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Spatial data structures kd-trees

- Jianping Fan
- Department of Computer Science
- UNC-Charlotte

Summary

- This lecture introduces multi-dimensional queries

in databases, as well as addresses how we can

query and represent multi-dimensional data

- A reasonable man adapts himself to his

environment. An unreasonable man persists in

attempting to adapt his environment to suit

himself Therefore, all progress depends on

unreasonable man - George Bernard Shaw

Contents

- Definitions
- Basic operations and construction
- Range queries on multi-attributes
- Variants
- Applications

Usage

- Rendering
- Surface reconstruction
- Collision detection
- Vision and machine learning
- Intel Interactive technology

KD tree definition

- A recursive space partitioning tree.
- Partition along x and y axis in an alternating

fashion. - Each internal node stores the splitting node

along x (or y).

K-d tree

- Used for point location and multiple database

quesries, k number of the attributes to perform

the search - Geometric interpretation to perform search in

2D space 2-d tree - Search components (x,y) interchange!

K-d tree example

d

d

e

c

f

b

f

c

a

e

b

a

Kd tree example

3D kd tree

Construction

- The canonical method of kd-tree construction is

the following - As one moves down the tree, one cycles through

the axes used to select the splitting planes.

(For example, the root would have an x-aligned

plane, the root's children would both have

y-aligned planes, the root's grandchildren would

all have z-aligned planes, the next level would

have an x-aligned plane, and so on.) - Points are inserted by selecting the median of

the points being put into the subtree, with

respect to their coordinates in the axis being

used to create the splitting plane. (Note the

assumption that we feed the entire set of points

into the algorithm up-front.)

Consruction

- This method leads to a balanced kd-tree, in which

each leaf node is about the same distance from

the root. However, balanced trees are not

necessarily optimal for all applications. - Note also that it is not required to select the

median point. In that case, the result is simply

that there is no guarantee that the tree will be

balanced. A simple heuristic to avoid coding a

complex linear-time median-finding algorithm or

using an O(n log n) sort is to use sort to find

the median of a fixed number of randomly selected

points to serve as the cut line

Kd tree mean vs median

kd-tree partitions of a uniform set of data

points, using the mean (left image) and the

median (right image) thresholding options.

Median The middle value of a set of values.

Mean The arithmetic average. (Andrea Vivaldi and

Brian Fulkersson) http//www.vlfeat.org/overview/k

dtree.html

Example of using Median

Additions

- One adds a new point to a kd-tree in the same way

as one adds an element to any other search tree. - First, traverse the tree, starting from the root

and moving to either the left or the right child

depending on whether the point to be inserted is

on the "left" or "right" side of the splitting

plane. - Once you get to the node under which the child

should be located, add the new point as either

the left or right child of the leaf node, again

depending on which side of the node's splitting

plane contains the new node. - Adding points in this manner can cause the tree

to become unbalanced, leading to decreased tree

performance

Deletions

- To remove a point from an existing kd-tree,

without breaking the invariant, the easiest way

is to form the set of all nodes and leaves from

the children of the target node, and recreate

that part of the tree. - Another approach is to find a replacement for the

point removed. First, find the node R that

contains the point to be removed. For the base

case where R is a leaf node, no replacement is

required. For the general case, find a

replacement point, say p, from the sub-tree

rooted at R. Replace the point stored at R with

p. Then, recursively remove p.

Balancing

- Balancing a kd-tree requires care. Because

kd-trees are sorted in multiple dimensions, the

tree rotation technique cannot be used to balance

them this may break the invariant. - Several variants of balanced kd-tree exists. They

include divided kd-tree, pseudo kd-tree,

K-D-B-tree, hB-tree and Bkd-tree. Many of these

variants are adaptive k-d tree.

Quering

- Kdtree query uses a best-bin first search

heuristic. This is a branch-and-bound technique

that maintains an estimate of the smallest

distance from the query point to any of the data

points down all of the open paths. - Kdtree query supports two important operations

nearest-neighbor search and k-nearest neighbor

search. The first returns nearest-neighbor to a

query point, the latter can be used to return the

k nearest neighbors to a given query point Q. For

instance

Nearest-neighbor search

- Starting with the root node, the algorithm moves

down the tree recursively (i.e. it goes right or

left depending on whether the point is greater or

less than the current node in the split

dimension). - Once the algorithm reaches a leaf node, it saves

that node point as the "current best" - The algorithm unwinds the recursion of the tree,

performing the following steps at each node

Recursion step

- If the current node is closer than the current

best, then it becomes the current best. - The algorithm checks whether there could be any

points on the other side of the splitting plane

that are closer to the search point than the

current best. In concept, this is done by

intersecting the splitting hyperplane with a

hypersphere around the search point that has a

radius equal to the current nearest distance. - If the hypersphere crosses the plane, there could

be nearer points on the other side of the plane,

so the algorithm must move down the other branch

of the tree from the current node looking for

closer points, following the same recursive

process as the entire search. - If the hypersphere doesn't intersect the

splitting plane, then the algorithm continues

walking up the tree, and the entire branch on the

other side of that node is eliminated.

