Recursive Spoke Darts: Local Hyperplane Sampling for Delaunay and Voronoi Meshing in Arbitrary Dimensions
|
|
- Job Jefferson
- 5 years ago
- Views:
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)
ScienceDirect. Recursive Spoke Darts: Local Hyperplane Sampling for Delaunay and Voronoi Meshing in Arbitrary Dimensions
Aville online t www.sciencedirect.com ScienceDirect Procedi Engineering 163 (2016 ) 110 122 25th Interntionl Meshing Roundtle (IMR25) Recursive Spoke Drts: Locl Hyperplne Smpling for Deluny nd Voronoi
More information2 Computing all Intersections of a Set of Segments Line Segment Intersection
15-451/651: Design & Anlysis of Algorithms Novemer 14, 2016 Lecture #21 Sweep-Line nd Segment Intersection lst chnged: Novemer 8, 2017 1 Preliminries The sweep-line prdigm is very powerful lgorithmic design
More informationCOMP 423 lecture 11 Jan. 28, 2008
COMP 423 lecture 11 Jn. 28, 2008 Up to now, we hve looked t how some symols in n lphet occur more frequently thn others nd how we cn sve its y using code such tht the codewords for more frequently occuring
More informationBefore We Begin. Introduction to Spatial Domain Filtering. Introduction to Digital Image Processing. Overview (1): Administrative Details (1):
Overview (): Before We Begin Administrtive detils Review some questions to consider Winter 2006 Imge Enhncement in the Sptil Domin: Bsics of Sptil Filtering, Smoothing Sptil Filters, Order Sttistics Filters
More informationPresentation Martin Randers
Presenttion Mrtin Rnders Outline Introduction Algorithms Implementtion nd experiments Memory consumption Summry Introduction Introduction Evolution of species cn e modelled in trees Trees consist of nodes
More informationP(r)dr = probability of generating a random number in the interval dr near r. For this probability idea to make sense we must have
Rndom Numers nd Monte Crlo Methods Rndom Numer Methods The integrtion methods discussed so fr ll re sed upon mking polynomil pproximtions to the integrnd. Another clss of numericl methods relies upon using
More informationLecture 10 Evolutionary Computation: Evolution strategies and genetic programming
Lecture 10 Evolutionry Computtion: Evolution strtegies nd genetic progrmming Evolution strtegies Genetic progrmming Summry Negnevitsky, Person Eduction, 2011 1 Evolution Strtegies Another pproch to simulting
More informationFrom Dependencies to Evaluation Strategies
From Dependencies to Evlution Strtegies Possile strtegies: 1 let the user define the evlution order 2 utomtic strtegy sed on the dependencies: use locl dependencies to determine which ttriutes to compute
More informationCS321 Languages and Compiler Design I. Winter 2012 Lecture 5
CS321 Lnguges nd Compiler Design I Winter 2012 Lecture 5 1 FINITE AUTOMATA A non-deterministic finite utomton (NFA) consists of: An input lphet Σ, e.g. Σ =,. A set of sttes S, e.g. S = {1, 3, 5, 7, 11,
More informationIn the last lecture, we discussed how valid tokens may be specified by regular expressions.
LECTURE 5 Scnning SYNTAX ANALYSIS We know from our previous lectures tht the process of verifying the syntx of the progrm is performed in two stges: Scnning: Identifying nd verifying tokens in progrm.
More informationSlides for Data Mining by I. H. Witten and E. Frank
Slides for Dt Mining y I. H. Witten nd E. Frnk Simplicity first Simple lgorithms often work very well! There re mny kinds of simple structure, eg: One ttriute does ll the work All ttriutes contriute eqully
More informationPARALLEL AND DISTRIBUTED COMPUTING
PARALLEL AND DISTRIBUTED COMPUTING 2009/2010 1 st Semester Teste Jnury 9, 2010 Durtion: 2h00 - No extr mteril llowed. This includes notes, scrtch pper, clcultor, etc. - Give your nswers in the ville spce
More information6.3 Volumes. Just as area is always positive, so is volume and our attitudes towards finding it.
