Transcript Spatial Index - CSE, IIT Bombay

Indexing Spatial Data
(Parts of Chapter 25+R-tree paper)
Database System Concepts
©Silberschatz, Korth and Sudarshan
See www.db-book.com for conditions on re-use
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Spatial and Geographic Databases
Spatial databases store information related to spatial locations, and
support efficient storage, indexing and querying of spatial data.
Special purpose index structures are important for accessing spatial
data, and for processing spatial queries.
Computer Aided Design (CAD) databases store design information
about how objects are constructed E.g.: designs of buildings, aircraft,
layouts of integrated-circuits
Geographic databases store geographic information (e.g., maps):
often called geographic information systems or GIS.
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Spatial/Geographic Data
Raster data consist of bit maps or pixel maps, in two or more
dimensions.
Example 2-D raster image: satellite image of cloud cover,
where each pixel stores the cloud visibility in a particular area.
Additional dimensions might include the temperature at
different altitudes at different regions, or measurements taken
at different points in time.
Design databases generally do not store raster data.
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Spatial/Geographic Data (Cont.)
Vector data are constructed from basic geometric objects: points,
line segments, triangles, and other polygons in two dimensions, and
cylinders, speheres, cuboids, and other polyhedrons in three
dimensions.
Vector format often used to represent map data.
Roads can be considered as two-dimensional and represented
by lines and curves.
Some features, such as rivers, may be represented either as
complex curves or as complex polygons, depending on whether
their width is relevant.
Features such as regions and lakes can be depicted as
polygons.
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Applications of Geographic Data
Examples of geographic data
map data for vehicle navigation
distribution network information for power, telephones, water
supply, and sewage
Vehicle navigation systems store information about roads and
services for the use of drivers:
Spatial data: e.g, road/restaurant/gas-station coordinates
Non-spatial data: e.g., one-way streets, speed limits, traffic
congestion
Global Positioning System (GPS) unit - utilizes information
broadcast from GPS satellites to find the current location of user
with an accuracy of tens of meters.
increasingly used in vehicle navigation systems as well as
utility maintenance applications.
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Applications of Geographic Data (Cont.)
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Applications of Geographic Data (Cont.)
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Representation of Vector Information

Various geometric constructs can be represented in a database in a normalized
fashion.

Represent a line segment by the coordinates of its endpoints.

Approximate a curve by partitioning it into a sequence of segments


Create a list of vertices in order, or

Represent each segment as a separate tuple that also carries with it the
identifier of the curve (2D features such as roads).
Closed polygons

List of vertices in order, starting vertex is the same as the ending vertex, or

Represent boundary edges as separate tuples, with each containing
identifier of the polygon, or

Use triangulation — divide polygon into triangles

Note the polygon identifier with each of its triangles.
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Representation of Vector Data
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Representation of Vector Data (Cont.)
Representation of points and line segment in 3-D similar to 2-D,
except that points have an extra z component
Represent arbitrary polyhedra by dividing them into tetrahedrons, like
triangulating polygons.
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Spatial Queries
Nearness queries request objects that lie near a specified location.
Nearest neighbor queries, given a point or an object, find the
nearest object that satisfies given conditions.
Region queries deal with spatial regions. e.g., ask for objects that
lie partially or fully inside a specified region.
Queries that compute intersections or unions of regions.
Spatial join of two spatial relations with the location playing the role
of join attribute.
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Spatial Queries (Cont.)
Spatial data is typically queried using a graphical query language;
results are also displayed in a graphical manner.
Graphical interface constitutes the front-end
Extensions of SQL with abstract data types, such as lines,
polygons and bit maps, have been proposed to interface with backend.
allows relational databases to store and retrieve spatial
information
Queries can use spatial conditions (e.g. contains or overlaps).
queries can mix spatial and nonspatial conditions
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Indexing of Spatial Data
k-d tree - early structure used for indexing in multiple dimensions.
Each level of a k-d tree partitions the space into two.
choose one dimension for partitioning at the root level of the tree.
choose another dimensions for partitioning in nodes at the next level
and so on, cycling through the dimensions.
In each node, approximately half of the points stored in the sub-tree fall
on one side and half on the other.
Partitioning stops when a node has less than a given maximum number
of points.
The k-d-B tree extends the k-d tree to allow multiple child nodes for
each internal node; well-suited for secondary storage.
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Division of Space by a k-d Tree
Each line in the figure (other than the outside box) corresponds to
a node in the k-d tree
the maximum number of points in a leaf node has been set to
1.
The numbering of the lines in the figure indicates the level of the
tree at which the corresponding node appears.
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Division of Space by Quadtrees
Quadtrees
Each node of a quadtree is associated with a rectangular region of space; the top
node is associated with the entire target space.
Each non-leaf node divides its region into four equal sized quadrants
correspondingly each such node has four child nodes corresponding to the four
quadrants and so on
Leaf nodes have between zero and some fixed maximum number of points (set to 1 in
example).
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Quadtrees (Cont.)
PR quadtree: stores points; space is divided based on regions, rather
than on the actual set of points stored.
Region quadtrees store array (raster) information.
A node is a leaf node is all the array values in the region that it
covers are the same. Otherwise, it is subdivided further into four
children of equal area, and is therefore an internal node.
Each node corresponds to a sub-array of values.
The sub-arrays corresponding to leaves either contain just a single
array element, or have multiple array elements, all of which have
the same value.
Extensions of k-d trees and PR quadtrees have been proposed to
index line segments and polygons
Require splitting segments/polygons into pieces at partitioning
boundaries

