SORTING IN SPACE HANAN SAMET

Transcription

1 SORTING IN SPACE HANAN SAMET COMPUTER SCIENCE DEPARTMENT AND CENTER FOR AUTOMATION RESEARCH AND INSTITUTE FOR ADVANCED COMPUTER STUDIES UNIVERSITY OF MARYLAND COLLEGE PARK, MARYLAND USA e mail: hjs@cs.umd.edu ul: These notes may not e epoduced y any means (mechanical o electonic o any othe) without the expess witten pemission of Hanan Samet Copyight 008 Hanan Samet

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3 Why Soting of Spatial Data is Impotant Most opeations invaialy involve seach Seach is sped up y soting the data sot - Definition: ve. to put in a cetain place o ank accoding to kind, class, o natue. to aange accoding to chaacteistics Examples. Wanock algoithm: soting ojects fo display vecto: hidden-line elimination aste: hidden-suface elimination. Back-to-font and font-to-ack algoithms. BSP tees fo visiility detemination 4. Acceleating ay tacing and ay casting y finding ay-oject intesections 5. Bounding ox hieachies aange space accoding to whethe occupied o unoccupied Copyight 008 y Hanan Samet

4 Soting Implies the Existence of an Odeing. Fine fo one-dimensional data sot people y weight and find closest in weight to Bill and can also find closest in weight to Lay sot cities y distance fom Chicago c and find closest to Chicago ut cannot find closest to New Yok unless a esot. Had fo two-dimensions as highe as notion of odeing does not exist unless a dominance elation holds point a = {a i i d} dominates point = { i i d} if a i i, i d.. Only solution is to lineaize data as in a space-filling cuve sot is explicit need implicit sot so no need to esot if efeence point changes Copyight 008 y Hanan Samet

5 Soting Implies the Existence of an Odeing. Fine fo one-dimensional data sot people y weight and find closest in weight to Bill and can also find closest in weight to Lay sot cities y distance fom Chicago c and find closest to Chicago ut cannot find closest to New Yok unless a esot. Had fo two-dimensions as highe as notion of odeing does not exist unless a dominance elation holds point a = {a i i d} dominates point = { i i d} if a i i, i d. a. Only solution is to lineaize data as in a space-filling cuve sot is explicit need implicit sot so no need to esot if efeence point changes Copyight 008 y Hanan Samet

6 Soting Implies the Existence of an Odeing. Fine fo one-dimensional data sot people y weight and find closest in weight to Bill and can also find closest in weight to Lay sot cities y distance fom Chicago c and find closest to Chicago ut cannot find closest to New Yok unless a esot. Had fo two-dimensions as highe as notion of odeing does not exist unless a dominance elation holds point a = {a i i d} dominates point = { i i d} if a i i, i d. a does not dominate. Only solution is to lineaize data as in a space-filling cuve sot is explicit need implicit sot so no need to esot if efeence point changes Copyight 008 y Hanan Samet

7 Soting Implies the Existence of an Odeing. Fine fo one-dimensional data sot people y weight and find closest in weight to Bill and can also find closest in weight to Lay sot cities y distance fom Chicago c and find closest to Chicago ut cannot find closest to New Yok unless a esot. Had fo two-dimensions as highe as notion of odeing does not exist unless a dominance elation holds point a = {a i i d} dominates point = { i i d} if a i i, i d. a does not dominate ut dominates c. Only solution is to lineaize data as in a space-filling cuve sot is explicit need implicit sot so no need to esot if efeence point changes Copyight 008 y Hanan Samet

8 PRINCE GEORGES COUNTY hi8 Copyight 008 y Hanan Samet

9 hi7 EXAMPLE QUERIES ON LINE SEGMENT DATABASES Queies aout line segments. All segments that intesect a given point o set of points. All segments that have a given set of endpoints. All segments that intesect a given line segment 4. All segments that ae coincident with a given line segment Poximity queies. The neaest line segment to a given point. All segments within a given distance fom a given point (also known as a ange o window quey) Queies involving attiutes of line segments. Given a point, find the closest line segment of a paticula type. Given a point, find the minimum enclosing polygon whose constituent line segments ae all of a given type. Given a point, find all the polygons that ae incident on it Copyight 008 y Hanan Samet

10 gs0 WHAT MAKES CONTINUOUS SPATIAL DATA DIFFERENT. Spatial extent of the ojects is the key to the diffeence. A ecod in a DBMS may e consideed as a point in a multidimensional space a line can e tansfomed (i.e., epesented) as a point in 4-d space with (x, y, x, y ) (x, y ) (x, y ) good fo queies aout the line segments not good fo poximity queies since points outside the oject ae not mapped into the highe dimensional space epesentative points of two ojects that ae physically close to each othe in the oiginal space (e.g., -d fo lines) may e vey fa fom each othe in the highe dimensional space (e.g., 4-d) Ex: polem is that the tansfomation only tansfoms the space occupied y the ojects and not the est of the space (e.g., the quey point) can ovecome y pojecting ack to oiginal space. Use an index that sots ased upon spatial occupancy (i.e., extent of the ojects) B A Copyight 008 y Hanan Samet 4

11 hi9. SPATIAL INDEXING REQUIREMENTS. Compatiility with the data eing stoed. Choose an appopiate zeo o efeence point. Need an implicit athe than an explicit index impossile to foesee all possile queies in advance cannot have an attiute fo evey possile spatial elationship a. deive adjacency elations. -d stings captue a suset of adjacencies all ows all columns implicit index is ette as an explicit index which, fo example, sots two-dimensional data on the asis of distance fom a given point is impactical as it is inapplicale to othe points implicit means that don't have to esot the data fo queies othe than updates Copyight 008 y Hanan Samet 5

12 gs SORTING ON THE BASIS OF SPATIAL OCCUPANCY Decompose the space fom which the data is dawn into egions called uckets (like hashing ut peseves ode) Inteested in methods that ae designed specifically fo the spatial data type eing stoed Basic appoaches to decomposing space. minimum ounding ectangles e.g., R-tee good at distinguishing empty and non-empty space dawacks: a. non-disjoint decomposition of space may need to seach entie space. inaility to coelate occupied and unoccupied space in two maps. disjoint cells dawack: ojects may e epoted moe than once unifom gid a. all cells the same size. dawack: possiility of many spase cells adaptive gid quadtee vaiants a. egula decomposition. all cells of width powe of patitions at aitay positions a. dawack: not a egula decomposition. e.g., R + -tee Can use as appoximations in filte/efine quey pocessing stategy Copyight 008 y Hanan Samet 6

13 MINIMUM BOUNDING RECTANGLES hi Ojects gouped into hieachies, stoed in a stuctue simila to a B-tee Dawack: not a disjoint decomposition of space Oject has single ounding ectangle, yet aea that it spans may e included in seveal ounding ectangles Examples include the R-tee and the R*-tee Ode (m,m ) R-tee. etween m M/ and M enties in each node except oot. at least enties in oot unless a leaf node a g h i e d f c Copyight 008 y Hanan Samet 7

14 MINIMUM BOUNDING RECTANGLES hi Ojects gouped into hieachies, stoed in a stuctue simila to a B-tee Dawack: not a disjoint decomposition of space Oject has single ounding ectangle, yet aea that it spans may e included in seveal ounding ectangles Examples include the R-tee and the R*-tee Ode (m,m ) R-tee. etween m M/ and M enties in each node except oot. at least enties in oot unless a leaf node R R4 g h R5 a i R6 e d f c R: a R4: d g h R5: c i R6: e f Copyight 008 y Hanan Samet 7

15 MINIMUM BOUNDING RECTANGLES hi Ojects gouped into hieachies, stoed in a stuctue simila to a B-tee Dawack: not a disjoint decomposition of space Oject has single ounding ectangle, yet aea that it spans may e included in seveal ounding ectangles Examples include the R-tee and the R*-tee Ode (m,m ) R-tee. etween m M/ and M enties in each node except oot. at least enties in oot unless a leaf node z R R R4 g h R5 R a i R6 e d f c R: R R4 R: R5 R6 R: a R4: d g h R5: c i R6: e f Copyight 008 y Hanan Samet 7

16 MINIMUM BOUNDING RECTANGLES hi Ojects gouped into hieachies, stoed in a stuctue simila to a B-tee Dawack: not a disjoint decomposition of space Oject has single ounding ectangle, yet aea that it spans may e included in seveal ounding ectangles Examples include the R-tee and the R*-tee Ode (m,m ) R-tee. etween m M/ and M enties in each node except oot. at least enties in oot unless a leaf node 4 g z R R R4 g h R5 R a i R6 e d f R0 c R0: R R R: R R4 R: R5 R6 R: a R4: d g h R5: c i R6: e f Copyight 008 y Hanan Samet 7

