Recursive Spoke Darts: Local Hyperplane Sampling for Delaunay and Voronoi Meshing in Arbitrary Dimensions

Transcription

1 Aville online t Procedi Engineering 124 (2016) th Interntionl Meshing Roundtle (IMR25) Recursive Spoke Drts: Locl Hyperplne Smpling for Deluny nd Voronoi Meshing in Aritrry Dimensions Mohmed S. Eeid, Ahmd A. Rushdi, Sndi Ntionl Lortories, Aluquerque NM 87185, U.S.A. Institute for Computtionl Engineering nd Sciences, University of Texs, Austin TX 78712, U.S.A Astrct We introduce Recursive Spoke Drts (RSD): recursive hyperplne smpling lgorithm tht exploits the full dulity etween Voronoi nd Deluny entities of vrious dimensions. Our lgorithm ndons the dependence on the empty sphere principle in the genertion of Deluny simplices providing the foundtion needed for sclle consistent meshing. The lgorithm relies on two simple opertions: line-hyperplne trimming nd sphericl rnge serch. Consequently, this pproch improves sclility s multiple processors cn operte on different seeds t the sme time. Moreover, generting consistent meshes cross processors elimintes the communiction needed etween them, improving sclility even more. We introduce simple twek to the lgorithm which mkes it possile not to visit ll vertices of Voronoi cell, generting lmost-exct Deluny grphs while voiding the nturl curse of dimensionlity in high dimensions. c 2016 The Authors. Pulished y Elsevier Ltd. Peer-review under responsiility of orgnizing committee of the 25 th Interntionl Meshing Roundtle (IMR25). Keywords: Deluny Tringultion, Voronoi Meshes, High-Dimensionl Spces. 1. Introduction Deluny nd Voronoi meshing techniques re very populr in severl computtionl geometry [1,2] nd meshing [3 5] prolems for their rich geometric structures nd ppeling element properties. For set of point seeds X in the d-dimensionl Eucliden spce D, the Deluny mesh is defined y the tringultion Del(X) where no point in X flls inside the circum-hypersphere of ny simplex in Del(X) (e.g., in 2d, every tringle in Deluny tringultion is ssocited with n empty circumcircle pssing through its vertices). The dul of Del(X) is the Voronoi mesh Vor(X) which splits D into cells where ech cell contins ll the points closer to its seed thn ny other seed in X. Fundmentlly, Domin Decomposition (DD) method reks down domin into finite numer of elements, with certin desired properties, e.g., positive Jcoins, convex elements, nd plnr fcets. A DD lso defines how point in the Corresponding uthor. Tel.: E-mil ddress: mseeid@sndi.gov c 2016 The Authors. Pulished y Elsevier Ltd. Peer-review under responsiility of orgnizing committee of the 25 th Interntionl Meshing Roundtle (IMR25).

2 2 / Procedi Engineering 124 (2016) domin is ssigned to n element (or more), nd how ech element in the DD identifies its neighor element(s) in its close proximity. For exmple, the Voronoi Domin Decomposition (VDD) decomposes domin into convex elements (cells) ounded y plnr convex fcets. VDD cells re ssocited with generting smple points. Ech cell is formed round given seed, contining ll points closer to tht seed thn ny other seed in the domin. Despite their undnce of desired fetures, VDDs re not yet fully exploited in mny pplictions. This cn e minly ttriuted to their unsolved computtionl chllenges; generting true (unclipped) Voronoi cells tht nturlly conform to prescried oundries is n open computtionl geometry prolem. In low-dimensions, stte-of-the-rt techniques rely on clipping, which lcks roustness nd genertes cells tht re non-convex nd not str-shped. See Figure 1 for 2d illustrtion of how clipping is employed into the evlution of points neighorhood nd consequently define Vor(X). Note the curse of dimensionlity ssocited with this clss of hyperplne-hyperplne trimming opertions. In high-dimensions, explicit genertion of Voronoi cells, regrdless of oundries, is intrctle. The numer of corners of ech Voronoi cell grows exponentilly with dimension, imposing lingering curse of dimensionlity. Therefore, severl Deluny nd Voronoi meshing pproches were tilored to specific prolems with fixed dimensionlity (e.g., 4d medicl imges [6] nd scientific visuliztion [2,7]) or for prticulr set of points (e.g., [8]) Relted Work Severl methods rely on the dulity etween Del(X) nd Vor(X) for their genertion. Note tht Del(X) is defined through the Deluny simplices covering the convex hull of the points in X. While cell in Vor(X) contins ll the points closer to its seed thn ny other seed