Transcript Trees for spatial indexing Tree (data structure)

Trees for spatial indexing
Tree (data structure)
• Introduction
• B-Tree,B+-Tree,B*-Tree
• Spatial Access Method (SAM) vs Point
Access Method (PAM)
• Buddy-Tree, UB-Tree (8 slides)
• R-Tree
• X-Tree, TV-Tree
Pantheon Problem
• 200’000’000 points are in a database.
• Indexing in a B-Tree is not suffisant. We want to optimize
the query range.
• Which indexing method should we use ?
• What is the best structure ?
Pantheon
What kind of data structure ?
Structur depends on what kind of data :
• point access method : A data structure to search for lines, polygons,
… etc.
– k-d tree
– quadtree
– UB-tree
– buddy tree
• Spatial access method : A data structure and associated algorithms
primarily to search for points defined in multidimensional space.
– D-tree
– P-tree
– R+-tree
– R-tree
– R*-tree
Types of queries in spatial data
'geometry' refers to a point, line, box or other two or three dimensional
shape, the kind of queries we need are :
• Distance(geometry, geometry)
• Equals(geometry, geometry)
• Disjoint(geometry, geometry)
• Intersects(geometry, geometry)
• Touches(geometry, geometry)
• Crosses(geometry, geometry)
• Overlaps(geometry, geometry)
• Contains(geometry, geometry)
• Intersects(geometry, geometry)
• Several other operations performed on only one geometry such as
length, area and centroid
Introduction
•
Some Definitions :
– Node : A node may contain a value or a condition or represent a
separate data structure or a tree of its own. Each node in a tree has 0 or
more child nodes. A node that has a child is called the child's parent
node (or ancestor node, or superior). A node has at most one parent.
– Root nodes : The topmost node in a tree is called the root node. Being
the topmost node, the root node will not have parents. Every node in a
tree can be seen as the root node of the subtree rooted at that node.
– Leaf nodes : Nodes at the bottom most level of the tree are called Leaf
nodes. Since they are at the bottom most level, they will not have any
children.
Tree of the trees
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B-Tree
B+
B*
…
R-Tree
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Buddy
UB-Tree
…
R*-Tree
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X
TV
UBU
Spatial Access Method (SAM) vs Point Access Method (PAM)
?
Common Operations
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Enumerating all the items
Searching for an item
Adding a new item at a certain position on the tree
Deleting an item
Removing a whole section of a tree (called pruning)
Adding a whole section to a tree (called grafting)
Finding the root for any node
B-Tree
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a B-tree is a tree data structure that keeps data sorted and allows insertions and
deletions in logarithmic amortized time. It is most commonly used in databases and
filesystems.
in a 2-3 B-tree (often simply 2-3 tree), each internal node may have only 2 or 3 child
nodes.
Each internal node's elements act as
separation values which divide its subtrees.
B+-Tree
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A B+ tree is a variation on a B-tree. In a B+ tree, in contrast to a B-tree, all
data is saved in the leaves. Internal nodes contain only keys and tree
pointers. All leaves are at the same lowest level. Leaf nodes are also linked
together as a linked list to make range queries easy.
R-Tree
• Extends the B+-Tree
• All non-leaf node contains entries of form
(cp,rectangle) where cp is the address of a
child node and rectangle is the minimum
bounding box rectangle (MBR).
• ~ Leaf nodes contain entries of the form
(dataObject,Rectangle).
• We use the term directory rectangle which is
the MBR of the underlying rectangles.
R-Tree properties
• Let M be the maximum number of entries that fit
in one node and let m be a parameter specifying
the minimum number of entries in a node (2 ≤ m
≤ M), an R-Tree statisfies the following
properties
– The root has at least two children unless it’s a leaf.
– Every non-leaf node has beetween m and M children
unless it’s a root.
– Every leaf node contains beetween m and M entries
unless it’s a root.
– All leaves appear on the same level.
PAM’s
• The basic principle of all multidimensional
PAMs is to partition the data space into page
regions. We classify PAMs according to 3
properties :
Rectangular
Avoid emptyspace
x
Disjoint
PAM
x
UB-Tree
x
x
Twin-grid file
x
x
Buddy-Tree
Buddy-Tree
• The Buddy-Tree uses similar concepts as
the R-Tree.
• But it is extended and has more interesting
properties :
– It does not partition empty space
– Insertion and deletion of a record is restricted
to exactly one path.
– It does not allow overlap in the directory
nodes.
Buddy-Tree : Formal Definition
• The nodes of the tree-directory consist of a
collection of entries {E1,…,Ek}, k ≥ 2.
• Each entry Ei, 1 ≤ i ≤ k, is given by a tuple
Ei=(Ri,pi) where Ri is a d-dimensional rectangle
and pi is a pointer referring to as subtree or to a
data page containing all the records of the file
which are in the rectangle Ri.
• The set of rectangles in a directory node must
be a regular B-partition
B-Rectangle, B-partition
• Given 2 d-dimensional rectangles R,S with R ≤
S, R is called a B-rectangle of S iff it can be
generated by successive halfing of S.
• A B-region of R, written B(R) is the smallest
rectangle such that R ≤ B.
• Such a B-region also exists for a union of
rectangles R1 U R2 U … U Rk, k ≥ 1.
