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Data Warehousing & Data MiningWolf-Tilo BalkeSilviu HomoceanuInstitut für InformationssystemeTechnische Universität Braunschweighttp://www.ifis.cs.tu-bs.de

6. Optimization6.1 Multidimensional storage

6.2 DW Optimization / Indexes

Tree based indexes

Bitmap indexesHash Based

Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 2

6. Optimization

• The basic storage structure is the multidimensional array

– Customized based upon

• The data e.g., sparse or dense

• Characteristics of the secondary memory e.g., block- or page- oriented

– Cube data cells are stored sequentially

• Multidimensional cubes are linearized


6.1 Multidimensional Storage

• Linearization– For n dimensions D1to Dn there are |D1| * |D2| *…* |Dn|

addressed cube cells

– E.g., 2D cube• |D1| = 5, |D2| = 4, cube cells = 20

– To access a measure in a cube we gothrough dimensions

• Sold Jackets in February are stored incube cell D1[4], D2[3]

• After linearization D1[4], D2[3] becomes array cell 14– This represents (Index(D2) – 1) * |D1| + Index(D1) (2 full rows + the

remaining 4 elements from the incomplete row)

– Linearized Index = 2 * 5 + 4 = 14



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Jan (1)

Feb(2)

Feb(3)

Feb(4)

D1

D2

• General idea of linearization

– Considering a cube C=((D1, D2, …, Dn), (M1:Type1, M2:Type2, …, Mm:Typem)), the index of a cube cell z with coordinates (x1, x2, …, xn) can be linearized as follows:

• Index(z) = x1 + (x2 - 1) * |D1| + (x3 - 1) * |D1| * |D2| + … + (xn - 1) * |D1| * … * |Dn-1| =

= 1+ ∑i=1

n ((xi - 1) * ∏j=1

i-1 |Di|)



• Problems in array-storage– Influence of the order of the dimensions in the

cube definition• In the cube the cells of D2 are ordered

one under the other, e.g., PantsA query of sales of all pants involves a column in the cube

• After linearization, the information isspread among more data blocks or pages

• If we consider a data block can hold 5 cells, a query over all products sold in January can be answered with just 1 block read, but a query of all sold pants, involves reading 4 blocks


6.1 Problems in Array-Storage

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• The problem of dimensions order can be diminished by using caching solutions

– Caching and swapping is performed alsoby the operating system

– MDBMS has to manage its cachessuch that the OS doesn’t perform anydamaging swaps



• Storage of dense cubes

– If cubes are dense, array storage is more efficient. However, operations suffer due to the large cubes

– The solution is to store dense cubes not linear but on 2 levels

• The first contains an indexes and the second the data cells stored in blocks

• Different optimization procedures like indexes (trees, bitmaps), physical partitioning, and compression (run-length-encoding) can be used



• Storage of sparse cubes– All the cells of a cube, including empty ones, have to

be stored– Sparseness leads to data being stored in many physical

blocks or pages• The query speed is affected by the large number of block

accesses on the secondary memory

– Solutions:• Do not store empty blocks or pages: if there are large empty

portions of the array, they will not be physically stored, but the index structure will be adapted

• 2 level data structure: upper layer holds all possible combinations of the sparse dimensions, lower layer holds dense dimensions



• 2 level cube storage



Marketing campaign

Customer

Product

TimeGeo

Dense low level

Sparse upper level

• Indexes are used to optimize queries. OLAP queries have an aggregation role– How many articles from product group washing

devices were sold in 2008 for each month in each region

• Very big detailed data set (lots of sales in the sales fact table)

• Such aggregation (2008, region) queries on big data sets are costly: e.g., consider 100 GB of sales data stored in a star schema; for this query the whole set needs to be read…at an average speed of 40 MB/s it still takes 43 minutes only to read the data


6.2 Indexes

• Restriction role

– Besides aggregation, restrictions are the most time-consuming

– Typologies of restrictions based on query types

• Range query

– Restricted through intervals in each dimension


6.2 Remember OLAP Queries?

Pro

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Art

icle

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Time (Days)

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• Partial range query– Some dimensions are not restricted

– Geometrically described as a sub-space

• Partial match query– Restricts more dimensions on a point,

while other dimensions remainunspecified

– Geometrically described as a hyper-level in

the data set

• Point query– Restricted to a point on all dimensions


6.2 Remember OLAP Queries?

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ct (

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Time (Days)

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6.2 Optimization Procedures

R1 R2

MV1

Base-layer of

the data

Layer of

materialization

Layer of

partitioning

Layer of

Index struct.