17 SEARCHING FOR A POINT OR LINE SEGMENT IN AN R-TREE hi Dawack is that may have to examine many nodes since a line segment can e contained in the coveing ectangles of many nodes yet its ecod is contained in only one leaf node (e.g., i in R, R, R4, and R5) Ex: Seach fo a line segment containing point Q R0: R R R: R R4 R: R5 R6 R: a R4: d g h R5: c i R6: e f R R h a g i R6 e R4 d Q f R0 R R5 c Copyight 008 y Hanan Samet 8

18 SEARCHING FOR A POINT OR LINE SEGMENT IN AN R-TREE v hi Dawack is that may have to examine many nodes since a line segment can e contained in the coveing ectangles of many nodes yet its ecod is contained in only one leaf node (e.g., i in R, R, R4, and R5) Ex: Seach fo a line segment containing point Q R0: R R R: R R4 R: R5 R6 R: a R4: d g h R5: c i R6: e f R R h a g i R6 e R4 d Q f R0 R R5 c Q is in R0 Copyight 008 y Hanan Samet 8

19 SEARCHING FOR A POINT OR LINE SEGMENT IN AN R-TREE hi Dawack is that may have to examine many nodes since a line segment can e contained in the coveing ectangles of many nodes yet its ecod is contained in only one leaf node (e.g., i in R, R, R4, and R5) Ex: Seach fo a line segment containing point Q v R0: R R R: R R4 R: R5 R6 R: a R4: d g h R5: c i R6: e f R R h a g i R6 e R4 d Q f R0 R R5 c Q is in R0 Q can e in oth R and R Copyight 008 y Hanan Samet 8

20 SEARCHING FOR A POINT OR LINE SEGMENT IN AN R-TREE hi Dawack is that may have to examine many nodes since a line segment can e contained in the coveing ectangles of many nodes yet its ecod is contained in only one leaf node (e.g., i in R, R, R4, and R5) Ex: Seach fo a line segment containing point Q 4 z v R0: R R R: R R4 R: R5 R6 R: a R4: d g h R5: c i R6: e f R R h a g i R6 e R4 d Q f R0 R R5 c Q is in R0 Q can e in oth R and R Seaching R fist means that R4 is seached ut this leads to failue even though Q is pat of i which is in R4 Copyight 008 y Hanan Samet 8

21 SEARCHING FOR A POINT OR LINE SEGMENT IN AN R-TREE hi Dawack is that may have to examine many nodes since a line segment can e contained in the coveing ectangles of many nodes yet its ecod is contained in only one leaf node (e.g., i in R, R, R4, and R5) Ex: Seach fo a line segment containing point Q 5 g 4 z v R0: R R R: R R4 R: R5 R6 R: a R4: d g h R5: c i R6: e f R R h a g i R6 e R4 d Q f R0 R R5 c Q is in R0 Q can e in oth R and R Seaching R fist means that R4 is seached ut this leads to failue even though Q is pat of i which is in R4 Seaching R finds that Q can only e in R5 Copyight 008 y Hanan Samet 8

22 DISJOINT CELLS hi Ojects decomposed into disjoint suojects; each suoject in diffeent cell Techniques diffe in degee of egulaity Dawack: in ode to detemine aea coveed y oject, must etieve all cells that it occupies R+-tee (also k-d-b-tee) and cell tee ae examples of this technique a g h d Q i e f c Copyight 008 y Hanan Samet 9

23 DISJOINT CELLS hi Ojects decomposed into disjoint suojects; each suoject in diffeent cell Techniques diffe in degee of egulaity Dawack: in ode to detemine aea coveed y oject, must etieve all cells that it occupies R+-tee (also k-d-b-tee) and cell tee ae examples of this technique R6 R g h R4 R5 a i e d Q f c R: d g h R4: c h i R5: c f i R6: a e i Copyight 008 y Hanan Samet 9

24 DISJOINT CELLS z hi Ojects decomposed into disjoint suojects; each suoject in diffeent cell Techniques diffe in degee of egulaity Dawack: in ode to detemine aea coveed y oject, must etieve all cells that it occupies R+-tee (also k-d-b-tee) and cell tee ae examples of this technique R6 R R R g h R4 R5 a i e d Q f c R: R R4 R: R5 R6 R: d g h R4: c h i R5: c f i R6: a e i Copyight 008 y Hanan Samet 9

25 DISJOINT CELLS 4 g z hi Ojects decomposed into disjoint suojects; each suoject in diffeent cell Techniques diffe in degee of egulaity Dawack: in ode to detemine aea coveed y oject, must etieve all cells that it occupies R+-tee (also k-d-b-tee) and cell tee ae examples of this technique R0 R6 R R R g h R4 R5 a i e d Q f c R0: R R R: R R4 R: R5 R6 R: d g h R4: c h i R5: c f i R6: a e i Copyight 008 y Hanan Samet 9

26 K-D-B-TREES hi. Rectangula emedding space is hieachically decomposed into disjoint ectangula egions No dead space in the sense that at any level of the tee, entie emedding space is coveed y one of the nodes Blocks of k-d tee patition of space ae aggegated into nodes of a finite capacity When a node oveflows, it is split along one of the axes Oiginally developed to stoe points ut may e extended to non-point ojects epesented y thei minimum ounding oxes Dawack: in ode to detemine aea coveed y oject, must etieve all cells that it occupies a g h d Q i e f c Copyight 008 y Hanan Samet

27 K-D-B-TREES hi. Rectangula emedding space is hieachically decomposed into disjoint ectangula egions No dead space in the sense that at any level of the tee, entie emedding space is coveed y one of the nodes Blocks of k-d tee patition of space ae aggegated into nodes of a finite capacity When a node oveflows, it is split along one of the axes Oiginally developed to stoe points ut may e extended to non-point ojects epesented y thei minimum ounding oxes Dawack: in ode to detemine aea coveed y oject, must etieve all cells that it occupies R R4 R6 a g h d Q R5 i e f c R: d g h R4: c h i R5: c f i R6: Copyight 008 y Hanan Samet a e i

28 K-D-B-TREES hi. Rectangula emedding space is hieachically decomposed into disjoint ectangula egions No dead space in the sense that at any level of the tee, entie emedding space is coveed y one of the nodes Blocks of k-d tee patition of space ae aggegated into nodes of a finite capacity When a node oveflows, it is split along one of the axes Oiginally developed to stoe points ut may e extended to non-point ojects epesented y thei minimum ounding oxes Dawack: in ode to detemine aea coveed y oject, must etieve all cells that it occupies z R R4 R6 R R a g h d Q R5 i e f c R: R R4 R: R5 R6 R: d g h R4: c h i R5: c f i R6: Copyight 008 y Hanan Samet a e i

29 K-D-B-TREES hi. Rectangula emedding space is hieachically decomposed into disjoint ectangula egions No dead space in the sense that at any level of the tee, entie emedding space is coveed y one of the nodes Blocks of k-d tee patition of space ae aggegated into nodes of a finite capacity When a node oveflows, it is split along one of the axes Oiginally developed to stoe points ut may e extended to non-point ojects epesented y thei minimum ounding oxes Dawack: in ode to detemine aea coveed y oject, must etieve all cells that it occupies 4 g z R R4 R6 R R R0 a g h d Q R5 i e f c R0: R R R: R R4 R: R5 R6 R: d g h R4: c h i R5: c f i R6: Copyight 008 y Hanan Samet a e i

30 hi4 UNIFORM GRID Ideal fo unifomly distiuted data Suppots set-theoetic opeations Spatial data (e.g., line segment data) is aely unifomly distiuted Copyight 008 y Hanan Samet 0

31 QUADTREES hi5 Hieachical vaiale esolution data stuctue ased on egula decomposition Many diffeent decomposition schemes and applicale to diffeent data types:. points. lines. egions 4. ectangles 5. sufaces 6. volumes 7. highe dimensions including time changes meaning of neaest a. neaest in time, OR. neaest in distance Can handle oth aste and vecto data as just a spatial index Shape is usually independent of ode of inseting data Ex: egion quadtee A decomposition into locks not necessaily a tee! Copyight 008 y Hanan Samet

