ISSUES IN SPATIAL DATABASES AND GEOGRAPHIC INFORMATION SYSTEMS (GIS) HANAN SAMET

Transcription

1 ISSUES IN SPATIAL DATABASES AND GEOGRAPHIC INFORMATION SYSTEMS (GIS) HANAN SAMET COMPUTER SCIENCE DEPARTMENT AND CENTER FOR AUTOMATION RESEARCH AND INSTITUTE FOR ADVANCED COMPUTER STUDIES UNIVERSITY OF MARYLAND COLLEGE PARK, MARYLAND USA Copyiht 007 Hanan Samet These notes may not e epoduced y any means (mechanical o electonic o any othe) without expess witten pemission of Hanan Samet

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3 PRINCE GEORGES COUNTY hi8 Copyiht 007 y Hanan Samet

4 zk8 BACKGROUND (A PERSONAL VIEW!). GIS oiinally focussed on pape map as output anythin is ette than dawin y hand no eat emphasis on execution time. Pape output suppots hih esolution display sceen is of limited esolution can admit less pecise aloithms Ex: uffe zone computation (spatial ane quey) a. usually use a Euclidean distance metic (L ) takes a lon time. can e sped up usin a quadtee and a Chessoad distance metic (L ) not as accuate as Euclidean ut may not e ale to peceive the diffeence on a display sceen! as much as 3 odes of manitude faste 3. Uses accustomed to speadsheets GIS should wok like a speadsheet fast esponse time aility to ask what if questions and see the esults incopoate a dataase fo seamless inteation of spatial and nonspatial (i.e., attiute data) Copyiht 007 y Hanan Samet 3

5 zk9 GENERAL SPATIAL DATABASE ISSUES. Why do we want a dataase? to stoe data so that it can e etieved efficiently should not lose siht of this pupose. How to inteate spatial data with nonspatial data 3. Lon fields in elational dataase ae not the answe a stopap solution as just a epositoy fo data does not aid in etievin the data if data is lae in volume, then eaks down as tuples et vey lae 4. A dataase is eally a collection of ecods with fields coespondin to attiutes of diffeent types ecods ae like points in hihe dimensional space a. some adaptations take advantae of this analoy. howeve, can act like a staiht jacket in case of elational model 5. Retieval is facilitated y uildin an index need to find a way to sot the data index should e compatile with data ein stoed choose an appopiate zeo o efeence point need an implicit athe than an explicit index a. impossile to foesee all possile queies in advance. explicit would sot two-dimensional points on the asis of distance fom a paticula point P impactical as sot is inapplicale to points diffeent fom P Copyiht 007 y Hanan Samet 4

6 zk0 6. Identify the possile queies and find thei analos in conventional dataases e.., a map in a spatial dataase is like a elation in a conventional dataase (also known as spatial elation) a. diffeence is the pesence of spatial attiute(s). also pesence of spatial output 7. How do we inteact with the dataase? SQL may not e easy to adapt aphical quey lanuae output may e visual in which case a owsin capaility (e.., an iteato) is useful 8. What statey do we use in answein a quey that mixes taditional data with nontaditional data? need quey optimization ules must define selectivity factos a. dependent on whethe index exists on nontaditional data. if no, then select on taditional data fist Ex: find all cities within 00 miles of the Mississippi Rive with population in excess of million a. spatial selection fist if eion is small (implies hih spatial selectivity). elational selection fist if vey few cities with a lae population (implies hih elational selectivity) Copyiht 007 y Hanan Samet 5

7 zk DATA IN SPATIAL DATABASES. Spatial infomation locations of ojects (ae discete, individual points in space) space occupied y ojects (ae continuous; have extent) a. example ojects lines (e.., oads, ives) eions (e.., uildins, cop maps, polyheda) othes.... ae ojects disjoint o may they ovelap? e.., seveal cop types may e own on a plot of land not concened hee with aste vs: vecto issues as these ae data epesentation issues athe than data type issues. Non-spatial infomation eion names, postal codes,... city population, yea founded,... oad names, speed limits,... Copyiht 007 y Hanan Samet 6

8 hi7 EXAMPLE QUERIES ON LINE SEGMENT DATABASES Queies aout line sements. All sements that intesect a iven point o set of points. All sements that have a iven set of endpoints 3. All sements that intesect a iven line sement 4. All sements that ae coincident with a iven line sement Poximity queies. The neaest line sement to a iven point. All sements within a iven distance fom a iven point (also known as a ane o window quey) Queies involvin attiutes of line sements. Given a point, find the closest line sement of a paticula type. Given a point, find the minimum enclosin polyon whose constituent line sements ae all of a iven type 3. Given a point, find all the polyons that ae incident on it Copyiht 007 y Hanan Samet 3

9 s0 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 ) ood fo queies aout the line sements not ood fo poximity queies since points outside the oject ae not mapped into the hihe dimensional space epesentative points of two ojects that ae physically close to each othe in the oiinal space (e.., -d fo lines) may e vey fa fom each othe in the hihe dimensional space (e.., 4-d) Ex: polem is that the tansfomation only tansfoms the space occupied y the ojects and not the est of the space (e.., the quey point) can ovecome y pojectin ack to oiinal space 3. Use an index that sots ased upon spatial occupancy (i.e., extent of the ojects) B A Copyiht 007 y Hanan Samet 4

10 hi9. SPATIAL INDEXING REQUIREMENTS. Compatiility with the data ein stoed. Choose an appopiate zeo o efeence point 3. 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 stins 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 iven 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 Copyiht 007 y Hanan Samet 5

