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Indexing Multidimensional Feature Spaces

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Indexing Multidimensional Feature Spaces

Overview of Multidimensional Index Structure

Hybrid Tree, Chakrabarti et. al. ICDE 1999

Local Dimensionality Reduction, Chakrabarti et. al. VLDB 2000

- Consider a d-dimensional feature space
- color histogram, texture, …

- Nature of Queries
- range queries: objects that reside within the region specified in the query
- K-nearest neighbor queries: objects that are closest to a query object based on a distance metric
- Approx. nearest neighbor queries: retrieved object is within (1+ epsilon) of the real nearest neighbor.
- All-pair (similarity join) queries: retrieve all pairs of objects within a epsilon threshold.

- A search algorithm may include:
- false positives: objects that do not meet the query condition, but are retrieved anyway. We tend to minimize false positives
- false negatives: objects that meet the query condition but are not returned. Usually, approaches avoid false negatives

- Index on attributes independently
- Project query range to each attribute determine pointers.
- Intersect pointers
- go to the database and retrieve objects in the intersection.

May result in very high I/O cost

- Index on one attribute provides pointers to an index on the other

- Cannot support partial match queries on second attribute
- performance of range search not much better compared to independent attribute approach
- the secondary indices may be of different sizes -- specifically some of them may be very small

Index on first attribute

Index on second attribute

- Extension of B-tree to multidimensional space.
- Paginated, balanced, guaranteed storage utilization.
- Can support both point data and data with spatial extent
- Groups objects into possibly overlapping clusters (rectangles in our case)
- Search for range query proceeds along all paths that overlap with the query.

- Step I1
- Chooseleaf L to Insert E /* find position to insert*/

- Step I2
- If L has room install E
- Else SplitNode(L)

- Step I3:
- Adjust Tree /* propagate Changes*/

- Step I4:
- if node split propagates to root adjust height of tree

- Step CL1:
- Set N to be root

- Step CL2:
- If N is a leaf return N

- Step CL3:
- If N is not a root, let F be an entry whose rectangle needs least enlargement to include object
- break ties by choosing smaller rectangle

- If N is not a root, let F be an entry whose rectangle needs least enlargement to include object
- Step CL4
- Set N to be child node pointed by entry F
- goto Step CL2

- Given a node split it into two nodes which are each atleast half full
- Multiple Objectives:
- minimize overlap
- minimize covered area

- R-tree minimizes covered area
- What is an optimal criteria???

Minimize covered area

Minimize overlap

- Group objects into 2 parts such that the covered area is minimized
- NP Hard!!
- Hence use heuritics
- Two heuristics explored
- quadratic and linear

- /* Divide the set of M+1 entries into 2 groups G1 and G2 */
- PickSeeds for G1 and G2
- Invoke PickNext to assign an object to a group recursively until either all objects assigned or one of the groups becomes half full.
- If one group gets half full assign rest of the objects to the other group.

- PickSeed:
- for each pair of entries E1 and E2 compose a rectangle J including E1.rect and E2.rect
- let d = area(J) - area(E1.rect) - area(E2.rect) /* d is wasted space */

- Choose the most wasteful pair with largest d as seeds for groups G1 and G2.

- for each pair of entries E1 and E2 compose a rectangle J including E1.rect and E2.rect
- PickNext /*select next entry to put in a group */
- Determine cost of putting each entry in the group G1 and G2
- for each unassigned entry calculate
- d1 = area increase required in the covering rectangle in Group G1 to include the entry
- d2= area increase required in the covering rectangle in Group G2 to include the entry.

- Select entry with greatest preference for a group
- choose any entry with the maximum difference between d1 and d2

- Determine cost of putting each entry in the group G1 and G2

- PickSeed
- find extreme rectangles along each dimension
- find entries with the highest low side and the lowest high side

- record the separation
- Normalize the separation by width of extent along the dimension
- Choose as seeds the pair that has the greatest normalized distance along any dimension

- find extreme rectangles along each dimension
- PickNext
- randomly choose entry to assign

- Start from root
- If node T is not leaf
- check entries E in T to determine if E.rectangle overlaps S
- for all overlapping entries invoke search recursively

- If T is leaf
- check each entry to see if it entry satisfies range query

- Step D1
- find the object and delete entry

- Step D2
- Condense Tree

- Step D3
- if root has 1 node shorten tree height

- If node is underful
- delete entry from parent and add to a set Q

- Adjust bounding rectangle of parent
- Do the above recursively for all levels
- Reinsert all the orphaned entries
- insert entries at the same level they were deleted.

