Spatial Indexing

1
Spatial Indexing

• SAMs

2
Spatial Access Methods

• PAMs
• Grid File
• kd-tree based (LSD-, hB- trees)
• Z-ordering B-tree
• R-tree
• Variations R-tree, Hilbert R-tree

3
R-tree
Multi-way external memory structure, indexes
MBRs Dynamic structure
P1
P3
I
C
A
G
H
F
B
J
E
P4
D
P2
4
R-tree properties

• 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.

5
R-tree

• The original R-tree tries to minimize the area of
  each enclosing rectangle in the index nodes.
• Is there any other property that can be
  optimized?

R-tree ? Yes!
6
R-tree

• Optimization Criteria
• (O1) Area covered by an index MBR
• (O2) Overlap between directory MBRs
• (O3) Margin of a directory rectangle
• (O4)Storage utilization
• Sometimes it is impossible to optimize all the
  above criteria at the same time!

7
R-tree

• ChooseSubtree
• If next node is a leaf node, choose the node
  using the following criteria
• Least overlap enlargement
• Least area enlargement
• Smaller area
• Else
• Least area enlargement
• Smaller area

8
R-tree

• SplitNode
• Choose the axis to split
• Choose the two groups along the chosen axis
• ChooseSplitAxis
• Along each axis, sort rectangles and break them
  into two groups (M-2m2 possible ways where one
  group contains at least m rectangles). Compute
  the sum S of all margin-values of each pair of
  groups. Choose the one that minimizes S
• ChooseSplitIndex
• Along the chosen axis, choose the grouping that
  gives the minimum overlap-value

9
R-tree

• Forced Reinsert
• 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

10
R-tree variations

• What about static datasets?
• (no ins/del) Hilbert
• What about other bounding shapes?

11
R-trees - variations

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

12
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

13
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

14
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?

15
R-trees - variations

• A plane-sweep on HILBERT curve!

16
R-trees - variations

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

17
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

18
Hilbert R-tree

• Data structure of a node?

x-low, ylow x-high, y-high
LHV
ptr
h-value gt LHV MBRs inside parent MBR
19
R-trees - variations

• Data structure of a node?

B-tree
x-low, ylow x-high, y-high
LHV
ptr
h-value gt LHV MBRs inside parent MBR
20
R-trees - variations

• Data structure of a node?

R-tree
x-low, ylow x-high, y-high
LHV
ptr
h-value gt LHV MBRs inside parent MBR
21
R-trees - variations

• What if we have regions, instead of points?
• I.e., how to impose a linear ordering (h-value)
  on rectangles?

22
R-trees - variations

• What if we have regions, instead of points?
• I.e., how to impose a linear ordering (h-value)
  on rectangles?
• A1 h-value of center
• A2 h-value of 4-d point
• (center, x-radius,
• y-radius)
• A3 ...

23
R-trees - variations

• What if we have regions, instead of points?
• I.e., how to impose a linear ordering (h-value)
  on rectangles?
• A1 h-value of center
• A2 h-value of 4-d point
• (center, x-radius,
• y-radius)
• A3 ...

24
R-trees - variations

• with h-values, we can have deferred splits,
  2-to-3 splits (3-to-4, etc)
• Instead of splitting a full node, find the
  siblings (using the h-values) and redistribute
  the rectangles among the nodes. Split only when
  all siblings are full.
• experimentally faster than R-trees
• (reference Kamel Faloutsos vldb 94)

25
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)

26
R-trees - variations

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

27
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

28
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)

29
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)

30
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)

31
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)

32
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)

33
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)

34
R-trees - NN search
35
R-trees - NN search

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

36
R-trees - NN search

• A1 depth-first search then, range query

P1
I
P3
C
A
G
H
F
B
J
E
P4
q
D
P2
37
R-trees - NN search

• A1 depth-first search then, range query

P1
P3
I
C
A
G
H
F
B
J
E
P4
q
D
P2
38
R-trees - NN search

• A1 depth-first search then, range query

P1
P3
I
C
A
G
H
F
B
J
E
P4
q
D
P2
39
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!

40
R-trees - NN search
consider only P2 and P4, for illustration
q
41
R-trees - NN search
best of P4
gt P4 is useless for 1-nn
worst of P2
H
J
E
P4
q
D
P2
42
R-trees - NN search

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

worst of P2
E
q
D
P2
43
R-trees - NN search

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

q
P2
44
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 (k1)-th, check the queue, and scan
  some more of the tree
• optimal (but, may need too much memory)