11 s SORTING ON THE BASIS OF SPATIAL OCCUPANCY Decompose the space fom which the data is dawn into eions called uckets (like hashin ut peseves ode) Inteested in methods that ae desined specifically fo the spatial data type ein stoed Basic appoaches to decomposin space. minimum oundin ectanles e.., R-tee ood at distinuishin 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 id a. all cells the same size. dawack: possiility of many spase cells adaptive id quadtee vaiants a. eula decomposition. all cells of width powe of patitions at aitay positions a. dawack: not a eula decomposition. e.., R + -tee Can use as appoximations in filte/efine quey pocessin statey Copyiht 007 y Hanan Samet 6

12 MINIMUM BOUNDING RECTANGLES hi3 Ojects ouped into hieachies, stoed in a stuctue simila to a B-tee Dawack: not a disjoint decomposition of space Oject has sinle oundin ectanle, yet aea that it spans may e included in seveal oundin ectanles 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 h i e d f c Copyiht 007 y Hanan Samet 7

13 MINIMUM BOUNDING RECTANGLES hi3 Ojects ouped into hieachies, stoed in a stuctue simila to a B-tee Dawack: not a disjoint decomposition of space Oject has sinle oundin ectanle, yet aea that it spans may e included in seveal oundin ectanles 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 R3 R4 h R5 a i R6 e d f c R3: a R4: d h R5: c i R6: e f Copyiht 007 y Hanan Samet 7

14 MINIMUM BOUNDING RECTANGLES hi3 Ojects ouped into hieachies, stoed in a stuctue simila to a B-tee Dawack: not a disjoint decomposition of space Oject has sinle oundin ectanle, yet aea that it spans may e included in seveal oundin ectanles 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 3 z R R3 R4 h R5 R a i R6 e d f c R: R3 R4 R: R5 R6 R3: a R4: d h R5: c i R6: e f Copyiht 007 y Hanan Samet 7

15 MINIMUM BOUNDING RECTANGLES hi3 Ojects ouped into hieachies, stoed in a stuctue simila to a B-tee Dawack: not a disjoint decomposition of space Oject has sinle oundin ectanle, yet aea that it spans may e included in seveal oundin ectanles 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 3 z R R3 R4 h R5 R a i R6 e d f R0 c R0: R R R: R3 R4 R: R5 R6 R3: a R4: d h R5: c i R6: e f Copyiht 007 y Hanan Samet 7

16 SEARCHING FOR A POINT OR LINE SEGMENT IN AN R-TREE hi3 Dawack is that may have to examine many nodes since a line sement can e contained in the covein ectanles of many nodes yet its ecod is contained in only one leaf node (e.., i in R, R3, R4, and R5) Ex: Seach fo a line sement containin point Q R0: R R R: R3 R4 R: R5 R6 R3: a R4: d h R5: c i R6: e f R R3 h a i R6 e R4 d Q f R0 R R5 c Copyiht 007 y Hanan Samet 8

17 SEARCHING FOR A POINT OR LINE SEGMENT IN AN R-TREE v hi3 Dawack is that may have to examine many nodes since a line sement can e contained in the covein ectanles of many nodes yet its ecod is contained in only one leaf node (e.., i in R, R3, R4, and R5) Ex: Seach fo a line sement containin point Q R0: R R R: R3 R4 R: R5 R6 R3: a R4: d h R5: c i R6: e f R R3 h a i R6 e R4 d Q f R0 R R5 c Q is in R0 Copyiht 007 y Hanan Samet 8

18 SEARCHING FOR A POINT OR LINE SEGMENT IN AN R-TREE hi3 Dawack is that may have to examine many nodes since a line sement can e contained in the covein ectanles of many nodes yet its ecod is contained in only one leaf node (e.., i in R, R3, R4, and R5) Ex: Seach fo a line sement containin point Q 3 v R0: R R R: R3 R4 R: R5 R6 R3: a R4: d h R5: c i R6: e f R R3 h a i R6 e R4 d Q f R0 R R5 c Q is in R0 Q can e in oth R and R Copyiht 007 y Hanan Samet 8

19 SEARCHING FOR A POINT OR LINE SEGMENT IN AN R-TREE hi3 Dawack is that may have to examine many nodes since a line sement can e contained in the covein ectanles of many nodes yet its ecod is contained in only one leaf node (e.., i in R, R3, R4, and R5) Ex: Seach fo a line sement containin point Q 4 z 3 v R0: R R R: R3 R4 R: R5 R6 R3: a R4: d h R5: c i R6: e f R R3 h a i R6 e R4 d Q f R0 R R5 c Q is in R0 Q can e in oth R and R Seachin R fist means that R4 is seached ut this leads to failue even thouh Q is pat of i which is in R4 Copyiht 007 y Hanan Samet 8

20 SEARCHING FOR A POINT OR LINE SEGMENT IN AN R-TREE hi3 Dawack is that may have to examine many nodes since a line sement can e contained in the covein ectanles of many nodes yet its ecod is contained in only one leaf node (e.., i in R, R3, R4, and R5) Ex: Seach fo a line sement containin point Q 5 4 z 3 v R0: R R R: R3 R4 R: R5 R6 R3: a R4: d h R5: c i R6: e f R R3 h a i R6 e R4 d Q f R0 R R5 c Q is in R0 Q can e in oth R and R Seachin R fist means that R4 is seached ut this leads to failue even thouh Q is pat of i which is in R4 Seachin R finds that Q can only e in R5 Copyiht 007 y Hanan Samet 8

21 DISJOINT CELLS hi33 Ojects decomposed into disjoint suojects; each suoject in diffeent cell Techniques diffe in deee of eulaity 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 h d Q i e f c Copyiht 007 y Hanan Samet 9

22 DISJOINT CELLS hi33 Ojects decomposed into disjoint suojects; each suoject in diffeent cell Techniques diffe in deee of eulaity 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 R3 h R4 R5 a i e d Q f c R3: d h R4: c h i R5: c f i R6: a e i Copyiht 007 y Hanan Samet 9

