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Introduction to The NSP-Tree: A Space-Partitioning Based Indexing Method. Gang Qian University of Central Oklahoma November 2006. Summary. Overview Motivation and Existing Work NSP-Tree Structure, Algorithms and Performance Conclusion and Future Work. Overview. - PowerPoint PPT Presentation
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Introduction to The NSP-Tree: A Space-Partitioning Based Indexing Method
Gang Qian
University of Central OklahomaNovember 2006
Summary
Overview Motivation and Existing Work NSP-Tree Structure, Algorithms and
Performance Conclusion and Future Work
Overview
The NSP-tree is a disk-based index structure Similar to B-tree/B+-tree
It is designed to index a large amount of vectors with non-ordered discrete components Domains with discrete values that are not naturally ordered
are very common E.g., gender, profession, genome bases, etc.
It is used to speed up similarity queries over the indexed data Unlike exact queries, a similarity query searches for data
items that are similar to the given query data item
Motivation
Traditional database technology is mature Data model: Relational Data Model Design: ER/EER Diagrams Query: SQL Data integrity: Transaction Processing Index: B-tree/B+-tree Some hard unsolved issues still exist
E.g., Multidimensional Query Optimization
New problems occur with the increasing demand for the management of non-traditional data types Multimedia data Scientific data Spatial data Temporal data Biological data, etc.
With the new data types, exact queries are no longer useful Similarity queries become more and more important
Vector Model
The Vector Model is one of the very useful tools to support these new data types Many non-traditional data types are vectors or can be
easily converted into vectors E.g., feature vectors for images
Vectors can be deemed as points in high dimensional data spaces
Therefore, the distance between a pair of vectors is a natural quantitative measure of (dis)similarity between two data objects that the two vectors represent E.g., Euclidean distance
The problem of managing non-traditional databases becomes the problem of managing vector databases
Designing index structures to support efficient similarity queries on vectors is an open research area of vector databases For example, the NSP-tree is designed to index
vectors with discrete and non-ordered components E.g., genome sequence data
Existing Work
A number of index structures are proposed for vectors with continuous numerical components E.g., R-tree and its variants:
SS-tree SR-tree X-tree Hybrid tree, etc.
Due to the volume of the data, almost all proposed index structures are disk-based
The basic structure of these indices are very similar to that of the B+-tree Hierarchical tree structure Each tree node occupies one and only one disk
block and has a minimum utilization requirement Vectors are stored in leaf nodes Non-leaf nodes contain routing information that is
used for tree construction and searching Routing information are usually represented by a certain
type of minimum bounding shapes Minimum Bounding Rectangle (MBR), Minimum Bounding
Sphere (MBS), etc.
Example: R-Tree Structure
Figure adopted from “The SR-tree: An Index Structure for High-Dimensional Nearest Neighbor Queries” (SIGMOD 1997).
Such an index tree grows in a bottom-up fashion Vectors are incrementally inserted into the tree When a leaf node is full, it is split into two leaves The split of a child in the tree may cause the split of a
parent Node split may propagate all the way up to the root,
when the root itself will be split to create a new root Search works top-down from the root
Search performance is usually measured in terms of the total number of disk blocks/nodes accessed
Search efficiency is derived from pruning branches that are not within the search range Unlike a brute force linear search, vectors in irrelevant
branches will not be visited
Unfortunately, those index trees mentioned in previous slides cannot be directly used for vectors with non-ordered discrete components
The ND-tree was proposed to index such vectors See “The ND-Tree: A Dynamic Indexing Techniqu
e for Multidimensional Non-ordered Discrete Data Spaces” (VLDB 2003)
Discrete Space Concepts
The structure of the ND-tree is very similar to those of the R-tree variants
However, all the underlying geometrical concepts are redefined to accommodate discrete vectors
Euclidean/Continuous Space Discrete Space
Vector Discrete Vector
Rectangle Discrete Rectangle
Area Discrete Area
Euclidean Distance Hamming Distance
… …
Example: Discrete Rectangles Introduced to bound vectors with non-ordered discrete
components Normal rectangle can be deemed as the Cartesian product
of ranges for every dimension in the data space E.g., [0.1, 0.2] [0.7, 0.8] is a two-dimensional rectangle
A discrete rectangle is defined as the Cartesian product of sets of discrete values from every dimension E.g., {a, g} {t, c, g} is a two-dimensional discrete rectangle
that covers vectors such as <a, c>, <g, t> and <g, g> Discrete Minimum Bounding Rectangles (DMBR) store the
routing information for the ND-Tree
Problem of The ND-tree
Overlap in an index tree may dramatically affect its search performance
The construction of the ND-tree cannot totally avoid the overlap among DMBRs in the tree The ND-tree works well when the data is randomly
distributed However, for certain data sets, overlap cannot be avoided
For example, the skewed data set based on the Zipf distribution
To guarantee the minimum disk utilization, the split algorithm may NOT be able to find an overlap-free split for an overflow node
Basic Idea of The NSP-Tree
There are three factors that affect search performance Disk utilization Overlap Fan-out
Maximum number of children of a tree node
Since overlap can not be totally avoided when there is a minimum disk utilization requirement, the design of the NSP-tree dropped the requirement so that overlap-free can be guaranteed
Space-Partitioning Indexing Methods Ideas of overlap-free index structures are not new
What makes the NSP-tree new is that it can handle non-ordered discrete data based on an overlap-free structure
There are a category of index trees that have such a feature KDB-tree hB-tree LSD-tree, etc.
