1. Temple University – CIS Dept. CIS616– Principles of Data Management V. Megalooikonomou Spatial Access Methods (SAMs) ( based on notes by Silberchatz,Korth, and Sudarshan and notes by C. Faloutsos at CMU )

2. General Overview Multimedia Indexing Spatial Access Methods (SAMs) k-d trees Point Quadtrees MX-Quadtree z-ordering R-trees

3. SAMs - Detailed outline spatial access methods problem dfn k-d trees point quadtrees MX-quadtrees z-ordering R-trees

4. Spatial Access Methods - problem Given a collection of geometric objects (points, lines, polygons,...) organize them on disk, to answer spatial queries (like??)

5. Spatial Access Methods - problem Given a collection of geometric objects (points, lines, polygons,...) organize them on disk, to answer point queries range queries k-nn queries spatial joins (‘all pairs’ queries)

6. Spatial Access Methods - problem Given a collection of geometric objects (points, lines, polygons,...) organize them on disk, to answer point queries range queries k-nn queries spatial joins (‘all pairs’ queries)

7. Spatial Access Methods - problem Given a collection of geometric objects (points, lines, polygons,...) organize them on disk, to answer point queries range queries k-nn queries spatial joins (‘all pairs’ queries)

8. Spatial Access Methods - problem Given a collection of geometric objects (points, lines, polygons,...) organize them on disk, to answer point queries range queries k-nn queries spatial joins (‘all pairs’ queries)

9. Spatial Access Methods - problem Given a collection of geometric objects (points, lines, polygons,...) organize them on disk, to answer point queries range queries k-nn queries spatial joins (‘all pairs’ within ε)

10. SAMs - motivation Q: applications?

11. SAMs - motivation salary age traditional DB GIS

12. SAMs - motivation salary age traditional DB GIS

13. SAMs - motivation CAD/CAM find elements too close to each other

14. SAMs - motivation CAD/CAM

15. day 1365 day 1365 S1 Sn F(S1) F(Sn) SAMs - motivation eg, avg eg,. std

16. SAMs: solutions K-d trees point quadtrees MX-quadtrees z-ordering R-trees (grid files) Q: how would you organize, e.g., n-dim points, on disk? (C points per disk page)

17. SAMs - Detailed outline spatial access methods problem dfn k-d trees point quadtrees MX-quadtrees z-ordering R-trees

18. k-d trees Used to store k dimensional point data It is not used to store region data A 2-d tree (i.e., for k=2) stores 2-dimensional point data while a 3-d tree stores 3- dimensional point data, etc.

19. 2-d trees – node structure Binary trees Info: information field Xval,Yval: coordinates of a point associated with the node Llink, Rlink: pointers to children Properties (N: node): If level N even -> for all nodes M in the subtree rooted at N.Llink: M.Xval < N.Xval for all nodes P in the subtree rooted at N.Rlink: P.Xval >= N.Xval If level N odd -> Similarly use Yvals

20. 2-d trees – Example

21. 2-d trees: Insertion/Search To insert a node N into the tree pointed by T If N and T agree on Xval, Yval then overwrite T Else, branch left if N.Xval < T.xval, right otherwise (even levels) Similarly for odd levels (branching on Yvals)

22. 2-d trees – Example of Insertion City(Xval, Yval) Banja Luka(19, 45) Derventa(40, 50) Toslic(38, 38) Tuzla(54, 35) Sinj(4, 4) Splitting of region by Banja LukaSplitting of region by Derventa Splitting of region by ToslicSplitting of region by Sinj

23. 2-d trees: Deletion Deletion of point (x,y) from T If N is a leaf node easy Otherwise either T l (left subtree) or T r (right subtree) is non-empty Find a “candidate replacement” node R in T l or T r Replace all of N’s non-link fields by those of R Recursively delete R from T i Recursion guaranteed to terminate - Why?