32 hi6 REGION QUADTREE Repeatedly sudivide until otain homogeneous egion Fo a inay image (BLACK and WHITE 0) Can also use fo multicoloed data (e.g., a landuse class map associating colos with cops) Can also define data stuctue fo gayscale images A collection of maximal locks of size powe of two and placed at pedetemined positions. could implement as a list of locks each of which has a unique pai of numes: concatenate sequence of it codes coesponding to the path fom the oot to the lock s node the level of the lock s node. does not have to e implemented as a tee tee good fo logaithmic access A vaiale esolution data stuctue in contast to a pyamid (i.e., a complete quadtee) which is a multiesolution data stuctue B F H G I J N O L M Q NW NE A SE SW B C D E K P F G H I J L M N O Q Copyight 008 y Hanan Samet

33 SPACE REQUIREMENTS. Rationale fo using quadtees/octees is not so much fo saving space ut fo saving execution time g7. Execution time of standad image pocessing algoithms that ae ased on tavesing the entie image and pefoming a computation at each image element is popotional to the nume of locks in the decomposition of the image athe than thei size aggegation of space leads diectly to execution time savings as the aggegate (i.e., lock) is visited just once instead of once fo each image element (i.e., pixel, voxel) in the aggegate (e.g., connected component laeling). If want to save space, then, in geneal, statistical image compession methods ae supeio dawack: statistical methods ae not pogessive as need to tansmit the entie image wheeas quadtees lend themselves to pogessive appoximation quadtees, though, do achieve compession as a esult of use of common suexpession elimination techniques a. e.g., checkeoad image. see also vecto quantization 4. Sensitive to positioning of the oigin of the decomposition fo an n x n image, the optimal positioning equies an O(n log n) dynamic pogamming algoithm (Li, Gosky, and Jain) Copyight 008 y Hanan Samet

34 DIMENSION REDUCTION g8. Nume of locks necessay to stoe a simple polygon as a egion quadtee is popotional to its peimete (Hunte) implies that many quadtee algoithms execute in O(peimete) time as they ae tee tavesals the egion quadtee is a dimension educing device as peimete (ignoing factal effects) is a onedimensional measue and we ae stating with twodimensional data genealizes to highe dimensions a. egion octee takes O (suface aea) time and space (Meaghe). d-dimensional data take time and space popotional to a O (d-)-dimensional quantity (Walsh). Altenatively, fo a egion quadtee, the space equiements doule as the esolution doules in contast with quadupling in the aay epesentation fo a egion octee the space equiements quaduple as the esolution doules ex. aay egion quadtee Copyight 008 y Hanan Samet

35 DIMENSION REDUCTION g8. Nume of locks necessay to stoe a simple polygon as a egion quadtee is popotional to its peimete (Hunte) implies that many quadtee algoithms execute in O(peimete) time as they ae tee tavesals the egion quadtee is a dimension educing device as peimete (ignoing factal effects) is a onedimensional measue and we ae stating with twodimensional data genealizes to highe dimensions a. egion octee takes O (suface aea) time and space (Meaghe). d-dimensional data take time and space popotional to a O (d-)-dimensional quantity (Walsh). Altenatively, fo a egion quadtee, the space equiements doule as the esolution doules in contast with quadupling in the aay epesentation fo a egion octee the space equiements quaduple as the esolution doules ex. aay egion quadtee Copyight 008 y Hanan Samet

36 DIMENSION REDUCTION g8. Nume of locks necessay to stoe a simple polygon as a egion quadtee is popotional to its peimete (Hunte) implies that many quadtee algoithms execute in O(peimete) time as they ae tee tavesals the egion quadtee is a dimension educing device as peimete (ignoing factal effects) is a onedimensional measue and we ae stating with twodimensional data genealizes to highe dimensions a. egion octee takes O (suface aea) time and space (Meaghe). d-dimensional data take time and space popotional to a O (d-)-dimensional quantity (Walsh). Altenatively, fo a egion quadtee, the space equiements doule as the esolution doules in contast with quadupling in the aay epesentation fo a egion octee the space equiements quaduple as the esolution doules ex. aay z egion quadtee Copyight 008 y Hanan Samet

37 DIMENSION REDUCTION g8. Nume of locks necessay to stoe a simple polygon as a egion quadtee is popotional to its peimete (Hunte) implies that many quadtee algoithms execute in O(peimete) time as they ae tee tavesals the egion quadtee is a dimension educing device as peimete (ignoing factal effects) is a onedimensional measue and we ae stating with twodimensional data genealizes to highe dimensions a. egion octee takes O (suface aea) time and space (Meaghe). d-dimensional data take time and space popotional to a O (d-)-dimensional quantity (Walsh). Altenatively, fo a egion quadtee, the space equiements doule as the esolution doules in contast with quadupling in the aay epesentation fo a egion octee the space equiements quaduple as the esolution doules ex. aay 4 g z egion quadtee Copyight 008 y Hanan Samet

38 DIMENSION REDUCTION g8. Nume of locks necessay to stoe a simple polygon as a egion quadtee is popotional to its peimete (Hunte) implies that many quadtee algoithms execute in O(peimete) time as they ae tee tavesals the egion quadtee is a dimension educing device as peimete (ignoing factal effects) is a onedimensional measue and we ae stating with twodimensional data genealizes to highe dimensions a. egion octee takes O (suface aea) time and space (Meaghe). d-dimensional data take time and space popotional to a O (d-)-dimensional quantity (Walsh). Altenatively, fo a egion quadtee, the space equiements doule as the esolution doules in contast with quadupling in the aay epesentation fo a egion octee the space equiements quaduple as the esolution doules ex. aay 5 4 g z egion quadtee easy to see dependence on peimete as decomposition only takes place on the ounday as the esolution inceases Copyight 008 y Hanan Samet

39 hi7 PYRAMID Intenal nodes contain summay of infomation in nodes elow them Useful fo avoiding inspecting nodes whee thee could e no elevant infomation c c c c4 c5 c6 {c,c,c,c4,c5,c6} {c6} {c,c,c6} {c,c,c4,c5} {c,c,c, c4,c5,c6} Copyight 008 y Hanan Samet

40 hi8 QUADTREES VS. PYRAMIDS Quadtees ae good fo location-ased queies. e.g., what is at location x?. not good if looking fo a paticula featue as have to examine evey lock o location asking ae you the one I am looking fo? Pyamid is good fo featue-ased queies e.g.,. does wheat exist in egion x? if wheat does not appea at the oot node, then impossile to find it in the est of the stuctue and the seach can cease. epot all cops in egion x just look at the oot. select all locations whee wheat is gown only descend node if thee is possiility that wheat is in one of its fou sons implies little wasted wok Ex: tuncated pyamid whee 4 identically-coloed sons ae meged {c,c,c,c4,c5,c6} {c6} {c,c,c6} {c,c,c4,c5} {c,c,c, c4,c5,c6} c c c c4 c5 c6 {c,c,c5} {c,c,c,c5} Can epesent as a list of leaf and nonleaf locks (e.g., as a linea quadtee) Copyight 008 y Hanan Samet 4

41 PR QUADTREE (Oenstein) Regula decomposition point epesentation hp9 Decomposition occus wheneve a lock contains moe than one point Useful when the domain of data points is not discete ut finite Maximum level of decomposition depends on the minimum sepaation etween two points if two points ae vey close, then decomposition can e vey deep can e ovecome y viewing locks as uckets with capacity c and only decomposing the lock when it contains moe than c points Ex: c = (0,00) (00,00) (5,4) Chicago (0,0) (00,0) Copyight 008 y Hanan Samet 5

42 PR QUADTREE (Oenstein) Regula decomposition point epesentation hp9 Decomposition occus wheneve a lock contains moe than one point Useful when the domain of data points is not discete ut finite Maximum level of decomposition depends on the minimum sepaation etween two points if two points ae vey close, then decomposition can e vey deep can e ovecome y viewing locks as uckets with capacity c and only decomposing the lock when it contains moe than c points Ex: c = (0,00) (00,00) (5,4) Chicago (0,0) (5,0) Moile (00,0) Copyight 008 y Hanan Samet 5

43 PR QUADTREE (Oenstein) Regula decomposition point epesentation hp9 Decomposition occus wheneve a lock contains moe than one point Useful when the domain of data points is not discete ut finite Maximum level of decomposition depends on the minimum sepaation etween two points if two points ae vey close, then decomposition can e vey deep can e ovecome y viewing locks as uckets with capacity c and only decomposing the lock when it contains moe than c points Ex: c = (0,00) (00,00) z (6,77) Toonto (5,4) Chicago (0,0) (5,0) Moile (00,0) Copyight 008 y Hanan Samet 5

44 PR QUADTREE (Oenstein) Regula decomposition point epesentation hp9 Decomposition occus wheneve a lock contains moe than one point Useful when the domain of data points is not discete ut finite Maximum level of decomposition depends on the minimum sepaation etween two points if two points ae vey close, then decomposition can e vey deep can e ovecome y viewing locks as uckets with capacity c and only decomposing the lock when it contains moe than c points Ex: c = (0,00) (00,00) 4 g z (6,77) Toonto (5,4) Chicago (8,65) Buffalo (0,0) (5,0) Moile (00,0) Copyight 008 y Hanan Samet 5