6.3 Volumes Just s re is lwys positive, so is volume nd our ttitudes towrds finding it. Let s review how to find the volume of regulr geometric prism, tht is, 3-dimensionl oject with two regulr fces seprted
More information9 Graph Cutting Procedures
9 Grph Cutting Procedures Lst clss we begn looking t how to embed rbitrry metrics into distributions of trees, nd proved the following theorem due to Brtl (1996): Theorem 9.1 (Brtl (1996)) Given metric
More informationCS143 Handout 07 Summer 2011 June 24 th, 2011 Written Set 1: Lexical Analysis
CS143 Hndout 07 Summer 2011 June 24 th, 2011 Written Set 1: Lexicl Anlysis In this first written ssignment, you'll get the chnce to ply round with the vrious constructions tht come up when doing lexicl
More informationComplete Coverage Path Planning of Mobile Robot Based on Dynamic Programming Algorithm Peng Zhou, Zhong-min Wang, Zhen-nan Li, Yang Li
2nd Interntionl Conference on Electronic & Mechnicl Engineering nd Informtion Technology (EMEIT-212) Complete Coverge Pth Plnning of Mobile Robot Bsed on Dynmic Progrmming Algorithm Peng Zhou, Zhong-min
More informationON THE DEHN COMPLEX OF VIRTUAL LINKS
ON THE DEHN COMPLEX OF VIRTUAL LINKS RACHEL BYRD, JENS HARLANDER Astrct. A virtul link comes with vriety of link complements. This rticle is concerned with the Dehn spce, pseudo mnifold with oundry, nd
More informationA dual of the rectangle-segmentation problem for binary matrices
A dul of the rectngle-segmenttion prolem for inry mtrices Thoms Klinowski Astrct We consider the prolem to decompose inry mtrix into smll numer of inry mtrices whose -entries form rectngle. We show tht
More informationTries. Yufei Tao KAIST. April 9, Y. Tao, April 9, 2013 Tries
Tries Yufei To KAIST April 9, 2013 Y. To, April 9, 2013 Tries In this lecture, we will discuss the following exct mtching prolem on strings. Prolem Let S e set of strings, ech of which hs unique integer
More informationWhat are suffix trees?
Suffix Trees 1 Wht re suffix trees? Allow lgorithm designers to store very lrge mount of informtion out strings while still keeping within liner spce Allow users to serch for new strings in the originl
More informationToday. CS 188: Artificial Intelligence Fall Recap: Search. Example: Pancake Problem. Example: Pancake Problem. General Tree Search.
CS 88: Artificil Intelligence Fll 00 Lecture : A* Serch 9//00 A* Serch rph Serch Tody Heuristic Design Dn Klein UC Berkeley Multiple slides from Sturt Russell or Andrew Moore Recp: Serch Exmple: Pncke
More informationFig.25: the Role of LEX
The Lnguge for Specifying Lexicl Anlyzer We shll now study how to uild lexicl nlyzer from specifiction of tokens in the form of list of regulr expressions The discussion centers round the design of n existing
More informationA Tautology Checker loosely related to Stålmarck s Algorithm by Martin Richards
A Tutology Checker loosely relted to Stålmrck s Algorithm y Mrtin Richrds mr@cl.cm.c.uk http://www.cl.cm.c.uk/users/mr/ University Computer Lortory New Museum Site Pemroke Street Cmridge, CB2 3QG Mrtin
More informationAnnouncements. CS 188: Artificial Intelligence Fall Recap: Search. Today. Example: Pancake Problem. Example: Pancake Problem
Announcements Project : erch It s live! Due 9/. trt erly nd sk questions. It s longer thn most! Need prtner? Come up fter clss or try Pizz ections: cn go to ny, ut hve priority in your own C 88: Artificil
More informationApproximate computations
Living with floting-point numers Stndrd normlized representtion (sign + frction + exponent): Approximte computtions Rnges of vlues: Representtions for:, +, +0, 0, NN (not numer) Jordi Cortdell Deprtment
More informationAgilent Mass Hunter Software
Agilent Mss Hunter Softwre Quick Strt Guide Use this guide to get strted with the Mss Hunter softwre. Wht is Mss Hunter Softwre? Mss Hunter is n integrl prt of Agilent TOF softwre (version A.02.00). Mss
More informationApproximation by NURBS with free knots
pproximtion by NURBS with free knots M Rndrinrivony G Brunnett echnicl University of Chemnitz Fculty of Computer Science Computer Grphics nd Visuliztion Strße der Ntionen 6 97 Chemnitz Germny Emil: mhrvo@informtiktu-chemnitzde
More informationAnnouncements. CS 188: Artificial Intelligence Fall Recap: Search. Today. General Tree Search. Uniform Cost. Lecture 3: A* Search 9/4/2007
CS 88: Artificil Intelligence Fll 2007 Lecture : A* Serch 9/4/2007 Dn Klein UC Berkeley Mny slides over the course dpted from either Sturt Russell or Andrew Moore Announcements Sections: New section 06:
More informationReducing a DFA to a Minimal DFA
Lexicl Anlysis - Prt 4 Reducing DFA to Miniml DFA Input: DFA IN Assume DFA IN never gets stuck (dd ded stte if necessry) Output: DFA MIN An equivlent DFA with the minimum numer of sttes. Hrry H. Porter,
More informationIf you are at the university, either physically or via the VPN, you can download the chapters of this book as PDFs.