- Many generalizations of R-tree
- different splitting criteria
- different shapes of clusters (e.g., d-dimensional spheres)
- adding redundancy to reduce search cost:
- store objects in multiple rectangles instead of a single rectangle to reduce cost of retrieval. But now insert has to store objects in many clusters. This strategy also increases overlap causing search performance to detoriate.

- Space Partitioning Data Structures
- unlike R-tree which group objects into possibly overlapping clusters, these methods attempt to partition space into non-overlapping regions.
- E.g., KD tree, quad tree, grid files, KD-Btree, HB-tree, hybrid tree.

- Space filling curves
- superimpose an ordering on multidimensional space that preserves proximity in multidimensional space. (Z-ordering, hilbert ordering)
- Use a B-tree as an index on that ordering

- A main memory data structure based on binary search trees
- can be adapted to block model of storage (KD-Btree)

- Levels rotate among the dimensions, partitioning the space based on a value for that dimension
- KD-tree is not necessarily balanced.

X=7

X=3

X=5

X=8

X=5

y=6

y=5

Y=6

x=8

x=7

x=3

Y=2

y=2

- Search:
- straightforward…. Just descend down the tree like binary search trees.

- Insertion:
- lookup record to be inserted, reaching the appropriate leaf.
- If room on leaf, insert in the leaf block
- Else, find a suitable value for the appropriate dimension and split the leaf block

- Similar to B-tree, tree nodes split many ways instead of two ways
- Risk:
- insertion becomes quite complex and expensive.
- No storage utilization guarantee since when a higher level node splits, the split has to be propagated all the way to leaf level resulting in many empty blocks.

- Risk:
- Pack many interior nodes (forming a subtree) into a block.
- Risk
- it may not be feasible to group nodes at lower level into a block productively.
- Many interesting papers on how to optimally pack nodes into blocks recently published.

- Risk

- Nodes split along all dimensions simultaneously
- Division fixed: by quadrants
- As with KD-tree we cannot make quadtree levels uniform

X=7

X=3

SW

NW

SE

NE

X=5

X=8

- Insert:
- Find Leaf node to which point belongs
- If room, put it there
- Else, make the leaf an interior node and give it leaves for each quadrant. Split the points among the new leaves.

- Search:
- straighforward… just descend down the right subtree

- Space Partitioning strategy but different from a tree.
- Select dividers along each dimension. Partition space into cells
- Unlike KD-tree dividers cut all the way.
- Each cell corresponds to 1 disk page.
- Many cells can point to the same page.
- Cell directory potentially exponential in the number of dimensions

- Maintain linear scales for each dimension that contain split positions for the dimension
- Cell directory implemented as a multidimensional array.
- /* can be large and may not fit in memory */

- Exact Match Search: at most 2 I/Os assuming linear scales fit in memory.
- First use liner scales to determine the index into the cell directory
- access the cell directory to retrieve the bucket address (may cause 1 I/O if cell directory does not fit in memory)
- access the appropriate bucket (1 I/O)

- Range Queries:
- use linear scales to determine the index into the cell directory.
- Access the cell directory to retrieve the bucket addresses of buckets to visit.
- Access the buckets.

- Determine the bucket into which insertion must occur.
- If space in bucket, insert.
- Else, split bucket
- how to choose a good dimension to split?

- If bucket split causes a cell directory to split do so and adjust linear scales.
- /* notice that cell directory split results in p^(d-1) new entries to be created in cell directory */
- insertion of these new entries potentially requires a complete reorganization of the cell directory--- expensive!!!

- Inserting a new split position will require the cell directory to increase by 1 column. In d-dim space, it will cause p^(d-1) new entries to be created

- Assumption
- finite precision in representing each coordinate.

B

A

Z(A) = shuffle(x_A, y_A) = shuffle(00,11)

= 0101 = 5

Z(B) = 11 = 3

(common prefix to all its blocks)

Z(C1) = 0010 = 2

Z(C2) = 1000 = 8

00 01 10 11

00 01 10 11

C

- Obtain a quad-tree decomposition of an object by recursively dividing it into blocks until blocks are homogeneous.

01

11

Objects representation

is

0001, 0011,01

00

10

00

11

01

00

11

- For disk storage, represent object based on its Z-value
- Use a B-tree index.
- Range Query:
- translate query range to Z values
- search B-tree with Z-values of data regions for matches

- Retrieve the nearest neighbor of query point Q
- Simple Strategy:
- convert the nearest neighbor search to range search.
- Guess a range around Q that contains at least one object say O
- if the current guess does not include any answers, increase range size until an object found.