45 PR QUADTREE (Oenstein) Regula decomposition point epesentation hp9 Decomposition occus wheneve a lock contains moe than one point Useful when the domain of data points is not discete ut finite Maximum level of decomposition depends on the minimum sepaation etween two points if two points ae vey close, then decomposition can e vey deep can e ovecome y viewing locks as uckets with capacity c and only decomposing the lock when it contains moe than c points Ex: c = (0,00) (00,00) 5 v 4 g z (6,77) Toonto (8,65) Buffalo (5,45) Denve (5,4) Chicago (0,0) (5,0) Moile (00,0) Copyight 008 y Hanan Samet 5

46 PR QUADTREE (Oenstein) Regula decomposition point epesentation hp9 Decomposition occus wheneve a lock contains moe than one point Useful when the domain of data points is not discete ut finite Maximum level of decomposition depends on the minimum sepaation etween two points if two points ae vey close, then decomposition can e vey deep can e ovecome y viewing locks as uckets with capacity c and only decomposing the lock when it contains moe than c points Ex: c = (0,00) (00,00) 6 g 5 v 4 g z (6,77) Toonto (8,65) Buffalo (5,45) Denve (5,4) Chicago (7,5) Omaha (0,0) (5,0) Moile (00,0) Copyight 008 y Hanan Samet 5

47 PR QUADTREE (Oenstein) Regula decomposition point epesentation hp9 Decomposition occus wheneve a lock contains moe than one point Useful when the domain of data points is not discete ut finite Maximum level of decomposition depends on the minimum sepaation etween two points if two points ae vey close, then decomposition can e vey deep can e ovecome y viewing locks as uckets with capacity c and only decomposing the lock when it contains moe than c points Ex: c = (0,00) (00,00) 7 z 6 g 5 v 4 g z (6,77) Toonto (8,65) Buffalo (5,45) Denve (5,4) Chicago (7,5) Omaha (85,5) Atlanta (0,0) (5,0) Moile (00,0) Copyight 008 y Hanan Samet 5

48 PR QUADTREE (Oenstein) Regula decomposition point epesentation hp9 Decomposition occus wheneve a lock contains moe than one point Useful when the domain of data points is not discete ut finite Maximum level of decomposition depends on the minimum sepaation etween two points if two points ae vey close, then decomposition can e vey deep can e ovecome y viewing locks as uckets with capacity c and only decomposing the lock when it contains moe than c points Ex: c = (0,00) (00,00) 8 7 z 6 g 5 v 4 g z (6,77) Toonto (8,65) Buffalo (5,45) Denve (5,4) Chicago (7,5) Omaha (85,5) Atlanta (0,0) (5,0) Moile (90,5) Miami (00,0) Copyight 008 y Hanan Samet 5

49 PR quadtee REGION SEARCH Ex: Find all points within adius of point A hp0 A Use of quadtee esults in puning the seach space Copyight 008 y Hanan Samet

50 PR quadtee REGION SEARCH Ex: Find all points within adius of point A hp A Use of quadtee esults in puning the seach space If a quadant sudivision point p lies in a egion l, then seach the quadants of p specified y l. SE 6. NE. All ut SW. SE, SW 7. NE, NW. All ut SE. SW 8. NW. All 4. SE, NE 9. All ut NW 5. SW, NW 0. All ut NE Copyight 008 y Hanan Samet

51 PR quadtee REGION SEARCH Ex: Find all points within adius of point A z hp p A Use of quadtee esults in puning the seach space If a quadant sudivision point p lies in a egion l, then seach the quadants of p specified y l. SE 6. NE. All ut SW. SE, SW 7. NE, NW. All ut SE. SW 8. NW. All 4. SE, NE 9. All ut NW 5. SW, NW 0. All ut NE Copyight 008 y Hanan Samet

52 PR quadtee REGION SEARCH Ex: Find all points within adius of point A 4 g z hp0 p A Use of quadtee esults in puning the seach space If a quadant sudivision point p lies in a egion l, then seach the quadants of p specified y l. SE 6. NE. All ut SW. SE, SW 7. NE, NW. All ut SE. SW 8. NW. All 4. SE, NE 9. All ut NW 5. SW, NW 0. All ut NE Copyight 008 y Hanan Samet

53 PR quadtee REGION SEARCH Ex: Find all points within adius of point A 5 v 4 g z hp0 9 0 p 5 4 A Use of quadtee esults in puning the seach space If a quadant sudivision point p lies in a egion l, then seach the quadants of p specified y l. SE 6. NE. All ut SW. SE, SW 7. NE, NW. All ut SE. SW 8. NW. All 4. SE, NE 9. All ut NW 5. SW, NW 0. All ut NE Copyight 008 y Hanan Samet

54 FINDING THE NEAREST OBJECT Ex: find the neaest oject to P E C hp 9 4 B 5 D P A 0 F Assume PR quadtee fo points (i.e., at most one point pe lock) Seach neighos of lock in counteclockwise ode Points ae soted with espect to the space they occupy which enales puning the seach space Algoithm: Copyight 008 y Hanan Samet 7

55 FINDING THE NEAREST OBJECT Ex: find the neaest oject to P E C hp 9 4 B 5 D P A 0 F Assume PR quadtee fo points (i.e., at most one point pe lock) Seach neighos of lock in counteclockwise ode Points ae soted with espect to the space they occupy which enales puning the seach space Algoithm:. stat at lock and compute distance to P fom A Copyight 008 y Hanan Samet 7

56 FINDING THE NEAREST OBJECT Ex: find the neaest oject to P E C z hp 9 4 B 5 D P A 0 F Assume PR quadtee fo points (i.e., at most one point pe lock) Seach neighos of lock in counteclockwise ode Points ae soted with espect to the space they occupy which enales puning the seach space Algoithm:. stat at lock and compute distance to P fom A. ignoe lock whethe o not it is empty as A is close to P than any point in Copyight 008 y Hanan Samet 7

57 FINDING THE NEAREST OBJECT Ex: find the neaest oject to P E C 4 g z hp 9 4 B 5 D P A 0 F Assume PR quadtee fo points (i.e., at most one point pe lock) Seach neighos of lock in counteclockwise ode Points ae soted with espect to the space they occupy which enales puning the seach space Algoithm:. stat at lock and compute distance to P fom A. ignoe lock whethe o not it is empty as A is close to P than any point in. examine lock 4 as distance to SW cone is shote than the distance fom P to A; howeve, eject B as it is futhe fom P than A Copyight 008 y Hanan Samet 7

58 FINDING THE NEAREST OBJECT Ex: find the neaest oject to P E C 5 v 4 g z hp 9 4 B 5 D P A 0 F Assume PR quadtee fo points (i.e., at most one point pe lock) Seach neighos of lock in counteclockwise ode Points ae soted with espect to the space they occupy which enales puning the seach space Algoithm:. stat at lock and compute distance to P fom A. ignoe lock whethe o not it is empty as A is close to P than any point in. examine lock 4 as distance to SW cone is shote than the distance fom P to A; howeve, eject B as it is futhe fom P than A 4. ignoe locks 6, 7, 8, 9, and 0 as the minimum distance to them fom P is geate than the distance fom P to A Copyight 008 y Hanan Samet 7

59 FINDING THE NEAREST OBJECT Ex: find the neaest oject to P E C 6 z 5 v 4 g z hp 9 4 B 5 D P A 0 F Assume PR quadtee fo points (i.e., at most one point pe lock) Seach neighos of lock in counteclockwise ode Points ae soted with espect to the space they occupy which enales puning the seach space Algoithm:. stat at lock and compute distance to P fom A. ignoe lock whethe o not it is empty as A is close to P than any point in. examine lock 4 as distance to SW cone is shote than the distance fom P to A; howeve, eject B as it is futhe fom P than A 4. ignoe locks 6, 7, 8, 9, and 0 as the minimum distance to them fom P is geate than the distance fom P to A 5. examine lock as the distance fom P to the southen ode of is shote than the distance fom P to A; howeve, eject F as it is futhe fom P than A Copyight 008 y Hanan Samet 7

