Temple University

1
Temple University CIS Dept.CIS661 Principles
of Data Management

• V. Megalooikonomou
• Spatial Access Methods (SAMs)
• (based on slides C. Faloutsos at CMU )

2
General Overview

• Advanced topics
• Distributed Databases
• Spatial Access Methods (SAMs)
• Multimedia Indexing
• Authorization / Stat. DB

3
SAMs - Detailed outline

• spatial access methods
• problem dfn
• 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 - Detailed outline

• spatial access methods
• problem dfn
• z-ordering
• R-trees

17
SAMs solutions

• z-ordering
• R-trees
• (grid files)
• Q how would you organize, e.g., n-dim points, on
  disk? (C points per disk page)

18
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?

19
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?

20
z-ordering

• Q2 how?
• A assume finite granularity z-ordering
  bit-shuffling N-trees Morton keys
  geo-coding ...

21
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?

22
z-ordering

• Q2.1 how to map n-d cells to 1-d cells?

23
z-ordering

• Q2.1 how to map n-d cells to 1-d cells?
• A row-wise
• Q is it good?

24
z-ordering

• Q is it good?
• A great for x axis bad for y axis

25
z-ordering

• Q How about the snake curve?

26
z-ordering

• Q How about the snake curve?
• A still problems

232
232
27
z-ordering

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

232
232
28
z-ordering

• Q solution? (w/ good clustering, and easy to
  compute, for 2-d and n-d?)

29
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

30
z-ordering

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

31
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)
32
z-ordering

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

order-2
order-1
33
z-ordering

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

Method 2?
34
z-ordering

• bit-shuffling

y
11 10 01 00
00
10
x
01
11
35
z-ordering

• bit-shuffling

y
11 10 01 00
How about the reverse (x,y) g(z) ?
00
10
x
01
11
36
z-ordering

• bit-shuffling

y
11 10 01 00
How about n-d spaces?
00
10
x
01
11
37
z-ordering

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

Method 3?
38
z-ordering

• linear-quadtrees assign N-gt1, S-gt0 e.t.c.

W E
1
N S
0
0
1
39
z-ordering

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

W E
11
00
1
N S
0
0
1
40
z-ordering

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

W E
1
N S
0
0
1
41
z-ordering

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

W E
method1 14 method2 shuffle(1110)
(1110)2 14
1
N S
0
0
1
42
z-ordering

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

W E
method1 14 method2 shuffle(1110)
(1110)2 14 method3 ENES ... 14
1
N S
0
0
1
43
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

44
z-ordering - usage algos

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

45
z-ordering - usage algos

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

PGH
SF
46
z-ordering - usage algos

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

47
z-ordering - usage algos

• Q2 queries? (eg. find city at (0,3) )?

PGH
SF
48
z-ordering - usage algos

• Q2 queries? (eg. find city at (0,3) )?
• A find z-value search B-tree

PGH
SF
49
z-ordering - usage algos

• Q2 range queries?

PGH
SF
50
z-ordering - usage algos

• Q2 range queries?
• A compute ranges of z-values use B-tree

PGH
9,11-15
SF
51
z-ordering - usage algos

• Q2 range queries - how to reduce of
  qualifying of ranges?

PGH
9,11-15
SF
52
z-ordering - usage algos

• Q2 range queries - how to reduce of
  qualifying of ranges?
• A Augment the query!

PGH
9,11-15 -gt 8-15
SF
53
z-ordering - usage algos

• Q2 range queries - how to break a query into
  ranges?

9,11-15
54
z-ordering - usage algos

• Q2 range queries - how to break a query into
  ranges?
• A recursively, quadtree-style decompose only
  non-full quadrants

12-15
9,11-15
55
z-ordering - usage algos

• Q2 range queries - how to break a query into
  ranges?
• A recursively, quadtree-style decompose only
  non-full quadrants

12-15
9,11-15
9, 11
56
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

57
z-ordering - usage algos
skip

• Q3 k-nn queries? (say, 1-nn)?