Lecture 5 Wlks, Trils, Pths nd Connectedness Reding: Some of the mteril in this lecture comes from Section 1.2 of Dieter Jungnickel (2008), Grphs, Networks nd Algorithms, 3rd edition, which is ville online
More informationTOWARDS GRADIENT BASED AERODYNAMIC OPTIMIZATION OF WIND TURBINE BLADES USING OVERSET GRIDS
TOWARDS GRADIENT BASED AERODYNAMIC OPTIMIZATION OF WIND TURBINE BLADES USING OVERSET GRIDS S. H. Jongsm E. T. A. vn de Weide H. W. M. Hoeijmkers Overset symposium 10-18-2012 Deprtment of mechnicl engineering
More informationAccelerating 3D convolution using streaming architectures on FPGAs
Accelerting 3D convolution using streming rchitectures on FPGAs Hohun Fu, Robert G. Clpp, Oskr Mencer, nd Oliver Pell ABSTRACT We investigte FPGA rchitectures for ccelerting pplictions whose dominnt cost
More informationMa/CS 6b Class 1: Graph Recap
M/CS 6 Clss 1: Grph Recp By Adm Sheffer Course Detils Adm Sheffer. Office hour: Tuesdys 4pm. dmsh@cltech.edu TA: Victor Kstkin. Office hour: Tuesdys 7pm. 1:00 Mondy, Wednesdy, nd Fridy. http://www.mth.cltech.edu/~2014-15/2term/m006/
More information10.5 Graphing Quadratic Functions
0.5 Grphing Qudrtic Functions Now tht we cn solve qudrtic equtions, we wnt to lern how to grph the function ssocited with the qudrtic eqution. We cll this the qudrtic function. Grphs of Qudrtic Functions
More information1 Quad-Edge Construction Operators
CS48: Computer Grphics Hndout # Geometric Modeling Originl Hndout #5 Stnford University Tuesdy, 8 December 99 Originl Lecture #5: 9 November 99 Topics: Mnipultions with Qud-Edge Dt Structures Scribe: Mike
More informationCOMBINATORIAL PATTERN MATCHING
COMBINATORIAL PATTERN MATCHING Genomic Repets Exmple of repets: ATGGTCTAGGTCCTAGTGGTC Motivtion to find them: Genomic rerrngements re often ssocited with repets Trce evolutionry secrets Mny tumors re chrcterized
More informationDr. D.M. Akbar Hussain
Dr. D.M. Akr Hussin Lexicl Anlysis. Bsic Ide: Red the source code nd generte tokens, it is similr wht humns will do to red in; just tking on the input nd reking it down in pieces. Ech token is sequence
More information4452 Mathematical Modeling Lecture 4: Lagrange Multipliers
Mth Modeling Lecture 4: Lgrnge Multipliers Pge 4452 Mthemticl Modeling Lecture 4: Lgrnge Multipliers Lgrnge multipliers re high powered mthemticl technique to find the mximum nd minimum of multidimensionl
More informationCone Cluster Labeling for Support Vector Clustering
Cone Cluster Lbeling for Support Vector Clustering Sei-Hyung Lee Deprtment of Computer Science University of Msschusetts Lowell MA 1854, U.S.A. slee@cs.uml.edu Kren M. Dniels Deprtment of Computer Science
More informationMA1008. Calculus and Linear Algebra for Engineers. Course Notes for Section B. Stephen Wills. Department of Mathematics. University College Cork
MA1008 Clculus nd Liner Algebr for Engineers Course Notes for Section B Stephen Wills Deprtment of Mthemtics University College Cork s.wills@ucc.ie http://euclid.ucc.ie/pges/stff/wills/teching/m1008/ma1008.html
More information1. SEQUENCES INVOLVING EXPONENTIAL GROWTH (GEOMETRIC SEQUENCES)
Numbers nd Opertions, Algebr, nd Functions 45. SEQUENCES INVOLVING EXPONENTIAL GROWTH (GEOMETRIC SEQUENCES) In sequence of terms involving eponentil growth, which the testing service lso clls geometric
More informationEECS150 - Digital Design Lecture 23 - High-level Design and Optimization 3, Parallelism and Pipelining
