Efficient Maximal Poisson-Disk Sampling

Transcription

1 Efficient Maximal Poisson-Disk Sampling Mohamed S. Ebeida Sandia National Laboatoies Andew A. Davidson Univesity of Califonia, Davis Anjul Patney Univesity of Califonia, Davis Patick M. Knupp Sandia National Laboatoies Scott A. Mitchell Sandia National Laboatoies John D. Owens Univesity of Califonia, Davis Figue 1: A Poisson-disk sampling of a non-convex domain (left). The gay-shaded disks show the sampling is maximal (ight). Abstact We solve the poblem of geneating a unifom Poisson-disk sampling that is both maximal and unbiased ove bounded non-convex domains. To ou knowledge this is the fist povably coect algoithm with time and space dependent only on the numbe of points poduced. Ou method has two phases, both based on classical datthowing. The fist phase uses a backgound gid of squae cells to apidly ceate an unbiased, nea-maximal coveing of the domain. The second phase completes the maximal coveing by calculating the connected components of the emaining uncoveed voids, and by using thei geomety to efficiently place unbiased samples that cove them. The second phase conveges quickly, ovecoming a common difficulty in dat-thowing methods. The deteministic memoy is O(n) and the expected unning time is O(n log n), whee n is the output size, the numbe of points in the final sample. Ou seial implementation veifies that the log n dependence is mino, and nealy O(n) pefomance fo both time and memoy is achieved in pactice. We also pesent a paallel implementation on GPUs to demonstate the paallel-fiendly natue of ou method, which achieves 2.4 the pefomance of ou seial vesion. CR Categoies: I.3.5 [Computing Methodologies]: Compute Gaphics Computational Geomety and Object Modeling; F.2.2 [Theoy of Computation]: Analysis of Algoithms and Poblem Complexity Nonnumeical Algoithms and Poblems Keywods: Poisson disk, maximal, povable convegence, linea complexity, sampling, blue noise Links: DL PDF msebeid@sandia.gov c ACM, 211. This is the autho s vesion of the wok. It is posted hee by pemission of ACM fo you pesonal use. Not fo edistibution. The definitive vesion appeas in ACM Tansactions on Gaphics, 3(4), August Intoduction Maximal Poisson-disk sampling distibutions ae useful in many applications. In compute gaphics these distibutions ae desiable because the andomness avoids aliasing, and they have the blue noise popety. Blue noise means the inte-sample distances follow a cetain powe law, with high fequencies moe common. The lack of low-fequency noise poduces visually pleasing esults fo endeing, imaging, and geomety pocessing [Pha and Humpheys 24]. The bias-fee popety is cucial in factue popagation simulations. In this pocess, a andom point cloud is equied to minimize the effect of the dynamic e-meshing on the diection of the cack gowth. Regula geometies tend to fom pefeential diections fo cack popagation. [Bolande and Saito 1998] A andomly geneated paticle system, on the othe hand, appoximates isotopic factue popeties well. [Jiásek and Bazant 1995] A maximal distibution impoves the quality bounds and pefomance of meshing methods such as Delaunay tiangulation [Attali and Boissonnat 24]. Poisson-disk sampling is a pocess that selects a andom set of points, X = {x i}, fom a given domain, D, in some K- dimensional space. The samples ae at least a minimum distance apat, satisfying an empty disk citeion. In this wok, we focus on the two-dimensional unifom case, whee the disk adius,, is constant egadless of location o iteation. Inseting a new point, x i, defines a smalle domain, D i D, available fo futue insetions, whee D o = D. The maximal condition equies that the sample disks ovelap, in the sense that they cove the whole domain leaving no oom to inset an additional point. This popety identifies the temination citeion of the associated sampling pocess. Biasfee o unbiased means that the likelihood of a sample being inside any subdomain is popotional to the aea of the subdomain, povided the subdomain is completely outside all pio samples disks. This is unifom sampling fom the uncoveed aea. This definition of unbiased is standad in the Poisson-disk context [Gamito and Maddock 29] and is equivalent to the Matén second pocess in statistics [196]. (And it goes by othe names in othe sciences.)

2 Bias-fee: x i X, Ω D i 1 : P (x i Ω) = Aea(Ω) Aea(D i 1) (1a) Empty disk: x i, x j X, x i x j : x i x j (1b) Maximal: x D, x i X : x x i < (1c) Despite the desiability of this distibution, it has been challenging fo the community to discove an efficient algoithm that satisfies all thee conditions. To ou knowledge, all pio methods elax the unbiased o maximal conditions, o equie potentially unbounded time o space. The elaxations may be quite small in pactice. The maximal condition may be esolved down to machine pecision. The bias may be unnoticeable in pai-wise distance spectum plots. But ou pesent wok appeas to be the fist method that povably meets all the conditions with time and space dependent only on the numbe of samples poduced. (It appeas that White et al. [27] and Gamito and Maddock [29] equie a tee whose depth is dependent on machine pecision; see below.) Ou main dawbacks ae the memoy equiements fo stoing polygons and the complication of coding geometic pimitives and tacking data stuctues. Ou implementations show that any dawbacks ae not ovewhelming, and the method woks well in pactice. Ou unning time is competitive with the best. Fo a detailed suvey of Poisson sampling methods, see Lagae and Dute [28]. Selecting an unbiased Poisson-disk sample point is known as dat-thowing in compute gaphics. The basic pocedue is to thow a dat, andom and unifomly by aea. If it is aleady coveed by a pio dat s disk, it is a miss and discaded; othewise it is a hit and kept. The challenge is that as the numbe of pio dats becomes lage, the uncoveed aea becomes smalle and its bounday becomes moe complex. The classic method [Dippé and Wold 1985; Cook 1986] is to sample unifomly fom the entie domain; when the faction of new thows that ae hits becomes vey small, the sampling is likely close to maximal, so the algoithm teminates. This is unbiased but also not maximal in finite time. To get close to maximal, we must take additional steps to tack the uncoveed domain and select new points fom it. Tile-based methods impove the pefomance, but elax the biasfee condition. Fo example, Wang tiles [Cohen et al. 23; Lagae and Duté 25] equie a biased Voonoi elaxation step to satisfy the empty-disk condition. Penose tiles [Ostomoukhov et al. 24; Ostomoukhov 27] have a single sample pe tile and equie Voonoi elaxation to educe sampling atifacts. Anothe class of methods impoves efficiency by computing samples on the fly [Mitchell 1987; Jones 26; Dunba and Humpheys 26; Bidson 27]. Howeve, these methods ae biased and equie elatively lage stoage. Dunba and Humpheys [26] poposed a linea-time advancing-font method whee each new sample is picked fom a egion nea to pio samples. Each new point has the same distance to its neaest neighbo, which violates the biasfee condition. Gid-based methods have emeged ecently and ae vey efficient. Wei [28] poposed a paallel sampling method that employs a sequence of multi-esolution unifom gids in the dat-thowing pocess. While quite pactical, the phase goups ae not completely bias-fee, and the algoithm teminates with only a nealy-maximal distibution. Bowes et al. [21] use a simila phase-goup-decomposition method to Wei but without a hieachy. To get close to a maximal distibution, White et al. [27] uses a tee to captue the emaining uncoveed aea and select new samples. The memoy equiements have been impoved by a vaiation due to Gamito and Maddock [29]. These methods ae vey effective in pactice, and ae unbiased. Howeve, it appeas to us that the authos do not claim