MV2

P1

P2

P3

B*-Baum

kB-Baum

R*-Baum

UB-Baum

Logical

access paths

Physical

access paths

• Why index?– Consider a 10 GB table; at 10 MB/s read speed we

need 17 minutes for a full table scan

– Consider an OLAP query: the number of Bosch S500 washing machines sold in Braunschweig last month?

• Applying restrictions (product, location) the selectivity would be strongly reduced

– If we have 30 location, 10000 products and 24 months

in the DW, the selectivity is

1/30 * 1/ 10000 * 1/24 = 0,00000014

– So…we read 10 GB for 1,4KB of data…not very smart


6.2 Index Structures

• Reduce the size of read pages to a

minimum with indexes



Pro

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Full table scan

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Cluster primary

index

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More secondary

Indexes, bitmap indexes

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Optimal multi-

dimensional index

Scanned data Selected data

• Types of queries

– k-nearest-neighbor-Search (k-NN-Search)

• Find the first K objects with the smallest distance to the query object

• Such a query is usually performed with approximations (with an error rate)

– Reverse-nearest-neighbor-Search

• Find all the objects whose nearest neighbor is the queried object



• Curse of Dimensionality (Richard Bellman)

– The volume increases exponentially with the dimensions number

– More dimensions mean more comparisons will be performed

• At the present time there is no real scalable indexing



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• Classification criteria for index structures– Clustering

• Data with high probability of being used together

– Dimensionality• Refers to the number of attributes used to calculate the index

key

– Symmetry• The order of the index attribute is not performance relevant

– Tuple identifier (TID)• TID s are position numbers pointing to the physical storage place

of the corresponding data

– Dynamical behavior• Effort needed for dynamical modifications can strongly vary from

index to index



• In the beginning…there were

B-Trees

– Data structures for storing sorted

data with amortized run times for

insertion and deletion

– Basic structure of a node


6.2 Trees

…

Tree Node

Node Pointers

Key Value Data Pointer

• B-Trees as primary indexes in OLTP


6.2 Trees

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$ 8

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2, B

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, $

19

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3, B

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5, M

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, $

68

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7, R

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, $

86

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8, S

mit

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$6

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9, T

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2 6 7

1 3 4 5 8 9

• What’s wrong with B-Trees?

– K-D-B trees are useful for point data only

• Exact-point lookup!

• Not good for storing geometrical data and multi-dimensional data

– The R-Tree provided a way to do that (thanks to Guttman ‘84)

• R-tree represents data objects in intervals in several dimensions

– Exact-point and range lookups!


6.2 B-Trees

• Demands

– Can store d-dimensional hypercubes

– Performs point, line, and box queries as fast as possible...

– ...but also keeps memory usage in check!


6.2 R-Trees

• R-Trees are recommended for lower dimensionality

– Up to 10 dimensions

• More scalable variants:

– R+-Trees, R*-Trees und X-Trees

– Each up to 20 dimensions


6.2 R-Trees

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• There is no total ordering of objects in the multidimensional space that preserves spatial proximity

– E.g. in time dimension it makes sense to keep data belonging to consecutive quarters clustered together


6.2 R-Trees

• R-Trees

– Can organize any-dimensional data

• Representing the data by a minimum bounding rectangle (MBR)

– Each node bounds it’s children

– A node can have many objects in it

• E.g.,

– node capacity of 3 (in praxis node

capacity is of 100s)

– 2 dimensions


6.2 R-Trees

• The leaves point to the actual objects (stored on disk probably)

• The height is always log n (it is height balanced)


6.2 R-Trees

R1 R2

R3 R4 R5 R6

R1

R4

R3

R5

R6R2

R3.1 R3.2 R3.3 R4.1 R4.2 R4.3

R4.1

…

Points to data tuplesleafs

Non-leafs

Root

• 2 types of nodes

– Non-leaf nodes, contain entries of the form (I, P)