60 FINDING THE NEAREST OBJECT 7 6 z 5 v 4 g z hp Ex: find the neaest oject to P E C 9 4 B 5 D P A 0 new F F Assume PR quadtee fo points (i.e., at most one point pe lock) Seach neighos of lock in counteclockwise ode Points ae soted with espect to the space they occupy which enales puning the seach space Algoithm:. stat at lock and compute distance to P fom A. ignoe lock whethe o not it is empty as A is close to P than any point in. examine lock 4 as distance to SW cone is shote than the distance fom P to A; howeve, eject B as it is futhe fom P than A 4. ignoe locks 6, 7, 8, 9, and 0 as the minimum distance to them fom P is geate than the distance fom P to A 5. examine lock as the distance fom P to the southen ode of is shote than the distance fom P to A; howeve, eject F as it is futhe fom P than A If F was moved, a ette ode would have stated with lock, the southen neigho of, as it is closest Copyight 008 y Hanan Samet 7

61 INCREMENTAL NEAREST NEIGHBORS (HJATASON/SAMET) Motivation hp. often don't know in advance how many neighos will need. e.g., want neaest city to Chicago with population > million Seveal appoaches. guess some aea ange aound Chicago and check populations of cities in ange if find a city with population > million, must make sue that thee ae no othe cities that ae close with population > million inefficient as have to guess size of aea to seach polem with guessing is we may choose too small a egion o too lage a egion a. if size too small, aea may not contain any cities with ight population and need to expand the seach egion. if size too lage, may e examining many cities needlessly. sot all the cities y distance fom Chicago impactical as we need to e-sot them each time pose a simila quey with espect to anothe city also soting is ovekill when only need fist few neighos. find k closest neighos and check population condition Copyight 008 y Hanan Samet

62 MECHANICS OF INCREMENTAL NEAREST NEIGHBOR ALGORITHM hpc Make use of a seach hieachy (e.g., tee) whee. ojects at lowest level. oject appoximations ae at next level (e.g., ounding oxes in an R-tee). nonleaf nodes in a tee-ased index Tavese seach hieachy in a est-fist manne simila to A*-algoithm instead of moe taditional depth-fist o eadth-fist mannes. at each step, visit element with smallest distance fom quey oject among all unvisited elements in the seach hieachy i.e., all unvisited elements whose paents have een visited. use a gloal list of elements, oganized y thei distance fom quey oject use a pioity queue as it suppots necessay inset and delete minimum opeations ties in distance: pioity to lowe type numes if still tied, pioity to elements deepe in seach hieachy Copyight 008 y Hanan Samet

63 INCREMENTAL NEAREST NEIGHBOR ALGORITHM Algoithm: INCNEAREST(q, S, T). Q NEWPRIORITYQUEUE(). e t oot of the seach hieachy induced y q, S, and T. ENQUEUE(Q, e t, 0) 4. while not ISEMPTY(Q) do 5. e t DEQUEUE(Q) 6. if t = 0 then /* e t is an oject */ 7. Repot e t as the next neaest oject 8. else 9. fo each child element e t of e t do 0. ENQUEUE(Q, e t, d t (q, e t )) hpd. Lines - initialize pioity queue with oot 4. In main loop take element e t closest to q off the queue epot e t as next neaest oject if e t is an oject othewise, inset child elements of e t into pioity queue Copyight 008 y Hanan Samet

64 PM QUADTREE cd Vetex-ased (one vetex pe lock) a DECOMPOSITION RULE: Patitioning occus when a lock contains moe than one segment unless all the segments ae incident at the same vetex which is also in the same lock Shape independent of ode of insetion Copyight 008 y Hanan Samet 8

65 PM QUADTREE cd Vetex-ased (one vetex pe lock) a DECOMPOSITION RULE: Patitioning occus when a lock contains moe than one segment unless all the segments ae incident at the same vetex which is also in the same lock Shape independent of ode of insetion Copyight 008 y Hanan Samet 8

66 PM QUADTREE z cd Vetex-ased (one vetex pe lock) a c DECOMPOSITION RULE: Patitioning occus when a lock contains moe than one segment unless all the segments ae incident at the same vetex which is also in the same lock Shape independent of ode of insetion Copyight 008 y Hanan Samet 8

67 PM QUADTREE 4 g z cd Vetex-ased (one vetex pe lock) a d c DECOMPOSITION RULE: Patitioning occus when a lock contains moe than one segment unless all the segments ae incident at the same vetex which is also in the same lock Shape independent of ode of insetion Copyight 008 y Hanan Samet 8

68 PM QUADTREE 5 v 4 g z cd Vetex-ased (one vetex pe lock) a e d c DECOMPOSITION RULE: Patitioning occus when a lock contains moe than one segment unless all the segments ae incident at the same vetex which is also in the same lock Shape independent of ode of insetion Copyight 008 y Hanan Samet 8

69 PM QUADTREE 6 5 v 4 g z cd Vetex-ased (one vetex pe lock) a e d f c DECOMPOSITION RULE: Patitioning occus when a lock contains moe than one segment unless all the segments ae incident at the same vetex which is also in the same lock Shape independent of ode of insetion Copyight 008 y Hanan Samet 8

70 PM QUADTREE 7 z 6 5 v 4 g z cd Vetex-ased (one vetex pe lock) a g e d f c DECOMPOSITION RULE: Patitioning occus when a lock contains moe than one segment unless all the segments ae incident at the same vetex which is also in the same lock Shape independent of ode of insetion Copyight 008 y Hanan Samet 8

71 PM QUADTREE 8 g 7 z 6 5 v 4 g z cd Vetex-ased (one vetex pe lock) a h g e d f c DECOMPOSITION RULE: Patitioning occus when a lock contains moe than one segment unless all the segments ae incident at the same vetex which is also in the same lock Shape independent of ode of insetion Copyight 008 y Hanan Samet 8

72 PM QUADTREE 9 v 8 g 7 z 6 5 v 4 g z cd Vetex-ased (one vetex pe lock) a h g e d i f c DECOMPOSITION RULE: Patitioning occus when a lock contains moe than one segment unless all the segments ae incident at the same vetex which is also in the same lock Shape independent of ode of insetion Copyight 008 y Hanan Samet 8

73 MX-CIF QUADTREE (Kedem)... Collections of small ectangles fo VLSI applications Each ectangle is associated with its minimum enclosing quadtee lock hp4 Like hashing: quadtee locks seve as hash uckets 4 5 B 6 E C A D 0 F A {,6,7,8,9,0} B {} C {} D {} E {,4,5} F {} Copyight 008 y Hanan Samet 4

74 MX-CIF QUADTREE (Kedem) Collections of small ectangles fo VLSI applications Each ectangle is associated with its minimum enclosing quadtee lock hp4 Like hashing: quadtee locks seve as hash uckets Collision = moe than one ectangle in a lock esolve y using two one-dimensional MX-CIF tees to stoe the ectangle intesecting the lines passing though each sudivision point 4 5 B 6 E C A D 0 F A {,6,7,8,9,0} B {} C {} D {} E {,4,5} F {} Copyight 008 y Hanan Samet 4

75 MX-CIF QUADTREE (Kedem) Collections of small ectangles fo VLSI applications Each ectangle is associated with its minimum enclosing quadtee lock hp4 Like hashing: quadtee locks seve as hash uckets Collision = moe than one ectangle in a lock esolve y using two one-dimensional MX-CIF tees to stoe the ectangle intesecting the lines passing though each sudivision point one fo y-axis g 4 5 Binay tee fo y- axis though A 7 B 8 6 A E C 9 Y Y 0 Y4 8 6 Y6 Y Y5 Y7 D 0 F A {,6,7,8,9,0} B {} C {} D {} E {,4,5} F {} Copyight 008 y Hanan Samet 4

76 MX-CIF QUADTREE (Kedem) Collections of small ectangles fo VLSI applications Each ectangle is associated with its minimum enclosing quadtee lock hp4 Like hashing: quadtee locks seve as hash uckets Collision = moe than one ectangle in a lock esolve y using two one-dimensional MX-CIF tees to stoe the ectangle intesecting the lines passing though each sudivision point one fo y-axis if a ectangle intesects oth x and y axes, then associate it with the y axis 4 v g 4 5 Binay tee fo y- axis though A 7 B 8 6 A E C 9 Y Y 0 Y4 8 6 Y6 Y Y5 Y7 D 0 F A {,6,7,8,9,0} B {} C {} D {} E {,4,5} F {} Copyight 008 y Hanan Samet 4

