Spatial%20Indexing

1
Spatial Indexing

• Many slides taken from George Kollios, Boston
  University

2
Spatial Indexing

• Point Access Methods can index only points. What
  about regions?
• Use the transformation technique
• Z-ordering and quadtrees
• New methods Spatial Access Methods SAMs
• R-tree and variations

3
Problem

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

4
R-tree

• z-ordering cuts regions to pieces -gt dup. elim.
• how could we avoid that?
• Idea try to extend/merge B-trees and k-d trees
• R-tree a generalization of the B-tree for
  multidimensional spaces

5
Kd-Btrees

• Robinson, 81 if f is the fanout, split
  point-set in f parts and so on, recursively

6
Kd-Btrees

• But insertions/deletions are tricky (splits may
  propagate downwards and upwards)
• no guarantee on space utilization

7
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)

8
R-trees

• A multi-way external memory tree
• Index nodes and data (leaf) nodes
• All leaf nodes appear on the same level
• Every node contains between m and M entries
• The root node has at least 2 entries (children)

9
Example

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

I
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F
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E
D
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Example

• F4

P1
P3
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C
A
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F
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E
P4
D
P2
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Example

• F4

P1
P3
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F
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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
...
13
R-trees - format of nodes

• (MBR node_ptr) for non-leaf nodes

x-low x-high y-low y-high ...
node ptr
...
14
R-treesSearch
P1
P3
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F
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E
P4
D
P2
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R-treesSearch
P1
P3
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E
P4
D
P2
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R-treesSearch

• Main points
• 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

17
R-treesInsertion
Insert X
P1
P3
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A
G
H
F
B
X
J
E
P4
D
P2
X
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R-treesInsertion
Insert Y
P1
P3
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C
A
G
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F
B
J
E
P4
Y
D
P2
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R-treesInsertion

• Extend the parent MBR

P1
P3
I
C
A
G
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F
B
J
E
P4
Y
D
P2
Y
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R-treesInsertion

• How to find the next node to insert the new
  object?
• Using ChooseLeaf Find the entry that needs the
  least enlargement to include Y. Resolve ties
  using the area (smallest)
• Other methods (later)

21
R-treesInsertion

• If node is full then Split ex. Insert w

P1
P3
K
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A
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W
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F
B
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K
E
P4
D
P2
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R-treesSplit

• Split node P1 partition the MBRs into two groups.

• (A1 plane sweep,
• until 50 of rectangles)
• A2 linear split
• A3 quadratic split
• A4 exponential split
• 2M-1 choices

P1
K
C
A
W
B
23
R-treesSplit

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

seed1
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R-treesSplit

• pick two rectangles as seeds
• assign each rectangle R to the closest
  seed
• closest the smallest increase in area

seed1
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R-treesSplit

• How to pick Seeds
• LinearFind the highest and lowest side in each
  dimension, normalize the separations, choose the
  pair with the greatest normalized separation
• Quadratic For each pair E1 and E2, calculate the
  rectangle JMBR(E1, E2) and d J-E1-E2. Choose
  the pair with the largest d

26
R-treesInsertion

• Use the ChooseLeaf to find the leaf node to
  insert an entry E
• If leaf node is full, then Split, otherwise
  insert there
• Propagate the split upwards, if necessary
• Adjust parent nodes

27
R-TreesDeletion

• Find the leaf node that contains the entry E
• Remove E from this node
• If underflow
• Eliminate the node by removing the node entries
  and the parent entry
• Reinsert the orphaned (other entries) into the
  tree using Insert
• Other method (later)

28
R-trees Variations

• R-tree DO not allow overlapping, so split the
  objects (similar to z-values)
• R-tree change the insertion, deletion
  algorithms (minimize not only area but also
  perimeter, forced re-insertion )
• Hilbert R-tree use the Hilbert values to insert
  objects into the tree

29
R-trees - variations

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

30
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

31
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

32
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?

33
R-trees - variations

• A plane-sweep on HILBERT curve!

34
R-trees - variations

• A plane-sweep on HILBERT curve!
• In fact, it can be made dynamic (how?), as well
  as to handle regions (how?)