in X, Deluny simplex in Del(X) connects d +1 seeds, nd is ssocited with n empty circumsphere (Deluny sphere) tht hs no seeds inside it. The center of Deluny sphere is corner of Voronoi cell (Voronoi vertex). This feture is termed: the empty sphere principle, nd is the sic rule to generte Del(X); see Figure 2 for 2d exmple of how empty spheres develop s points re dded to the domin nd the resulting Del(X). An importnt remrk t this point is tht Vor(X) is uniquely defined for given X, while Del(X) is not due to potentil miguous connectivity scenrios (e.g., when more thn d + 1 seeds fll on the surfce of the sme Deluny sphere). An miguous Deluny simplex corresponds to degenerte Voronoi entity of some dimension. There exist severl lgorithms to generte Vor(X): Direct lgorithms: compute the convex hull of X, e.g., plne cutting [9], Fortune s [10], nd Quickhull [11]. Indirect lgorithms: strt y generting Del(X) nd otin Vor(X) s its dul, e.g., Bowyer Wtson [12 14]. In nutshell, ny exct lgorithm must visit ll Voronoi vertices, or equivlently visit the corresponding Deluny spheres in order to construct VDD or its dul Deluny simplices. Note tht the numer of Voronoi vertices grows with dimensions introducing nturl curse of dimensionlity. Therefore, oth Voronoi nd Deluny DDs re completely ndoned in high-dimensionl prolems. Severl computtionl chllenges fce ech of the Voronoi nd Deluny decomposition pproches. On the direct methods front, plne cutting techniques go the extr mile of constructing the entire fcets (not just vertices) of ech Voronoi cell vi successive hyperplne-hyperplne trimming opertions. This pproch enles the independent construction of different Voronoi cells; n ppeling feture for sclle prllel implementtions. However, s the numer of dimensions grows, these opertions ecome more complex nd would require intrctle extr storge cpcity to trck ll ounding entities of the hyperplne eing trimmed. This dd-on lgorithmic curse of dimensionlity limits the implementtion of plne cutting pproches to low dimensions (e.g., d = 2, 3) [15,16]. The Quickhull lgorithm voids the successive trimming opertions y lifting the seeds onto d + 1 proloid, deepening the curse of dimensionlity nd therefore incresing the computtionl nd storge requirements exponentilly. The implementtion of Quickhull, Qhull, is the stndrd tool to uild Voronoi nd Deluny meshes in ritrry dimensions. Indirect methods, on the other hnd, rely on the empty sphere principle [17 20] nd hve een well-studied in the lst few decdes. However, most of the reserch efforts here were directed towrds low dimensionl prolems (e.g., d = 2, 3). Tools using the empty sphere principle re usully dimension-specific. Furthermore, from n implementtion perspective, stte of the rt tetrhedrl Deluny meshing softwre tools re inherently sequentil, i.e., they insert points one fter the other, which leds to poor prllel sclility. Moreover, chnging the order of point insertions typiclly genertes inconsistent meshes.

3 / Procedi Engineering 124 (2016) c c d () Identifying the Voronoi neighors of seed. () Seed is declred s cndidte neighor of. (c) Seed c trims the cndidte fcet ssocited w/. (d) Seed d further trims the cndidte fcet. c c e f f e h d g i d (e) Trimming using ll seeds, is neighor of. (f) Seed c is declred new cndidte neighor of. (g) Trimming with f, c cn not e neighor of. (h) After considering ll seeds, Vor() is formed. Figure 1: Constructing 2-d Vor(X) vi successive plne cutting opertions. To find the exct cell edges round seed, we need to identify its neighor cells, which in turn requires successive hyperplne-hyperplne trimming opertions, dding n lgorithmic curse of dimensionlity. () Lrgest empty circumcircle pssing through 3 points in D. () As point is dded, three empty circumcircles re creted. (c) Adding more points chnges the circumcircles rrngement. (d) All points re dded, nd circumcircles define Del(X). Figure 2: Constructing 2-d Del(X) vi the empty sphere principle: the circumcircles of ll vlid Deluny tringles hve empty interiors. () A rndom line smples point from Voronoi fcet. () A redundnt point smpled from the sme fcet. (c) Locl MC line smpling to find Deluny neighors. (d) Neighors shring reltively smll fcets missed. Figure 3: Constructing 2-d Del(X) vi locl hyperplne smpling: identifying the significnt neighors of given Voronoi cell cn e done without imposing the curse of dimensionlity ssocited with the hyperplne-hyperplne trimming opertions.