Same segment/polygon may be represented at several leaf
nodes
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R-Trees
R-trees are a N-dimensional extension of B+-trees, useful for
indexing sets of rectangles and other polygons.
Supported in many modern database systems
Basic idea: generalize the notion of a one-dimensional interval
associated with each B+ -tree node to an
N-dimensional interval, that is, an N-dimensional rectangle.
Will consider only the two-dimensional case (N = 2)
generalization for N > 2 is straightforward
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Bounding Rectangle
 Suppose we have a cluster of points in 2-D space...

We can build a “box” around points. The smallest box (which is
axis parallel) that contains all the points is called a Minimum
Bounding Rectangle (MBR)

also known as minimum bounding box
MBR = {(L.x,L.y)(U.x,U.y)}
Note that we only need two points to describe an MBR, we typically use
lower left, and upper right.
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Clustering Points
We can group clusters of datapoints into MBRs
Can also handle line-segments, rectangles, polygons, in addition
to points
R1
R2
R4
R5
R3
R6
We can further recursively group
MBRs into larger MBRs….
R9
R7
R8
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R-Tree Structure
Nested MBRs are organized as a tree
R10
R11
R10 R11 R12
R1 R2 R3
R12
R4 R5 R6
R7 R8 R9
Data nodes containing points
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R Trees (Cont.)
A rectangular bounding box (a.k.a. MBR) is associated with each
tree node.
Bounding box of a leaf node is a minimum sized rectangle that
contains all the rectangles/polygons associated with the leaf node.
The bounding box associated with a non-leaf node contains the
bounding box associated with all its children.
Bounding box of a node serves as its key in its parent node (if any)
Bounding boxes of children of a node are allowed to overlap
A polygon is stored only in one node, and the bounding box of the
node must contain the polygon
The storage efficiency or R-trees is better than that of k-d trees or
quadtrees since a polygon is stored only once
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Example R-Tree
A set of rectangles (solid line) and the bounding boxes (dashed line) of the nodes of
an R-tree for the rectangles. The R-tree is shown on the right.
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Search in R-Trees
To find data items (rectangles/polygons) intersecting (overlaps) a
given query point/region, do the following, starting from the root node:
If the node is a leaf node, output the data items whose keys
intersect the given query point/region.
Else, for each child of the current node whose bounding box
overlaps the query point/region, recursively search the child
Can be very inefficient in worst case since multiple paths may need to
be searched
but works acceptably in practice.
Simple extensions of search procedure to handle
predicates contained-in and contains
Find data items nearest to a given point
Find data items within a given distance of a given point
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Nearest Neighbor Search
Given an MBR, we can compute lower bounds on nearest object
Once we know there IS an item within some distance d, we can prune
away all items/MBRs at distance > d
Even if we haven’t actually found the nearest item yet
Similar technique possible for k-d trees and quadtrees as well
Q
R10
R11
R10 R11 R12
R1 R2 R3 R4 R5 R6 R7 R8 R9
R12
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Data nodes containing points
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MBR: Distance Calculation
The formula for the distance between a point and the closest possible
point within an MBR
MBR = {(L.x,L.y)(U.x,U.y)}
Q = (x,y)
MINDIST(Q,MBR)
if L.x < x < U.x and L.y < y < U.y then 0
elseif L.x < x < U.x then min( abs(L.y -y) , abs(U.y -y) )
elseif ….
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MBR: Distance Example
Two examples of MINDIST(point, MBR), calculations…
MINDIST(point, MBR) = 5
MINDIST(point, MBR) = 0
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MBR: Distance Bounds
Distance bounds on bounding boxes
Suppose we have a query point Q and one known point R
Could any of the points in the MBR be closer to Q than R is?
R = (1,7)
Q = (3,5)
MBR = {(6,1),(8,4)}
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Insertion in R-Trees
To insert a data item E:
Find a leaf to store it, and add it to the leaf