to povide a povably maximal distibution with a data-stuctue size independent of numeical pecision. The issue is the tee depth. The tee must be deep enough to epesent the geometic gap between non-ovelapping disks. In theoy, this gap may be infinitely small, and thus thei tee-based methods may be quite deep. In pactice, they assume that dats ae placed on a discete numeical-pecision gid, athe than in continuous eal space. So the gap distance that needs to be epesented is bounded by some function of machine pecision, and the methods bound the tee depth by a pedetemined constant. White et al. state, In theoy the numbe of active squae levels could be unbounded, but in pactice we only need enough levels fo the pecision of the numbe being used. Gamito and Maddock state, A maximal-subdivisionlevel condition is impotant to pevent the algoithm fom becoming locked in an infinite loop and uncoveed gaps that ae too small to be captued by that level ae discaded, so the maximality condition is elaxed somewhat. Figue 7 of Gamito and Maddock shows thee nealy-ovelapping cicles with a vey small uncoveed egion. This example is poblematic fo both Gamito and Maddock, and White et al., but ou method handles it with no special consideation. Gamito and Maddock, and Wei, povide algoithm desciptions fo domains in abitay dimensions. Thei papes show implementation esults up to fou and six dimensions espectively. In this pape, we pesent an effective and povably coect algoithm to solve the two-dimensional maximal Poisson-disk sampling poblem. Ou algoithm inheits fom Wei s algoithm [28] many desiable popeties, such as efficient paallel implementation using GPUs. Ou algoithm is moe complicated than Wei s, but ou output is unbiased. We geneate maximal distibutions ove nonconvex domains while consuming limited esouces. To ou knowledge, this is the fist pactical algoithm that simultaneously satisfies all the equiements of a maximal Poisson-disk sampling, with complexity E(n log n) time and O(n) space dependent only on the numbe of output points. Highe numeical pecision inheently equies moe memoy fo epesenting numbes and moe time fo aithmetic opeations; howeve, ou method equies nothing beyond that. Ou sampling pocess has two phases. Fo efficiency, we use a backgound gid of squae cells coveing the whole domain. Each cell can accommodate a single sample. In the fist phase, dats ae thown into these cells. The initial dats ae unlikely to ovelap so the algoithm pogesses quickly at fist, but slows down as moe dats ae placed. Thus, we switch to a second phase. The fist phase leaves many small empty voids, the pat of a gid cell outside all cicles. These ae appoximated by convex polygons. Duing the second phase, dats ae thown diectly into the voids, with pobability popotional to the elative aeas of the voids, which maintains the bias-fee condition. (Special cae is needed because of the polygonal appoximation.) The algoithm is capable of tacking the emaining voids in the domain up to ound-off eo using only O(n) size data. A maximal distibution is achieved when the domain is completely coveed, leaving no oom fo new points to be selected. Ou algoithm is capable of handling non-convex domains with holes, which ae typical in many meshing applications. Most MPS methods focus on the unit squae. To ou knowledge, no pio MPS method consides non-convex domains, with o without holes. The seial implementation of ou algoithm is capable of geneating one million samples fom a squae domain in less than 1 seconds on a moden CPU. Ou paallel implementation on a GPU also poduces unbiased maximal distibutions and is about 2.4 faste than ou CPU implementation. In the est of this pape, we pesent ou algoithm in gadual steps.

3 In Section 2 we descibe the vaious steps of the two-dimensional sequential algoithm, with poofs and analysis in Section 2.5. We then pesent ou paallel implementation in Section 3. Finally, we descibe application examples in Section 4 that demonstate the efficiency of the poposed method and the quality of the output distibutions. 2 Sequential Sampling Ou algoithm consists of the following steps: 1. Geneate a backgound gid; mak inteio and bounday cells. 2. Phase I. Thow dats into squae cells; emove hit cells. 3. Geneate polygonal appoximations to the emaining voids. 4. Phase II. Thow dats into voids; update emaining aeas. While Phase I is not equied fo the coectness of the algoithm, it is a significant time and memoy optimization. In Phase I, successful dat thowing is vey fast initially, then slows down as moe dats ae inseted. Phase II equies computing and stoing the intesection of geometic objects, and a dat is thown into a polygon selected using a binay seach. This esults in an additional cost in the dat thowing pocedue; howeve, it does not slow down as dats ae inseted. (It actually speeds up!) Ou two phase-algoithm utilizes an active pool of cells to guide the dat thowing pocess. Duing Phase I, each cell in the active pool is squae-shaped and can accommodate a single sample. Once a dat is thown successfully into a andomly-selected cell, this cell is invalidated and emoved fom the active pool. An invalid cell is a cell that does not have any oom fo a new sample. If a thown dat violates the empty-disk condition o lies outside the domain, the associated sample is ejected and that tial is consideed a miss. In Phase I, we thow A C dats whee C is the set of initial cells intesecting the inteio of the domain. In addition, as a speedup, afte B C thows we teminate ealy if thee ae M consecutive misses. A = 5, B = 1/16 and M = 4 woks well fo the seial implementation. The unning-time of Phase I is linea, O( C ). The egions of cells that emain uncoveed by disks afte Phase I ae called voids. They ae usually small and scatteed all ove the domain. We loop ove the emaining valid cells and calculate thei intesection with pio dats to get a moe accuate epesentation of thei emaining voids (if any). Fo simplicity we bound the voids by chods of cicles athe than the tue cicula acs; this is a good appoximation since the cicles ae lage compaed to the squaes. A polygonal void is an oute appoximation to an ac-bounded void. A polygonal void is selected based on its aea elative to the sum of the aeas of all the polygonal voids. A dat is thown into the selected void, based on a unifom sampling of the polygon. As in Phase I, a dat is a miss if it is inside a pio sample, i.e. it is inside the polygonal void but outside the ac-bounded void. The chance of success is bounded below by a constant, because the aea of a polygonal void is at most a constant facto geate than the ac-bounded void it appoximates. Selecting a void and updating selection pobabilities can be done using a tee in O(log V) time, whee V is the set of emaining voids. Fo memoy efficiency, in ou implementation we use a flat aay and a lazy update of the selection pobabilities, updating only when many misses occu in a ow. This is called a stage. In pactice, despite the non-linea theoetical complexity, void selection is fast compaed to the lineatime geometic calculations. The expected un-time of Phase II is O V + V log V with a lage constant in font of the linea tem. Vitually all the methods in the liteatue have a had time selecting points fom small voids, doing wose as voids get smalle. We do not have this poblem. Voids ae selected popotional to thei polygonal aea. If a void is selected, the chance of successfully thowing a dat into it is actually bette if its aea is smalle, because the chod is a bette appoximation fo shote acs. 2.1 Geneation of a backgound gid We geneate the backgound gid using a vaiant of flood-fill (a.k.a. seed-fill, bounday-fill), a ubiquitous technique in gaphics. The input to ou algoithm is a set of edges defining a polygon with holes. Bounday edges ae oiented so the domain