4 4 / Procedi Engineering 124 (2016) Locl Hyperplne Smpling The ide of smpling prmeter spce using rndom well-spced pointset is very ppeling for high-dimensionl pplictions, e.g., Uncertinty Quntifiction (UQ) nd optimiztion. However, the stte-of-the-rt lgorithms to generte it rely on ckground grids to trck the uncovered regions for future smpling. This introduces curse of dimensionlity, ounding their implementtions to out 6-d in prctice. A non-trditionl smpling pproch uses hyperplne smples (e.g., lines, plnes, etc) rther thn simple 0-d points. For exmple, Mximl-Poisson disk smpling (MPS) technique [21] ims to generte well-spced smples tht cn chieve mximlity in covering D. Using hyperplne smples ws found to hve superior performnce in finding the voids etween MPS disks, nd hence rech mximl disk pcking in high dimensions without suffering from the curse of dimensionlity [22]. Coupling this hyperplne smpling pproch with the dulity etween Voronoi fcets nd Deluny edges enled the first implicit Voronoi domin decomposition, where Voronoi cell is not explicitly constructed vi its vertices. Insted, domin point is clssified into cell using closest neighor serch over ll seeds. This opertion is not costly is it is independent of dimensionlity. In the Spoke Drts lgorithm [23], hyperplne smpling ws used to solve complex glol optimiztion prolems in high dimensions. This pproch relies on the dulity etween Voronoi fcet F v nd the dul Deluny edge E d. The locl hyperplne smpling pproch does not visit ny Voronoi vertex. Insted, it rndomly smples points from Voronoi fcet to witness the dul Deluny edge. The curse of dimensionlity ssocited with visiting ll Voronoi vertices is therefore roken. However, some Deluny edges might e missed if their dul Voronoi fcets re reltively smll, hence the resulting Deluny grph is only pproximte. See Figure 3 for 2d illustrtion of how the significnt Deluny neighors with reltively-lrge shred fcets re found vi successive Monte Crlo (MC) smpling opertions. 2. Algorithm Overview The locl hyperplne smpling technique using y Spoke Drts in 1.2 only exploits the dulity etween Voronoi fcets nd Deluny edges. This indeed reks the curse of dimensionlity nd voids the usge of the empty sphere principle, yet the generted Deluny grph is only pproximte. For n exct solution in ritrry numer of dimensions, we introduce recursive locl hyperplne smpling pproch: Recursive Spoke Drts (RSD), which exploits the full dulity etween the entities of the Voronoi nd Deluny meshes. RSD uses the reltions etween Voronoi s fcets F v nd edges E v ) nd Deluny s edges E d nd fcets F d to find oth Vor(X) nd Del(X) in ritrry dimensions nd generte their exct meshes in roust fshion. In rief, our lgorithm strts t n ritrry rndom Voronoi seed, smples rndom spokes (rys) from series of d, d 1,..., 1-dimensionl spces (ligned with vrious Voronoi entities) in order to identify vlid Voronoi vertex t which we smple d deterministic lines long the Voronoi edges connected to it to identify its neighor vertices. For illustrtion nd without loss of generlity, we present the steps of the RSD lgorithm in 3-dimensionl spce. Similr steps re employed in other dimensions. 1. Initil Rndom Seed Selection: Pick n ritrry Voronoi seed x 0 to strt from d Rndom Spoke Smpling: Smple rndom spoke in the 3-d spce strting from x 0. This ry identifies point x 1 on Voronoi fcet F v which in turn identifies the dul Deluny edge E d of F v d Rndom Spoke Smpling: Strting from point x 1, smple nother rndom spoke from the 2-d spce of F v, identifying point x 2 long Voronoi edge E v tht ounds F v. Point x 2 identifies the Deluny fcet F d ; which is the dul of the Voronoi edge E v d Rndom Spoke Smpling: Smple nother rndom spoke strting t point x 2 long the 1-d edge E v to identify Voronoi vertex x 3. Vertex x 3, in turn, identifies its dul Deluny simplex connecting x 0 nd 3 neighor seeds. 5. Deterministic Line Smpling: Once vertex is identified, 3 lines re smpled long the Voronoi edges connected to it in order to retrieve its neighor vertices in the sme cell, nd their corresponding dul Deluny simplices. 6. Propgtion: Continue this propgtion until ech Voronoi vertex identifies its neighor vertices on the sme cell, completing the exct construction of tht cell nd its dul Deluny simplices. See Figure 4 nd Figure 5 for illustrtive exmples in 2-d nd 3-d spces, respectively.