23 DISJOINT CELLS 3 z hi33 Ojects decomposed into disjoint suojects; each suoject in diffeent cell Techniques diffe in deee of eulaity 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 R3 R h R4 R5 a i e d Q f c R: R3 R4 R: R5 R6 R3: d h R4: c h i R5: c f i R6: a e i Copyiht 007 y Hanan Samet 9

24 DISJOINT CELLS 4 3 z hi33 Ojects decomposed into disjoint suojects; each suoject in diffeent cell Techniques diffe in deee of eulaity 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 R3 R h R4 R5 a i e d Q f c R0: R R R: R3 R4 R: R5 R6 R3: d h R4: c h i R5: c f i R6: a e i Copyiht 007 y Hanan Samet 9

25 K-D-B-TREES hi33. Rectanula emeddin space is hieachically decomposed into disjoint ectanula eions No dead space in the sense that at any level of the tee, entie emeddin space is coveed y one of the nodes Blocks of k-d tee patition of space ae aeated into nodes of a finite capacity When a node oveflows, it is split alon one of the axes Oiinally developed to stoe points ut may e extended to non-point ojects epesented y thei minimum oundin oxes Dawack: in ode to detemine aea coveed y oject, must etieve all cells that it occupies a h d Q i e f c Copyiht 007 y Hanan Samet

26 K-D-B-TREES hi33. Rectanula emeddin space is hieachically decomposed into disjoint ectanula eions No dead space in the sense that at any level of the tee, entie emeddin space is coveed y one of the nodes Blocks of k-d tee patition of space ae aeated into nodes of a finite capacity When a node oveflows, it is split alon one of the axes Oiinally developed to stoe points ut may e extended to non-point ojects epesented y thei minimum oundin oxes Dawack: in ode to detemine aea coveed y oject, must etieve all cells that it occupies R3 R4 R6 a h d Q R5 i e f c R3: d h R4: c h i R5: c f i R6: Copyiht 007 y Hanan Samet a e i

27 K-D-B-TREES hi33. Rectanula emeddin space is hieachically decomposed into disjoint ectanula eions No dead space in the sense that at any level of the tee, entie emeddin space is coveed y one of the nodes Blocks of k-d tee patition of space ae aeated into nodes of a finite capacity When a node oveflows, it is split alon one of the axes Oiinally developed to stoe points ut may e extended to non-point ojects epesented y thei minimum oundin oxes Dawack: in ode to detemine aea coveed y oject, must etieve all cells that it occupies 3 z R3 R4 R6 R R a h d Q R5 i e f c R: R3 R4 R: R5 R6 R3: d h R4: c h i R5: c f i R6: Copyiht 007 y Hanan Samet a e i

28 K-D-B-TREES hi33. Rectanula emeddin space is hieachically decomposed into disjoint ectanula eions No dead space in the sense that at any level of the tee, entie emeddin space is coveed y one of the nodes Blocks of k-d tee patition of space ae aeated into nodes of a finite capacity When a node oveflows, it is split alon one of the axes Oiinally developed to stoe points ut may e extended to non-point ojects epesented y thei minimum oundin oxes Dawack: in ode to detemine aea coveed y oject, must etieve all cells that it occupies 4 3 z R3 R4 R6 R R R0 a h d Q R5 i e f c R0: R R R: R3 R4 R: R5 R6 R3: d h R4: c h i R5: c f i R6: Copyiht 007 y Hanan Samet a e i

29 hi34 UNIFORM GRID Ideal fo unifomly distiuted data Suppots set-theoetic opeations Spatial data (e.., line sement data) is aely unifomly distiuted Copyiht 007 y Hanan Samet 0

30 QUADTREES hi35 Hieachical vaiale esolution data stuctue ased on eula decomposition Many diffeent decomposition schemes and applicale to diffeent data types:. points. lines 3. eions 4. ectanles 5. sufaces 6. volumes 7. hihe dimensions includin time chanes meanin 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 insetin data Ex: eion quadtee A decomposition into locks not necessaily a tee! Copyiht 007 y Hanan Samet

31 hi36 REGION QUADTREE Repeatedly sudivide until otain homoeneous eion Fo a inay imae (BLACK and WHITE 0) Can also use fo multicoloed data (e.., a landuse class map associatin colos with cops) Can also define data stuctue fo ayscale imaes 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 coespondin 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 ood fo loaithmic 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 Copyiht 007 y Hanan Samet

32 hi37 PYRAMID Intenal nodes contain summay of infomation in nodes elow them Useful fo avoidin inspectin nodes whee thee could e no elevant infomation c c c3 c4 c5 c6 {c,c,c3,c4,c5,c6} {c6} {c,c3,c6} {c,c3,c4,c5} {c,c,c3, c4,c5,c6} Copyiht 007 y Hanan Samet 3

33 hi38 QUADTREES VS. PYRAMIDS Quadtees ae ood fo location-ased queies. e.., what is at location x?. not ood if lookin fo a paticula featue as have to examine evey lock o location askin ae you the one I am lookin fo? Pyamid is ood fo featue-ased queies e..,. does wheat exist in eion 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 eion x just look at the oot 3. select all locations whee wheat is own 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 meed {c,c,c3,c4,c5,c6} {c6} {c,c3,c6} {c,c3,c4,c5} {c,c,c3, c4,c5,c6} c c c3 c4 c5 c6 {c,c3,c5} {c,c,c3,c5} Can epesent as a list of leaf and nonleaf locks (e.., as a linea quadtee) Copyiht 007 y Hanan Samet 4

34 PR QUADTREE (Oenstein) Reula 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 viewin locks as uckets with capacity c and only decomposin the lock when it contains moe than c points Ex: c = (0,00) (00,00) (35,4) Chicao (0,0) (00,0) Copyiht 007 y Hanan Samet 5

35 PR QUADTREE (Oenstein) Reula 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 viewin locks as uckets with capacity c and only decomposin the lock when it contains moe than c points Ex: c = (0,00) (00,00) (35,4) Chicao (0,0) (5,0) Moile (00,0) Copyiht 007 y Hanan Samet 5