- Compute distance d’ between Q and O
- re-execute the range query with the distance d’ around Q.
- Compute distance of Q from each retrieved object. The object at minimum distance is the nearest neighbor!!! Why?
- Issues: how to guess range, the retrieval may be sub-optimal if incorrect range guessed. Becomes a problem in high dimensional spaces.

Distance between

Q and A

b

Initial range search

Q

A

Revised range search

A optimal strategy that results in minimum number of I/Os possible

using priority queues.

Mindist(Q,B)

- Let Q be the query point.
- Traverse nodes in the data structure in the order of MINDIST(Q,N), where
- MINDIST(Q,N) = dist(Q,N), if N is an object.
- MINDIST(Q,N) = minimum distance between Q and any object in N, if N is an interior node.

B

Q

A

Mindist(Q, A)

Mindist(Q,C)

C

Q

Q

T

Q

S

- Motivation:
- disparate applications require different data structures and access methods.
- Requires separate code for each data structure to be integrated with the database code
- too much effort.
- Vendors will not spend time and energy unless application very important or data structure has general applicability.

- Generalized search trees abstract the notion of data structure into a template.
- Basic observation: most data structures are similar and a lot of book keeping and implementation details are the same.
- Different data structures can be seen as refinements of basic GiST structure. Refinements specified by providing a registering a bunch of functions per data structure to the GiST.

- GiST is like a “template” - it defines its interface in terms of ADT rather than physical elements (like nodes, pointers etc.)
- The access method (AM) can customize GiST by defining his or her own ADT class i.e. you just define the ADT class, you have your access method implemented!
- No concern about search/insertion/deletion, structural modifications like node splits etc.

x=5

x+y=12

y=5

y=4

y=3

x=3

x=4

x=6

Generalized Search Trees (GiSTs)

x>5

and

y>4

x>4

and

y£3

x£3

x£6

x+y

£12

x+y

>12

y£5

x>6

y>5

Data nodes containing points

- Very high dimensionality of feature spaces -- e.g., shape may define a 100-D space.
- Traditional multidim. data structures perform worse than linear scan at such high dimensionality. (dimensionality curse)

- Arbitrary distance functions-- e.g., distance functions may change across iterations of relevance feedback.
- Traditional multidim. data structures support a fixed distance measure -- usually euclidean (L2) or Lmax.

- No support for Multi-point Queries -- as in query expansion.
- Executing K-NN for each query point and merging results to generate K-NN for multi-point query is very expensive.

- No Support for Refinement
- query in the following iterations do not diverge greatly from query in previous iterations. Effort spent in previous iterations should be exploited for evaluating K-NN in future iterations

Multidim. Data Structures

- design data structures that scale to high dim. spaces
- Existing proposals perform worse than linear scan over >= 10 dim. Spaces [Weber, et al., VLDB 98]
- Fundamental Limitation dimensionality beyond which linear scan wins over indexing! (approx. 610)

Dimensionality Reduction

- transform points in high dim. space to low dim. space
- works well when data correlated into a few dimensions only
- difficult to manage in dynamic environments

Data Partitioning (DP)

Bounding Region (BR) Based e.g., R-tree, X-tree, SS-tree, SR-tree, M-tree

All k dim. used to represent partitioning

Poor scalability to dimensionality due to high degree of overlap and low fanout at high dimensions

seq. scan wins for > 10D

Space Partitioning(SP)

Based on disjoint partitioning of space e.g., KDB-tree, hB-tree, LSDh-tree, VP tree, MVP tree

no overlap and fanout independent of dimensions

Poor scalability to dimensionality due to either poor storage utilization or redundant information storage requirements.

D

B

Dim2

3

2

A

C

0,0

3

Dim1

Data Points

R1

R2

R3

R4

R1 R2 R3 R4

dim=1

pos=3

Non-leaf

nodes of

hybrid tree

organized

as kd-tree

dim=2

pos=2

dim=2

pos=3

Data Points

Data Points

Data Points

A

B

C

D

F

F

B

C

B

C

4

3

Clean split possible

without violating

node utilization

D

E

D

E

A

A

(0,0)

2

4

6

dim=1

pos=4

dim=1

pos=4

dim=2

pos=3

dim=2

pos=4

dim=2

pos=3

dim=2

pos=4

A

dim=1

pos=2

F

dim=1

pos=6

A

dim=1

pos=2

F

dim=1

pos=6

D

B

C

E

D

B

C

E

dim=1

pos=4

dim=2

pos=3

dim=2

pos=4

dim=1

pos=2

dim=1

pos=6

dim=2

pos=5

A

dim=1

pos=7

dim=2

pos=2

B

C

D

G

E

F

I

H

Clean split not possible

without violating node util.