Nearest-neighbor search

- kd-trees are not suitable for efficiently finding

the nearest neighbour in high dimensional spaces. - In very high dimensional spaces, the curse of

dimensionality causes the algorithm to need to

visit many more branches than in lower

dimensional spaces. In particular, when the

number of points is only slightly higher than the

number of dimensions, the algorithm is only

slightly better than a linear search of all of

the points. - The algorithm can be improved. It can provide the

k-Nearest Neighbors to a point by maintaining k

current bests instead of just one. Branches are

only eliminated when they can't have points

closer than any of the k current bests.

Range search

- Kd tree provide convenient tool for range search

query in databases with more than one key. The

search might go down the root in both directions

(left and right), but can be limited by strict

inequality on key value at each tree level. - Kd tree is the only data structure that allows

easy multi-key search.

Kd tree

http//upload.wikimedia.org/wikipedia/en/9/9c/KDTr

ee-animation.gif

Complexity

- Building a static kd-tree from n points takes O(n

log 2 n) time if an O(n log n) sort is used to

compute the median at each level. - The complexity is O(n log n) if a linear

median-finding algorithm such as the one

described in Cormen et al. is used. - Inserting a new point into a balanced kd-tree

takes O(log n) time. - Removing a point from a balanced kd-tree takes

O(log n) time. - Querying an axis-parallel range in a balanced

kd-tree takes O(n1-1/k m) time, where m is the

number of the reported points, and k the

dimension of the kd-tree.

Kd tree of rectangles

- Instead of points, a kd-tree can also contain

rectangles. - A 2D rectangle is considered a 4D object (xlow,

xhigh, ylow, yhigh). - Thus range search becomes the problem of

returning all rectangles intersecting the search

rectangle. - The tree is constructed the usual way with all

the rectangles at the leaves. In an orthogonal

range search, the opposite coordinate is used

when comparing against the median. For example,

if the current level is split along xhigh, we

check the xlow coordinate of the search

rectangle. If the median is less than the xlow

coordinate of the search rectangle, then no

rectangle in the left branch can ever intersect

with the search rectangle and so can be pruned.

Otherwise both branches should be traversed. - Note that interval tree is a 1-dimensional

special case.

Applications

- Query processing in sensor networks
- Nearest-neighbor searchers
- Optimization
- Ray tracing
- Database search by multiple keys

Examples of applications

Progressive Meshes

Developed by Hugues Hoppe, Microsoft Research

Inc. Published first in SIGGRAPH 1996.

Terrain visualization applications

Geometric subdivision

Problems with Geometric Subdivisions

ROAM principle

The basic operating principle of ROAM

Quad-tree and Bin-tree for ROAM (real-time

adaptive mesh)

Review questions

- Define kd tree
- What is the difference from B tree? R tree? Quad

tree? Grid file? Interval tree? - Define complexity of basic operations
- What is the difference between mean and median kd

tree? - List typical queries nearest-neighbor, k

nearest neighbors - Provide examples of kd tree applciations

Sources

- In-line references to current research in the

area and variety of research papers and web

sources and applications.

Decision Tree

- Database indexing structure is built for decision

making and tries to make the decision as fast as

possible!

Color Green?

yes

no

Size Big?

Color Yellow?

yes

yes

no

no

Shape Round?

Size small?

watermelon

Size Medium?

yes

no

yes

no

no

yes

apple

Size Big?

banana

Taste sweet?

apple

yes

no

yes

no

Grape

grapefruit

lemon

cherry

grape

Decision Tree

- How to obtain decision for a database?

- Obtain a set of labeled training data set from

the database. - Calculate the entropy impurity

c. Classifier is built by

KD-tree

- By treating query as a decision making procedure,

we can use decision to build more effective

database indexing!

Database root node

no

Salary gt 75000?

Age gt 60?

yes

no

yes

no

Data table

Age gt 60?

no

yes

KD-tree

- Each inter-node, only one attribute is used!
- It is not balance! Search from different node may

have different I/O cost! - It can support multiple attribute database

indexing like R-tree! - It has integrated decision making and database

query!

KD-tree

Tree levels N Leaf nodes M Number of data

entries for leaf node K The inter-nodes for

kd-tree at the same level are stored on the same

page.

- Equal query N M
- Range query N M
- Insert N M 1
- Delete N M 1

Storage Management for High-Dimensional Indexing

Structures

We want to put the similar data in the same page

or neighboring pages!

UNCLUSTERED

Index entries

direct search for

CLUSTERED

UNCLUSTERED

data entries

Index entries

CLUSTERED

direct search for

data entries

Data entries

Data entries

(Index File)

Data entries

Data entries

(Data file)

(Index File)

(Data file)

Data Records

Data Records

Data Records

Data Records

Storage Management for High-Dimensional Indexing

Structures

It is very hard to do multi-dimensional data

sorting!

Hilbert Curve scale multi-dimensional data into

one dimension.

00 01 10 11

Storage Management for High-Dimensional Indexing

Structures

0

14

15

1

2

3

12

13

4

7

11

8

5

9

6

10

From multi-dimensional indexing to

one-dimensional storage in disk!

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