5 / Procedi Engineering 124 (2016) () A rndom ry smples point from 2-d spce. () A point is identified s edge nd neighor witness. (c) A rndom ry smples point from 1-d edge. (d) A deluny simplex round the retrieved vertex. (e) Two line smples from vertex long connected edges. (f) Neighor vertices nd dul simplices identified. (g) More line smples to identify other verteces. (h) All vertices nd their dul simplices identified. Figure 4: Constructing exct 2-d Del(X) nd Vor(X) using our Recursive Spoke Drts (RSD) pproch. x 1 x 1 x 2 x 0 x 0 x 0 () Initil seed selection. () Rndom 3-d spoke identifies x 1. (c) Rndom 2-d spoke identifies x 2. x 3 x 1 x 3 x 1 x 3 x 1 x 2 x 2 x 2 x 0 x 0 x 0 (d) Rndom 1-d spoke identifies x 3. (e) 3 lines identify neighor vertices. (f) Further line propgtion. Figure 5: Constructing exct 3-d Del(X) nd Vor(X) using our Recursive Spoke Drts (RSD) pproch.

6 6 / Procedi Engineering 124 (2016) Algorithm Significnce The RSD lgorithm hs following significnt properties, which mkes it ppeling especilly for sclle prllel meshing pplictions: 1. Andoning the dependence on the empty sphere principle. The empty sphere principle hs serious limittions. In specific, it imposes sequentil seed insertion, nd results in different meshes in cse of Deluny miguities if the order of insertion chnges. RSD ndons the usge of the empty sphere principle in the genertion of Deluny simplices providing the foundtion needed for sclle consistent meshing. 2. Sclle Prllel Meshing. The RSD pproch gurntees sclility s multiple processors cn operte on different seeds t the sme time. Moreover, generting consistent meshes cross processors vi unified opertions elimintes the communiction needed etween them even t interfces, improving sclility even more. 3. Roust Dimension-Independent Implementtion. From n implementtion point of view, the RSD lgorithm relies on two simple opertions: ) line-hyperplne trimming, nd ) sphericl rnge serch. The first opertion recursively exploits the full Voronoi-Deluny dulity, elimintes the need for intrctle hyperplne-hyperplne trimming ssocited with direct methods, nd enles dimension-independent implementtions. The second opertion spots the equidistnt seeds from retrieved vertex to roustly construct the dul Deluny simplices in cnonicl form. 4. Consistent Deluny Meshing. Conflicts etween miguous Deluny entities re hndled using integer opertors on seed indices. This gurntees mesh consistency regrdless of the numer of processors/prtitions utilized, s long s the order of seeds indices is loclly preserved in ech prtition. Similr opertors eliminte slivers on the fly, ensuring high-qulity mesh. 5. Simplifiction for Trctility in High-Dimensions. To void the nturl curse of dimensionlity ssocited with Deluny nd Voronoi meshing in high dimensions, RSD mkes it possile not to visit ll vertices. This cn e simply chieved y limiting the propgtion from Voronoi vertex to others, still improving the ccurcy of the pproximte Deluny grph over tht resulting from the non-recursive version. 4. Implementtion Detils In this section, we list further detils out the implementtion of our lgorithm. In specific, we present our initil dt structure ( k-d tree) for neighorhood serches, illustrte how we use it to trim rndom spokes, nd how we resolve miguities in consistent wy so tht ll processors would mke the sme resolution decision with no need for communiction etween them Initil Dt Structure We use lnced k-d trees [24] for our initil domin decomposition s they do not suffer from the curse of dimensionlity nd cn lso e implemented in prllel [25,26]. On