24. 2-d trees: Deletion Finding candidate replacement nodes for deletion Replacement node R must bear same spatial relation to all nodes in T l and T r as node N

25. 2-d trees: Range Queries Q: Given a point (x c, y c ) and a distance r find all points in the 2-d tree that lie within the circle A: Each node N in a 2-d tree implicitly represents a region R N – If the circle (specified by the query) has no intersection with R N then there is no point in searching the subtree rooted at node N

26. SAMs - Detailed outline spatial access methods problem dfn k-d trees point quadtrees z-ordering R-trees

27. Point Quadtrees Represent point data Always split regions into 4 parts 2-d tree: a node N splits a region into two by drawing one line through the point (N.xval, N.yval) Point quadtree: a node N splits a region by drawing a horizontal and a vertical line through the point (N.xval, N.yval) Four parts: NW, SW, NE, and SE quadrants Q: Quadtree nodes have 4 children?

28. Point Quadtrees Nodes in point quadtrees represent regions

29. Point quadtrees - Insertion City(Xval, Yval) Banja Luka(19, 45) Derventa(40, 50) Toslic(38, 38) Tuzla(54, 35) Sinj(4, 4) Splitting of region by Banja LukaSplitting of region by Derventa Splitting of region by ToslicSplitting of region by SinjSplitting of region by Tuzla

30. Point Quadtrees - Insertion

31. Point quadtrees: Deletion Deletion of point (x,y) from T If N is a leaf node easy Otherwise a subtree (N.NW, N.SW, N.NE. N.SE) is non- empty Find a “candidate replacement” node R in one of the subtrees such that: Every other node R1 in N.NW is to the NW of R Every other node R2 in N.SW is to the SW of R etc… Replace all of N’s non-link fields by those of R Recursively delete R from T i In general, it may not always be possible to find such as replacement node Q: What happens in the worst case?

32. Point quadtrees: Deletion Deletion of point (x,y) from T If N is a leaf node easy Otherwise a subtree (N.NW, N.SW, N.NE. N.SE) is non- empty Find a “candidate replacement” node R in one of the subtrees such that: Every other node R1 in N.NW is to the NW of R Every other node R2 in N.SW is to the SW of R etc… Replace all of N’s non-link fields by those of R Recursively delete R from T i In general, it may not always be possible to find such as replacement node Q: What happens in the worst case? May require all nodes to be reinserted

33. Point quadtrees: Range Searches Each node in a point quadtree represents a region Do not search regions that do not intersect the circle defined by the query

34. SAMs - Detailed outline spatial access methods problem dfn k-d trees point quadtrees MX-quadtrees z-ordering R-trees

35. MX-Quadtrees Drawbacks of 2-d trees, point quadtrees: shape of tree depends upon the order in which objects are inserted into the tree splits may be uneven depending upon where the point (N.xval, N.yval) is located inside the region (represented by N) MX-quadtrees: shape (and height) of tree independent of number of nodes and order of insertion

36. MX-Quadtrees Assumption: the map is represented as a grid of size (2 k x 2 k ) for some k When a region gets “split” it splits down the middle

37. MX-Quadtrees - Insertion After insertion of A, B, C, and D respectively

38. MX-Quadtrees - Insertion After insertion of A, B, C, and D respectively

39. MX-Quadtrees - Deletion Fairly easy – why? All point are represented at the leaf level Total time for deletion: O(k)

40. MX-Quadtrees –Range Queries Same as in point quadtrees One difference: Checking to see if a point is in the circle defined by the range query needs to be performed at the leaf level (points are stored at the leaf level)

41. SAMs - Detailed outline spatial access methods problem dfn k-d trees point quadtrees MX-quadtrees z-ordering R-trees

42. z-ordering Q: how would you organize, e.g., n-dim points, on disk? (C points per disk page) Hint: reduce the problem to 1-d points(!!) Q1: why? A: Q2: how?

43. z-ordering Q: how would you organize, e.g., n-dim points, on disk? (C points per disk page) Hint: reduce the problem to 1-d points (!!) Q1: why? A: B-trees! Q2: how?

