇnkåE [fk]∫ 10αk(1− α)n−kdα=n∑k=1ÇnkåE [fk]1(n+ 1)(nk) = 1n+1 n∑k=1E [fk]Then, we can prove that the expected complexity of localizing the last point in the Delaunaytriangulation of a uniform sample of size k is O(log n).We can write∫ 10E[gα]α dα =n∑k=1nÇn− 1k − 1åE[fk]k∫ 10αk−1(1− α)n−kdα=n∑k=1nÇn− 1k − 1åE[fk]k(n− k)!(k − 1)!n!=n∑k=1E[fk]k .To compute∫ 10E[gα]α dα we can use E [gα] = O(1) (Theorem 4) except in the neighborhood of 0where we use a separate, naive boundE [gα] ≤ Eî] (Y)dó≤ dd E [] (Y)]d = dd(αn)dusing again Lemma 2.2 from [5].∫ 10E[gα]α dα =∫ 1n0E[gα]α dα+∫ 11nE[gα]α dα≤∫ 1n0ddαd−1nddα+∫ 11nO(1)α dα= dd−1 +O(1) · log n = O (log n) .Acknowledgements. This work follows a question asked by Jean-Daniel Boissonnat and was solved during the16th INRIA–McGill–Victoria Workshop on Computational Geometry at the Bellairs Research Institute. Theauthors wish to thank all the participants for creating a pleasant and stimulating atmosphere.RR n° 90828 O. Devillers & M. GlisseReferences[1] Nina Amenta, Sunghee Choi, and Günter Rote. Incremental constructions con BRIO. In Proc. 19th AnnualSymposium on Computational geometry, pages 211–219, 2003. doi:10.1145/777792.777824.[2] Jean-Daniel Boissonnat and Monique Teillaud. On the randomized construction of the Delaunay tree. TheoreticalComputer Science, 112:339–354, 1993. doi:10.1016/0304-3975(93)90024-N.[3] Keneth L. Clarkson and Peter W. Shor. Applications of random sampling in computational geometry, II.Discrete & Computational Geometry, 4:387–421, 1989. doi:10.1007/BF02187740.[4] Olivier Devillers. The Delaunay hierarchy. International Journal of Foundations of Computer Science,13:163–180, 2002. hal:inria-00166711. doi:10.1142/S0129054102001035.[5] Olivier Devillers, Marc Glisse, Xavier Goaoc, and Rémy Thomasse. Smoothed complexity of convex hullsby witnesses and collectors. Journal of Computational Geometry, 7(2):101–144, 2016. hal:hal-01285120.doi:10.20382/jocg.v7i2a6.InriaRESEARCH CENTRESACLAY – ÎLE-DE-FRANCEParc Orsay Université4 rue Jacques Monod91893 Orsay CedexPublisherInriaDomaine de Voluceau - RocquencourtBP 105 - 78153 Le Chesnay Cedexinria.frISSN 0249-6399

La triangulation de Delaunay d'un échantillon aléatoire d'un  bon échantillon a une taille linéaire

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La triangulation de Delaunay d'un échantillon aléatoire d'un bon échantillon a une taille linéaire

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