inteio is to thei left. Ou algoithm can also accommodate point set inputs and pescibed sample points embedded in a bounded domain. Pescibed intenal edges have no elevance fo sampling, but ae allowed because they ae useful fo some mesh geneation algoithms. We stat the algoithm by geneating a unifom gid coveing a bounding box of the input domain. Cell sides ae length / 2 and cell diagonals ae length, so each cell can only accommodate a single sample. Each cell has two attibutes. A valid cell is one that might accept a sample, and a bounday cell intesects one o moe of the input edges. These two classifications ae stoed using two boolean aays to get the best pefomance. Sampling a million points in a squae takes thee million cells, which consumes less than 1. MB of memoy. Afte geneating the unifom gid, we identify the bounday cells. Along each edge we geneate a unifom set of tempoay points, and locate the hit-cell containing each point. Some cells might not have a point because an edge could gaze its cone and have a shot length inside the cell. These missed cells ae neighbos of hit cells, and ae ecoveed by checking the intesection of the edge with the sides of the hit cell, using only intege opeations. The cells that ae inteio to the domain ae distinguished fom those exteio to the domain using a flood-fill algoithm. Exteio cells ae invalid, inteio and bounday cells ae valid. Fo the puposes of poving that we do not spend too much time on extenal cells when building the backgound gid, we assume that the numbe of extenal cells is bounded by a constant times the numbe of intenal cells. Fo the puposes of poving constant complexity pe void in Section 2.5.2, we assume that is chosen small enough that a cell contains at most one domain bounday vetex and two domain bounday edges. These assumptions can be ovecome using known pepocessing techniques but we skip the tedious details. 2.2 Phase I. Thowing dats into squae cells Since all the cells have the same aea, 2 /2, the dat-thowing pocedue selects a cell unifomly fom the valid pool. We then daw a unifom sample fom its squae. If the selected sample violates the empty-disk condition o lies outside the domain bounday, that iteation is a miss. Othewise, the dat hits a still-uncoveed egion. We accept the selected sample, and the cell is maked as invalid since it can no longe accept a sample. The checks ae executed in constant time, since we only need to check fo pio disks lying in a constant numbe of neaby cells and the numbe of bounday edges in a cell is at most two; see Section We iteate at least C /16 times, whee C is the initial set of valid cells. We continue iteating up to 5 C times, stopping ealy if 4 consecutive misses occu. At this point, the chance of successfully accepting a new sample is usually quite low, and we switch to Phase II.

4 2.3 Polygonal appoximations to ac-voids (a) Find the bounday cells (dak) (b) Flood-fill to find valid cells (dak) Afte Phase I, fo each valid cell we gathe the connected components of its disk-fee egion inside the domain. In Phase II these components take the place of cells in Phase I. Each component is a void, V. It is an ac-gon, V, a closed 2D egion bounded by staight segments and acs of cicles. We constuct a polygonal oute appoximation to it, V p, epesenting acs by chods. We shall pove in Section that V p is convex. A cone of V p is a vetex with inteio angle < 18. We epesent the polygon by an odeed list of its cones. The following constuction algoithm is illustated in Figue 3: 1. Initialize V p to the cell s squae, then intesect it with any bounday edges to etain just the domain inteio. 2. Fo evey disk d in a neaby cell, subtact it fom V p: fo evey cone c of V p it contains, (a) Stat fom c and tavese the edges of V p in both diections, to find the fist two edges intesecting the disk s cicle. (b) If no edges intesect the cicle, then V is completely coveed and the void is deleted. (c) If the chod between the points of intesection sepaates c fom the cente of d, then the disk cuts V p into multiple connected components. Find the additional cicleedge intesections and split V p. (d) Othewise, inset two cones at the points of intesection, and emove all the intevening cones and edges fom V p, since they ae coveed by the disk. (e) Adjust the location of the new cones to be at the intesection of the ac-gon V and the cicle. 3. If acs of V intesect at any point othe than a polygon cone, split the polygon into connected components, as illustated in Figue 4. We shall show in Section that fo each cell, the numbe of neaby disks, cones and connected components ae bounded by constants. The unning time of constucting each void is constant, and the unning time of constucting all voids is O( V ). Because of the geometic opeations, the unning time of this step is a significant potion of the whole. (c) Phase I dats (dots) land in many of the cells (light) Figue 3: Geneation of the polygonal void (dak) bounded by thee cicles, fom left to ight. The polygon is initialized to the cell bounday. The ed, blue, then geen disks ae intesected with the polygon. We get a polygon by using the chods instead of the acs, but we update vetices at cicle-polygon intesections with ciclecicle intesections. 2.4 Phase II. Thowing dats into polygonal cells (d) Only the uncoveed cells (dak) ae passed to Phase II Figue 2: Ou algoithm though Phase I. Phase II is simila to Phase I, with polygonal voids taking the place of squae cells. When selecting a void we must take into account the elative aeas of voids to maintain the bias-fee popety. Afte selecting a void, we choose a unifom andom point inside it; see Section and Gaphics Gems [Tuk 1993]. We use the ac-gon to detemine if the selected point satisfies the empty-disk condition.

5 Figue 4: Geneation of two voids (dak) entapped between two cicles and a bounday edge. Fist the squae is updated to espect the bounday edge. Next it is intesected with the cicles. We detect ovelapping cicles containing no othe polygon vetices and split the polygon in Step 3. If so, the pocess was a success, and we etain that sample. Updating the elative pobabilities can be expensive, so we do that in a lazy fashion. Let V be the initial set of voids, and Vi the set at stage i. Simila to phase I, we thow at least Vi /16 dats. We thow at most 3 Vi dats, quitting ealie if 1 consecutive misses occu. The expected numbe of hits will be a constant faction of Vi. We then update all the polygonal voids that the inseted disks ovelap, using the algoithm of Section 2.3. We then compess the list of emaining voids by emoving the coveed ones, ecomputing the elative pobabilities, and continuing with pool Vi+1. This continues until we have an empty pool, i.e. all voids ae filled, and the distibution is maximal. We shall pove in Section that, at each thow, the pobability of success is bounded below by a constant. At each stage we will fill in a constant faction of the emaining voids. This ecusion gives the total amount of wok as a constant times the total amount of wok in the fist stage, fo V. Placing a dat and checking if it is disk-fee is a constant-time opeation. Updating the polygonal voids is a linea amount of wok in going fom stage i to i + 1, so this is an amotized constant. The only non-constant opeation is selecting which void to thow the dat in, which we next show is O(log Vi ). Thus the total time is O(V log V) and the total space is O(V). Using a tee to keep tack of the emaining uncoveed egions as in Dunba and Humpheys [26] and Jones [26] is a good appoach if the aeas ae constantly updated. Howeve, we use a simple aay with half the size of a tee, and that woks well fo ou lazy updating scheme. In pactice, it appeas that many cells and voids ae completely coveed ove in the couse of selecting new samples, so we do not think it is woth