44. z-ordering Q2: how? A: assume finite granularity; z-ordering = bit-shuffling = N-trees = Morton keys = geo-coding =...

45. z-ordering Q2: how? A: assume finite granularity (e.g., 2 32 x2 32 ; 4x4 here) Q2.1: how to map n-d cells to 1-d cells?

46. z-ordering Q2.1: how to map n-d cells to 1-d cells?

47. z-ordering Q2.1: how to map n-d cells to 1-d cells? A: row-wise Q: is it good?

48. z-ordering Q: is it good? A: great for ‘x’ axis; bad for ‘y’ axis

49. z-ordering Q: How about the ‘snake’ curve?

50. z-ordering Q: How about the ‘snake’ curve? A: still problems: 2^32

51. z-ordering Q: Why are those curves ‘bad’? A: no distance preservation (~ clustering) Q: solution? 2^32

52. z-ordering Q: solution? (w/ good clustering, and easy to compute, for 2-d and n-d?)

53. z-ordering Q: solution? (w/ good clustering, and easy to compute, for 2-d and n-d?) A: z-ordering/bit-shuffling/linear- quadtrees ‘looks’ better: few long jumps; scoops out the whole quadrant before leaving it a.k.a. space filling curves

54. z-ordering z-ordering/bit-shuffling/linear-quadtrees Q: How to generate this curve (z = f(x,y) )? A: 3 (equivalent) answers!

55. z-ordering z-ordering/bit-shuffling/linear-quadtrees Q: How to generate this curve (z = f(x,y))? A1: ‘z’ (or ‘N’) shapes, RECURSIVELY order-1 order-2... order (n+1)

56. z-ordering Notice: self similar (we’ll see about fractals, soon) method is hard to use: z =? f(x,y) order-1 order-2... order (n+1)

57. z-ordering z-ordering/bit-shuffling/linear-quadtrees Q: How to generate this curve (z = f(x,y) )? A: 3 (equivalent) answers! Method #2?

58. z-ordering bit-shuffling 00 01 10 11 10 01 00 x y x 0 y 1 z =( 0 1 0 1 ) 2 = 5

59. z-ordering bit-shuffling 00 01 10 11 10 01 00 x y x 0 y 1 z =( 0 1 0 1 ) 2 = 5 How about the reverse: (x,y) = g(z) ?

60. z-ordering bit-shuffling 00 01 10 11 10 01 00 x y x 0 y 1 z =( 0 1 0 1 ) 2 = 5 How about n-d spaces?

61. z-ordering z-ordering/bit-shuffling/linear-quadtrees Q: How to generate this curve (z = f(x,y) )? A: 3 (equivalent) answers! Method #3?

62. z-ordering linear-quadtrees : assign N->1, S->0 e.t.c. 0 1 0 1 00... 01... 10... 11... W E NSNS

63. z-ordering... and repeat recursively. Eg.: z gray-cell = WN;WN = (0101) 2 = 5 0 1 0 1 00... 01... 10... 11... W E NSNS 00 11

64. z-ordering Drill: z-value of grey cell, with the three methods? 0 1 0 1 W E NSNS

65. z-ordering Drill: z-value of grey cell, with the three methods? 0 1 0 1 W E NSNS method#1: 14 method#2: shuffle(11;10)= (1110) 2 = 14

66. z-ordering Drill: z-value of grey cell, with the three methods? 0 1 0 1 W E NSNS method#1: 14 method#2: shuffle(11;10)= (1110) 2 = 14 method#3: EN;ES =... = 14

67. z-ordering - Detailed outline spatial access methods z-ordering main idea - 3 methods use w/ B-trees; algorithms (range, knn queries...) non-point (eg., region) data analysis; variations R-trees

68. z-ordering - usage & algo’s Q1: How to store on disk? A: Q2: How to answer range queries etc

69. z-ordering - usage & algo’s Q1: How to store on disk? A: treat z-value as primary key; feed to B-tree SF PGH

70. z-ordering - usage & algo’s MAJOR ADVANTAGES w/ B-tree: already inside commercial systems (no coding /debugging!) concurrency & recovery is ready SF PGH

71. z-ordering - Detailed outline spatial access methods z-ordering main idea - 3 methods use w/ B-trees; algorithms (range, knn queries...) non-point (eg., region) data analysis; variations R-trees

72. z-ordering - variations Q: is z-ordering the best we can do?