EECS150 - Digitl Design Lecture 23 - High-level Design nd Optimiztion 3, Prllelism nd Pipelining Nov 12, 2002 John Wwrzynek Fll 2002 EECS150 - Lec23-HL3 Pge 1 Prllelism Prllelism is the ct of doing more
More informationA New Learning Algorithm for the MAXQ Hierarchical Reinforcement Learning Method
A New Lerning Algorithm for the MAXQ Hierrchicl Reinforcement Lerning Method Frzneh Mirzzdeh 1, Bbk Behsz 2, nd Hmid Beigy 1 1 Deprtment of Computer Engineering, Shrif University of Technology, Tehrn,
More informationa < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1
Mth 33 Volume Stewrt 5.2 Geometry of integrls. In this section, we will lern how to compute volumes using integrls defined by slice nlysis. First, we recll from Clculus I how to compute res. Given the
More informationINTRODUCTION TO SIMPLICIAL COMPLEXES
INTRODUCTION TO SIMPLICIAL COMPLEXES CASEY KELLEHER AND ALESSANDRA PANTANO 0.1. Introduction. In this ctivity set we re going to introduce notion from Algebric Topology clled simplicil homology. The min
More informationthis grammar generates the following language: Because this symbol will also be used in a later step, it receives the
LR() nlysis Drwcks of LR(). Look-hed symols s eplined efore, concerning LR(), it is possile to consult the net set to determine, in the reduction sttes, for which symols it would e possile to perform reductions.
More informationNetwork Interconnection: Bridging CS 571 Fall Kenneth L. Calvert All rights reserved
Network Interconnection: Bridging CS 57 Fll 6 6 Kenneth L. Clvert All rights reserved The Prolem We know how to uild (rodcst) LANs Wnt to connect severl LANs together to overcome scling limits Recll: speed
More informationcisc1110 fall 2010 lecture VI.2 call by value function parameters another call by value example:
cisc1110 fll 2010 lecture VI.2 cll y vlue function prmeters more on functions more on cll y vlue nd cll y reference pssing strings to functions returning strings from functions vrile scope glol vriles
More informationSAPPER: Subgraph Indexing and Approximate Matching in Large Graphs
SAPPER: Sugrph Indexing nd Approximte Mtching in Lrge Grphs Shijie Zhng, Jiong Yng, Wei Jin EECS Dept., Cse Western Reserve University, {shijie.zhng, jiong.yng, wei.jin}@cse.edu ABSTRACT With the emergence
More informationII. THE ALGORITHM. A. Depth Map Processing
Lerning Plnr Geometric Scene Context Using Stereo Vision Pul G. Bumstrck, Bryn D. Brudevold, nd Pul D. Reynolds {pbumstrck,brynb,pulr2}@stnford.edu CS229 Finl Project Report December 15, 2006 Abstrct A
More informationPlanning with Reachable Distances: Fast Enforcement of Closure Constraints
Plnning with Rechle Distnces: Fst Enforcement of Closure Constrints Xinyu Tng, Shwn Thoms, nd Nncy M. Amto Astrct Motion plnning for closed-chin systems is prticulrly difficult due to dditionl closure
More informationA Sparse Grid Representation for Dynamic Three-Dimensional Worlds
A Sprse Grid Representtion for Dynmic Three-Dimensionl Worlds Nthn R. Sturtevnt Deprtment of Computer Science University of Denver Denver, CO, 80208 sturtevnt@cs.du.edu Astrct Grid representtions offer
More informationApproximation of Two-Dimensional Rectangle Packing
pproximtion of Two-imensionl Rectngle Pcking Pinhong hen, Yn hen, Mudit Goel, Freddy Mng S70 Project Report, Spring 1999. My 18, 1999 1 Introduction 1-d in pcking nd -d in pcking re clssic NP-complete
More informationMa/CS 6b Class 1: Graph Recap