the computational time to constantly update voids. Let ai be the aea of the ith polygonal void, so pi = ai /A is the pobability we should P select void i. Aay enty i points to the ith void, and stoes fi = ij=1 pi, the sum of the pobabilities of the pio voids. At each iteation we select u [, 1] unifomly. Using binay seach on the aay, we find the cell with the ith pecentile elative aea, i.e. the i such that fi u > fi 1. This binay seach takes O(log Vi ) time. If the dat thow is successful, we mak the void as filled to avoid futhe geometic computations, but leave it in the aay to avoid bias. A pactical heuistic is to decide when to update the aay dynamically, based on the hit ate. Updating the aay afte 1 consecutive misses, o when the aea of the invalidated polygons exceeds.7a, seems to wok well. 2.5 Coectness and complexity analysis The uninteested eade may skip ahead to Section 3. We povide some explicit values fo the constants affecting the size of the data stuctues. Knowing the wost case allowed us to use small, fixedsize aays in ou implementation. Fo the constants affecting the expected unning time of the algoithm, we did not ty to find explicit values because they ae not vey useful. Instead we tuned the Figue 5: Maximal Poisson samples of a unit squae at fou densities: =.1,.5,.1,.5. Fo the two coasest densities we also show thei disks. algoithm empiically. Let n be the numbe of dats in the domain afte the algoithm teminates. We fist show that we do not have too many cells. Theoem 1 The total numbe of cells C intesecting the inteio of the domain is Θ(n) in any maximal Poisson sampling. Poof C = Ω(n) because each cell contains at most one dat. Fo the othe diection, an empty cell can only be touched by a constant numbe of disks, because the disks have constant adius Bias Fee In eithe phase, let Ck fo k J denote a paticula cell o polygonal void. Let Ω be any domain subegion Ω Di 1. Assume fo now Ω Ck. The pobability that the next point xi will be taken fom Ω is the pobability of selecting Ck times the pobability of selecting Ω within Ck, compounded by e-thows if the dat misses the emaining domain Di 1 entiely. Let A( ) denote aea, so A(Ck ) A(Ω) P (xi Ω) = P (1+P (miss)+p 2 (miss)+...). A(C ) A(C ) j k J P Since the miss pobability P P is 1 A(Di 1 )/ A(Cj ), we have m A(Cj ) /A(Di 1 ). Simplifying yields m= P (miss) = P (xi Ω) = A(Ω), A(Di 1 )

6 which is pecisely the equiement fo the bias-fee equation (1a). To extend to Ω C k, i.e. Ω spanning seveal cells o voids, simply patition subsets of Ω among cells. (Fo Phase I, A(C k )/ A(C j) = 1/ J.) Void complexity and convexity Fo the puposes of assigning a disk cente to a unique squae, squaes ae consideed open on thei minimal extemes, as in Figue 6. We call such squaes half-open squaes. Figue 6: Left, any -disk intesecting the cental half-open squae is assigned to a unique squae within this template. Right, some chod-angle identities. Lemma 2 ( Civilization template) -disks that intesect a halfopen squae ae assigned to one of 21 squaes, up to two squaes away, within a 5 5 gid of squaes with the cone squaes emoved. See Figue 6 and the compute game Sid Meie s Civilization pio to vesion V. This shows that checking if a dat is in any disk is a constanttime opeation, that any void is bounded by a constant numbe of disks, and that any squae contains a constant numbe of voids. The bounds povided by Lemma 2 may be tightened by using aea, angle, and distance aguments. Fo lineaity in Phase II, it emains to show that a constant faction of each polygonal void is outside any cicle. Lemma 3 (disks in squae by aea) No moe than 15 -disks can intesect a squae. Poof An empty-disk system of -disks induces a system of nonintesecting /2-disks with the same centes but half adius. An aea agument shows that only 15 of these disks can have centes close enough (< 3/4) to touch the squae. Fo voids we will impove the 15-disk bound to 9. Figue 8 is a constuction showing that 8 disks can bound a void. A void V is one connected component of the non-empty intesection of a squae, togethe with the closed complement of some adius- disks. V is an ac-gon. A polygonal void V p is the convex hull of V. Fo convenience in the poofs, we etain flat vetices v of V as vetices of V p when the angle of V p at v is 18. Note that V is closed and bounded, and V p is an oute appoximation to V. The polygonal angle at a vetex y of V will mean the inteio angle between segments xy and yz whee x, y, and z ae consecutive vetices of V. xy denotes the line segment between x and y and xy denotes the staight-line distance between x and y. Ou next seies of lemmas shows that all the vetices of V appea as vetices of V p, and we bound the size and shape of voids. Fo simplicity we assume that a squae is completely inteio to the domain. At the end we elax this assumption and note that the changes to the esults ae slight. Lemma 4 (kites) The angle subtended by a chod xy is twice the angle α between the cicle s tangent at x and the chod. And α = acsin ( xy /2). See Figue 6. x! " "! y c z C z " 6 y " 3 "! 3 " 6 " 6 c! x x C x Figue 7: Left, an uppe bound on chod lengths implies an uppe bound on polygonal angles. Right, only one cicle bounding a void can intesect a given squae side twice. Theoem 5 (convex cones) The polygonal angle at a vetex of V is at most 18. Poof We have two cases. In the fist case vetex y is at the intesection of a cicle C and squae edge e. At wost x and z ae also on e, in which case the angle is 18. In the second case y is at the intesection of two cicles C x and C z; see Figue 7. The angle between the tangents of C x and the line between the points of intesection between the two cicles is at most 6, achieved when c x lies on C z. The angle between the tangent of C x at y and the chod xy is acsin( xy /4) fom Lemma 4. Since the chod must lie inside the squae, xy, and this angle is at most 3. The angles between the cicles-intesection line and yz ae also at most 9, so the sum of these angles is 18. Coollay 6 (natually convex) All vetices of V ae on the bounday of V p. Bounday edges of V p ae chods of cicles and sub-segments of squae sides. Coollay 7 (one ac) A cicle contibutes at most one ac to V. Lemma 8 (convex centes) The centes of cicles bounding V must be in convex position. Poof See Figue 7 left. Suppose that thee cicles C x, C y, and C z touch V, but that cente c y is not in convex position with espect to centes c x and c z. That is, fo some point p of V on C y, c zc yp + pc yc x > 18. Assume C y intesects both of the othe cicles. Conside the ac of C y between its intesection points touching V. (This ac is unique by Coollay 7.) Then the ac s chod is longe than, the diagonal of the squae, a contadiction. Hence the othe two cicles ae too fa away fom one anothe to both intesect V. If C y does not intesect the othe two cicles, then they ae even fathe away. Lemma 9 (1 ac sides) Fewe than 1 cicles bound V. Poof We show that fo -disks bounding V, the distance between the centes of the two fathest-apat cicles c x and c y must be > 3, so not both can ovelap a squae with diagonal. By symmety and Lemma 8, the closest c x and c y can be is if all cicles ae aanged on a egula n-gon, with side length. By Lemma 4 the cicumscibed cicle C fo this n-gon has adius R = /(2 sin(18/n)). Checking distances between opposite vetices fo odd and even n completes the poof. Figue 8 shows that 8 cicles is possible. Lemma 1 (8 squae sides) The squae contibutes at most 8 sides to V, and only 4 if flat sides ae ignoed. Poof Any cicle intesecting a side exactly once (and non-tangent) contains one of the cones of the squae, and so does not incease the numbe of subsegments of the side bounding V. Thee can only be one cicle that intesects one of the fou squae sides twice, o is tangent to it; see Figue 7 ight.!!