36 PR QUADTREE (Oenstein) Reula 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 viewin locks as uckets with capacity c and only decomposin the lock when it contains moe than c points Ex: c = (0,00) (00,00) 3 z (6,77) Toonto (35,4) Chicao (0,0) (5,0) Moile (00,0) Copyiht 007 y Hanan Samet 5

37 PR QUADTREE (Oenstein) Reula 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 viewin locks as uckets with capacity c and only decomposin the lock when it contains moe than c points Ex: c = (0,00) (00,00) 4 3 z (6,77) Toonto (35,4) Chicao (8,65) Buffalo (0,0) (5,0) Moile (00,0) Copyiht 007 y Hanan Samet 5

38 PR QUADTREE (Oenstein) Reula 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 viewin locks as uckets with capacity c and only decomposin the lock when it contains moe than c points Ex: c = (0,00) (00,00) 5 v 4 3 z (6,77) Toonto (8,65) Buffalo (5,45) Denve (35,4) Chicao (0,0) (5,0) Moile (00,0) Copyiht 007 y Hanan Samet 5

39 PR QUADTREE (Oenstein) Reula 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 viewin locks as uckets with capacity c and only decomposin the lock when it contains moe than c points Ex: c = (0,00) (00,00) 6 5 v 4 3 z (6,77) Toonto (8,65) Buffalo (5,45) Denve (35,4) Chicao (7,35) Omaha (0,0) (5,0) Moile (00,0) Copyiht 007 y Hanan Samet 5

40 PR QUADTREE (Oenstein) Reula 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 viewin locks as uckets with capacity c and only decomposin the lock when it contains moe than c points Ex: c = (0,00) (00,00) 7 z 6 5 v 4 3 z (6,77) Toonto (8,65) Buffalo (5,45) Denve (35,4) Chicao (7,35) Omaha (85,5) Atlanta (0,0) (5,0) Moile (00,0) Copyiht 007 y Hanan Samet 5

41 PR QUADTREE (Oenstein) Reula 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 viewin locks as uckets with capacity c and only decomposin the lock when it contains moe than c points Ex: c = (0,00) (00,00) 8 7 z 6 5 v 4 3 z (6,77) Toonto (8,65) Buffalo (5,45) Denve (35,4) Chicao (7,35) Omaha (85,5) Atlanta (0,0) (5,0) Moile (90,5) Miami (00,0) Copyiht 007 y Hanan Samet 5

42 REGION SEARCH hp0 Ex: Find all points within adius of point A A Use of quadtee esults in punin the seach space Copyiht 007 y Hanan Samet 6

43 REGION SEARCH hp0 Ex: Find all points within adius of point A A Use of quadtee esults in punin the seach space If a quadant sudivision point p lies in a eion l, then seach the quadants of p specified y l. SE 6. NE. All ut SW. SE, SW 7. NE, NW. All ut SE 3. SW 8. NW 3. All 4. SE, NE 9. All ut NW 5. SW, NW 0. All ut NE Copyiht 007 y Hanan Samet 6

44 REGION SEARCH 3 z hp0 Ex: Find all points within adius of point A A Use of quadtee esults in punin the seach space If a quadant sudivision point p lies in a eion l, then seach the quadants of p specified y l. SE 6. NE. All ut SW. SE, SW 7. NE, NW. All ut SE 3. SW 8. NW 3. All 4. SE, NE 9. All ut NW 5. SW, NW 0. All ut NE Copyiht 007 y Hanan Samet 6

45 FINDING THE NEAREST OBJECT Ex: find the neaest oject to P E C hp B 5 D P A 3 0 F Assume PR quadtee fo points (i.e., at most one point pe lock) Seach neihos of lock in counteclockwise ode Points ae soted with espect to the space they occupy which enales punin the seach space Aloithm: Copyiht 007 y Hanan Samet 7

46 FINDING THE NEAREST OBJECT Ex: find the neaest oject to P E C hp B 5 D P A 3 0 F Assume PR quadtee fo points (i.e., at most one point pe lock) Seach neihos of lock in counteclockwise ode Points ae soted with espect to the space they occupy which enales punin the seach space Aloithm:. stat at lock and compute distance to P fom A Copyiht 007 y Hanan Samet 7

47 FINDING THE NEAREST OBJECT Ex: find the neaest oject to P E C 3 z hp B 5 D P A 3 0 F Assume PR quadtee fo points (i.e., at most one point pe lock) Seach neihos of lock in counteclockwise ode Points ae soted with espect to the space they occupy which enales punin the seach space Aloithm:. stat at lock and compute distance to P fom A. inoe lock 3 whethe o not it is empty as A is close to P than any point in 3 Copyiht 007 y Hanan Samet 7

48 FINDING THE NEAREST OBJECT Ex: find the neaest oject to P E C 4 3 z hp B 5 D P A 3 0 F Assume PR quadtee fo points (i.e., at most one point pe lock) Seach neihos of lock in counteclockwise ode Points ae soted with espect to the space they occupy which enales punin the seach space Aloithm:. stat at lock and compute distance to P fom A. inoe lock 3 whethe o not it is empty as A is close to P than any point in 3 3. 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 Copyiht 007 y Hanan Samet 7

49 FINDING THE NEAREST OBJECT Ex: find the neaest oject to P E C 5 v 4 3 z hp B 5 D P A 3 0 F Assume PR quadtee fo points (i.e., at most one point pe lock) Seach neihos of lock in counteclockwise ode Points ae soted with espect to the space they occupy which enales punin the seach space Aloithm:. stat at lock and compute distance to P fom A. inoe lock 3 whethe o not it is empty as A is close to P than any point in 3 3. 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. inoe locks 6, 7, 8, 9, and 0 as the minimum distance to them fom P is eate than the distance fom P to A Copyiht 007 y Hanan Samet 7