M/CS 6 Clss 1: Grph Recp By Adm Sheffer Course Detils Instructor: Adm Sheffer. TA: Cosmin Pohot. 1pm Mondys, Wednesdys, nd Fridys. http://mth.cltech.edu/~2015-16/2term/m006/ Min ook: Introduction to Grph
More informationEfficient K-NN Search in Polyphonic Music Databases Using a Lower Bounding Mechanism
Efficient K-NN Serch in Polyphonic Music Dtses Using Lower Bounding Mechnism Ning-Hn Liu Deprtment of Computer Science Ntionl Tsing Hu University Hsinchu,Tiwn 300, R.O.C 886-3-575679 nhliou@yhoo.com.tw
More information8.2 Areas in the Plane
39 Chpter 8 Applictions of Definite Integrls 8. Ares in the Plne Wht ou will lern out... Are Between Curves Are Enclosed Intersecting Curves Boundries with Chnging Functions Integrting with Respect to
More informationMobile IP route optimization method for a carrier-scale IP network
Moile IP route optimiztion method for crrier-scle IP network Tkeshi Ihr, Hiroyuki Ohnishi, nd Ysushi Tkgi NTT Network Service Systems Lortories 3-9-11 Midori-cho, Musshino-shi, Tokyo 180-8585, Jpn Phone:
More informationCS481: Bioinformatics Algorithms
CS481: Bioinformtics Algorithms Cn Alkn EA509 clkn@cs.ilkent.edu.tr http://www.cs.ilkent.edu.tr/~clkn/teching/cs481/ EXACT STRING MATCHING Fingerprint ide Assume: We cn compute fingerprint f(p) of P in
More informationDistributed Systems Principles and Paradigms
Distriuted Systems Principles nd Prdigms Chpter 11 (version April 7, 2008) Mrten vn Steen Vrije Universiteit Amsterdm, Fculty of Science Dept. Mthemtics nd Computer Science Room R4.20. Tel: (020) 598 7784
More informationLECT-10, S-1 FP2P08, Javed I.
A Course on Foundtions of Peer-to-Peer Systems & Applictions LECT-10, S-1 CS /799 Foundtion of Peer-to-Peer Applictions & Systems Kent Stte University Dept. of Computer Science www.cs.kent.edu/~jved/clss-p2p08
More informationA Formalism for Functionality Preserving System Level Transformations
A Formlism for Functionlity Preserving System Level Trnsformtions Smr Abdi Dniel Gjski Center for Embedded Computer Systems UC Irvine Center for Embedded Computer Systems UC Irvine Irvine, CA 92697 Irvine,
More informationVideo-rate Image Segmentation by means of Region Splitting and Merging
Video-rte Imge Segmenttion y mens of Region Splitting nd Merging Knur Anej, Florence Lguzet, Lionel Lcssgne, Alin Merigot Institute for Fundmentl Electronics, University of Pris South Orsy, Frnce knur.nej@gmil.com,
More informationSystems I. Logic Design I. Topics Digital logic Logic gates Simple combinational logic circuits
Systems I Logic Design I Topics Digitl logic Logic gtes Simple comintionl logic circuits Simple C sttement.. C = + ; Wht pieces of hrdwre do you think you might need? Storge - for vlues,, C Computtion
More informationZZ - Advanced Math Review 2017
ZZ - Advnced Mth Review Mtrix Multipliction Given! nd! find the sum of the elements of the product BA First, rewrite the mtrices in the correct order to multiply The product is BA hs order x since B is
More informationSolving Problems by Searching. CS 486/686: Introduction to Artificial Intelligence Winter 2016
Solving Prolems y Serching CS 486/686: Introduction to Artificil Intelligence Winter 2016 1 Introduction Serch ws one of the first topics studied in AI - Newell nd Simon (1961) Generl Prolem Solver Centrl
More informationAn evolutionary approach to inter-session network coding
An evolutionry pproch to inter-session network coding The MIT Fculty hs mde this rticle openly ville. Plese shre how this ccess enefits you. Your story mtters. Cittion As Pulished Pulisher Kim, M. et l.