verge, lnced k-d trees use O(log N) to find the nerest neighor within tree of N nodes. Our k-d trees re lnced in the sense tht t ny level in the tree: pivot hyperplne in some su-domin is chosen to pss through the point t or closest to the medin of the points in tht su-domin. See Figure 6 for 2-d exmple of their construction. We rely on this clss of inry trees to find the closest point to the end of the rndom spoke nd hence mke the right spoke-trimming decision s we explin next. We use qsort [27] to speed up the sorting process long the tree construction Trimming Spokes At ny Voronoi seed x i in dimension d, whenever rndom spoke #» s (x i ) is smpled, it needs to e trimmed t the oundries of tht Voronoi cell. For this purpose, we use the k-d tree to collect the closest neighors {x j } of tht Voronoi seed nd use them to trim the spoke. At lest d + 1 neighors re needed. Our lgorithm collects neighors using neighorhood sphere centered t x i, nd the rdius of tht sphere is prmetrized y α; the glol trgeted spect rtio of Voronoi cells, nd r min ; the distnce etween x i nd its closest neighor x. The rdius of the

7 / Procedi Engineering 124 (2016) () Binry tree splitting () k-d tree connections. 16 Figure 6: Building lnced k-d tree s n initil domin prtitioning dt structure for our lgorithm. Exmple shows 2-d tree of 16 nodes. neighorhood sphere round x i is then given y r i = α r min. The spoke is then trimmed using the Voronoi hyperplne etween x i nd the closest point in the sphere to the end of the spoke x c. The process goes on until no point in the sphere cn trim the spoke nymore; t which the dimensionlity of the next spoke is reduced y 1. To reduce the dimensionlity t ech spoke step, we employ the Grhm-Schmidt pproch [28]: given set of orthogonl sis vectors {u 0, u 1,..., u d 1 } in the spce R d, we cn construct nother orthogonl sis set {v 0, v 1,..., v d 1 } y successive projection: v 0 = u 0, k 1 v k = u k proj v j u k j=1 Using this pproch, we construct the sis vectors D of the upcoming spoke, orthogonl to the sis of prior spokes D. This continues until the dimensionlity of future spoke is 0; indicting Voronoi vertex, t which the propgtion step kick in long ssocited Voronoi edges until ll Voronoi vertices round x i re visited. Oviously, the point distriution lso impcts the construction of Deluny simplices. Figure 7 shows the impct of incresing α during the successive phses of our lgorithm for set of well-spced points. Similrly, Figure 8 shows the impct of incresing α on the construction of the Deluny grph when the initil 2-d pointset is Monte-Crlo smpled. In generl, our lgorithm progressively increses α s illustrted in Figure 7 nd Figure 8. Initilly, α strts t the vlue of 2 (sufficient for uniform well-spced pointset). The vlue of α gets douled in ech phse, serching for Voronoi vertices of cells with mtching (or less) spect rtios. Experimentlly, we chose termintion vlue of α mx = This limiting vlue of α cn e incresed to hndle suspected cses on Voronoi cells with higher spect rtios, with no significnt increse in meshing time. Even with Monte Crlo pointsets, the vlue of α mx ws sufficient in ll our experiments.

8 8 / Procedi Engineering 124 (2016) () Pointset X () α=2 (c) α=4 (d) α=8 (e) α=16 (f) α=32 (g) α=64 (h) α=128 Figure 7: RSD construction of 2-d Deluny meshes on well-spced pointset X s the trget spect rtio prmeter α is incresed. () Pointset X () α=2 (c) α=4 (d) α=8 (e) α=16 (f) α=32 (g) α=64 (h) α=128 Figure 8: RSD construction of 2-d Deluny meshes on Monte Crlo pointset X s the trget spect rtio prmeter α is incresed.