Always clean split;

Downward

cascading splits

(empty nodes)

Complex splits

(space overhead;

tree becomes large)

Allow Overlap

(avoid by relaxing node util,

otherwise minimize overlap)

(Hybrid Tree)

H

I

H

I

B

C

G

B

C

G

F

F

D

D

A

A

E

E

5

4

dim=2

pos=3,4

3

2

(0,0)

2

4

6

7

dim=1

pos=4,4

dim=1

pos=4,4

dim=1

pos=4,4

dim=2

pos=3,3

dim=2

pos=4,4

dim=1

pos=6,6

dim=1

pos=2,2

dim=2

pos=5,5

A

dim=1

pos=2,2

dim=1

pos=6,6

dim=2

pos=5,5

A

dim=2

pos=2,2

dim=1

pos=7,7

D

B

C

G

dim=1

pos=7,7

dim=2

pos=2,2

B

C

D

G

E

F

H

I

E

F

H

I

Consider a range (cube) query, side length r along each dimension

Split node along this

Node BR

Node BR

expanded

by (r/2) on

each side

along each

dimension

(Minkowski

Sum)

r

w

w+r

Prob. of range query accessing

node (assuming (0,1) space

and uniform query distribution)

Prob. of range query accessing

both nodes after split

(increase in EDA)

Choose split dimension and position that minimizes increase in EDA

- Data Node Splitting
- Spilt dimension: split along maximum spread dimension
- Split position: split as close to the middle as possible (without violating node utilization)

- Index Node Splitting:
- Split dimension: argminjò P(r) (wj + r)/ (sj + r) dr
- depends of the distribution of the query size
- argminj (wj + R)/ (sj + R)when all queries are cubes with side length R

- Split position: avoid overlap if possible, else minimize as much overlap as possible without violating utilization constraints

- Split dimension: argminjò P(r) (wj + r)/ (sj + r) dr

R1

R2

Space Partitioning (Hybrid tree): Without

dead space elimination

D

B

R3

R4

A

C

Data Partitioning (R-tree): No dead space

Space Partitioning (Hybrid tree): With dead space elimination

D

B

A

C

- Live space encoding using 3 bit precision (ELSPRECISION=3)
- Encoded Live Space (ELS) BR = (001,001,101,111)
- Bits required = 2*numdims*ELSPRECISION
- Compression = ELSPRECISION/32
- Only applied to leaf nodes

111

110

101

100

011

010

001

000

000 001 010 011 100 101 110 111

- Search:
- Point, Range, NN-search, distance-based search as in DP-techniques
- Reason: BR representation can be derived from kd-tree representation
- Exploit tree organization (pruning) for fast intra-node search

- Insertion:
- recursively choose space partition that contains the point
- break tries arbitrarily
- no volume computation (otherwise floating point exception at high dims)

- Deletion:
- details in thesis

H

I

B

C

G

F

D

A

E

Search algorithms developed for R-tree can be used directly

dim=2

pos=3,4

dim=1

pos=4,4

dim=1

pos=4,4

A

dim=1

pos=6,6

dim=1

pos=2,2

dim=2

pos=5,5

D

dim=2

pos=2,2

dim=1

pos=7,7

B

C

G

E

F

H

I

2

3

1

1

2

3

Range Queries

k-NN queries

1

3

2

Euclidean distance

Weighted Euclidean

Weighted Manhattan

- More scalable to high dimensionalities than:
- DP techniques (R-tree like index structures)
- Fanout independent of dimensionality: high fanout even at high dims
- Faster intranode search due to kd-tree-based organization
- No overlap at lowest level, low overlap at higher levels

- SP techniques
- Guaranteed nodeutilization
- No costly cascading splits
- EDA-optimal choice of splits

- DP techniques (R-tree like index structures)
- Supports arbitrary distance functions

- Effect of ELS encoding
- Test scalability of hybrid tree to high dimensionalities
- Compare performance of hybrid tree with SR-tree (data partitioning), hB-tree (space partitioning) and sequential scan

- Data Sets
- Fourier Data set (16-d Fourier vectors, 1.2 million)
- Color Histograms for COREL images (64-d color histograms from 70K images)

Factor of Sequential IO

to Random IO accounted for

- Hybrid Tree scales well to high dimensionalities
- Outperforms linear scan even at 64-d (mainly due to significantly lower CPU cost)