More informationDuality in linear interval equations
Aville online t http://ijim.sriu..ir Int. J. Industril Mthemtis Vol. 1, No. 1 (2009) 41-45 Dulity in liner intervl equtions M. Movhedin, S. Slhshour, S. Hji Ghsemi, S. Khezerloo, M. Khezerloo, S. M. Khorsny
More informationCS-184: Computer Graphics. Today. Lecture #10: Clipping and Hidden Surfaces ClippingAndHidden.key - October 27, 2014.
1 CS184: Computer Grphics Lecture #10: Clipping nd Hidden Surfces!! Prof. Jmes O Brien University of Cliforni, Berkeley! V2013F101.0 Tody 2 Clipping Clipping to view volume Clipping ritrry polygons Hidden
More informationGrade 7/8 Math Circles Geometric Arithmetic October 31, 2012
Fculty of Mthemtics Wterloo, Ontrio N2L 3G1 Grde 7/8 Mth Circles Geometric Arithmetic Octoer 31, 2012 Centre for Eduction in Mthemtics nd Computing Ancient Greece hs given irth to some of the most importnt
More informationRanking of Hexagonal Fuzzy Numbers for Solving Multi Objective Fuzzy Linear Programming Problem
Interntionl Journl of Computer pplictions 097 8887 Volume 8 No 8 Decemer 0 nking of egonl Fuzzy Numers Solving ulti Ojective Fuzzy Liner Progrmming Prolem jrjeswri. P Deprtment of themtics Chikknn Government
More informationF. R. K. Chung y. University ofpennsylvania. Philadelphia, Pennsylvania R. L. Graham. AT&T Labs - Research. March 2,1997.
Forced convex n-gons in the plne F. R. K. Chung y University ofpennsylvni Phildelphi, Pennsylvni 19104 R. L. Grhm AT&T Ls - Reserch Murry Hill, New Jersey 07974 Mrch 2,1997 Astrct In seminl pper from 1935,
More informationCS412/413. Introduction to Compilers Tim Teitelbaum. Lecture 4: Lexical Analyzers 28 Jan 08
CS412/413 Introduction to Compilers Tim Teitelum Lecture 4: Lexicl Anlyzers 28 Jn 08 Outline DFA stte minimiztion Lexicl nlyzers Automting lexicl nlysis Jlex lexicl nlyzer genertor CS 412/413 Spring 2008
More informationRay surface intersections
Ry surfce intersections Some primitives Finite primitives: polygons spheres, cylinders, cones prts of generl qudrics Infinite primitives: plnes infinite cylinders nd cones generl qudrics A finite primitive
More informationFig.1. Let a source of monochromatic light be incident on a slit of finite width a, as shown in Fig. 1.