35
R-trees - variations

• Dynamic (Hilbert R-tree)
• each point has an h-value (hilbert value)
• insertions like a B-tree on the h-value
• but also store MBR, for searches

36
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)

37
R-trees - variations

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

38
R-trees - conclusions

• Popular method like multi-d B-trees
• guaranteed utilization
• good search times (for low-dim. at least)
• Informix ships DataBlade with R-trees

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Spatial Queries
40
Spatial Queries

• 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)

41
Spatial Queries

• 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)

42
Spatial Queries

• 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)

43
Spatial Queries

• 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)

44
Spatial Queries

• 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)

45
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)

46
R-trees - NN search
47
R-trees - NN search

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

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R-trees - NN search

• A1 depth-first search then, range query

P1
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P3
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A
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F
B
J
E
P4
q
D
P2
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R-trees - NN search

• A1 depth-first search then, range query

P1
P3
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C
A
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F
B
J
E
P4
q
D
P2
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R-trees - NN search

• A1 depth-first search then, range query

P1
P3
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C
A
G
H
F
B
J
E
P4
q
D
P2
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R-trees - NN search

• A2 Roussopoulos, sigmod95
• priority queue, with promising MBRs, and their
  best and worst-case distance
• main idea Every face of any MBR contains at
  least one point of an actual spatial object!

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R-trees - NN search
consider only P2 and P4, for illustration
q
53
R-trees - NN search
best of P4
gt P4 is useless for 1-nn
worst of P2
H
J
E
P4
q
D
P2
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R-trees - NN search

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

worst of P2
E
q
D
P2
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R-trees - NN search

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

q
P2
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Nearest-neighbor searching

• Branch and bound strategy
• Compute MINDIST and MINMAXDIST RKV95
• MINDIST(p,R) is the minimum distance between p
  and R with corner points l and u
• the closest point in R is at least this distance
  away

R
u
Sqrt sum (pi-ri)2 where
ri li if pi lt li ui if pi gt ui pi
otherwise
p
p
MINDIST 0
l
p
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Nearest-neighbor searching

• MINMAXDIST(p,R) is the minimum of the maximum
  distance to each pair of faces of R
• MaxDistanceToFace(p,R,k) distance between p and
  Maxk (M1,M2,..,Mk-1mk,MK1,..,Mn)
• mi closer of the two boundary points along i
  axis
• Mi farther of the two boundary points along i
  axis

Max2
u
p
l
Max1
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Nearest-neighbor searching

• MINMAXDIST(p,R) is the minimum of the maximum
  distance to each pair of faces of R
• MaxDistanceToFace(p,R,k) distance between p and
  Maxk (M1,M2,..,Mk-1mk,MK1,..,Mn)
• mi closer of the two boundary points along i
  axis
• Mi farther of the two boundary points along i
  axis

Max1
u
p
l
Max2
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Pruning

• ESTIMATE smallest MINMAXDIST(p,R)
• Prune an MBR R for which MINDIST(p,R) is
  greater than ESTIMATE.
• Generalize to k-nearest neighbor searching
• Maintain kth largest MINMAXDIST
• Prune an MBR if MINDIST to it is larger than the
  current estimate of kth MINMAXDIST
• Can use objects to refine estimate

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Order of searching

• Depth first order
• Inspect children in MINDIST order
• For each node in the tree keep a list of nodes to
  be visited
• Prune some of these nodes in the list
• Continue until the lists are empty

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Another NN search

• Global order HS99
• Maintain distance to all entries in a common list
• Order the list by MINDIST
• Repeat
• Inspect the next MBR in the list
• Add the children to the list and reorder
• Until all remaining MBRs can be pruned

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Spatial Join

• Find all parks in a city
• Find all trails that go through a forest
• Basic operation
• find all pairs of objects that overlap
• Single-scan queries
• nearest neighbor queries, range queries
• Multiple-scan queries
• spatial join

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Algorithms

• No existing index structures
• Transform data into 1-d space O89
• z-transform sensitive to size of pixel
• Partition-based spatial-merge join PW96
• partition into tiles that can fit into memory
• plane sweep algorithm on tiles
• Spatial hash joins LR96, KS97
• Sort data BBKK01
• With index structures BKS93, HJR97
• k-d trees and grid files
• R-trees

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R-tree based Join BKS93
S
R
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Join1(R,S)

• Repeat
• Find a pair of intersecting entries E in R and F
  in S
• If R and S are leaf pages then add (E,F) to
  result-set
• Else Join1(E,F)
• Until all pairs are examined
• CPU and I/O bottleneck