50 FINDING THE NEAREST OBJECT Ex: find the neaest oject to P E C 6 z 5 v 4 3 z hp B 5 D P A 3 0 F Assume PR quadtee fo points (i.e., at most one point pe lock) Seach neihos of lock in counteclockwise ode Points ae soted with espect to the space they occupy which enales punin the seach space Aloithm:. stat at lock and compute distance to P fom A. inoe lock 3 whethe o not it is empty as A is close to P than any point in 3 3. 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. inoe locks 6, 7, 8, 9, and 0 as the minimum distance to them fom P is eate 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 Copyiht 007 y Hanan Samet 7

51 FINDING THE NEAREST OBJECT 7 6 z 5 v 4 3 z hp Ex: find the neaest oject to P E C B 5 D P A 3 0 new F F Assume PR quadtee fo points (i.e., at most one point pe lock) Seach neihos of lock in counteclockwise ode Points ae soted with espect to the space they occupy which enales punin the seach space Aloithm:. stat at lock and compute distance to P fom A. inoe lock 3 whethe o not it is empty as A is close to P than any point in 3 3. 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. inoe locks 6, 7, 8, 9, and 0 as the minimum distance to them fom P is eate 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 neiho of, as it is closest Copyiht 007 y Hanan Samet 7

52 PM QUADTREE cd3 Vetex-ased (one vetex pe lock) a DECOMPOSITION RULE: Patitionin occus when a lock contains moe than one sement unless all the sements ae incident at the same vetex which is also in the same lock Shape independent of ode of insetion Copyiht 007 y Hanan Samet 8

53 PM QUADTREE cd3 Vetex-ased (one vetex pe lock) a DECOMPOSITION RULE: Patitionin occus when a lock contains moe than one sement unless all the sements ae incident at the same vetex which is also in the same lock Shape independent of ode of insetion Copyiht 007 y Hanan Samet 8

54 PM QUADTREE 3 z cd3 Vetex-ased (one vetex pe lock) a c DECOMPOSITION RULE: Patitionin occus when a lock contains moe than one sement unless all the sements ae incident at the same vetex which is also in the same lock Shape independent of ode of insetion Copyiht 007 y Hanan Samet 8

55 PM QUADTREE 4 3 z cd3 Vetex-ased (one vetex pe lock) a d c DECOMPOSITION RULE: Patitionin occus when a lock contains moe than one sement unless all the sements ae incident at the same vetex which is also in the same lock Shape independent of ode of insetion Copyiht 007 y Hanan Samet 8

56 PM QUADTREE 5 v 4 3 z cd3 Vetex-ased (one vetex pe lock) a e d c DECOMPOSITION RULE: Patitionin occus when a lock contains moe than one sement unless all the sements ae incident at the same vetex which is also in the same lock Shape independent of ode of insetion Copyiht 007 y Hanan Samet 8

57 PM QUADTREE 6 5 v 4 3 z cd3 Vetex-ased (one vetex pe lock) a e d f c DECOMPOSITION RULE: Patitionin occus when a lock contains moe than one sement unless all the sements ae incident at the same vetex which is also in the same lock Shape independent of ode of insetion Copyiht 007 y Hanan Samet 8

58 PM QUADTREE 7 z 6 5 v 4 3 z cd3 Vetex-ased (one vetex pe lock) a e d f c DECOMPOSITION RULE: Patitionin occus when a lock contains moe than one sement unless all the sements ae incident at the same vetex which is also in the same lock Shape independent of ode of insetion Copyiht 007 y Hanan Samet 8

59 PM QUADTREE 8 7 z 6 5 v 4 3 z cd3 Vetex-ased (one vetex pe lock) a h e d f c DECOMPOSITION RULE: Patitionin occus when a lock contains moe than one sement unless all the sements ae incident at the same vetex which is also in the same lock Shape independent of ode of insetion Copyiht 007 y Hanan Samet 8

60 PM QUADTREE 9 v 8 7 z 6 5 v 4 3 z cd3 Vetex-ased (one vetex pe lock) a h e d i f c DECOMPOSITION RULE: Patitionin occus when a lock contains moe than one sement unless all the sements ae incident at the same vetex which is also in the same lock Shape independent of ode of insetion Copyiht 007 y Hanan Samet 8

61 PMR QUADTREE cd35 Ede-ased Avoids havin to split many times when two vetices o lines ae vey close as in PM quadtee Poailistic splittin and mein ules Uses a splittin theshold value say N Ex: N = a DECOMPOSITION RULE: Split a lock once if upon insetion the nume of sements intesectin a lock exceeds N Mee a lock with its silins if the total nume of line sements intesectin them is less than N Mees can e pefomed moe than once Does not uaantee that each lock will contain at most N line sements Splittin theshold is not the same as ucket capacity Shape depends on ode of insetion Copyiht 007 y Hanan Samet 9

62 cd35 PMR QUADTREE Ede-ased Avoids havin to split many times when two vetices o lines ae vey close as in PM quadtee Poailistic splittin and mein ules Uses a splittin theshold value say N Ex: N = a DECOMPOSITION RULE: Split a lock once if upon insetion the nume of sements intesectin a lock exceeds N Mee a lock with its silins if the total nume of line sements intesectin them is less than N Mees can e pefomed moe than once Does not uaantee that each lock will contain at most N line sements Splittin theshold is not the same as ucket capacity Shape depends on ode of insetion Copyiht 007 y Hanan Samet 9

63 cd35 PMR QUADTREE Ede-ased Avoids havin to split many times when two vetices o lines ae vey close as in PM quadtee Poailistic splittin and mein ules Uses a splittin theshold value say N Ex: N = 3 z a c DECOMPOSITION RULE: Split a lock once if upon insetion the nume of sements intesectin a lock exceeds N Mee a lock with its silins if the total nume of line sements intesectin them is less than N Mees can e pefomed moe than once Does not uaantee that each lock will contain at most N line sements Splittin theshold is not the same as ucket capacity Shape depends on ode of insetion Copyiht 007 y Hanan Samet 9