7 Putting the pio lemmas togethe we have the following theoem. Figue 8 shows that 8 cicles is possible. Theoem 11 (few ac-gon sides) The numbe Lemma 6 (8 squae of sides) The squae contibutes of Vat mostis 8 sidesat to V, and only 4 if flat sides ae ignoed. Poof Any cicle intesecting a side exactly once (and non-tangent) most 17, and at most 13 if flat sides ae emoved. contains one of the cones Figue of the squae, and 8so does ealizes not incease the numbe of subsegments of the side bounding V. Thee can only be one cicle that intesects one of the fou squae sides twice, o is a void with 16 sides, and Figue 8 shows tangent a to void it; see Figuewith 7 ight. 12 non-flat Putting the pio lemmas togethe we have the following theoem. sides. Theoem 3 (few ac-gon sides) The numbe of sides of V is at most 17, and at most 13 if flat sides ae emoved. Figue 8 ealizes a void with 16 sides, and Figue 8 shows a void with 12 non-flat sides. Figue 8: Voids with 16 sides (left) and a closeup (top-left). A slight tweak yeilds a void with 12 nonflat sides (bottom-left). (todo: put the two close-ups into one pdf to get the layout I want) Figue 8: A void with 16 sides (left) and a closeup (top-ight). A slight tweak yields a void with 12 nonflat sides (bottom-ight). The peceding consides only squaes that did not contain the bounday of the input domain, but most of the poofs only ely on squaes being contained in a cicle of adius /2. Fo bounday squaes, we note that they may have at most two edges of the input domain (shaing a common vetex), and thei segments on the bounday of V ae of length <, so that the numbe of sides inceases by at most Phase II geomety and complexity Aea atio of polygonal void to ac-gon void. We now conside the atio of the aea of V to V p, since that detemines the expected numbe of dat-misses in Phase II. Theoem 12 A(V )/A(V p) is at least a constant. Poof Conside the cicles bounding a void, including cicles intesecting a squae side twice. Conside the weighted Voonoi egion of the cicles [Edelsbunne and Shah 1992]. Assume fo now that the emainde egion is bounded entiely by -cicles, and tuncate the Voonoi cells at the polygonal void V p. Fo any cicle C, its Voonoi cell will contain the cicle chod on the polygon bounday χ, the ac-bounday s, and a pat of the inteio of V. The easons ae as follows. Let VC be the cicle s tuncated Voonoi cell, and VS ext its patition outside C, and VS int its pat inside C. Recall fom Lemma 8 the cicle centes ae in convex position and can be consideed in ode aound the bounday of the void. Since only the consecutive cicles aound the void may ovelap with C (else the void would not be connected), the chod is not inside any othe cicle, so it is in VC. Also, the Voonoi line of equal distances between C and a non-consecutive cicle lies stictly outside C. Since by Lemma 9 thee ae at most a constant numbe of cicles (< 1), thee ae a constant numbe of staight sides bounding VC. All of these bounding sides lie outside VS ext as well. At wost these sides appoach tangency with C, and fom a 9- sided polygonal oute appoximation to the ac. Since the ac s has constant cuvatue, the aea of VS ext is at least a constant faction of VS int. We do not wok out the exact constant because this bound is not vey tight; fo example, many fewe than 9 cicles can be packed close enough to be nealy tangent with C. Now elax the assumption that the emainde egion is bounded entiely by cicles. Teat the lines suppoting the squae sides o domain bounday as infinite-adius cicles centeed at infinity, and all the aguments of the pio paagaph still hold. The aea atio bound constant can be epoduced by assigning the Voonoi egions of the infinite-adius cicles to the closest -cicle, since fo the infinite-adius cicles the ac-gon and polygon ae identical. The peceding consides only squaes that did not contain the bounday of the input domain, but most of the poofs only ely on squaes being contained in a cicle of adius /2. Fo bounday squaes, we note that they may have at most two edges of the input domain (shaing a common vetex), and thei segments on the bounday of V ae of length <, so that the numbe of sides inceases by at most Phase II geomety and complexity Aea atio of polygonal void to ac-gon void. We now conside the atio of the aea of V to Vp, since that detemines the expected numbe of dat-misses in Phase II. Theoem 4 A(V)/A(Vp) is at least a constant. Poof Conside the cicles bounding a void, including cicles intesecting a squae side twice. Conside the weighted Voonoi egion of the cicles [Edelsbunne and Shah 1992]. Assume fo now that the emainde egion is bounded entiely by -cicles, and tuncate the Voonoi cells at the polygonal void Vp. Constant faction pogess pe stage. Theoem 12 poves that the fist dat thown in a stage i has a constant pobability of being a hit. Howeve, thee is a technical difficulty fo subsequent dats. The fist disk may cove othe polygonal voids, pehaps completely. We update the polygons lazily, so those voids educe the pobability of a successful hit. This is esolved by ecalling that any inseted disk can affect only a constant numbe of othe voids. Afte c 1 V i hits, c 2 V i voids emain unchanged, so the pobability of a dat being a success is at least c 2 times what it was at the stat of stage i. Hee c 1 is something smalle than 1/6, and c 2 = (1 6c 1). The 6 aises fom Lemma 3 whee each placed disk intesects at most 15 othe cells, and by Theoem 21 each cell has at most 4 voids. Thus a lowe bound on the expected numbe of hits in stage i is c 2 times the constant fom Theoem 12 times the numbe of thows c 1 V i : the point is this is O( V i ). In pactice, many moe voids ae filled than these constants suggest, but the above is sufficient to pove the following theoem. Fo any cicle C, its Voonoi cell will contain the cicle chod on the polygon bounday χ, the ac-bounday s, and a pat of the inteio of V. The easons ae as follows. Let VC be the cicle s tuncated Voonoi cell, and VSext its patition outside C, and VSint its pat inside C. Recall fom Lemma 4 the cicle centes ae in convex position and can be consideed in ode aound the bounday of the void. Since only the consecutive cicles aound the void may ovelap with C (else the void would not be connected), the chod is not inside any othe cicle, so it is in VC. Also, the Voonoi line of equal distances between C and a non-consecutive cicle lies stictly outside C. Since by Lemma 5 thee ae at most a constant numbe of cicles (< 1), thee ae a constant numbe of staight sides bounding VC. All of these bounding sides lie outside VSext as well. At wost these sides appoach tangency with C, and fom a 9- sided polygonal oute appoximation to the ac. Since the ac s has constant cuvatue, the aea of VSext is at least a constant faction of VSint. We do not wok out the exact constant because this bound is not vey tight; fo example, many fewe than 9 cicles can be packed close enough to be nealy tangent with C. Now elax the assumption that the emainde egion is bounded entiely by cicles. Teat the lines suppoting the squae sides o domain bounday as infinite-adius cicles centeed at infinity, and all the aguments of the pio paagaph still hold. The aea atio bound constant can be epoduced by assigning the Voonoi egions of the infinite-adius cicles to the closest -cicle, since fo the infinite-adius cicles the ac-gon and polygon ae identical. Constant faction pogess pe stage. Theoem 4 poves that the fist dat thown in a stage i has a constant pobability of being a hit. Howeve, thee is a technical difficulty fo subsequent dats. The fist disk may cove othe polygonal voids, pehaps completely. We update the