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## Spatial data structures – kd -trees

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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!

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 balancedkd-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/kdtree.html

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/KDTree-animation.gif

Complexity

- Building a static kd-tree from n points takes O(n log 2n) 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 Cormenet 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

Progressive Meshes

Developed by Hugues Hoppe, Microsoft Research Inc. Published first in SIGGRAPH 1996.

Geometric subdivision

Problems with Geometric Subdivisions

ROAM principle

The basic operating principle of ROAM

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 > $75000?

Age > 60?

yes

no

yes

no

Age > 60?

Data table

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

CLUSTERED

direct search for

UNCLUSTERED

Index entries

data entries

CLUSTERED

direct search for

data entries

Data entries

Data entries

Data entries

(Index File)

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.

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Storage Management for High-Dimensional Indexing Structures

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From multi-dimensional indexing to one-dimensional storage in disk!

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