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Reducing CPU bottleneck
S
R
67
Join2(R,S,IntersectedVol)

• Repeat
• Find a pair of intersecting entries E in R and F
  in S that overlap with IntersectedVol
• If R and S are leaf pages then add (E,F) to
  result-set
• Else Join2(E,F,CommonEF)
• Until all pairs are examined
• 146 comparisons instead of 49
• In general, number of comparisons equals
• size(R) size(S) relevant(R)relevant(S)
• Reduce the product term

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Using Plane Sweep
S
R
s1
s2
r1
r2
r3
Consider the extents along x-axis Start with the
first entry r1 sweep a vertical line
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Using Plane Sweep
S
R
s1
s2
r1
r2
r3
Check if (r1,s1) intersect along y-dimension Add
(r1,s1) to result set
70
Using Plane Sweep
S
R
s1
s2
r1
r2
r3
Check if (r1,s2) intersect along y-dimension Add
(r1,s2) to result set
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Using Plane Sweep
S
R
s1
s2
r1
r2
r3
Reached the end of r1 Start with next entry r2
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Using Plane Sweep
S
R
s1
s2
r1
r2
r3
Reposition sweep line
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Using Plane Sweep
S
R
s1
s2
r1
r2
r3
Check if r2 and s1 intersect along y Do not add
(r2,s1) to result
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Using Plane Sweep
S
R
s1
s2
r1
r2
r3
Reached the end of r2 Start with next entry s1
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Using Plane Sweep
S
R
s1
s2
r1
r2
r3
Total of 2(r1) 1(r2) 0 (s1) 1(s2) 0(r3)
4 comparisons
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Reducing I/O

• Read schedule r1, s1, s2, r3
• Every subtree examined only once
• Consider a slightly different layout

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Reducing I/O
S
R
s1
r2
r1
s2
r3
Read schedule is r1, s2, r2, s1, s2, r3
Subtree s2 is examined twice
78
Pinning of nodes

• After examining a pair (E,F), compute the degree
  of intersection of each entry
• degree(E) is the number of intersections between
  E and unprocessed rectangles of the other dataset
• If the degrees are non-zero, pin the pages of the
  entry with maximum degree
• Perform spatial joins for this page
• Continue with plane sweep

79
Reducing I/O
S
R
s1
r2
r1
s2
r3
After computing join(r1,s2), degree(r1)
0 degree(s2) 1 So, examine s2 next Read
schedule r1, s2, r3, r2, s1 Subtree s2
examined only once
80
References

• SK98 Optimal multi-step k-nearest neighbor
  search, T. Seidl and H. Kriegel, SIGMOD 1998
  154--165.
• BBKK01 Epsilon Grid Order An Algorithm for the
  Similarity Join on Massive High-Dimensional Data,
  C. Bohm, B. Braunmüller, F. Krebs and H.-P.
  Kriegel, SIGMOD 2001.
• RKV95 Roussopoulos N., Kelley S., Vincent F.
  Nearest Neighbor Queries. Proceedings of the
  ACM-SIGMOD International Conference on Management
  of Data, pages 71-79, 1995.

81
References

• HS99 G. R. Hjaltason and H. Samet, Distance
  browsing in spatial databases, ACM Transactions
  on Database Systems 24, 2 (June 1999), 265-318
• O89 Jack A. Orenstein Redundancy in Spatial
  Databases. SIGMOD Conference 1989 294-305
• PW96 Jignesh M. Patel, David J. DeWitt
  Partition Based Spatial-Merge Join. SIGMOD
  Conference 1996 259-270
• LR96 Ming-Ling Lo, Chinya V. Ravishankar
  Spatial Hash-Joins. SIGMOD Conference 1996
  247-258

82
References

• KS97 Nick Koudas, Kenneth C. Sevcik Size
  Separation Spatial Join. SIGMOD Conference 1997
  324-335
• HJR97 Yun-Wu Huang, Ning Jing, Elke A.
  Rundensteiner Spatial Joins Using R-trees
  Breadth-First Traversal with Global
  Optimizations. VLDB 1997, 396-405
• BKS93 Thomas Brinkhoff, Hans-Peter Kriegel,
  Bernhard Seeger Efficient Processing of Spatial
  Joins Using R-Trees. SIGMOD Conference 1993
  237-246