polygons lazily, so those voids educe the pobability of a successful hit. This is esolved by ecalling that any inseted disk can affect only a constant numbe of othe voids. Afte c1 Vi hits, c2 Vi voids emain unchanged, so the pobability of a dat being a success is at least c2 times what it was at the stat of stage i. Hee c1 is something smalle than 1/6, and c2 = (1 6c1). The 6 aises fom Lemma 2 whee each placed disk intesects at most 15 othe cells, and by Theoem 9 each cell has at most 4 voids. Thus a lowe bound on the expected numbe of hits in stage i is c2 times the constant fom Theoem 4 times the numbe of thows c1 Vi : the point is this is O( Vi ). In pactice, many moe voids ae filled than these constants suggest, but the above is sufficient to pove the following theoem. Theoem 13 In each Phase II stage i, a constant faction of the V i voids ae filled with dats. Success ate in pactice. The constants given fom the poofs of Theoem 12 and Theoem 13 ae not tight. These constants do not affect any data stuctues in ou algoithm, only the miss ate. Thei impotance is in the tuning of the algoithm paametes fo when to move to the next stage. Ou implementation shows the aea atio as in Theoem 12 is usually lage, about 1. It is about.93 in the beginning stages of Phase II, and eaches about.999 in the last stage. It tends to incease but is not monotonic. See Figue 9. Also, the faction of voids filled pe stage is much bette than the weak constants fom Theoem 13 might suggest, as many fewe than 6 voids ae touched by a new disk on aveage. Also the faction goes up as the domain gets filled, as the voids become moe isolated. See Figue 1. Ave. Polygon Void / Cell Aea log 1 scale Polygon & Ac gon Void Aea Phase II Stage Sum of Ac gon Voids Aeas / Sum of Polygon Voids Aeas Figue 9: The atio of the polygon to ac-gon aea is always lage, almost 1. The aveage atio tends to incease by stage as voids become smalle in aea. Running time. We now conside the unning time R( V i ) of Phase II fo stage i and all subsequent stages. We showed in Section 2.4 that R( V i ) = V i log V i + R( V i+1 ). Since V i+1 < c V i fo some constant c < 1, we have R( V ) < ( ) i= ci V log c i V < ( V log V ) i= ci = 1 V log V. 1 c with Theoem 1 we have Combining this

8 Numbe of Voids log 1 scale Voids Coveed pe Stage Invalida3on Ra3o Numbe of Voids Phase II Stage Fac%on of Voids Coveed in Cuent Stage A xz C x c x a xz = t xz b xy V c z a yz A a xy xy χxy t xy α c y C y C z c z V xz a xz c y C y C x c x V xy a xy Figue 1: The faction of voids filled in each stage is usually lage. C z A yz Theoem 14 Phase II unning time is O(n log n). Unifom sampling fom polygons. Recall that we needed to sample unifomly fom the polygonal voids. We ecap the method we use, adapted fom Gaphics Gems [Tuk 1993]. We tiangulate the polygon. Since it is convex, we could simply pick any one vetex and connect it to all the othes. But fo numeical easons, it is bette to keep angles away fom 18. We instead intoduce a new vetex inside the polygon, located at the aveage of all the othe vetices, and connect it to all the polygon vetices. We select one of these tiangles with pobability popotional to its aea elative to the polygon aea. Within abc, we sample unifomly fom it by picking unifomly fom a ight tiangle and linealy tansfoming to ou tiangle. Pick u, v [, 1] unifomly. This picks a point fom the squae; if u + v > 1, then eflect back into the tiangle by assigning u := 1 u and v := 1 v. The esulting sample point p = u ab + v ac is unifom fom abc Constant numbe of voids pe cell Since at most a constant numbe of cicles intesect a squae, combinatoics implies the numbe of voids in a given cell is constant. Impoving it is inteesting and allows some implementation efficiencies, but is not essential. These obsevations also hint at the obseved sepaation distances between voids as the stage inceases. We fist conside voids bounded entiely by cicle acs, then we shall see that allowing voids to be bounded by the sides of the squae inceases the numbe of voids pe squae by at most one. Two voids ae adjacent if they ae bounded by the same pai of cicles C x and C y. The vetices of cicle intesection ae labeled vetex a xy and b xy, whee b lies inside the efeence void. Ovelapping cicles bounding a void ae consecutive. We fist conside thee-sided emainde egions, and label thei featues as in Figue 11. Consecutive adjacent voids ae two voids in the same cell adjacent to the thid efeence void though its adjacent consecutive cicles C xc y and C xc z: e.g. two voids inside egions A xy and A xz in Figue 11, povided they and some pat of V ae in the same squae cell. Lemma 15 Fo consecutive adjacent voids V xy and V xz to V, thei closest pai of points ae no close than the cicle intesection points, a xy and a xz. Poof See Figue 11. Pick some point q of V in the cell. Since q is in a void, its distance to c x is at least. In paticula, all of V xy and V xz must be on the same side as q of the line though c x pependicula to qc x. All of V xy and V xz must be within of q, inside the ed cicle. The closest pai of points inside the ed cicle ae a xy and a xz. Conside moving one of thee pai-wise ovelapping cicles. We obseve the following invese elationships about the distances between pais of cicle centes and pais of void vetices. Figue 11: Top left, labeling of a thee-sided void. If C x and C z ae tangent, then t xz = a xz = b xz coincide. Top ight, the closest points of consecutive adjacent voids shaing cicle C x ae not close than a xya xz. Bottom left, the 3-sided void with smallest distance between an adjacent pai of voids. Bottom cente, the 3- sided void with the smallest distance between the second-closest pai of adjacent voids. Bottom left, fou voids in a squae. Lemma 16 (cicle void distances) If a xy, a yz and a xz ae in the same cell, then c yc z = a xya xz, a xya yz, a xza yz c xc y = a xza yz, a xza xy, a yza xy c xc z = a xya yz, a xya xz, a yza xz We omit the poof because of space limitations. The poof is based on Lemma 4, Lemma 8, and bounding distances by. Lemma 16 can now be used to pove the exteme cases in Figue 11 and the following lemma. Lemma 17 Fo thee-sided voids, the distance between consecutive adjacent voids is at least /2. Fo othe voids, the distance between consecutive adjacent voids is at least. Coollay 18 Fo a void with fou o moe sides, only two adjacent voids can be in the same cell as the void, and only one stictly inside. Poof The squae diagonal is, so fo thee consecutive adjacent voids, only one pai of consecutive adjacent voids can be placed inside the squae. Fo non-consecutive voids V 12 and V 34 adjacent to the void V, conside the two pais of cicles sepaating them fom V, C 1, C 2 and C 3, C 4. If the length c 1c 2 and c 3c 4 ae both 2, and c 1c 4 and c 2c 3 ae both, then we have a paallelogam and the distance between the cicle-cente midpoints t 12 and t 34 is. This is a slight vaiation of Figue 12 whee the paallelogam diagonals must be stictly geate than 2. We next ague that this paallelogam is the wost case. If c 1c 3 o c 2c 4 is geate than, this meely makes V 12 and V 34 futhe apat. A vaiation of Lemma 16 and Lemma 17 shows that this is the wost case. If t 12 is close enough to t 34 to be of inteest, then since the lengths c 1c 2 and c 3c 4 ae bounded between and 2,