77 MX-CIF QUADTREE (Kedem) Collections of small ectangles fo VLSI applications Each ectangle is associated with its minimum enclosing quadtee lock hp4 Like hashing: quadtee locks seve as hash uckets Collision = moe than one ectangle in a lock esolve y using two one-dimensional MX-CIF tees to stoe the ectangle intesecting the lines passing though each sudivision point one fo y-axis if a ectangle intesects oth x and y axes, then associate it with the y axis one fo x-axis 5 z 4 v g 4 5 Binay tee fo y- axis though A 7 B 8 6 A E D C 9 Y Y 0 Y4 8 6 Y6 Y Y5 Y7 Binay tee fo x- axis though A 0 F X4 X X 9 X5 X A {,6,7,8,9,0} X6 7 B {} C {} D {} E {,4,5} F {} Copyight 008 y Hanan Samet 4

78 Loose Quadtee (Octee)/Cove Fieldtee Ovecomes dawack of MX-CIF quadtee that the width w of the minimum enclosing quadtee lock of a ectangle o is not a function of the size of o B 4 5 E 6 C A D F A {,6,7,8,9,0} {} B E {,4,5} C{} D{} F{ } Copyight 008 y Hanan Samet

79 Loose Quadtee (Octee)/Cove Fieldtee Ovecomes dawack of MX-CIF quadtee that the width w of the minimum enclosing quadtee lock of a ectangle o is not a function of the size of o Instead, it depends on the position of the centoid of o and often considealy lage than o B E C A D F A {,6,7,8,9,0} {} B E {,4,5} C{} D{} F{ } Copyight 008 y Hanan Samet

80 Loose Quadtee (Octee)/Cove Fieldtee Ovecomes dawack of MX-CIF quadtee that the width w of the minimum enclosing quadtee lock of a ectangle o is not a function of the size of o Instead, it depends on the position of the centoid of o and often considealy lage than o B E C A D F A {,6,7,8,9,0} {} B E {,4,5} C{} D{} F{ } Copyight 008 y Hanan Samet

81 Loose Quadtee (Octee)/Cove Fieldtee Ovecomes dawack of MX-CIF quadtee that the width w of the minimum enclosing quadtee lock of a ectangle o is not a function of the size of o Instead, it depends on the position of the centoid of o and often considealy lage than o Solution: expand size of space spanned y each quadtee lock of width w y expansion facto p (p > 0) so expanded lock is of width ( + p)w B 6 E A 4 5 C D F A {,6,7,8,9,0} {} B E {,4,5} C{} D{} F{ } Copyight 008 y Hanan Samet

82 Loose Quadtee (Octee)/Cove Fieldtee Ovecomes dawack of MX-CIF quadtee that the width w of the minimum enclosing quadtee lock of a ectangle o is not a function of the size of o Instead, it depends on the position of the centoid of o and often considealy lage than o Solution: expand size of space spanned y each quadtee lock of width w y expansion facto p (p > 0) so expanded lock is of width ( + p)w B 6 E. p = A 4 5 C 9 0 D F A {,6,7,8,9,0} B {} E {,4,5} C{,9} {6} {7,8,0} D{} F{,} Copyight 008 y Hanan Samet

83 Loose Quadtee (Octee)/Cove Fieldtee Ovecomes dawack of MX-CIF quadtee that the width w of the minimum enclosing quadtee lock of a ectangle o is not a function of the size of o Instead, it depends on the position of the centoid of o and often considealy lage than o Solution: expand size of space spanned y each quadtee lock of width w y expansion facto p (p > 0) so expanded lock is of width ( + p)w B 6 E. p = 0.. p = A 4 5 C 9 0 D F A {} B{} E C{,9} {7,8,0} {,4} {5} {} {6} {9} {7} {8} {0} D{} {,} F {} {} Copyight 008 y Hanan Samet

84 Loose Quadtee (Octee)/Cove Fieldtee Ovecomes dawack of MX-CIF quadtee that the width w of the minimum enclosing quadtee lock of a ectangle o is not a function of the size of o Instead, it depends on the position of the centoid of o and often considealy lage than o Solution: expand size of space spanned y each quadtee lock of width w y expansion facto p (p > 0) so expanded lock is of width ( + p)w B 6 E. p = 0.. p = A 4 5 C 9 Maximum w (i.e., minimum depth of minimum enclosing quadtee lock) is a function of p and adius of o and independent of position of centoid of o. Range of possile atios w/ : /( + p) w/ < /p. Fo p, esticting w and to powes of, w/ takes on at most values and usually just {} B{} E C{,9} 0 {,4} {5} {} {6} {9} {7} {8} {0} A {7,8,0} D F D{} {,} F {} {} Copyight 008 y Hanan Samet

85 Patition Fieldtee Altenative to loose quadtee (octee)/cove fieldtee at ovecoming dawack of MX-CIF quadtee that the width w of the minimum enclosing quadtee lock of a ectangle o is not a function of the size of o Copyight 008 y Hanan Samet

86 Patition Fieldtee Altenative to loose quadtee (octee)/cove fieldtee at ovecoming dawack of MX-CIF quadtee that the width w of the minimum enclosing quadtee lock of a ectangle o is not a function of the size of o Achieves simila esult y shifting positions of the centoid of quadtee locks at successive levels of the sudivision y one half of the width of the lock that is eing sudivided Copyight 008 y Hanan Samet

87 Patition Fieldtee Altenative to loose quadtee (octee)/cove fieldtee at ovecoming dawack of MX-CIF quadtee that the width w of the minimum enclosing quadtee lock of a ectangle o is not a function of the size of o Achieves simila esult y shifting positions of the centoid of quadtee locks at successive levels of the sudivision y one half of the width of the lock that is eing sudivided Copyight 008 y Hanan Samet

88 Patition Fieldtee Altenative to loose quadtee (octee)/cove fieldtee at ovecoming dawack of MX-CIF quadtee that the width w of the minimum enclosing quadtee lock of a ectangle o is not a function of the size of o Achieves simila esult y shifting positions of the centoid of quadtee locks at successive levels of the sudivision y one half of the width of the lock that is eing sudivided Copyight 008 y Hanan Samet

89 Patition Fieldtee Altenative to loose quadtee (octee)/cove fieldtee at ovecoming dawack of MX-CIF quadtee that the width w of the minimum enclosing quadtee lock of a ectangle o is not a function of the size of o Achieves simila esult y shifting positions of the centoid of quadtee locks at successive levels of the sudivision y one half of the width of the lock that is eing sudivided Copyight 008 y Hanan Samet

90 Patition Fieldtee Altenative to loose quadtee (octee)/cove fieldtee at ovecoming dawack of MX-CIF quadtee that the width w of the minimum enclosing quadtee lock of a ectangle o is not a function of the size of o Achieves simila esult y shifting positions of the centoid of quadtee locks at successive levels of the sudivision y one half of the width of the lock that is eing sudivided Sudivision ule guaantees that width of minimum enclosing quadtee lock fo ectangle o is ounded y 8 times the maximum extent of o Copyight 008 y Hanan Samet

91 Patition Fieldtee Altenative to loose quadtee (octee)/cove fieldtee at ovecoming dawack of MX-CIF quadtee that the width w of the minimum enclosing quadtee lock of a ectangle o is not a function of the size of o Achieves simila esult y shifting positions of the centoid of quadtee locks at successive levels of the sudivision y one half of the width of the lock that is eing sudivided Sudivision ule guaantees that width of minimum enclosing quadtee lock fo ectangle o is ounded y 8 times the maximum extent of o o Copyight 008 y Hanan Samet

92 Patition Fieldtee Altenative to loose quadtee (octee)/cove fieldtee at ovecoming dawack of MX-CIF quadtee that the width w of the minimum enclosing quadtee lock of a ectangle o is not a function of the size of o Achieves simila esult y shifting positions of the centoid of quadtee locks at successive levels of the sudivision y one half of the width of the lock that is eing sudivided Sudivision ule guaantees that width of minimum enclosing quadtee lock fo ectangle o is ounded y 8 times the maximum extent of o o Copyight 008 y Hanan Samet

93 Patition Fieldtee Altenative to loose quadtee (octee)/cove fieldtee at ovecoming dawack of MX-CIF quadtee that the width w of the minimum enclosing quadtee lock of a ectangle o is not a function of the size of o Achieves simila esult y shifting positions of the centoid of quadtee locks at successive levels of the sudivision y one half of the width of the lock that is eing sudivided Sudivision ule guaantees that width of minimum enclosing quadtee lock fo ectangle o is ounded y 8 times the maximum extent of o o Copyight 008 y Hanan Samet

94 Patition Fieldtee Altenative to loose quadtee (octee)/cove fieldtee at ovecoming dawack of MX-CIF quadtee that the width w of the minimum enclosing quadtee lock of a ectangle o is not a function of the size of o Achieves simila esult y shifting positions of the centoid of quadtee locks at successive levels of the sudivision y one half of the width of the lock that is eing sudivided Sudivision ule guaantees that width of minimum enclosing quadtee lock fo ectangle o is ounded y 8 times the maximum extent of o o Copyight 008 y Hanan Samet