PGH
SF
58
z-ordering - usage algos
skip

• Q3 k-nn queries? (say, 1-nn)?
• A traverse B-tree find nn wrt z-values and ...

PGH
SF
59
z-ordering - usage algos
skip

• ... ask a range query.

PGH
SF
nn wrt z-value
12
5
3
60
z-ordering - usage algos
skip

• ... ask a range query.

PGH
SF
nn wrt z-value
12
5
3
61
z-ordering - usage algos
skip

• Q4 all-pairs queries? ( all pairs of cities
  within 10 miles from each other? )

PGH
SF
(well see spatial joins later find all PA
counties that intersect a lake)
62
z-ordering - Detailed outline
skip

• 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
• ...

63
z-ordering - regions
skip

• Q z-value for a region?

zB ?? zC ??
B
A
C
64
z-ordering - regions
skip

• Q z-value for a region?
• A 1 or more z-values by quadtree decomposition

zB ?? zC ??
65
z-ordering - regions
skip
dont care

• Q z-value for a region?

zB 11 zC ??
W E
11
00
1
N S
0
0
1
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z-ordering - regions
skip
dont care

• Q z-value for a region?

zB 11 zC 0010 1000
W E
11
00
1
N S
0
0
1
67
z-ordering - regions
skip

• Q How to store in B-tree?
• Q How to search (range etc queries)

68
z-ordering - regions
skip

• Q How to store in B-tree? A sort (lt0lt1)
• Q How to search (range etc queries)

69
z-ordering - regions
skip

• Q How to search (range etc queries) - eg red
  range query

70
z-ordering - regions
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• Q How to search (range etc queries) - eg red
  range query
• A break query in z-values check B-tree

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z-ordering - regions
skip

• Almost identical to range queries for point data,
  except for the dont cares - i.e.,

1100 ?? 11
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z-ordering - regions
skip

• Almost identical to range queries for point data,
  except for the dont cares - i.e.,
• z1 1100 ?? 11 z2
• Specifically does z1 contain/avoid/intersect z2?
• Q what is the criterion to decide?

73
z-ordering - regions
skip

• z1 1100 ?? 11 z2
• Specifically does z1 contain/avoid/intersect z2?
• Q what is the criterion to decide?
• A Prefix property let r1, r2 be the
  corresponding regions, and let r1 be the smallest
  (gt z1 has fewest s). Then

74
z-ordering - regions
skip

• r2 will either contain completely, or avoid
  completely r1.
• it will contain r1, if z2 is the prefix of z1

1100 ?? 11
region of z1 completely contained in region of z2
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z-ordering - regions
skip

• Drill (True/False). Given
• z1 011001
• z2 01
• z3 0100
• T/F r2 contains r1
• T/F r3 contains r1
• T/F r3 contains r2

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z-ordering - regions
skip

• Drill (True/False). Given
• z1 011001
• z2 01
• z3 0100
• T/F r2 contains r1 - TRUE (prefix property)
• T/F r3 contains r1 - FALSE (disjoint)
• T/F r3 contains r2 - FALSE (r2 contains r3)

77
z-ordering - regions
skip

• Drill (True/False). Given
• z1 011001
• z2 01
• z3 0100

z2
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z-ordering - regions
skip

• Drill (True/False). Given
• z1 011001
• z2 01
• z3 0100

z2
z3
T/F r2 contains r1 - TRUE (prefix property) T/F
r3 contains r1 - FALSE (disjoint) T/F r3 contains
r2 - FALSE (r2 contains r3)
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z-ordering - regions
skip

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

80
z-ordering - regions
skip

• Spatial joins find (quickly) all
• counties intersecting lakes
• Naive algorithm O( N M)
• Something faster?

81
z-ordering - regions
skip

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

82
z-ordering - regions
skip

• Spatial joins find (quickly) all
• counties intersecting lakes
• Solution merge the lists of (sorted) z-values,
  looking for the prefix property
• footnote1 needs careful treatment
• footnote2 need dup. elimination

83
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

84
z-ordering - variations

• Q is z-ordering the best we can do?