To find leaf, starting from root, at each step:
follow a child (if any) whose bounding box contains bounding
box of data item, else child whose bounding box needs least
enlargement to contain data item E

Handle overflows by splits (as in B+ -trees)

Split procedure is different though (see below)
Adjust bounding boxes starting from the leaf upwards
Split procedure:
Goal: divide entries of an overfull node into two sets such that the
bounding boxes have minimum total area

This is a heuristic. Alternatives like minimum overlap are
possible
Finding the “best” split is expensive, use heuristics instead

See next slide
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Splitting a Node
(Heuristic) goal: find split whose total area is maximized.
Minimizes chances that the nodes will be searched
Finding best split is expensive (not exponential time, but costly)
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Splitting an R-Tree Node
Quadratic split divides the entries in a node into two new nodes as
follows
1.
Find pair of entries with “maximum separation”

that is, the pair such that the bounding box of the two would
have the maximum wasted space (area of bounding box – sum
of areas of two entries)
2.
Place these entries in two new nodes
3.
Repeatedly find the entry with “maximum preference” for one of the
two new nodes, and assign the entry to that node
Preference of an entry to a node is the increase in area of
bounding box if the entry is added to the other node
4.
Stop when half the entries have been added to one node
Then assign remaining entries to the other node
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Linear Split
Cheaper linear split heuristic works in time linear in number of
entries,
Cheaper but generates slightly worse splits.
Idea: pick extreme elements by choosing dimension with maximum
(normalized) separation between rectangle with highest low side and
lowest high side
Normalize each dimension by scaling to 1
Choose entries in any order
For each one assign to side that results in least area increase
Until one side has > ½ entries, then assign all to other side
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Deleting in R-Trees
Deletion of an entry in an R-tree can be done much like a B+-tree
deletion.
In case of underfull node, borrow entries from a sibling if possible,
else merging sibling nodes
Alternative approach removes all entries from the underfull node,
deletes the node, then reinserts all entries

Goal: improve locality of R-tree by reinserting nodes

Described in detail in paper in Algorithm CondenseTree
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End of Chapter
Database System Concepts
©Silberschatz, Korth and Sudarshan
See www.db-book.com for conditions on re-use
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Design Databases
Represent design components as objects (generally geometric
objects); the connections between the objects indicate how the
design is structured.
Simple two-dimensional objects: points, lines, triangles,
rectangles, polygons.
Complex two-dimensional objects: formed from simple objects via
union, intersection, and difference operations.
Complex three-dimensional objects: formed from simpler objects
such as spheres, cylinders, and cuboids, by union, intersection,
and difference operations.
Wireframe models represent three-dimensional surfaces as a set
of simpler objects.
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Representation of Geometric Constructs
(a) Difference of cylinders
(b) Union of cylinders
Design databases also store non-spatial information about objects (e.g.,
construction material, color, etc.)
Spatial integrity constraints are important.
E.g., pipes should not intersect, wires should not be too close to
each other, etc.
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R10
{(1,0),(7,9)}
R1
R2
R3
{(1,3),(5,4)} {(2,0),(7,9)} {(1,1),(7,8)}
R12
At the leaf nodes we have
the location, and a pointer to
the record in question.
(3,4) 77
(1,3) 88
(2,3) 22
(5,4) 13
(2,2) 47
(3,0) 86
(7,9) 52
(5,1) 32
(1,4) 45
(5,6) 27
(7,8) 73
At the internal nodes, we just
have MBR information.
Data nodes
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