95 Patition Fieldtee Altenative to loose quadtee (octee)/cove fieldtee at ovecoming dawack of MX-CIF quadtee that the width w of the minimum enclosing quadtee lock of a ectangle o is not a function of the size of o Achieves simila esult y shifting positions of the centoid of quadtee locks at successive levels of the sudivision y one half of the width of the lock that is eing sudivided Sudivision ule guaantees that width of minimum enclosing quadtee lock fo ectangle o is ounded y 8 times the maximum extent of o Same atio is otained fo the loose quadtee (octee)/cove fieldtee when p =/4, and thus patition fieldtee is supeio to the cove fieldtee when p </4 o Copyight 008 y Hanan Samet

96 Patition Fieldtee Altenative to loose quadtee (octee)/cove fieldtee at ovecoming dawack of MX-CIF quadtee that the width w of the minimum enclosing quadtee lock of a ectangle o is not a function of the size of o Achieves simila esult y shifting positions of the centoid of quadtee locks at successive levels of the sudivision y one half of the width of the lock that is eing sudivided Sudivision ule guaantees that width of minimum enclosing quadtee lock fo ectangle o is ounded y 8 times the maximum extent of o Same atio is otained fo the loose quadtee (octee)/cove fieldtee when p =/4, and thus patition fieldtee is supeio to the cove fieldtee when p </4 Summay: cove fieldtee expands the width of the quadtee locks while the patition fieldtee shifts the positions of thei centoids o Copyight 008 y Hanan Samet

97 HIERARCHICAL RECTANGULAR DECOMPOSITION Simila to tiangula decomposition Good when data points ae the vetices of a ectangula gid Dawack is asence of continuity etween adjacent patches of unequal width (temed the alignment polem) sf Ovecoming the pesence of cacks. use the intepolated point instead of the tue point (Baea and Hinjosa). tiangulate the squaes (Von Hezen and Ba) can split into, 4, o 8 tiangles depending on how many lines ae dawn though the midpoint if split into tiangles, then cacks still emain no cacks if split into 4 o 8 tiangles Copyight 008 y Hanan Samet 5

98 RESTRICTED QUADTREE (VON HERZEN/BARR) sf All 4-adjacent locks ae eithe of equal size o of atio : Note: also used in finite element analysis to adptively efine an element as well as to achieve element compatiility (temed h-efinement y Kela, Peucchio, and Voelcke) Copyight 008 y Hanan Samet 6

99 RESTRICTED QUADTREE (VON HERZEN/BARR) sf All 4-adjacent locks ae eithe of equal size o of atio : Note: also used in finite element analysis to adptively efine an element as well as to achieve element compatiility (temed h-efinement y Kela, Peucchio, and Voelcke) Copyight 008 y Hanan Samet 6

100 RESTRICTED QUADTREE (VON HERZEN/BARR) z sf All 4-adjacent locks ae eithe of equal size o of atio : Note: also used in finite element analysis to adptively efine an element as well as to achieve element compatiility (temed h-efinement y Kela, Peucchio, and Voelcke) 8-tiangle decomposition ule. decompose each lock into 8 tiangles (i.e., tiangles pe edge). unless the edge is shaed y a lage lock. in which case only tiangle is fomed Copyight 008 y Hanan Samet 6

101 RESTRICTED QUADTREE (VON HERZEN/BARR) sf All 4-adjacent locks ae eithe of equal size o of atio : Note: also used in finite element analysis to adptively efine an element as well as to achieve element compatiility (temed h-efinement y Kela, Peucchio, and Voelcke) 4 g z 8-tiangle decomposition ule. decompose each lock into 8 tiangles (i.e., tiangles pe edge). unless the edge is shaed y a lage lock. in which case only tiangle is fomed 4-tiangle decomposition ule. decompose each lock into 4 tiangles (i.e., tiangle pe edge). unless the edge is shaed y a smalle lock. in which case tiangles ae fomed along the edge Copyight 008 y Hanan Samet 6

102 RESTRICTED QUADTREE (VON HERZEN/BARR) sf All 4-adjacent locks ae eithe of equal size o of atio : Note: also used in finite element analysis to adptively efine an element as well as to achieve element compatiility (temed h-efinement y Kela, Peucchio, and Voelcke) 5 v 4 g z 8-tiangle decomposition ule. decompose each lock into 8 tiangles (i.e., tiangles pe edge). unless the edge is shaed y a lage lock. in which case only tiangle is fomed 4-tiangle decomposition ule. decompose each lock into 4 tiangles (i.e., tiangle pe edge). unless the edge is shaed y a smalle lock. in which case tiangles ae fomed along the edge Pefe 8-tiangle ule as it is ette fo display applications (shading) Copyight 008 y Hanan Samet 6

103 sf4 PROPERTY SPHERES (FEKETE) Appoximation of spheical data Uses icosahedon which is a Platonic solid. 0 faces each is a egula tiangle. lagest possile egula polyhedon Copyight 008 y Hanan Samet 7

104 sf5 ALTERNATIVE SPHERICAL APPROXIMATIONS Could use othe Platonic solids. all have faces that ae egula polygons tetahedon: 4 equilateal tiangula faces hexahedon: 6 squae faces octahedon: 8 equilateal tiangula faces dodecahedon: pentagonal faces. octahedon is nice fo modeling the gloe it can e aligned so that the poles ae at opposite vetices the pime meidian and the equato intesect at anothe vetex one sudivision line of each face is paallel to the equato Decompose on the asis of latitude and longitude values. not so good if want a patition into units of equal aea as geat polems aound the poles. poject sphee onto plane using Lamet s cylindical pojection which is locally aea peseving Instead of appoximating sphee with the solids, poject the faces of the solids on the sphee (Scott). all edges ecome su-acs of a geat cicle. use egula decomposition on tiangula, squae, o pentagonal spheical suface patches Copyight 008 y Hanan Samet 8

105 hi60 OCTREES. Inteio (voxels) analogous to egion quadtee appoximate oject y aggegating simila voxels good fo medical images ut not fo ojects with plana faces Ex: 4 5 A B Bounday adaptation of PM quadtee to thee-dimensional data decompose until each lock contains a. one face. moe than one face ut all meet at same edge c. moe than one edge ut all meet at same vetex impose a spatial index on a ounday model (BRep) Copyight 008 y Hanan Samet 9

106 hi9 EXAMPLE QUADTREE-BASED QUERY Quey: find all cities with population in excess of 5,000 in wheat gowing egions within 0 miles of the Mississippi Rive. assume ive is a linea featue use a line map could e a egion if asked fo sandas in the ive. egion map fo the wheat. assume cities ae points point map fo cities could e egion is asked fo high income aeas Comines spatial and non-spatial (i.e., attiute) data Many possile execution plans - e.g.,. compute uffe o coido aound ive. extact wheat aea. intesect with 4. intesect city map with 5. etieve value of population attiute fo cities in 4 fom the nonspatial dataase (e.g., elational) Regula decomposition hieachical data stuctues such as the quadtee. all maps ae in egistation all locks ae in the same positions not tue fo R+-tees and BSP tees disjoint decomposition of space - unlike R-tee. can pefom set-theoetic opeations on diffeent featue types (e.g., and 4) Copyight 008 y Hanan Samet

107 FURTHER READING f. F. Baec and H. Samet, Client ased spatial owsing on the wold wide we. IEEE Intenet Computing, ():5 59, Jan/Fe H. Samet, Foundations of Multidimensional and Metic Data Stuctues, Mogan Kaufmann, San Fancisco, 006. [ ook flye.pdf]. H. Samet, Applications of Spatial Data Stuctues: Compute Gaphics, Image Pocessing, and GIS, Addison Wesley, Reading, MA, H. Samet, Design and Analysis of Spatial Data Stuctues, Addison Wesley, Reading, MA, Spatial Data Applets at Copyight (c) 008 y Hanan Samet