64 cd35 PMR QUADTREE Ede-ased Avoids havin to split many times when two vetices o lines ae vey close as in PM quadtee Poailistic splittin and mein ules Uses a splittin theshold value say N Ex: N = 4 3 z a d c DECOMPOSITION RULE: Split a lock once if upon insetion the nume of sements intesectin a lock exceeds N Mee a lock with its silins if the total nume of line sements intesectin them is less than N Mees can e pefomed moe than once Does not uaantee that each lock will contain at most N line sements Splittin theshold is not the same as ucket capacity Shape depends on ode of insetion Copyiht 007 y Hanan Samet 9

65 cd35 PMR QUADTREE Ede-ased Avoids havin to split many times when two vetices o lines ae vey close as in PM quadtee Poailistic splittin and mein ules Uses a splittin theshold value say N Ex: N = 5 v 4 3 z a e d c DECOMPOSITION RULE: Split a lock once if upon insetion the nume of sements intesectin a lock exceeds N Mee a lock with its silins if the total nume of line sements intesectin them is less than N Mees can e pefomed moe than once Does not uaantee that each lock will contain at most N line sements Splittin theshold is not the same as ucket capacity Shape depends on ode of insetion Copyiht 007 y Hanan Samet 9

66 cd35 PMR QUADTREE Ede-ased Avoids havin to split many times when two vetices o lines ae vey close as in PM quadtee Poailistic splittin and mein ules Uses a splittin theshold value say N Ex: N = 6 5 v 4 3 z a e d f c DECOMPOSITION RULE: Split a lock once if upon insetion the nume of sements intesectin a lock exceeds N Mee a lock with its silins if the total nume of line sements intesectin them is less than N Mees can e pefomed moe than once Does not uaantee that each lock will contain at most N line sements Splittin theshold is not the same as ucket capacity Shape depends on ode of insetion Copyiht 007 y Hanan Samet 9

67 cd35 PMR QUADTREE Ede-ased Avoids havin to split many times when two vetices o lines ae vey close as in PM quadtee Poailistic splittin and mein ules Uses a splittin theshold value say N Ex: N = 7 z 6 5 v 4 3 z a e d f c DECOMPOSITION RULE: Split a lock once if upon insetion the nume of sements intesectin a lock exceeds N Mee a lock with its silins if the total nume of line sements intesectin them is less than N Mees can e pefomed moe than once Does not uaantee that each lock will contain at most N line sements Splittin theshold is not the same as ucket capacity Shape depends on ode of insetion Copyiht 007 y Hanan Samet 9

68 cd35 PMR QUADTREE Ede-ased Avoids havin to split many times when two vetices o lines ae vey close as in PM quadtee Poailistic splittin and mein ules Uses a splittin theshold value say N Ex: N = 8 7 z 6 5 v 4 3 z h a e d f c DECOMPOSITION RULE: Split a lock once if upon insetion the nume of sements intesectin a lock exceeds N Mee a lock with its silins if the total nume of line sements intesectin them is less than N Mees can e pefomed moe than once Does not uaantee that each lock will contain at most N line sements Splittin theshold is not the same as ucket capacity Shape depends on ode of insetion Copyiht 007 y Hanan Samet 9

69 cd35 PMR QUADTREE Ede-ased Avoids havin to split many times when two vetices o lines ae vey close as in PM quadtee Poailistic splittin and mein ules Uses a splittin theshold value say N Ex: N = 9 v 8 7 z 6 5 v 4 3 z h a e d i f c DECOMPOSITION RULE: Split a lock once if upon insetion the nume of sements intesectin a lock exceeds N Mee a lock with its silins if the total nume of line sements intesectin them is less than N Mees can e pefomed moe than once Does not uaantee that each lock will contain at most N line sements Splittin theshold is not the same as ucket capacity Shape depends on ode of insetion Copyiht 007 y Hanan Samet 9

70 cd46 ADVANTAGES OF EDGE-BASED METHODS The decomposition is focussed whee the line sements ae the densest Handles the situation that seveal non-intesectin lines ae vey close to each othe Example: PM PMR (N=) Good aveae ehavio in tems of node occupancy Ale to epesent nonplana line sement data Copyiht 007 y Hanan Samet 0

71 CONSISTENCY OF PM APPROACH cd47 Stoes lines exactly. not a diitized epesentation. no thickness associated with line sements Updates can e made in a consistent manne - i.e., when a vecto featue is deleted, the dataase can e estoed to the state it would have een in had the deleted featue neve een inseted Each line sement is epesented y a pointe to a ecod containin its endpoints Uses the decomposition induced y the quadtee to specify what pats of the line sements (i.e., q-edes) ae actually pesent The line sement descipto stoed in a lock only implies the pesence of the coespondin q-ede - it does not mean that the entie line sement is pesent as a lineal featue Useful fo epesentin faments of lines such as those that esult fom the intesection of a line map with an aea map Ex: Copyiht 007 y Hanan Samet

72 CONSISTENCY OF PM APPROACH cd47 Stoes lines exactly. not a diitized epesentation. no thickness associated with line sements Updates can e made in a consistent manne - i.e., when a vecto featue is deleted, the dataase can e estoed to the state it would have een in had the deleted featue neve een inseted Each line sement is epesented y a pointe to a ecod containin its endpoints Uses the decomposition induced y the quadtee to specify what pats of the line sements (i.e., q-edes) ae actually pesent The line sement descipto stoed in a lock only implies the pesence of the coespondin q-ede - it does not mean that the entie line sement is pesent as a lineal featue Useful fo epesentin faments of lines such as those that esult fom the intesection of a line map with an aea map Ex: Copyiht 007 y Hanan Samet