85
z-ordering - variations

• Q is z-ordering the best we can do?
• A probably not - occasional long jumps
• Q then?

86
z-ordering - variations

• Q is z-ordering the best we can do?
• A probably not - occasional long jumps
• Q then? A1 Gray codes

87
z-ordering - variations

• A2 Hilbert curve! (a.k.a. Hilbert-Peano curve)

88
z-ordering - variations

• Looks better (never long jumps). How to derive
  it?

89
z-ordering - variations

• Looks better (never long jumps). How to derive
  it?

...
order (n1)
order-1
order-2
90
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)

91
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)

2
1
92
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
• ...

93
z-ordering - analysis

• Q How many pieces (quad-tree blocks) per
  region?
• A proportional to perimeter (surface etc)

94
z-ordering - analysis

• (How long is the coastline, say, of England?
• Paradox The answer changes with the yard-stick
  -gt fractals ...)

95
z-ordering - analysis

• Q Should we decompose a region to full detail
  (and store in B-tree)?

96
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

97
z-ordering - analysis

• Q how to measure the goodness of a curve?

98
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)
99
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)

100
z-ordering - fun observations

• Hilbert and z-ordering curves space filling
  curves eventually, they visit every point
• in n-d space - therefore

101
z-ordering - fun observations

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

102
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

103
z-ordering - fun observations

• In general, Hilbert curve is great for preserving
  distances, clustering, vector quantization etc

104
SAMs - Detailed outline

• spatial access methods
• problem dfn
• z-ordering
• R-trees

105
Conclusions

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

106
SAMs - Detailed outline

• spatial access methods
• problem dfn
• z-ordering
• R-trees

107
SAMs - more detailed outline

• R-trees
• main idea file structure
• (algorithms insertion/split)
• (deletion)
• (search range, nn, spatial joins)
• variations (packed hilbert...)

108
Reminder problem

• Given a collection of geometric objects (points,
  lines, polygons, ...)
• organize them on disk, to answer spatial queries
  (range, nn, etc)

109
R-trees

• z-ordering cuts regions to pieces -gt dup. elim.
• how could we avoid that?
• Idea Minimum Bounding Rectangles

110
R-trees

• Guttman 84 Main idea allow parents to overlap!
• gt guaranteed 50 utilization
• gt easier insertion/split algorithms.
• (only deal with Minimum Bounding Rectangles -
  MBRs)

111
R-trees

• eg., w/ fanout 4 group nearby rectangles to
  parent MBRs each group -gt disk page

I
C
A
G
H
F
B
J
E
D
112
R-trees

• eg., w/ fanout 4

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

• eg., w/ fanout 4

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

• (MBR obj-ptr) for leaf nodes

x-low x-high y-low y-high ...
obj ptr
...
115
R-trees - format of nodes

• (MBR node-ptr) for non-leaf nodes

116
R-trees - range search?
P1
P3
I
C
A
G
H
F
B
J
E
P4
D
P2
117
R-trees - range search?
P1
P3
I
C
A
G
H
F
B
J
E
P4
D
P2
118
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.
  (ie., no need for dup. elim.)
• a point query may follow multiple branches.
• everything works for any dimensionality

119
SAMs - more detailed outline

• R-trees
• main idea file structure
• algorithms insertion/split
• deletion
• search range, nn, spatial joins
• performance analysis
• variations (packed hilbert...)

120
R-trees - insertion

• eg., rectangle X

P1
P3
I
C
A
G
H
F
B
X
J
E
P4
D
P2
121
R-trees - insertion

• eg., rectangle X

P1
P3
I
C
A
G
H
F
B
X
J
E
P4
D
P2
X
122
R-trees - insertion
skip

• eg., rectangle Y

P1
P3
I
C
A
G
H
F
B
J
E
P4
Y
D
P2
123
R-trees - insertion
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• eg., rectangle Y extend suitable parent.