73. z-ordering - variations Q: is z-ordering the best we can do? A: probably not - occasional long ‘jumps’ Q: then?

74. z-ordering - variations Q: is z-ordering the best we can do? A: probably not - occasional long ‘jumps’ Q: then? A1: Gray codes

75. z-ordering - variations A2: Hilbert curve! (a.k.a. Hilbert-Peano curve)

76. z-ordering - variations ‘Looks’ better (never long jumps). How to derive it?

77. z-ordering - variations ‘Looks’ better (never long jumps). How to derive it? order-1order-2... order (n+1)

78. z-ordering - variations Q: function for the Hilbert curve ( h = f(x,y) )? A: bit-shuffling, followed by post-processing, to account for rotations. Linear on # bits. See textbook, for pointers to code/algorithms (eg., [Jagadish, 90])

79. z-ordering - variations Q: how about Hilbert curve in 3-d? n-d? A: Exists (and is not unique!). Eg., 3-d, order-1 Hilbert curves (Hamiltonian paths on cube) #1 #2

80. z-ordering - Detailed outline spatial access methods z-ordering main idea - 3 methods use w/ B-trees; algorithms (range, knn queries...) non-point (eg., region) data analysis; variations R-trees...

81. z-ordering - analysis Q: How many pieces (‘quad-tree blocks’) per region? A: proportional to perimeter (surface etc)

82. z-ordering - analysis (How long is the coastline, say, of England? Paradox: The answer changes with the yard- stick -> fractals...)

83. z-ordering - analysis Q: Should we decompose a region to full detail (and store in B-tree)?

84. z-ordering - analysis Q: Should we decompose a region to full detail (and store in B-tree)? A: NO! approximation with 1-3 pieces/z- values is best [Orenstein90]

85. z-ordering - analysis Q: how to measure the ‘goodness’ of a curve?

86. z-ordering - analysis Q: how to measure the ‘goodness’ of a curve? A: e.g., avg. # of runs, for range queries 4 runs3 runs (#runs ~ #disk accesses on B-tree)

87. z-ordering - analysis Q: So, is Hilbert really better? A: 27% fewer runs, for 2-d (similar for 3-d) Q: are there formulas for #runs, #of quadtree blocks etc? A: Yes ([Jagadish; Moon+ etc] see textbook)

88. z-ordering - fun observations Hilbert and z-ordering curves: “space filling curves”: eventually, they visit every point in n-d space - therefore: order-1 order-2... order (n+1)

89. z-ordering - fun observations... they show that the plane has as many points as a line (-> headaches for 1900’s mathematics/topology). (fractals, again!) order-1 order-2... order (n+1)

90. z-ordering - fun observations Observation #2: Hilbert (like) curve for video encoding [Y. Matias+, CRYPTO ‘87]: Given a frame, visit its pixels in randomized hilbert order; compress; and transmit

91. z-ordering - fun observations In general, Hilbert curve is great for preserving distances, clustering, vector quantization etc

92. Conclusions z-ordering is a great idea (n-d points -> 1-d points; feed to B-trees) used by TIGER system and (most probably) by other GIS products works great with low-dim points

93. SAMs – Detailed Outline spatial access methods problem dfn k-d trees point quadtrees MX-quadtrees z-ordering R-trees

94. SAMs - more detailed outline R-trees main idea; file structure (algorithms: insertion/split) (deletion) (search: range, nn, spatial joins) variations (packed; hilbert;...)