73 CONSISTENCY OF PM APPROACH 3 z cd47 Stoes lines exactly. not a diitized epesentation. no thickness associated with line sements Updates can e made in a consistent manne - i.e., when a vecto featue is deleted, the dataase can e estoed to the state it would have een in had the deleted featue neve een inseted Each line sement is epesented y a pointe to a ecod containin its endpoints Uses the decomposition induced y the quadtee to specify what pats of the line sements (i.e., q-edes) ae actually pesent The line sement descipto stoed in a lock only implies the pesence of the coespondin q-ede - it does not mean that the entie line sement is pesent as a lineal featue Useful fo epesentin faments of lines such as those that esult fom the intesection of a line map with an aea map Ex: Copyiht 007 y Hanan Samet

74 cd58 MAXIMAL SEARCH RADIUS Popeties of the PM quadtee family (PM, PMR, etc.) eatly localize the seach aea fo neaest line sement Assume that the quey point P falls in the SW cone of the small hihlihted lock P By vitue of the existence of a lock of this size, we ae uaanteed that at least one of the emainin silins contains a line sement Copyiht 007 y Hanan Samet

75 MAXIMAL SEARCH RADIUS cd58 Popeties of the PM quadtee family (PM, PMR, etc.) eatly localize the seach aea fo neaest line sement Assume that the quey point P falls in the SW cone of the small hihlihted lock P By vitue of the existence of a lock of this size, we ae uaanteed that at least one of the emainin thee silins contains a line sement Copyiht 007 y Hanan Samet

76 MAXIMAL SEARCH RADIUS 3 z cd58 Popeties of the PM quadtee family (PM, PMR, etc.) eatly localize the seach aea fo neaest line sement Assume that the quey point P falls in the SW cone of the small hihlihted lock P By vitue of the existence of a lock of this size, we ae uaanteed that at least one of the emainin thee silins contains a line sement Copyiht 007 y Hanan Samet

77 NEAREST LINE SEGMENT ALGORITHM A fou stae intellient seach pocess cd59 Maximal seach adius equal to lenth of paent node's diaonal Basic aloithm:. Seach the lock containin the quey point Copyiht 007 y Hanan Samet 3

78 NEAREST LINE SEGMENT ALGORITHM A fou stae intellient seach pocess cd59 Maximal seach adius equal to lenth of paent node's diaonal Basic aloithm:. Seach the lock containin the quey point. Seach the thee silins Copyiht 007 y Hanan Samet 3

79 NEAREST LINE SEGMENT ALGORITHM A fou stae intellient seach pocess 3 z cd59 Maximal seach adius equal to lenth of paent node's diaonal Basic aloithm:.. 3. Seach the lock containin the quey point Seach the thee silins Seach the thee eions of size equal to that of the paent that ae incident to the lock containin the quey point Copyiht 007 y Hanan Samet 3

80 NEAREST LINE SEGMENT ALGORITHM A fou stae intellient seach pocess 4 3 z cd59 Maximal seach adius equal to lenth of paent node's diaonal Basic aloithm: Seach the lock containin the quey point Seach the thee silins Seach the thee eions of size equal to that of the paent that ae incident to the lock containin the quey point Seach the final fou oups of two adjacent locks to the pevious step Copyiht 007 y Hanan Samet 3

81 MX-CIF QUADTREE (Kedem).. 3. Collections of small ectanles fo VLSI applications Each ectanle is associated with its minimum enclosin quadtee lock hp4 Like hashin: quadtee locks seve as hash uckets 4 5 B 3 6 E C A D 0 F A {,6,7,8,9,0} B {} C {} D {} E {3,4,5} F {} Copyiht 007 y Hanan Samet 4

82 MX-CIF QUADTREE (Kedem) Collections of small ectanles fo VLSI applications Each ectanle is associated with its minimum enclosin quadtee lock hp4 Like hashin: quadtee locks seve as hash uckets Collision = moe than one ectanle in a lock esolve y usin two one-dimensional MX-CIF tees to stoe the ectanle intesectin the lines passin thouh each sudivision point 4 5 B 3 6 E C A D 0 F A {,6,7,8,9,0} B {} C {} D {} E {3,4,5} F {} Copyiht 007 y Hanan Samet 4

83 MX-CIF QUADTREE (Kedem) Collections of small ectanles fo VLSI applications Each ectanle is associated with its minimum enclosin quadtee lock hp4 Like hashin: quadtee locks seve as hash uckets Collision = moe than one ectanle in a lock esolve y usin two one-dimensional MX-CIF tees to stoe the ectanle intesectin the lines passin thouh each sudivision point one fo y-axis Binay tee fo y- axis thouh A 7 B A E C 9 Y Y 0 Y4 8 6 Y6 Y3 Y5 Y7 D 0 F A {,6,7,8,9,0} B {} C {} D {} E {3,4,5} F {} Copyiht 007 y Hanan Samet 4

84 MX-CIF QUADTREE (Kedem) Collections of small ectanles fo VLSI applications Each ectanle is associated with its minimum enclosin quadtee lock hp4 Like hashin: quadtee locks seve as hash uckets Collision = moe than one ectanle in a lock esolve y usin two one-dimensional MX-CIF tees to stoe the ectanle intesectin the lines passin thouh each sudivision point one fo y-axis if a ectanle intesects oth x and y axes, then associate it with the y axis 4 v Binay tee fo y- axis thouh A 7 B A E C 9 Y Y 0 Y4 8 6 Y6 Y3 Y5 Y7 D 0 F A {,6,7,8,9,0} B {} C {} D {} E {3,4,5} F {} Copyiht 007 y Hanan Samet 4