Answer on Question #5692, Physics, Optics Stte slient fetures of single slit Frunhofer diffrction pttern. The slit is verticl nd illuminted by point source. Also, obtin n expression for intensity distribution
More informationΕΠΛ323 - Θεωρία και Πρακτική Μεταγλωττιστών
ΕΠΛ323 - Θωρία και Πρακτική Μταγλωττιστών Lecture 3 Lexicl Anlysis Elis Athnsopoulos elisthn@cs.ucy.c.cy Recognition of Tokens if expressions nd reltionl opertors if è if then è then else è else relop
More informationA Fast Imaging Algorithm for Near Field SAR
Journl of Computing nd Electronic Informtion Mngement ISSN: 2413-1660 A Fst Imging Algorithm for Ner Field SAR Guoping Chen, Lin Zhng, * College of Optoelectronic Engineering, Chongqing University of Posts
More informationCS201 Discussion 10 DRAWTREE + TRIES
CS201 Discussion 10 DRAWTREE + TRIES DrwTree First instinct: recursion As very generic structure, we could tckle this problem s follows: drw(): Find the root drw(root) drw(root): Write the line for the
More informationEfficient Rerouting Algorithms for Congestion Mitigation
2009 IEEE Computer Society Annul Symposium on VLSI Efficient Rerouting Algorithms for Congestion Mitigtion M. A. R. Chudhry*, Z. Asd, A. Sprintson, nd J. Hu Deprtment of Electricl nd Computer Engineering
More informationAI Adjacent Fields. This slide deck courtesy of Dan Klein at UC Berkeley
AI Adjcent Fields Philosophy: Logic, methods of resoning Mind s physicl system Foundtions of lerning, lnguge, rtionlity Mthemtics Forml representtion nd proof Algorithms, computtion, (un)decidility, (in)trctility
More informationA Heuristic Approach for Discovering Reference Models by Mining Process Model Variants
A Heuristic Approch for Discovering Reference Models by Mining Process Model Vrints Chen Li 1, Mnfred Reichert 2, nd Andres Wombcher 3 1 Informtion System Group, University of Twente, The Netherlnds lic@cs.utwente.nl
More informationIMAGE QUALITY OPTIMIZATION BASED ON WAVELET FILTER DESIGN AND WAVELET DECOMPOSITION IN JPEG2000. Do Quan and Yo-Sung Ho
IMAGE QUALITY OPTIMIZATIO BASED O WAVELET FILTER DESIG AD WAVELET DECOMPOSITIO I JPEG2000 Do Qun nd Yo-Sung Ho School of Informtion & Mechtronics Gwngju Institute of Science nd Technology (GIST) 26 Cheomdn-gwgiro
More informationOrthogonal line segment intersection
Computtionl Geometry [csci 3250] Line segment intersection The prolem (wht) Computtionl Geometry [csci 3250] Orthogonl line segment intersection Applictions (why) Algorithms (how) A specil cse: Orthogonl
More informationObject and image indexing based on region connection calculus and oriented matroid theory
Discrete Applied Mthemtics 147 (2005) 345 361 www.elsevier.com/locte/dm Oject nd imge indexing sed on region connection clculus nd oriented mtroid theory Ernesto Stffetti, Antoni Gru, Frncesc Serrtos c,
More informationInternational Journal of Scientific & Engineering Research, Volume 4, Issue 12, December ISSN
Interntionl Journl of Scientific & Engineering Reserch, Volume 4, Issue 1, December-1 ISSN 9-18 Generlised Gussin Qudrture over Sphere K. T. Shivrm Abstrct This pper presents Generlised Gussin qudrture
More informationOUTPUT DELIVERY SYSTEM
Differences in ODS formtting for HTML with Proc Print nd Proc Report Lur L. M. Thornton, USDA-ARS, Animl Improvement Progrms Lortory, Beltsville, MD ABSTRACT While Proc Print is terrific tool for dt checking
More informationUNIT 11. Query Optimization
UNIT Query Optimiztion Contents Introduction to Query Optimiztion 2 The Optimiztion Process: An Overview 3 Optimiztion in System R 4 Optimiztion in INGRES 5 Implementing the Join Opertors Wei-Png Yng,
More informationCS-184: Computer Graphics. Today. Clipping. Hidden Surface Removal. Tuesday, October 7, Clipping to view volume Clipping arbitrary polygons