9 / Procedi Engineering 124 (2016) () Initil Deluny simplex constructed. () An miguity sphere is detected with other points on it. (c) The center of mss of ll points is dded. (d) Connect point with lowest index to ll others. Figure 9: Consistent resolution of Deluny miguities. If n miguity is detected, the center of mss of the miguous points is temporrily dded, resolving the miguity nd deducing the connections etween these points. Removing the point nd keeping the connections, further connections re dded y connecting the point with the lowest index to ll others. Monte-Crlo Structured RSD: Del Mesh RSD: Approximte Del Grph Spoke-Drts: Approximte Del Grph Figure 10: Using 121 structured nd Monte-Crlo points in 2-d, we distinguish the construction of the Deluny mesh Del(X) using RSD, n lmost-exct pproximtion of the Deluny grph using the pproximte version of RSD, nd n pproximtion of the Deluny grph using Spoke-Drts [23] with severl missing Deluny edges Resolving Amiguities nd Conflicts Our lgorithm cn consistently resolve Deluny miguities without ny communiction lod etween processors. If n miguity is detected, we dd point in the center of the miguous points, temporrily resolving the miguity nd deducing the connections etween these points. Removing the center of mss point nd keeping the connections, we dd further connections y connecting the point with the lowest index to ll others. Since this rule is known hed

10 10 / Procedi Engineering 124 (2016) of time, different processors would resolve n miguity in the sme exct wy, reching the sme exct mesh. See Figure 9 for n illustrtive exmple Approximte RSD Implementtion in High Dimensions In high dimensions, the numer of Voronoi vertices nd Deluny simplices grow exponentilly; mjor ottleneck for discretizing high dimensionl spces. However, the numer of Deluny edges relies only on the numer of input points (nd not dimension), hence cn e enumerted efficiently. Similrly, lthough the numer of Voronoi vertices ecomes intrctle in high dimension, the numer of Voronoi cells hs one-to-one mpping to the numer of input points (Voronoi seeds). Coupling these oservtions mkes it possile to decompose high dimensionl spces in trctle wy. In this cse, our RSD lgorithm cn still proceed to some precision nd stop the propgtion step efore ll vertices re collected. Once Voronoi vertex is retrieved, the equi-distnt Voronoi seeds to tht vertex re connected in the pproximte Deluny grph. Visiting Voronoi vertex tht dds no new Deluny edges to the pproximte grph is considered miss. We stop the pproximte RSD procedure round some seed when some numer of successive misses occur (e.g., 100 in our experiments). This pproximte version of RSD smples suset of the Voronoi vertices, rther thn visit ll of them. Our experiments show tht the pproximte RSD lgorithm generted n lmost exct Deluny grph despite not visiting ll vertices. See Figure 10 for comprisons etween Spoke Drts, RSD, nd RSD pproximte version. Using 121 structured nd Monte-Crlo points in 2-d, we distinguish the construction of the Deluny mesh Del(X) using RSD, n lmostexct pproximtion of the Deluny grph using the pproximte version of RSD, nd n pproximtion of the Deluny grph using Spoke-Drts [23] with severl missing Deluny edges. Note tht Deluny meshing technique hs to resolve Deluny miguities when more thn d +1 points fll on the sme Deluny sphere. Some of the Deluny edges hve to e discrded to void self-intersecting elements. RSD does tht successfully nd consistently s explined in section 4.3 nd illustrted in Figure 9. The Deluny grph, however, does not need these conflict resolutions, s shown in the structured cse of the pproximte version of RSD of Figure Experimentl Results In this section, we compre the performnce of RSD to stndrd Deluny nd Voronoi meshing techniques in different dimensions in terms of the meshing runtime. In specific, we compre RSD to the stte-of-the-rt