9 Figue 12: The configuation minimizing the distance to an adjacent void fo two voids simultaneously. The cicle centes fom a paallelogam, with vetices fom fou voids along the diagonal of a squae. Only thee cicle-bounded voids can fit stictly inside a cell. the angle between lines c 1c 2 and c 3c 4 is small, meaning the lines ae close enough to paallel, so that deceasing c 3c 4 makes a 12 fathe fom a 34 athe than close. Theoem 19 Fo a thee-sided void, only two adjacent voids can be stictly inside the same cell. Poof We conside the configuation minimizing the secondclosest pai of adjacent egions, and show that this distance is lage. WLOG a xya xz a xza yz a yza xy. We seek to show the fist inequality is equality by contadiction. Suppose the fist inequality is stict. Then Lemma 16 shows that inceasing c yc z inceases a xza yz and deceases a xza yz and a xya yz, a contadiction to the configuation being optimal. So a xya xz = a xza yz a yza xy. Again Lemma 16 shows that inceasing c xc z will decease both a xza xy and a xza yz, so the optimal configuation has maximal c xc z, o c xc z = 2. Hence all the disks ae at distance 2 fom one anothe, and each of the thee adjacent cell vetex pais ae distance apat; see Figue 11. Since the squae only has diagonal length, not all thee of these vetices can fit. Using Lemma 17 fo thee-sided voids and Coollay 18 fo fouo-moe sided voids poves the following. Theoem 2 A cell can contain at most thee cicle-bounded voids. Theoem 21 A cell can contain at most fou voids. Poof In the pio theoems, at most two ovelapping cicle pais can sepaate the squae into multiple connected components. The main idea of this theoem s poof is that a single cicle can also sepaate one connected component of a squae into two, by ovelapping with a squae side. The squae is too small compaed to the cicle adius fo thee to be two such cicles. 3 Paallel Algoithm While ou seial appoach is faily fast, a paallel implementation povides the oppotunity to speed up sample geneation, and to deploy ou algoithm in GPU-based envionments, whee paallelism is citical fo pefomance. Fotunately, the appoach is staightfowad to paallelize. In this section we descibe the paallel-specific details and ou GPU implementation. We use the same two phases as in ou seial algoithm. /2 /2 Phase I Ou fist phase, the dat-thowing phase (Section 2.2), equies special cae to eliminate bias. Doing this in paallel with no bias has many pitfalls; the use of phase goups by Wei [28], fo instance, makes sample choices biased towads the simultaneous acceptance of cells in the same phase goup, and hence picks candidates non-unifomly fom the input domain. Ou algoithm needs to allow samples to have an equal pobability of being selected eveywhee. Thus we (must) allow samples to be chosen in paallel such that conflicts between them ae possible, but we then eliminate the conflicts in an unbiased fashion. In ou implementation, we achieve this though two sample buffes, candidates and final, to stoe the locations of points. We also stoe a state pe cell, whee each cell is eithe empty, test, accepted, o done. All cells ae initialized to empty. We then implement ou algoithm as thee successive kenels, and epeatedly iteate ove these thee steps until we move to Phase 2. Fist, in paallel, we geneate candidates by picking a andom point [Tzeng and Wei 28] while checking fo done points in its vicinity. We check against all neaby cells that might contain a conflicting point. We check a 5 5 neighbohood centeed aound the point s cell fo othe cells that ae aleady maked as done. Ou cell is maked as test only if no conflicts ae found in this neighbohood; it then becomes a candidate fo addition to the distibution. Second, we test fo conflicts again, this time fo each test sample against othe test candidates. We look at all test cells in paallel, looking fo cells in the neighbohood with eithe test o accepted states. If we find no conflicts with othe candidates, we mak the cell as accepted. Thid, again in paallel, we pomote all accepted cells to done and copy thei positions fom the candidate buffe to the final buffe, and eset all emaining test (conflicted) cells to empty. In summay, on each iteation, we fist test against points aleady in the final distibution; next test against candidate points geneated in this iteation; and finally add successful candidates to the final distibution. Both the seial and paallel algoithms shae a simila citeion fo switching to Phase II. Howeve, while the seial algoithm simply casts one candidate point pe iteation until it eaches this citeion, the paallel algoithm must decide how many points to cast on each iteation. Too few and the algoithm will take too many iteations to convege; too many and many candidates will knock each othe out. Fo n gid cells, we have found a good balance by casting n/5 points on each iteation. Phase II Phase II has a simila stuctue to Phase I, with fou steps: constucting void polygons, geneating candidates in the void polygons, testing conflicts, and updating states. Fo this Phase, we also add an additional cell state, ejected, fo cells that ae completely coveed by neaby disks and hence can neve contain a point. We iteate ove these steps and teminate when no void polygons emain. We constuct void polygons by visiting all empty cells in paallel, looping ove thei local neighbohood to find all points whose discs intesect with this cell. The pocedue is simila to the seial algoithm: begin with a polygon that is the size of the cell, then successively subtact coveed aeas. We split polygons if necessay. If an unsplit polygon has zeo aea, the cell is labeled ejected and no polygons ae geneated. Since each polygon is convex, we convet it into tiangles and add it to a tiangle buffe. We also stoe its aea in anothe buffe. The

10 latte aea buffe then undegoes a paallel exclusive pefix-sum opeation [Sengupta et al. 27] to obtain the cumulative aea fo each tiangle. At the end of this step we also know the total aea of all void polygons. In the second step, we geneate test points in some of the void tiangles. This is simila to geneating candidates in Phase I, but now we must pick tiangles with a pobability popotional to thei aeas. We un seveal theads in paallel fo this pupose. Each thead picks a andom numbe between and 1, and pefoms a binay seach ove the aea and cumulative aea buffes to identify the tiangle whose aea faction coves this numbe. The thead then ties to inset a test sample unifomly in this tiangle, but backs off if anothe thead has aleady picked the associated gid cell. Each thead pefoms seveal ties fo diffeent andom numbes befoe giving up. The thid and fouth steps ae identical to Phase I s second and thid steps, checking each test point against neaby ones. We iteate Phase II until all polygons ae consumed. Vey small polygons last many iteations because thei aeas ae a small faction of the oveall polygon aea, but as lage polygons ae emoved, thei aea factions ise and they will all be visited befoe completion. 