CS184: Computer Grphics Lecture #10: Clipping nd Hidden Surfces Prof. Jmes O Brien University of Cliforni, Berkeley V2008S101.0 1 Tody Clipping Clipping to view volume Clipping ritrry polygons Hidden Surfce
More informationarxiv: v1 [cs.gr] 19 Jan 2019
Automtic norml orienttion in point clouds of uilding interiors Sestin Ochmnn 1 nd Reinhrd Klein 1 1 Institute of Computer Science II, University of Bonn, Germny rxiv:1901.06487v1 [cs.gr] 19 Jn 2019 Astrct
More informationComputing offsets of freeform curves using quadratic trigonometric splines
Computing offsets of freeform curves using qudrtic trigonometric splines JIULONG GU, JAE-DEUK YUN, YOONG-HO JUNG*, TAE-GYEONG KIM,JEONG-WOON LEE, BONG-JUN KIM School of Mechnicl Engineering Pusn Ntionl
More informationAgenda & Reading. Class Exercise. COMPSCI 105 SS 2012 Principles of Computer Science. Arrays
COMPSCI 5 SS Principles of Computer Science Arrys & Multidimensionl Arrys Agend & Reding Agend Arrys Creting & Using Primitive & Reference Types Assignments & Equlity Pss y Vlue & Pss y Reference Copying
More informationAn Efficient Divide and Conquer Algorithm for Exact Hazard Free Logic Minimization
An Efficient Divide nd Conquer Algorithm for Exct Hzrd Free Logic Minimiztion J.W.J.M. Rutten, M.R.C.M. Berkelr, C.A.J. vn Eijk, M.A.J. Kolsteren Eindhoven University of Technology Informtion nd Communiction
More informationCHAPTER III IMAGE DEWARPING (CALIBRATION) PROCEDURE
CHAPTER III IMAGE DEWARPING (CALIBRATION) PROCEDURE 3.1 Scheimpflug Configurtion nd Perspective Distortion Scheimpflug criterion were found out to be the best lyout configurtion for Stereoscopic PIV, becuse
More informationMultiresolution Interactive Modeling with Efficient Visualization
Multiresolution Interctive Modeling with Efficient Visuliztion Jen-Dniel Deschênes Ptrick Héert Philippe Lmert Jen-Nicols Ouellet Drgn Tuic Computer Vision nd Systems Lortory heert@gel.ulvl.c, Lvl University,
More informationAlignment of Long Sequences. BMI/CS Spring 2012 Colin Dewey
Alignment of Long Sequences BMI/CS 776 www.biostt.wisc.edu/bmi776/ Spring 2012 Colin Dewey cdewey@biostt.wisc.edu Gols for Lecture the key concepts to understnd re the following how lrge-scle lignment
More informationQubit allocation for quantum circuit compilers
Quit lloction for quntum circuit compilers Nov. 10, 2017 JIQ 2017 Mrcos Yukio Sirichi Sylvin Collnge Vinícius Fernndes dos Sntos Fernndo Mgno Quintão Pereir Compilers for quntum computing The first genertion
More informationRethinking Virtual Network Embedding: Substrate Support for Path Splitting and Migration
Rethinking Virtul Network Emedding: Sustrte Support for Pth Splitting nd Migrtion Minln Yu, Yung Yi, Jennifer Rexford, Mung Ching Computer Science, Princeton University, Emil: {minlnyu,jrex}@cs.princeton.edu
More informationMidterm 2 Sample solution
Nme: Instructions Midterm 2 Smple solution CMSC 430 Introduction to Compilers Fll 2012 November 28, 2012 This exm contins 9 pges, including this one. Mke sure you hve ll the pges. Write your nme on the
More informationUT1553B BCRT True Dual-port Memory Interface
UTMC APPICATION NOTE UT553B BCRT True Dul-port Memory Interfce INTRODUCTION The UTMC UT553B BCRT is monolithic CMOS integrted circuit tht provides comprehensive MI-STD- 553B Bus Controller nd Remote Terminl
More informationsuch that the S i cover S, or equivalently S
MATH 55 Triple Integrls Fll 16 1. Definition Given solid in spce, prtition of consists of finite set of solis = { 1,, n } such tht the i cover, or equivlently n i. Furthermore, for ech i, intersects i
More informationEvolutionary Approaches To Minimizing Network Coding Resources
This full text pper ws peer reviewed t the direction of IEEE Communictions Society suject mtter experts for puliction in the IEEE INFOCOM 2007 proceedings. Evolutionry Approches To Minimizing Network Coding
More information