Temple University CIS Dept. CIS616 Principles of Data Management

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 e)

10
SAMs - motivation

• Q applications?

11
SAMs - motivation
traditional DB
GIS
age
salary
12
SAMs - motivation
traditional DB
GIS
age
salary
13
SAMs - motivation
CAD/CAM
find elements too close to each other
14
SAMs - motivation
CAD/CAM
15
SAMs - motivation
eg,. std
S1
F(S1)
1
365
day
F(Sn)
Sn
eg, avg
1
365
day
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 k2) 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
• 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 otherwise (even levels)
• Similarly for odd levels (branching on Yvals)

22
2-d trees Example of Insertion
Splitting of region by Banja Luka
Splitting of region by Derventa
Splitting of region by Toslic
Splitting 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 Tl (left subtree) or Tr (right
  subtree) is non-empty
• Find a candidate replacement node R in Tl or
  Tr
• Replace all of Ns non-link fields by those of R
• Recursively delete R from Ti
• 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 Tl and Tr as node N

25
2-d trees Range Queries

• Q Given a point (xc, yc) 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 RN If the circle (specified
  by the query) has no intersection with RN 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
Splitting of region by Banja Luka
Splitting of region by Derventa
Splitting of region by Toslic
Splitting of region by Sinj
Splitting 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 Ns non-link fields by those of R
• Recursively delete R from Ti
• 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 Ns non-link fields by those of R
• Recursively delete R from Ti
• 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 (2k x 2k) 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., 232x232 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

232
232
51
z-ordering

• Q Why are those curves bad?
• A no distance preservation ( clustering)
• Q solution?

232
232
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-2
order-1
...
order (n1)
56
z-ordering

• Notice
• self similar (well see about fractals, soon)
• method is hard to use z ? f(x,y)

order-2
order-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

y
11 10 01 00
00
10
x
01
11
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z-ordering

• bit-shuffling

y
11 10 01 00
How about the reverse (x,y) g(z) ?
00
10
x
01
11
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z-ordering

• bit-shuffling

y
11 10 01 00
How about n-d spaces?
00
10
x
01
11
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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.

W E
1
N S
0
0
1
63
z-ordering

• ... and repeat recursively. Eg. zgray-cell
• WNWN (0101)2 5

W E
11
00
1
N S
0
0
1
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z-ordering

• Drill z-value of grey cell, with the three
  methods?

W E
1
N S
0
0
1
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z-ordering

• Drill z-value of grey cell, with the three
  methods?

W E
method1 14 method2 shuffle(1110)
(1110)2 14
1
N S
0
0
1
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z-ordering

• Drill z-value of grey cell, with the three
  methods?

W E
method1 14 method2 shuffle(1110)
(1110)2 14 method3 ENES ... 14
1
N S
0
0
1
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 algos

• Q1 How to store on disk?
• A
• Q2 How to answer range queries etc

69
z-ordering - usage algos

• Q1 How to store on disk?
• A treat z-value as primary key feed to B-tree

PGH
SF
70
z-ordering - usage algos

• MAJOR ADVANTAGES w/ B-tree
• already inside commercial systems (no coding
  /debugging!)
• concurrency recovery is ready

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-1
order-2
order (n1)
...
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 runs
3 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

89
z-ordering - fun observations

• ... they show that the plane has as many points
  as a line (- headaches for 1900s
  mathematics/topology). (fractals, again!)

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

I
C
A
G
H
F
B
J
E
D
98
R-trees

• eg., w/ fanout 4

P1
P3
I
C
A
G
H
F
B
J
E
P4
D
P2
99
R-trees

• eg., w/ fanout 4

P1
P3
I
C
A
G
H
F
B
J
E
P4
D
P2
100
R-trees - format of nodes

• (MBR obj-ptr) for leaf nodes

x-low x-high y-low y-high ...
obj ptr
...
101
R-trees - format of nodes

• (MBR node-ptr) for non-leaf nodes

102
R-trees - range search?
P1
P3
I
C
A
G
H
F
B
J
E
P4
D
P2
103
R-trees - range search?
P1
P3
I
C
A
G
H
F
B
J
E
P4
D
P2
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

P1
P3
I
C
A
G
H
F
B
X
J
E
P4
D
P2
107
R-trees - insertion

• eg., rectangle X

P1
P3
I
C
A
G
H
F
B
X
J
E
P4
D
P2
X
108
R-trees - insertion

• eg., rectangle Y

P1
P3
I
C
A
G
H
F
B
J
E
P4
Y
D
P2
109
R-trees - insertion

• eg., rectangle Y extend suitable parent.

P1
P3
I
C
A
G
H
F
B
J
E
P4
Y
D
P2
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

P1
P3
K
I
C
A
G
W
H
F
B
J
K
E
P4
D
P2
113
R-trees - insertion

• eg., rectangle W - focus on P1 - how to
  split?

P1
K
C
A
W
B
114
R-trees - insertion

• eg., rectangle W - focus on P1 - how to
  split?

P1

• (A1 plane sweep,
• until 50 of rectangles)
• A2 linear split
• A3 quadratic split
• A4 exponential split

K
C
A
W
B
115
R-trees - insertion split

• pick two rectangles as seeds
• assign each rectangle R to the closest seed

seed1
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
119
R-trees - insertion split

• pick two rectangles as seeds
• assign each rectangle R to the closest seed

seed1
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
skip
129
R-trees - nn search
skip

• Q How? (find near neighbor refine...)

130
R-trees - nn search
skip

• A1 depth-first search then, range query

P1
P3
I
C
A
G
H
F
B
J
E
P4
q
D
P2
131
R-trees - nn search
skip

• A1 depth-first search then, range query

P1
P3
I
C
A
G
H
F
B
J
E
P4
q
D
P2
132
R-trees - nn search
skip

• A1 depth-first search then, range query

P1
P3
I
C
A
G
H
F
B
J
E
P4
q
D
P2
133
R-trees - nn search
skip

• A2 Roussopoulos, sigmod95
• priority queue, with promising MBRs, and their
  best and worst-case distance
• main idea

134
R-trees - nn search
skip
consider only P2 and P4, for illustration
q
135
R-trees - nn search
skip
best of P4
P4 is useless for 1-nn
worst of P2
H
J
E
P4
q
D
P2
136
R-trees - nn search
skip

• what is really the worst of, say, P2?

worst of P2
E
q
D
P2
137
R-trees - nn search
skip

• what is really the worst of, say, P2?
• A the smallest of the two red segments!

q
P2
138
R-trees - nn search
skip

• 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 (k1)-th, check the queue, and scan
  some more of the tree
• optimal (but, may need too much memory)

139
SAMs - more detailed outline
skip

• 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
skip

• Spatial joins find (quickly) all
• counties intersecting lakes

141
R-trees - spatial joins
skip

• Assume that they are both organized in R-trees

142
R-trees - spatial joins
skip

• 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
skip

• 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

• Guttmans 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

• Guttmans 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

• Guttmans 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

• Guttmans 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, contd

• 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.