95. R-trees z-ordering: cuts regions to pieces -> dup. elim. how could we avoid that? Idea: Minimum Bounding Rectangles

96. R-trees [Guttman 84] Main idea: allow parents to overlap! => guaranteed 50% utilization => easier insertion/split algorithms. (only deal with Minimum Bounding Rectangles - MBRs)

97. R-trees eg., w/ fanout 4: group nearby rectangles to parent MBRs; each group -> disk page A B C D E F G H I J

98. R-trees eg., w/ fanout 4: A B C D E F G H I J P1 P2 P3 P4 FGDE HIJ ABC

99. R-trees eg., w/ fanout 4: A B C D E F G H I J P1 P2 P3 P4 P1P2P3P4 FGDEHIJABC

100. R-trees - format of nodes {(MBR; obj-ptr)} for leaf nodes P1P2P3P4 ABC x-low; x-high y-low; y-high... obj ptr...

101. R-trees - format of nodes {(MBR; node-ptr)} for non-leaf nodes P1P2P3P4 ABC x-low; x-high y-low; y-high... node ptr...

102. R-trees - range search? A B C D E F G H I J P1 P2 P3 P4 P1P2P3P4 FGDEHIJABC

103. R-trees - range search? A B C D E F G H I J P1 P2 P3 P4 P1P2P3P4 FGDEHIJABC

104. R-trees - range search Observations: every parent node completely covers its ‘children’ a child MBR may be covered by more than one parent - it is stored under ONLY ONE of them. (i.e., no need for dup. elim.) a point query may follow multiple branches. everything works for any dimensionality

105. SAMs - more detailed outline R-trees main idea; file structure algorithms: insertion/split deletion search: range, nn, spatial joins performance analysis variations (packed; hilbert;...)

106. R-trees - insertion eg., rectangle ‘X’ A B C D E F G H I J P1 P2 P3 P4 P1P2P3P4 FGDEHIJABC X

107. R-trees - insertion eg., rectangle ‘X’ A B C D E F G H I J P1 P2 P3 P4 P1P2P3P4 FGDEHIJABC X X

108. R-trees - insertion eg., rectangle ‘Y’ A B C D E F G H I J P1 P2 P3 P4 P1P2P3P4 FGDEHIJABC Y

109. R-trees - insertion eg., rectangle ‘Y’: extend suitable parent. A B C D E F G H I J P1 P2 P3 P4 P1P2P3P4 FGDEHIJABC Y Y

110. R-trees - insertion eg., rectangle ‘Y’: extend suitable parent. Q: how to measure ‘suitability’?

111. R-trees - insertion eg., rectangle ‘Y’: extend suitable parent. Q: how to measure ‘suitability’? A: by increase in area (volume) (more details: later, under ‘performance analysis’) Q: what if there is no room? how to split?

112. R-trees - insertion eg., rectangle ‘W’ A B C D E F G H I J P1 P2 P3 P4 P1P2P3P4 FGDEHIJABC W K K

113. R-trees - insertion eg., rectangle ‘W’ - focus on ‘P1’ - how to split? A B C P1 W K

114. R-trees - insertion eg., rectangle ‘W’ - focus on ‘P1’ - how to split? A B C P1 W K (A1: plane sweep, until 50% of rectangles) A2: ‘linear’ split A3: quadratic split A4: exponential split

115. R-trees - insertion & split pick two rectangles as ‘seeds’; assign each rectangle ‘R’ to the ‘closest’ ‘seed’ seed1 seed2 R

116. R-trees - insertion & split pick two rectangles as ‘seeds’; assign each rectangle ‘R’ to the ‘closest’ ‘seed’ Q: how to measure ‘closeness’?