108 MORGAN KAUFMANN PUBLISHERS Foundations of Multidimensional and Metic Data Stuctues By Hanan Samet, Univesity of Mayland at College Pak 04 pages August 006 ISBN Hadcove $ $ The field of multidimensional and metic data stuctues is lage and gowing vey quickly. Hee, fo the fist time, is a thoough teatment of multidimensional point data, oject and image-ased oject epesentations, intevals and small ectangles, high-dimensional datasets, as well as datasets fo which we only know that they eside in a metic space. 0% OFF! The ook includes a thoough intoduction; a compehensive suvey of multidimensional (including spatial) and metic data stuctues and algoithms; and implementation details fo the most useful data stuctues. Along with the hundeds of woked execises and hundeds of illustations, the esult is an excellent and valuale efeence tool fo pofessionals in many aeas, including compute gaphics and visualization, dataases, geogaphic infomation systems (GIS), and spatial dataases, game pogamming, image pocessing and compute vision, patten ecognition, solid modelling and computational geomety, similaity etieval and multimedia dataases, and VLSI design, and seach aspects of ioinfomatics. Featues Fist compehensive wok on multidimensional and metic data stuctues availale, a thoough and authoitative teatment. An algoithmic athe than mathematical appoach, with a lieal use of examples that allows the eades to easily see the possile implementation and use. Each section includes a lage nume of execises and solutions to self-test and confim the eade's undestanding and suggest futue diections. Witten y a well-known authoity in the aea of multidimensional (including spatial) data stuctues who has made many significant contiutions to the field. Hanan Samet is the dean of "spatial indexing"... This ook is encyclopedic... this ook will e invaluale fo those of us who stuggle with spatial data, scientific datasets, gaphics, vision polems involving volumetic queies, o with highe dimensional datasets common in data mining. - Fom the foewod y Jim Gay, Micosoft Reseach Samet's ook on multidimensional and metic data stuctues is the most complete and thoough pesentation on this topic. It has oad coveage of mateial fom computational geomety, dataases, gaphics, GIS, and similaity etieval liteatue. Witten y the leading authoity on hieachical spatial epesentations, this ook is a "must have" fo all instucto, eseaches, and developes woking and teaching in these aeas. - Dinesh Manocha, Univesity of Noth Caolina at Chapel Hill To summaize, this ook is excellent! It s a vey compehensive suvey of spatial and multidimensional data stuctues and algoithms, which is adly needed. The eadth and depth of coveage is astounding and I would conside seveal pats of it equied eading fo eal time gaphics and game developes. - Betton Wade, Univesity of Washington and Micosoft Cop. Ode fom Mogan Kaufmann Pulishes and eceive 0% off! Please efe to code 855. Mail: Elsevie Science, Ode Fulfillment, 80 Westline Industial D., St. Louis, MO 646 Phone: US/Canada , (Intl.) Fax: , (Intl.) uskinfo@elsevie.com Visit Mogan Kaufmann on the We: Volume discounts availale, contact: NASpecialSales@elsevie.com

109 Foundations of Multidimensional and Metic Data Stuctues By Hanan Samet, Univesity of Mayland at College Pak Availale August 006 ISBN pages Hadcove $59.95 $47.96 Chapte : Multidimensional Point Data. Intoduction. Range Tees. Pioity Seach Tees.4 Quadtees.4. Point Quadtees.4. Tie-Based Quadtee.4. Compaison of Point and Tie-Based Quadtees.5 K-d Tees.5. Point K-d Tees.5. Tie-Based K-d Tees.5. Conjugation Tee.6 One-Dimensional Odeings.7 Bucket Methods.7. Tee Diectoy Methods (K-d-B-Tee, Hyid Tee, LSD Tee, hb-tee, K-d-B-Tie, BV- Tee).7. Gid Diectoy Methods (Gid File, EXCELL, Linea Hashing, Spial Hashing).7. Stoage Utilization.8 PK-Tee.9 Conclusion Chapte Oject-ased and Image-ased Image Repesentations. Inteio-Based Repesentations.. Unit-Size Cells.. Blocks (Medial Axis Tansfom, Region Quadtee and Octee, Bintee, X-Y Tee, Teemap, Puzzletee).. Nonothogonal Blocks (BSP Tee, Layeed DAG)..4 Aitay Ojects (Loose Octee, Field Tee, PMR Quadtee)..5 Hieachical Inteio-Based Repesentations (Pyamid, R-Tee, Hilet R- tee, R*-Tee, Packed R-Tee,R+-Tee, Cell Tee, Bulk Loading). Bounday-Based Repesentations.. The Bounday Model (CSG,BREP, Winged Edge, Quad Edge,Lath, Voonoi Diagam, Delaunay Tiangulation, Tetaheda, Tiangle Tale, Cone Tale.. Image-Based Bounday Repesentations (PM Quadtee and Octee, Adaptively Sampled Distance Field).. Oject-ased Bounday Repesentation (LOD, Stip Tee, Simplification Methods)..4 Suface-Based Bounday Repesentations (TIN). Diffeence-Based Compaction Methods.. Runlength Encoding.. Chain Code.. Vetex Repesentation.4 Histoical Oveview MORGAN KAUFMANN PUBLISHERS an impint of Elsevie Tale of Contents and Topics Chapte Intevals and Small Rectangles. Plane-Sweep Methods and the Rectangle Intesection Polem.. Segment Tee.. Inteval Tee.. Pioity Seach Tee..4 Altenative Solutions and Related Polems. Plane-sweep Methods and the Measue Polem. Point-Based Methods.. Repesentative Points.. Collections of Repesentative Points.. LSD Tee..4 Summay.4 Aea-Based Methods.4. MX-CIF Quadtee.4. Altenatives to the MX-CIF Quadtee (HV/VH Tee).4. Multiple Quadtee Block Repesentations Chapte 4 High-Dimensional Data 4. Best-Fist Incemental Neaest Neigho Finding (Ranking) 4.. Motivation 4.. Seach Hieachy 4.. Algoithm 4..4 Duplicate Ojects 4..5 Spatial Netwoks 4..6 Algoithm Extensions (Fathest Neigho, Skylines) 4..7 Related Wok 4. The Depth-Fist K-Neaest Neigho Algoithm 4.. Basic Algoithm 4.. Puning Rules 4.. Effects of Clusteing Methods on Puning 4..4 Odeing the Pocessing of the Elements of the Active List 4..5 Impoved Algoithm 4..6 Incopoating MaxNeaestDist in a Best- Fist Algoithm 4..7 Example 4..8 Compaison 4. Appoximate Neaest Neigho Finding 4.4 Multidimensional Indexing Methods 4.4. X-Tee 4.4. Bounding Sphee Methods: Sphee Tee, SS-Tee, Balltee, and SR-Tee 4.4. Inceasing the Fanout: TV-Tee, Hyid Tee, and A-Tee Methods Based on the Voonoi Diagam: OS-Tee Appoximate Voonoi Diagam (AVD) Avoiding Ovelapping All of the Leaf Blocks Pyamid Technique Sequential Scan Methods (VA-File, IQ- Tee,VA+-File) 4.5 Distance-Based Indexing Methods 4.5. Distance Metic and Seach Puning 4.5. Ball Patitioning Methods (VP-Tee, MVP-Tee) 4.5. Genealized Hypeplane Patitioning Methods (GH-Tee, GNAT, MB-Tee) M-Tee Sa-Tee knn Gaph Distance Matix Methods SASH - Indexing Without Using the Tiangle Inequality 4.6 Dimension-Reduction Methods 4.6. Seaching in the Dimensionally- Reduced Space 4.6. Using Only One Dimension 4.6. Repesentative Point Methods Tansfomation into a Diffeent and Smalle Featue Set (SVD,DFT) Summay 4.7 Emedding Methods 4.7. Intoduction 4.7. Lipschitz Emeddings 4.7. FastMap Locality Sensitive Hashing (LSH) Appendix : Oveview of B-Tees Appendix : Linea Hashing Appendix : Spial Hashing Appendix 4: Desciption of Pseudo-Code Language Solutions to Execises Biliogaphy Name and Cedit Index Index Keywod Index Aout the Autho Ode fom Mogan Kaufmann Pulishes To eceive 0% off, please efe to code 855 Mail: Elsevie Science, Ode Fulfillment, 80 Westline Industial D., St. Louis, MO 646 Phone: US/Canada , (Intl.) Fax: , (Intl.) uskinfo@elsevie.com Visit Mogan Kaufmann on the We: Volume discounts availale, contact: NASpecialSales@elsevie.com Hanan Samet is Pofesso in the Depatment of Compute Science at the Univesity of Mayland at College Pak, and a meme of the Cente fo Automation Reseach and the Institute fo Advanced Compute Studies. He is widely pulished in the fields of spatial dataases and data stuctues, compute gaphics, image dataases and image pocessing, and geogaphic infomation systems (GIS), and is consideed an authoity on the use and design of hieachical spatial data stuctues such as the quadtee and octee fo geogaphic infomation systems, image pocessing, and compute gaphics. He is the autho of the fist two ooks on spatial data stuctues: The Design and Analysis of Spatial Data Stuctues and Applications of Spatial Data Stuctues: Compute Gaphics, Image Pocessing and GIS. He holds a Ph.D. in compute science fom Stanfod Univesity