85 MX-CIF QUADTREE (Kedem) Collections of small ectanles fo VLSI applications Each ectanle is associated with its minimum enclosin quadtee lock hp4 Like hashin: quadtee locks seve as hash uckets Collision = moe than one ectanle in a lock esolve y usin two one-dimensional MX-CIF tees to stoe the ectanle intesectin the lines passin thouh each sudivision point one fo y-axis if a ectanle intesects oth x and y axes, then associate it with the y axis one fo x-axis 5 z 4 v Binay tee fo y- axis thouh A 7 B A E D C 9 Y Y 0 Y4 8 6 Y6 Y3 Y5 Y7 Binay tee fo x- axis thouh A 0 F X4 X X 9 X5 X3 A {,6,7,8,9,0} X6 7 B {} C {} D {} E {3,4,5} F {} Copyiht 007 y Hanan Samet 4

86 HIERARCHICAL RECTANGULAR DECOMPOSITION Simila to tianula decomposition Good when data points ae the vetices of a ectanula id Dawack is asence of continuity etween adjacent patches of unequal width (temed the alinment polem) sf Ovecomin the pesence of cacks. use the intepolated point instead of the tue point (Baea and Hinjosa). tianulate the squaes (Von Hezen and Ba) can split into, 4, o 8 tianles dependin on how many lines ae dawn thouh the midpoint if split into tianles, then cacks still emain no cacks if split into 4 o 8 tianles Copyiht 007 y Hanan Samet 5

87 RESTRICTED QUADTREE (VON HERZEN/BARR) sf3 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) Copyiht 007 y Hanan Samet 6

88 RESTRICTED QUADTREE (VON HERZEN/BARR) sf3 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) Copyiht 007 y Hanan Samet 6

89 RESTRICTED QUADTREE (VON HERZEN/BARR) 3 z sf3 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-tianle decomposition ule. decompose each lock into 8 tianles (i.e., tianles pe ede). unless the ede is shaed y a lae lock 3. in which case only tianle is fomed Copyiht 007 y Hanan Samet 6

90 RESTRICTED QUADTREE (VON HERZEN/BARR) sf3 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 3 z 8-tianle decomposition ule. decompose each lock into 8 tianles (i.e., tianles pe ede). unless the ede is shaed y a lae lock 3. in which case only tianle is fomed 4-tianle decomposition ule. decompose each lock into 4 tianles (i.e., tianle pe ede). unless the ede is shaed y a smalle lock 3. in which case tianles ae fomed alon the ede Copyiht 007 y Hanan Samet 6

91 RESTRICTED QUADTREE (VON HERZEN/BARR) sf3 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 3 z 8-tianle decomposition ule. decompose each lock into 8 tianles (i.e., tianles pe ede). unless the ede is shaed y a lae lock 3. in which case only tianle is fomed 4-tianle decomposition ule. decompose each lock into 4 tianles (i.e., tianle pe ede). unless the ede is shaed y a smalle lock 3. in which case tianles ae fomed alon the ede Pefe 8-tianle ule as it is ette fo display applications (shadin) Copyiht 007 y Hanan Samet 6

92 sf4 PROPERTY SPHERES (FEKETE) Appoximation of spheical data Uses icosahedon which is a Platonic solid. 0 faces each is a eula tianle. laest possile eula polyhedon Copyiht 007 y Hanan Samet 7

93 sf5 ALTERNATIVE SPHERICAL APPROXIMATIONS Could use othe Platonic solids. all have faces that ae eula polyons tetahedon: 4 equilateal tianula faces hexahedon: 6 squae faces octahedon: 8 equilateal tianula faces dodecahedon: pentaonal faces. octahedon is nice fo modelin the loe it can e alined 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 lonitude values. not so ood if want a patition into units of equal aea as eat polems aound the poles. poject sphee onto plane usin Lamet s cylindical pojection which is locally aea pesevin Instead of appoximatin sphee with the solids, poject the faces of the solids on the sphee (Scott). all edes ecome su-acs of a eat cicle. use eula decomposition on tianula, squae, o pentaonal spheical suface patches Copyiht 007 y Hanan Samet 8

94 hi60 OCTREES. Inteio (voxels) analoous to eion quadtee appoximate oject y aeatin simila voxels ood fo medical imaes 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 ede c. moe than one ede ut all meet at same vetex impose a spatial index on a ounday model (BRep) Copyiht 007 y Hanan Samet 9

95 hi39 EXAMPLE QUADTREE-BASED QUERY Quey: find all cities with population in excess of 5,000 in wheat owin eions within 0 miles of the Mississippi Rive. assume ive is a linea featue use a line map could e a eion if asked fo sandas in the ive. eion map fo the wheat 3. assume cities ae points point map fo cities could e eion is asked fo hih income aeas Comines spatial and non-spatial (i.e., attiute) data Many possile execution plans - e..,. compute uffe o coido aound ive. extact wheat aea 3. intesect with 4. intesect city map with 3 5. etieve value of population attiute fo cities in 4 fom the nonspatial dataase (e.., elational) Reula decomposition hieachical data stuctues such as the quadtee. all maps ae in eistation 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.., 3 and 4) Copyiht 007 y Hanan Samet

96 zk6 SAND BROWSER: A SPATIO-RELATIONAL BROWSER Assume a elational dataase Relations have spatial and nonspatial attiutes Bowse thouh tuples o ojects (oups of tuples with simila attiute values) of a elation one at a time accodin to values within anes of the. nonspatial attiutes. undelyin space in which the ojects coespondin to the spatial attiutes ae emedded Make use of indexes to facilitate viewin (temed ankin) tuples in ode of neaness to a efeence attiute value (e.., zeo, oiin, etc.) and otain tuples in this ode Gaphical use inteface instead of SQL ut functionally equivalent Gaphical esult of spatial and nonspatial queies Output. display tuples satisfyin the quey one tuple o one oject at a time show the values of all of the attiutes of the most ecently eneated tuple cuso points at this tuple. cumulative display of spatial attiutes Can save the esult of an opeation as a elation fo futue opeations (SAVE GROUP) Copyiht 007 y Hanan Samet 3