• A set of d-dimensional rectangles {R1,…,Rk}, k ≥
1, is called a B-partition of the data space D, iff
B(Ri) ∩ B(Rj) = Ø
The Buddies
• Let V = {R1,…,Rk} a B-partition, k > 1, and
let S,T Є V, S ≠T.
• The rectangles S,T are called buddies iff
B(S U T) ∩ B(R) = Ø For all R Є V\{S,T}
S
S
T
S,T are Buddies
T
S,T are NOT Buddies
Dynamic behavior
• To obtain an efficient dynamic behavior it must be
possible to merge without destroying the order
preservation.
• For this the regions of the pages must be buddies.
• In the buddy-tree the set of rectangles in a directory
node must be a regular B-partition.
• We say that a B-parition is regular iff all B-rectangles
B(Ri) 1 ≤ i ≤ k can be represented in a kd-trie.
• A kd-trie is a binary tree where the internal ndoes consist
of an axis and 2 pointers referring to subtrees.
Example
• Here we say a regular B-Partition because we
can represent it by a kd-trie
t3
t1
t1
t2
s
s
t3
t2
Kd-trie
B-Partition
UB-Tree (Universal B-Tree)
• Methods with good performance are
guaranted for only 1 dimension. UB-Tree
can handle multidimensional data.
• We can implement the UB-Tree on top of
any database system. ( by preprocessing
techniques )
UB-Tree (Universal B-Tree)[2]
• Basic Concepts
– Area : First we Partition a cube C of dimension n into 2n
subcubes numbered : sc(i) for i=1,2,…,2n.
– For example : in 2 dimensions.
Sc(1)
Sc(2)
Sc(3)
Sc(4)
AreaC(k) := Ui=1 to k, sc(i) for k =
0,1,…,2n
AreaC(k.j) := AreaC(k) U Areasc(k+1)(J)
Area(3)
Concept of Address
An address α is a sequence
I1,i2,… il where ij Є 0,1,… 2n
For example this
area has address
0.3, noted alpha(A)
= 0.3
Definitions and lemmas
• Region : is the difference of 2 areas.
• Address of pixel : is the address of the
area defined by including the pixel as the
last and smallest subcube contained in
this Area.
• There is a one-to-one map beetween
Cartesian coordinates (x1,x2,…,xn) of a ndimensional pixel and its address α.
• Alpha(cart(α)) = α
Definitions and lemmas[2]
• A point (x1,x2,…xn) has address
region(β,δ), Γ = alpha(x1,x2,…,xn), it
belong to the unique region(β,δ) with the
condition β< Γ.
region(0.1,3)
Range Queries
• The query is defined by an interval for
each dimension. Each dimension can be
beetween (-∞,+∞).
• The query is the cartesian product of the
intervals for all dimensions, called the
query box.
Range queries (2)
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Definition : we call all subcubes of level s of a cube brothers.
Those with a smaller address are younger and those with a larger are
older.
Range queries (3)
Complexity of UB-Tree
• N is the number of objects, k = 1/2M.
Let Q be the number of objects intersecting the querybox q. Let r be
the number of regions intersecting q.
• Point-Query : O(logk(N))
• Range Query : r * O(logk(N)),
For points only it’s : (N*Q/M) * O(logk(N))
• Point insertion : O(logk(N))
Spatial Access Method
• Spatial indexes are used by spatial
databases to optimize spatial queries.
Indexes used by non-spatial databases
cannot effectively handle features such as
how far two points differ and whether
points fall within a spatial area of interest.
• TV-Tree
• X-Tree
TV-Tree (Telescopic-Vector tree)
• The basis of the tv-tree is to use
dynamically contracting and extending
feature vectors. ( Like in classification )
TV-tree
• We have also a hierarchical structure:
• The objects are clustered into leaf nodes
of the tree, and the (MBR), minimum
bounding region is stored in the parent
node.
• Parents are recursively grouped, until the
root is formed.
• At the top levels it’s optimal because it
uses only a few basic features.
TV-tree
• The TV-tree can be applied to a tree with
nodes that describe bounding regions of
any shape (cubes,spheres,rectangles, …
etc ).
Telescoping function
• The telescoping problem can be described as
follows.
• Given an n x 1 feature vector x and m x n (m≤n)
contraction matrix Am.
• The Amx is an m-contraction of x.
• A sequence of such matrices Am with m=1,…
describes a telescoping function provided that
the following condition is satisfied : If the m1contractions of the 2 vectors x and y are equal,
then so are their respective m2-contractions, for
every m2 ≤ m1.
Multiple shapes
• We can use for example a sphere,
because it’s only a center and a radius r.
Represents the set of points with
euclidean distance ≤ r.
• ~the euclidean distance is a special case
of the Lp metrics with p=2.
• For L1 metric (manhattan distance) it
defines a diamond shape.
• The TV-tree is working with any Lp-sphere.
TMBR (Telescopic Minimum
Bounding Region)
• Each node in the TV-Tree represents the
MBR (an Lp-sphere) of all its
descendents.
• Each region is represented by a center,
which is a vector determined by the
telescoping vectors representing the
objects and a scalar radius.
• We use the term TMBR to denote an MBR
with such a telescopic vector as a center.