- Order of magnitude better than SR-tree (DP) and hB-tree (SP) both in terms of I/O and CPU costs at all dimensionalities
- Performance gap increases with the increase in dimensionality

- Efficiently supportsarbitrary distance functions

- Dimensionality curse persists
- To achieve further scalability, dimensionality reduction (DR) commonly used in conjuction with index structures
- Exploit correlations in high dimensional data

Expected graph (hand drawn)

- First performPrincipal Component Analysis (PCA), then build index on reduced space
- Distances in reduced space lower bound distances in original space
- Range queries:
- map point, range query with same range, eliminate false positives
- k-NN query (a bit more complex)

- DR increases efficiency, not quality of answers

First Principal Component (PC)

r

Reduced space

r

First Principal

Component (PC)

First PC

- works well only when data is globally correlated
- otherwise too many false positives result in high query cost
- solution: find local correlations instead of global correlation

Cluster1

First PC of Cluster1

Cluster2

First PC of Cluster2

GDR

LDR

First PC

- Identify Correlated Clusters in dataset
- Definition of correlated clusters
- Bounding loss of information
- Clustering Algorithm

- Indexing the Clusters
- Index Structure
- Point Search, Range search and k-NN search
- Insertion and deletion

Mean of all

points in cluster

Centroid of cluster

(projection of mean on eliminated dim)

First PC

(retained dim.)

Second PC

(eliminated

dim.)

A set of locally correlated points = <PCs, subspace dim, centroid, points>

Centroid of cluster

Projection of Q on eliminated dim

Point Q

First PC

(retained dim)

Reconstruction

Distance(Q,S)

Second PC

(eliminated dim)

Centroid

£ MaxReconDist

First PC

(retained dim)

£ MaxReconDist

Second PC

(eliminated dim)

ReconDist(P, S) £ MaxReconDist, " P in S

- Dimensionality bound: A cluster must not retain any more dimensions necessary and subspace dimensionality £ MaxDim
- Size bound: number of points in the cluster ³ MinSize

- Choose a set of well-scattered points as centroids (piercing set) from random sample
- Group each point P in the dataset with its closest centroid C if the Dist(P,C) £ e

- Compute PCs

- Assign each point to cluster that needs min dim. to accommodate it
- Subspace dim. for each cluster is the min # dims to retain to keep most points

- Assign each point P to the cluster S such that ReconDist(P,S) £ MaxReconDist
- If multiple such clusters, assign to first cluster (overcomes “splitting” problem)

Empty

clusters

- Eliminate small clusters
- Map each point to subspace (also store reconstruction dist.)

Map

- Iterate for more clusters as long as new clusters are being found among outliers
- Overall Complexity: 3 passes, O(ND2K)

- Precision Experiments:
- Compare information loss in GDR and LDR for same reduced dimensionality
- Precision = |Orig. Space Result|/|Reduced Space Result| (for range queries)
- Note: precision measures efficiency, not answer quality

- Synthetic dataset:
- 64-d data, 100,000 points, generates clusters in different subspaces (cluster sizes and subspace dimensionalities follow Zipf distribution), contains noise

- Real dataset:
- 64-d data (8X8 color histograms extracted from 70,000 images in Corel collection), available at http://kdd.ics.uci.edu/databases/CorelFeatures

Root containing pointers to root of each cluster index (also stores PCs and subspace dim.)

Set of outliers (no index: sequential scan)

Index

on

Cluster 1

Index

on

Cluster K

Properties: (1) disk based

(2) height £ 1 + height(original space index)

(3) almost balanced

- Cost Experiments:
- Compare linear scan, Original Space Index(OSI), GDR and LDR in terms of I/O and CPU costs. We used hybridtree index structure for OSI, GDR and LDR.

- Cost Formulae:
- Linear Scan: I/O cost (#rand accesses)=file_size/10, CPU cost
- OSI:I/O cost=num index nodes visited, CPU cost
- GDR:I/O cost=index cost+post processing cost (to eliminate false positives), CPU cost
- LDR:I/O cost=index cost+post processing cost+outlier_file_size/10, CPU cost

- LDR is a powerful dimensionality reduction technique for high dimensional data
- reduces dimensionality with lower loss in distance information compared to GDR
- achieves significantly lower query cost compared to linear scan, original space index and GDR

- LDR is a general technique to deal with high dimensionality
- our experience shows high dimensional datasets often have local correlations - LDR is the only technique that can discover/exploit it
- applications beyond indexing: selectivity estimation, data mining etc. on high dimensional data (currently exploring)