P1
P3
I
C
A
G
H
F
B
J
E
P4
Y
D
P2
Y
124
R-trees - insertion
skip

• eg., rectangle Y extend suitable parent.
• Q how to measure suitability?

125
R-trees - insertion
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• 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?

126
R-trees - insertion
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• eg., rectangle W

P1
P3
K
I
C
A
G
W
H
F
B
J
K
E
P4
D
P2
127
R-trees - insertion
skip

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

P1
K
C
A
W
B
128
R-trees - insertion
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• 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
129
R-trees - insertion split
skip

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

seed1
130
R-trees - insertion split
skip

• pick two rectangles as seeds
• assign each rectangle R to the closest seed
• Q how to measure closeness?

131
R-trees - insertion split
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• pick two rectangles as seeds
• assign each rectangle R to the closest seed
• Q how to measure closeness?
• A by increase of area (volume)

132
R-trees - insertion split
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• pick two rectangles as seeds
• assign each rectangle R to the closest seed

seed1
133
R-trees - insertion split
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• pick two rectangles as seeds
• assign each rectangle R to the closest seed

seed1
134
R-trees - insertion split
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• 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!)

135
R-trees - insertion - pseudocode
skip

• 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.

136
R-trees - insertion - observations
skip

• many more split algorithms exist (next!)

137
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...)

138
R-trees - deletion
skip

• delete rectangle
• if underflow
• ??

139
R-trees - deletion
skip

• delete rectangle
• if underflow
• temporarily delete all siblings (!)
• delete the parent node and
• re-insert them

140
SAMs - more detailed outline

• R-trees
• main idea file structure
• algorithms insertion/split
• deletion
• search range, nn, spatial joins
• performance analysis
• variations (packed hilbert...)

141
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)

142
R-trees - nn search
skip
143
R-trees - nn search
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• Q How? (find near neighbor refine...)

144
R-trees - nn search
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• A1 depth-first search then, range query

P1
P3
I
C
A
G
H
F
B
J
E
P4
q
D
P2
145
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
146
R-trees - nn search
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• A1 depth-first search then, range query

P1
P3
I
C
A
G
H
F
B
J
E
P4
q
D
P2
147
R-trees - nn search
skip

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

148
R-trees - nn search
skip
consider only P2 and P4, for illustration
q
149
R-trees - nn search
skip
best of P4
gt P4 is useless for 1-nn
worst of P2
H
J
E
P4
q
D
P2
150
R-trees - nn search
skip

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

worst of P2
E
q
D
P2
151
R-trees - nn search
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• what is really the worst of, say, P2?
• A the smallest of the two red segments!

q
P2
152
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)

153
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...)

154
R-trees - spatial joins
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• Spatial joins find (quickly) all
• counties intersecting lakes

155
R-trees - spatial joins
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• Assume that they are both organized in R-trees

156
R-trees - spatial joins
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• 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)

157
R-trees - spatial joins
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• 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.)

158
SAMs - more detailed outline

• R-trees
• main idea file structure
• algorithms insertion/split
• deletion
• search range, nn, spatial joins
• variations (packed hilbert...)

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

• Guttmans R-trees sparked much follow-up work
• can we do better splits?
• i.e, defer splits?

161
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?

162
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

163
R-trees - variations

• Q Other ways to defer splits?

164
R-trees - variations

• Q Other ways to defer splits?
• A Push a few keys to the closest sibling node
• (closest ??)

165
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.

166
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?

167
R-trees - variations

• what about static datasets (no ins/del/upd)?
• Q Best way to pack points?

168
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

169
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

170
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?

171
R-trees - variations

• A plane-sweep on HILBERT curve!

172
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

173
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?

174
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)

175
R-trees - variations

• A3 L-shapes holes (hB-tree)
• A4 TV-trees Lin, VLDB-Journal 1994
• A5 SR-trees Katayama, SIGMOD97 (used in
  Informedia)

176
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
• Informix ships DataBlade with R-trees

177
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.

178
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.