They are called space-partitioning indexing methods R-tree variants are called data-partitioning indexing method
s All previous space-partitioning indices support only v
ectors with continuous numeric components
0 1
1
0.60.4
0.3
0.2
0.6
0.2 0.75
d:1v: 0.6
d:2v: 0.3
d:2v: 0.6
d:1v: 0.2
d:1v: 0.4
d:2v: 0.2
d:1v: 0.75
<= >
<= > <= >
d: Split dimension
v: Split point on the split dimension
Space-partitioning InformationPartitioned Data Space
Space-Partitioning vs. Data-Partitioning
Space-Partitioning Data-Partitioning
Objects that can be indexed
Vectors onlyVectors and spatial
objects
Minimum Utilization Requirement
No Yes
Guaranteed Overlap-free
Yes No
Fan-out Large Small
NSP-Tree Structure
Similar to those of the B+-tree and the R-tree, but with no minimum disk utilization requirement Each node occupies one disk block Vectors are stored in leaf nodes Space-partitioning information are stored in non-leaf nodes
The space concept in the NSP-tree is discrete A discrete data space is defined as the Cartesian product
of the sets of all possible values on every dimension Due to the non-ordered nature of the values, a split point
on a split dimension is no long enough to describe a split Need to explicitly record how each values on a dimension are
separated into two groups
Structure of The NSP-Tree
Routing Information: Split History Tree (SPT)
Conceptually, each node corresponds to a subspace of the discrete data space A subspace is defined as the Cartesian product of
the subsets of values on every dimension There is no overlap among the subspaces of the
children on the same level The subspace of a parent node contains the
subspaces of all its children
Eliminating Dead Space
One disadvantage of a pure space-partitioning approach is that the subspaces do not necessarily minimally bound the vectors in the space See next slide
To further improve the pruning power, DMBRs are used as additional routing information in tree
However, the use of DMBRs reduces the fan-out of tree More space in a node is needed to store the DMBRs We found that the benefits of using DMBRs are usually
greater than the disadvantage of the decrease of the fan-out
0 1
1
0.60.4
0.3
0.2
0.6
0.2 0.75
Actual Minimum Bounding Rectangle
Subspace is not minimum bounding
Dead space
r
Q
Tree Construction Algorithms An NSP-tree grows incrementally
Vectors are inserted one by one Insertion starts from the root and goes down the
tree until a suitable leaf node is found for the new vector
The tree grows in a bottom-up fashion There are two import algorithms used in the
insertion procedure ChooseSubtree SplitNode
ChooseSubtree Starting from the root, it is invoked on non-leaf nodes Given the vector to insert, the algorithm decides which child
nodes to follow based on whether a child’s subspace contains the new vector or not Due to the overlap-free property, there exists at most one
child that can contain the new vector SplitNode
Splits an overflow node into two nodes The split guarantees overlap-free It also tries to maximize disk utilization by choosing the
most balanced split
There are other algorithms for the NSP-tree Generating and maintaining DMBRS Query Deletion, etc.
Query Performance
Disk Utilization
Summary
The NSP-tree is the first indexing method that uses the space-partitioning approach to index vectors with non-ordered discrete components
The benefit of using an overlap-free tree structure is obvious when data distribution is skewed
With proper heuristics, the disadvantage of the removal of the minimum disk utilization requirement can be minimized
In general, the benefit of using DMBRs to eliminate dead space (hence, increasing the pruning power) overrides the disadvantage of the fan-out decrease
Future Work
Bulkloading the NSP-tree and the ND-tree Insert more than one vector at a time
Support approximate similarity queries Beat the Curse of High Dimensionality
Support queries based on the Editor Distance Besides the Hamming distance, the Editor
distance is another widely-used distance measure for discrete vectors
Aggregate all the technology into a viable bioinformatics search engine