117. R-trees - insertion & split pick two rectangles as ‘seeds’; assign each rectangle ‘R’ to the ‘closest’ ‘seed’ Q: how to measure ‘closeness’? A: by increase of area (volume)

118. R-trees - insertion & split pick two rectangles as ‘seeds’; assign each rectangle ‘R’ to the ‘closest’ ‘seed’ seed1 seed2 R

119. R-trees - insertion & split pick two rectangles as ‘seeds’; assign each rectangle ‘R’ to the ‘closest’ ‘seed’ seed1 seed2 R

120. R-trees - insertion & split pick two rectangles as ‘seeds’; assign each rectangle ‘R’ to the ‘closest’ ‘seed’ smart idea: pre-sort rectangles according to delta of closeness (ie., schedule easiest choices first!)

121. R-trees - insertion - pseudocode decide which parent to put new rectangle into (‘closest’ parent) if overflow, split to two, using (say,) the quadratic split algorithm propagate the split upwards, if necessary update the MBRs of the affected parents.

122. R-trees - insertion - observations many more split algorithms exist (next!)

123. SAMs - more detailed outline R-trees main idea; file structure algorithms: insertion/split deletion search: range, nn, spatial joins performance analysis variations (packed; hilbert;...)

124. R-trees - deletion delete rectangle if underflow ??

125. R-trees - deletion delete rectangle if underflow temporarily delete all siblings (!); delete the parent node and re-insert them

126. SAMs - more detailed outline R-trees main idea; file structure algorithms: insertion/split deletion search: range, nn, spatial joins performance analysis variations (packed; hilbert;...)

127. R-trees - range search pseudocode: check the root for each branch, if its MBR intersects the query rectangle apply range-search (or print out, if this is a leaf)

128. R-trees - nn search A B C D E F G H I J P1 P2 P3 P4 q

129. R-trees - nn search Q: How? (find near neighbor; refine...) A B C D E F G H I J P1 P2 P3 P4 q

130. R-trees - nn search A1: depth-first search; then, range query A B C D E F G H I J P1 P2 P3 P4 q

131. R-trees - nn search A1: depth-first search; then, range query A B C D E F G H I J P1 P2 P3 P4 q

132. R-trees - nn search A1: depth-first search; then, range query A B C D E F G H I J P1 P2 P3 P4 q

133. R-trees - nn search A2: [Roussopoulos+, sigmod95]: priority queue, with promising MBRs, and their best and worst-case distance main idea:

134. R-trees - nn search A B C D E F G H I J P1 P2 P3 P4 q consider only P2 and P4, for illustration

135. R-trees - nn search D E H J P2 P4 q worst of P2 best of P4 => P4 is useless for 1-nn

136. R-trees - nn search D E P2 q worst of P2 what is really the worst of, say, P2?

137. R-trees - nn search P2 q what is really the worst of, say, P2? A: the smallest of the two red segments!

138. R-trees - nn search variations: [Hjaltason & Samet] incremental nn: build a priority queue scan enough of the tree, to make sure you have the k nn to find the (k+1)-th, check the queue, and scan some more of the tree ‘optimal’ (but, may need too much memory)

139. SAMs - more detailed outline R-trees main idea; file structure algorithms: insertion/split deletion search: range, nn, spatial joins performance analysis variations (packed; hilbert;...)

140. R-trees - spatial joins Spatial joins: find (quickly) all counties intersecting lakes

141. R-trees - spatial joins Assume that they are both organized in R- trees:

142. R-trees - spatial joins for each parent P1 of tree T1 for each parent P2 of tree T2 if their MBRs intersect, process them recursively (ie., check their children)

143. R-trees - spatial joins Improvements - variations: - [Seeger+, sigmod 92]: do some pre-filtering; do plane-sweeping to avoid N1 * N2 tests for intersection - [Lo & Ravishankar, sigmod 94]: ‘seeded’ R-trees (FYI, many more papers on spatial joins, without R-trees: [Koudas+ Sevcik], e.t.c.)

144. SAMs - more detailed outline R-trees main idea; file structure algorithms: insertion/split deletion search: range, nn, spatial joins variations (packed; hilbert;...)