4 Implementation Pefomance In this section we show the pefomance of ou seial and paallel implementations of the algoithm. The seial implementation is tested using a Intel Coe i7 CPU M62 with 4 GB of DRAM unning a 64-bit Windows 7 opeating system. We stat by showing the significance of Phase II in achieving a maximal distibution with easonable pefomance. In ou algoithm, Phase I mimics an impoved vesion of the classical dat thowing algoithm. This povides a useful method to distibute an initial set of bias-fee andom points coveing most of the domain. Howeve, the capability of this phase to inset new points deteioates as moe points ae inseted. Hence such an appoach cannot by itself achieve maximal distibutions. This fact is demonstated in Figue 13, whee 7, dats wee thown into a unit squae domain duing Phase I. At the beginning of Phase I, the pecentage of successful dats is close to 1%, and as moe points ae inseted, this pecentage deceases significantly. Afte Phase I ends, only 594 points wee distibuted in the domain, consuming about 3 ms and coveing about 98% of the total aea of the domain. Phase II was able to each a maximal distibution by inseting an additional 2175 points in the emaining 2% of the aea. Limiting the numbe of the dats thown in Phase I in a typical implementation of ou algoithm achieves a simila esult in less than 1 ms. We compae ou times fo maximal sampling to White s [27] and Gamito and Mattock s [29] times fo nealy-maximal sampling, with tuncated tee depth. Ou sequential implementation samples 1k points/second, on pa with White s low-memoy algoithm, and ou method does not slow down as much when the numbe of points inceases. Gamito epots 1k points in 1.9 seconds. Figue 14 shows the memoy consumption ove the two phases of the algoithm. The domain was a unit squae. We geneated 8,269,89 points. The memoy equied was 1.97 GB, of which about 66 MB was fo the output point cloud. The peak memoy was when we built the polygonal voids at the beginning of Phase II. This suggests that memoy could be educed, at the cost of slowe pefomance, by focing Phase I to thow moe dats. The saw-tooth in the figue aises because the memoy jumps at the beginning of a stage when we compute the void polygons, is mostly flat duing sampling, then dops at the end of a stage when we discad voids. Voids ae ecomputed fom scatch at the beginning of each stage. Numbe of Hits Classic Dat Thowing 1% 8% 6% 4% 2% % Numbe of Dats Thown Figue 13: The capability of classic dat-thowing to inset a new point deteioates as the numbe of pio dats inceases. At 7, dats thown, 9% ae ejected. 8% of the accepted dats wee thown duing the fist 2, thows. Memoy (MB) Phase I Memoy Use Phase II Time (seconds) Figue 14: Seial memoy use while sampling 8 million points in a squae. The memoy amps up at the stat of each Phase II stage when the geomety of polygonal voids is calculated. Late stages have fewe voids. In this example thee teeth coesponding to the fist thee Phase II stages ae visible. The oughly-flat egion afte the thid tooth is actually compised of about 1 stages that consume little memoy. This figue also illustates that geometic void calculations ae a lage pat of the unning time. This avoids the cost of updating voids, many of which no longe exist. Compaed to White [27], we consume moe memoy. Polygonal voids appea moe expensive to epesent than tuncated-depth quadtees. Asymptotically, Gamito and Mattock [29] equie O(n log n) space vs. ou O(n). Figue 15 shows the untime of the algoithm. Note the binay seach in Phase II has a negligible effect on pefomance in pactice. The memoy consumption in Phase II is popotional to the numbe of the emaining voids afte Phase I. As illustated in this figue, the elation between the numbe of voids and the numbe of points in the final distibution is almost linea. Moeove, moe than 7% of the points ae inseted duing Phase I. Note that these esults may vay accoding to the input geomety as well as the temination citeion of Phase I. Hee we ae using a unit squae poblem with no holes, whee Phase I teminates afte 4 successive misses. GPU Implementation. Ou GPU implementation was built on the NVIDIA CUDA platfom and uns on an NVIDIA GeFoce GTX 46 with 1 GB of on-chip memoy. The algoithms used Pecentage of Thows that Hit

11 Time (seconds) 1 CPU Running Time 8 Phase I Phase I + Phase II Linea (Phase I) Linea (Phase I + Phase II) k points/s tendline 358k points/s tendline 2 E+ 1E+6 2E+6 3E+6 4E+6 5E+6 6E+6 7E+6 8E+6 9E+6 6E+6 5E+6 4E+6 3E+6 (a) 1K samples in the squae # Voids at Phase II stat # Points in Phase I Linea (# Voids at Phase II stat) Linea (# Points in Phase I) Phase I Points.73 Total Points Radial Mean Powe Ini$al Voids.73 Total Points.14 1E+6 1E+6 2E+6 3E+6 4E+6 5E+6 6E+6 7E+6 8E+6 9E+6 Time (seconds) Fequency 8 1 (c) Radial avg. powe spectum Fequency 8 1 (d) Anisotopy Figue 17: A paallel sampling of the squae (a) and its pai-wise distance spectum (b). Thee is no visible bias: the spectum is visually identical to the esults of pue dat-thowing. The adial powe spectum (c) and anisotopy (d) ove ten tials also show no bias. GPU Running Time Phase I Phase I + Phase II Linea (Phase I) Linea (Phase I + Phase II) 5 224k points/s tendline 1,484k points/s tendline 1 E+.1. Figue 15: CPU pefomance while sampling a squae at diffeent densities. The uppe figue shows nea-linea untime. The lowe figue shows that Phase I insets about a constant faction of the total numbe of points, and Phase I ceates nealy the same numbe of points as voids Numbe of Points Geneated (Phase I + II) 4 Radial Anisotopy.16 2E+6 E+ E+ (b) Fouie spectum Numbe of Points and Voids in Phase I vs. II Anisotopy (db) 7E+6 Mean powe # Phase I Points o Voids Numbe of Points Geneated 2E+5 4E+5 6E+5 8E+5 1E+6 Numbe of Points Geneated Figue 16: GPU pefomance. We obtain oughly linea pefomance in the numbe of points, and a 2.4 speedup ove ou CPU implementation. Peak pefomance is about 24K samples/second. Due to memoy constaints we wee unable to geneate moe than 2 million points. Conclusions We pesent an efficient algoithm fo maximal Poisson-disk sampling in two-dimensions. The algoithm is notable because it simultaneously achieves the following: (1) the final esult is povably maximal, (2) the sampling is unbiased (in the sense of datthowing), (3) it is O(n log n) in expected time, and (4) it is O(n) in deteministic memoy equied. The algoithm is not limited to convex domains, and has been efficiently implemented in both sequential and paallel foms. Empiical investigations of the sequential algoithm suggests that the ode O(n log n) dependence is weak and that pactical pefomance is vey close to ode n. We ae able to geneate 1, samples/second on the unit squae on a moden laptop. In the paallel implementation, we must choose samples in paallel. Conflicts that occu must be eliminated in an unbiased fashion. fo Phases I and II ae descibed in Section 3. Figue 16 shows the obtained unning times of ou implementation fo a vaying numbes of final samples. We obseve a linea gowth in the time taken fo the two phases, which is consistent with the algoithm and the CPU implementation. We also obseve a 2.4 speedup ove ou CPU implementation, eaching a peak geneation ate of about 24k points pe second. Although this is not a huge speedup, ou goal is to demonstate a poof-of-concept appoach that attains easonable pefomance. The obtained speedup demonstates the paallel-fiendly natue of ou appoach. Output Quality. Figue 17(b) shows the fequency spectum of a set of 1K samples, and figues 17(c) and 17(d) show the adial mean and anisotopy of the spectum acoss 1 tials. These measues match the expected unbiased behavio seen in pevious liteatue [Lagae and Dute 28; Wei 28], which clealy indicates the absence of any noticeable bias in ou appoach. We have begun the implementation of a 3D maximal Poisson-disk sampling algoithm following the ideas in this wok. We ae optimistic that simila esults will hold fo the 3D case. We ae also optimistic that the algoithm may be extended to non-unifom disk adii. Acknowledgements We thank Joseph E. Bishop, Mike Stone, Laua Swile, David G. Robinson, and Vitus Leung (all Sandia) fo useful discussions. We thank Thouis Ray Jones fo discussing weighted sampling algoithms with us. The Sandia authos thank the U.S. Depatment of Enegy and Sandia s Compute Science Reseach Institute fo suppoting this wok. This wok was also funded in pat by DOE s Office of Advanced Scientific Computing Reseach, SC-21, SciDAC-e. The UC Davis authos thank the SciDAC Institute fo Ultascale Visualization, the National Science Foundation (gant