Deluny meshing techniques: Tringle [18,29], TetGen [30], nd Qhull [31]. Tringle genertes exct Deluny tringultions nd Voronoi digrms in 2-d. TetGen is the stte of the rt method to generte tetrhedrl meshes on polyhedrl domins in 3-d. Qhull computes the convex hull using the Quickhull lgorithm, nd hence finds the Deluny tringultion nd Voronoi digrm in ritrry dimensions. Figure 11 shows comprison results illustrting how RSD would perform with vrile numer of processors, compred to Tringle, Qhull, nd TetGen in 2-d, 3-d, nd 4-d. Plots show the runtime of ech method ginst the output numer of Deluny simplices. For well-spced set of points with numerous Deluny miguities, RSD ws slightly slower thn Tringle in 2-d, however, it successfully outperformed TetGen nd Qhull in 2-, 3-, nd 4-d. Resolving miguities turned out to e chllenge for Qhull in vrious dimensions. In 2-d, Qhull did not produce vlid meshes with numer of simplices lrger thn 10M. To demonstrte the consistent meshing cpility of RSD, we lso rn our seril implementtion on numer of processors p with no communiction etween them. Ech processor loded nd constructed the k-d tree for the entire input pointset nd generted the output Deluny simplices ssocited with suset of the input vertices; s illustrted in Figure 12. We show the performnce of p = 10 nd p = 50 in Figure 11. Applying domin sudivision to ssign points to these processors would further improve the prllel performnce. For the cse of Monte-Crlo set of points in the right column of Figure 11, the spect rtio of the elements deteriorte significntly. Although this impcted the runtime performnce of RSD, it ws still cple of generting exct Deluny meshes in liner time. A thorough nlysis of this cpility including domin sudivisions is to e ddressed in follow-up pper. To vlidte nd check our finl meshes, we rely on the Empty Sphere Principle to check our meshes nd vlidte the finl results. For exmple, we would construct the circumsphere of every output Deluny simplex nd check tht its center mtches the corresponding dul Voronoi vertex retrieved y our lgorithm.

11 Runtime Runtime Runtime Runtime Runtime Runtime / Procedi Engineering 124 (2016) d 10 0 RSD - 1 RSD - 10 RSD - 50 Tringle Qhull Numer of Simplices RSD - 1 Tringle Qhull Numer of Simplices d RSD - 1 RSD - 10 RSD - 50 TetGen Qhull Numer of Simplices RSD - 1 TetGen Qhull Numer of Simplices d Numer of Simplices () Structured RSD - 1 RSD - 10 RSD - 50 Qhull Numer of Simplices () Monte-Crlo RSD - 1 Qhull Figure 11: Runtime comprisons etween Tringle, Qhull, TetGen, nd RSD with vrile numer of processors (where RSD-p indictes the utiliztion of p processors). Rows present different dimensionlity: 2-d, 3-d, nd 4-d, while columns present different point distriution: wellspced structured nd Monte-Crlo. () Pointset X () α=2 (c) α=4 (d) α=8 Figure 12: RSD prllel construction of 2-d Deluny meshes on 10 processors, with 100 MC points s α is incresed.

12 12 / Procedi Engineering 124 (2016) Conclusions We introduced novel recursive hyperplne smpling lgorithm to construct exct Deluny meshes in lowdimensions nd pproximte Deluny grphs in high-dimensions. Our pproch exploits the full dulity etween Voronoi nd Deluny entities of vrious dimensions, nd ndons the dependence on the empty sphere principle in the genertion of Deluny simplices providing the foundtion needed for sclle consistent meshing, resulting in communiction-free implementtion. We demonstrted the cpilities of our lgorithm when compred to stte-ofthe-rt Deluny meshing tools (Tringle, TetGen, nd Qhull). We showed tht the construction of n lmost-exct Deluny grph is trctle using the pproximte version of our lgorithm. Our future step include complete nlysis of the ccurcy of tht pproximte version in high dimensions. We will lso investigte vrious techniques to improve the RSD performnce when hndling rndom pointsets. References [1] C. D. Toth, J. O Rourke, J. E. Goodmn, Hndook of discrete nd computtionl geometry, CRC press, [2] J.