145. R-trees - variations Guttman’s R-trees sparked much follow-up work can we do better splits? what about static datasets (no ins/del/upd)? what about other bounding shapes?

146. R-trees - variations Guttman’s R-trees sparked much follow-up work can we do better splits? i.e, defer splits?

147. R-trees - variations A: R*-trees [Kriegel+, SIGMOD90] defer splits, by forced-reinsert, i.e.: instead of splitting, temporarily delete some entries, shrink overflowing MBR, and re-insert those entries Which ones to re-insert? How many?

148. R-trees - variations A: R*-trees [Kriegel+, SIGMOD90] defer splits, by forced-reinsert, i.e.: instead of splitting, temporarily delete some entries, shrink overflowing MBR, and re-insert those entries Which ones to re-insert? How many? A: 30%

149. R-trees - variations Q: Other ways to defer splits?

150. R-trees - variations Q: Other ways to defer splits? A: Push a few keys to the closest sibling node (closest = ??)

151. R-trees - variations R*-trees: Also try to minimize area AND perimeter, in their split. Performance: higher space utilization; faster than plain R-trees. One of the most successful R-tree variants.

152. R-trees - variations Guttman’s R-trees sparked much follow-up work can we do better splits? what about static datasets (no ins/del/upd)? Hilbert R-trees what about other bounding shapes?

153. R-trees - variations what about static datasets (no ins/del/upd)? Q: Best way to pack points?

154. R-trees - variations what about static datasets (no ins/del/upd)? Q: Best way to pack points? A1: plane-sweep great for queries on ‘x’; terrible for ‘y’

155. R-trees - variations what about static datasets (no ins/del/upd)? Q: Best way to pack points? A1: plane-sweep great for queries on ‘x’; bad for ‘y’

156. R-trees - variations what about static datasets (no ins/del/upd)? Q: Best way to pack points? A1: plane-sweep great for queries on ‘x’; terrible for ‘y’ Q: how to improve?

157. R-trees - variations A: plane-sweep on HILBERT curve!

158. R-trees - variations A: plane-sweep on HILBERT curve! In fact, it can be made dynamic (how?), as well as to handle regions (how?) A: [Kamel+, VLDB94]

159. R-trees - variations Guttman’s R-trees sparked much follow-up work can we do better splits? what about static datasets (no ins/del/upd)? what about other bounding shapes?

160. R-trees - variations what about other bounding shapes? (and why?) A1: arbitrary-orientation lines (cell-tree, [Guenther] A2: P-trees (polygon trees) (MB polygon: 0, 90, 45, 135 degree lines)

161. R-trees - variations A3: L-shapes; holes (hB-tree) A4: TV-trees [Lin+, VLDB-Journal 1994] A5: SR-trees [Katayama+, SIGMOD97] (used in Informedia)

162. R-trees - conclusions Popular method; like multi-d B-trees guaranteed utilization good search times (for low-dim. at least) R*-, Hilbert- and SR-trees: still used IBM (Informix) ships DataBlade with R-trees

163. References Guttman, A. (June 1984). R-Trees: A Dynamic Index Structure for Spatial Searching. Proc. ACM SIGMOD, Boston, Mass. Jagadish, H. V. (May 23-25, 1990). Linear Clustering of Objects with Multiple Attributes. ACM SIGMOD Conf., Atlantic City, NJ. Lin, K.-I., H. V. Jagadish, et al. (Oct. 1994). “The TV- tree - An Index Structure for High-dimensional Data.” VLDB Journal 3: 517-542.

164. References, cont’d Pagel, B., H. Six, et al. (May 1993). Towards an Analysis of Range Query Performance. Proc. of ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems (PODS), Washington, D.C. Robinson, J. T. (1981). The k-D-B-Tree: A Search Structure for Large Multidimensional Dynamic Indexes. Proc. ACM SIGMOD. Roussopoulos, N., S. Kelley, et al. (May 1995). Nearest Neighbor Queries. Proc. of ACM-SIGMOD, San Jose, CA.