-R. Sck, J. Urruti, Hndook of computtionl geometry, Elsevier, [3] S.-W. Cheng, T. K. Dey, J. Shewchuk, Deluny mesh genertion, CRC Press, [4] P. Frey, P.-L. George, Mesh genertion, John Wiley & Sons, [5] H. Edelsrunner, Geometry nd topology for mesh genertion, Cmridge University Press, [6] P. Foteinos, N. Chrisochoides, 4d spce time deluny meshing for medicl imges, Engineering with Computers 31 (2015) [7] C. Wng, J. To, Grphs in scientific visuliztion: A survey, in: Computer Grphics Forum, Wiley Online Lirry, [8] E. Krels, M. Neumüller, Generting dmissile spce-time meshes for moving domins in d + 1-dimensions, rxiv preprint rxiv: (2015). [9] C. H. Rycroft, G. S. Grest, J. W. Lndry, M. Z. Bznt, Anlysis of grnulr flow in pele-ed nucler rector, Physicl review E 74 (2006) [10] S. Fortune, A sweepline lgorithm for Voronoi digrms, Algorithmic 2 (1987) [11] C. B. Brer, D. P. Dokin, H. Huhdnp, The quickhull lgorithm for convex hulls, ACM Trnsctions on Mthemticl Softwre (TOMS) 22 (1996) [12] A. Bowyer, Computing Dirichlet tesselltions, The Computer Journl 24 (1981) [13] D. F. Wtson, Computing the n-dimensionl deluny tesselltion with ppliction to Voronoi polytopes, The Computer Journl 24 (1981) [14] H. Si, Tetgen, deluny-sed qulity tetrhedrl mesh genertor, ACM Trnsctions on Mthemticl Softwre (TOMS) 41 (2015) 11. [15] C. Rycroft, Voro++: A three-dimensionl Voronoi cell lirry in C++, Lwrence Berkeley Ntionl Lortory (2009). [16] M. S. Eeid, S. A. Mitchell, Uniform rndom Voronoi meshes, in: Proceedings of the 20th Interntionl Meshing Roundtle, Springer, 2012, pp [17] L. P. Chew, Gurnteed-qulity Deluny meshing in 3d (short version), in: Proceedings of the thirteenth nnul symposium on Computtionl geometry, ACM, 1997, pp [18] J. R. Shewchuk, Deluny refinement lgorithms for tringulr mesh genertion, Computtionl geometry 22 (2002) [19] T. K. Dey, J. A. Levine, Deluny meshing of piecewise smooth complexes without expensive predictes, Algorithms 2 (2009) [20] J. R. Shewchuk, Tringle: Engineering 2D qulity mesh genertor nd Deluny tringultor, Applied computtionl geometry towrds geometric engineering (1996) [21] M. S. Eeid, S. A. Mitchell, A. Ptney, A. A. Dvidson, J. D. Owens, A simple lgorithm for mximl Poisson-disk smpling in high dimensions, Computer Grphics Forum 31 (2012) [22] M. S. Eeid, A. Ptney, S. A. Mitchell, K. R. Dley, A. A. Dvidson, J. D. Owens, k-d drts: Smpling y k-dimensionl flt serches, ACM Trnsctions on Grphics (2014). In press. Also t rxiv: [cs.gr]; URL [23] M. S. Eeid, S. A. Mitchell, M. A. Awd, C. Prk, L. P. Swiler, D. Mnoch, L.-Y. Wei, Spoke drts for efficient high dimensionl lue noise smpling, rxiv preprint rxiv: (2014). [24] J. L. Bentley, Multidimensionl inry serch trees used for ssocitive serching, Communictions of the ACM 18 (1975) [25] L. Hu, S. Nooshdi, M. Ahmdi, Mssively prllel kd-tree construction nd nerest neighor serch lgorithms, in: Circuits nd Systems (ISCAS), 2015 IEEE Interntionl Symposium on, IEEE, 2015, pp [26] N. Rjni, K. McArdle, I. S. Dhillon, Prllel k nerest neighor grph construction using tree-sed dt structures, in: 1st High Performnce Grph Mining workshop, Sydney, 10 August 2015, [27] C. Allison, Sorting with qsort, C Users J. 11 (1993) 107 ff. [28] L. N. Trefethen, D. Bu III, Numericl liner lger, volume 50, Sim, [29] J. R. Shewchuk, Tringle: Engineering 2D Qulity Mesh Genertor nd Deluny Tringultor, in: M. C. Lin, D. Mnoch (Eds.), Applied Computtionl Geometry: Towrds Geometric Engineering, volume 1148 of Lecture Notes in Computer Science, Springer-Verlg, 1996, pp From the First ACM Workshop on Applied Computtionl Geometry. [30] H. Si, Tetgen, deluny-sed qulity tetrhedrl mesh genertor, ACM Trns. Mth. Softw. 41 (2015) 11:1 11:36. [31] C. B. Brer, D. P. Dokin, H. Huhdnp, The quickhull lgorithm for convex hulls, ACM Trns. Mth. Softw. 22 (1996)