• I represents a vector I=(I1, I2, …, In) describing the boundaries of the element on the n dimensions, e.g., time and location, for R1 we have I=([08 Qtr1, 09 Qtr1], [a, b])

• P represents a pointer(e.g., disk page address)to the node with itschildren


6.2 R-Trees

R1 R2

R3 R4

R1

R4

R3

R5

R6R2

08 Qtr1

08 Qtr2

08 Qtr3

08 Qtr4

09 Qtr1

Time

Locationa b c d e f g

• Leaf nodes, contain entries of the form (I, RID)

– I same as for non-leaf

– RID represents an unique tupleidentifier


6.2 R-Trees

R3.1 R3.2 R3.3

Points to data tuples

• Operations

– Let:

• EI be the rectangle part of an index entry E

• Ep be the tupel-identifier or pointer

• S be the search rectangle

– E.g., ([08 Qtr3, 09 Qtr1], [a, b])

• T be the root of the R-Tree

– Search:

• Start from the root node

• If multiple sub-trees contain the point of interest then follow all


6.2 R-Trees - Search

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• Search (T, S)– If T is not a leaf

• Check each entry E to determine whether EI overlaps S

• For all overlapping entries, invoke Search(Ep, S)

– If T is a leaf• Check all entries E to determine whether EI overlaps S

– If so, E is a qualifying record

• No good performance guarantees– In worst case all paths must be searched (due to

overlapping)

• Search algorithms try to cut out irrelevant regions („Pruning“)



• Search on sales on last 2 quarters of 08, locations e and f



R1

R3 R4

R3.1 R3.2 R3.3 R4.1 R4.2 R4.3

R1

R4

R3

R5

R6R2

08 Qtr1

08 Qtr2

08 Qtr3

08 Qtr4

09 Qtr1

Time

a b c d e f g

R5.3 R6.3R6.2

• Insert, general idea

– New index records are added to the leaves!

• Nodes that overflow are split

• Splits propagate up the tree

• Node splitting is not trivial


6.2 R-Trees - Insert

• Insert (T, E)– Find position for new record

• Invoke ChooseLeaf to select a leaf node L in which to place E

– Add record to leaf node: • If L has room for E then insert E and return

• Otherwise, invoke SplitNode to obtain L and LL containing E and all the old entries of L

– Propagate changes upwards• Invoke AdjustTree on L, also passing LL if a split was performed

– Grow tree taller• If node split propagation caused the root to split, create a new

root whose children are the two resulting nodes.



• ChooseLeaf(E)– Initialize

• Set N to be the root node

– Leaf check• If N is a leaf, return N

– Choose sub-tree• Let F be the entry in N whose rectangle FI needs least enlargement to include E

• Resolve ties by choosing the entry with the rectangle of smallest area

– Descend until a leaf is reached• Set N to be the child node pointed to by Fp and repeat from Leaf

check


6.2 R-Trees

• Node splitting

– A full node contains M entries

– Divide the collection of M+1 entries between

2 nodes.

– Objective: Make it as unlikely as possible for the resulting two new nodes to be examined on subsequent searches.

– Heuristic: The total area of two covering rectangles after a split should be minimized



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• Node splitting



Bad split Good split

• Node splitting methods

– Exhaustive algorithm

• Generate all possible groups and choose the best with minimum area

• Number of possibilities ~ 2M-1

• For M ~ 50 Number of possibilities ~ 600 Trillion

– Which is even more than theObama administration spentwith the crisis!!!



• Quadratic-cost algorithm

– A heuristic to find a small-area split

– Cost is quadratic in M and linear in the number of dimensions

– Pick two of the M+1 entries to be the first elements of the two new groups

• Calculate the MBR for each pair, and choose the one with the largest MBR

• These 2 objects are the new starting points for the resulting 2 nodes



• Quadratic-cost algorithm

– Assign remaining entries to groups one at a time

• Calculate d1 as the necessary volume difference to include the current entry in MBR1 and d2 for MBR2

• Calculate d1 and d2 for all the remaining entries

• Insert the entry with the highest preference into the corresponding node

– Preference = max(d1, d2) – min(d1, d2)



• Linear cost algorithm

– Identical to Quadratic with the following differences:

• Uses a linear procedure to identify the starting entries

– Find in each dimension 2 rectangles

» the rectangle with the highest minimum coordinates

» and the rectangle with the lowest maximum coordinates



C

AD

B E

On X:

highest minimum coordinates -> E

lowest maximum coordinates -> A

On Y:

highest minimum coordinates -> D

lowest maximum coordinates -> C

– Calculate the difference on the corresponding dimension between the coordinates of corresponding rectangles, and normalize it by the maximum on that dimension

– The two starting points are the two entries with the highest normalized difference

– Order the next entries so that the volume growth is the smallest from one step to another



C

AD

B E

Dx = (Ex – Ax)/max(x)

Dy = (Dy - Cy)/max(y)

If (Dx > Dy) then choose E and A as starting points

Else choose D and C

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• Quadratic vs. Linear cost algorithm

– Quadratic:

• Choose two objects that create as much empty space as possible

– Linear:

• Choose two objects that are furthest apart

– Linear node-split is simple, fast, and as good as quadratic!

– Quality of the splits is slightly worse!



• Delete entry

– Start a normal search of the entry to delete (FindLeaf)

– Delete the record from the leaf (DeleteRecord)

– Condense Tree if needed (if there are now nodes which only have few entries)

• At condensation the node to be condensed is deleted as a whole and the entries which should remain are then inserted

• If the root has just one child, it will be the new root


6.2 R-Trees - Delete

• Update

– If the datasets are updated the existent rectangles can be changed

– In this case the index entry must be deleted updated and inserted


6.2 R-Trees - Update

• Let

– M = Maximum number of entries in a node.

– m <= M/2

– N = Number of records

• Properties

– Every leaf node contains between m and M index records

• Root node is the exception

– For each index record (I, RID) in a leaf node, I is the smallest rectangle that spatially contains the n dimensional data object represented in the indicated tuple


6.2 R-Trees

– Every non-leaf node has between m and M children• Root node is the exception

– For each entry (I, P) in a non-leaf node, I is the smallest rectangle that spatially contains the rectangles in the child node

– The root node has at least two children unless it is a leaf

– All leaves appear on the same level

– Height of a tree = ceiling(logm N)-1

– Worst case utilization for all nodes except the root is m/M


6.2 R-Trees

• Advantages

– Efficient for non-point queries

– No downward cascading splits

– Guaranteed utilization

• Disadvantages

– Dimension dependent fan-out

– Overlapping regions - search performance problem


6.2 R-Trees

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• R+-Trees enhances retrieval performance by avoiding visiting multiple pathswhen searching for point queries

– No overlap for MBRs at the samelevel (internal nodes)

– Specific object’s entry might beduplicated

– Insertions might lead to a series of update operations in a chain-reaction


6.2 R-Tree Variations

• Compared to R-trees,

– Nodes are not guaranteed to be at least half filled

– The entries of any internal node do not overlap

– An object ID may be stored in more than one leaf node



A

B

C A, B and C do not overlap

• R+-Trees

– Advantages

• Because nodes are not overlapped with each other, point query performance benefits, e.g., a single path is followedand fewer nodes are visited than with the R-tree

– Disadvantages

• Since rectangles are duplicated, an R+-Tree can be larger than an R-Tree built on same data set

• Construction and maintenance of R+ trees is more complex than the construction and maintenance of R-Trees



• R*-Trees

– Node split is more sophisticated

• When a node overflows, p entries are extracted and reinserted in the tree (p might be 25%)

– Considers minimization of:

• Overlapping between minimum bounding rectangles at the same level

• Perimeter of the produced minimum bounding rectangles

– Insertion is more expensive while retrievals are faster



• Combination of B*-Tree and Z-curve = Universal B-Tree (UB-tree)

• Z-curve is used to map multidimensional points to one-dimensional values (Z-values)

• Z-values are used as keys in B*-Tree


6.2 UB-Trees

Data part

8 178 17 39 5139 51

2828Index part

• Concept of Z-Regions

– To create a disjunctive partitioning of the multidimensional space

– This allows for very efficient processing of multidimensional range queries


6.2 UB-Trees

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• Z-Regions

– The space covered by an interval on the Z-Curve

– Defined by two Z-Addresses a and b

• We call b the region address of [a : b]

– Each Z-Region maps exactly onto one page on secondary storage

• I.e., to one leaf page of the B*-Tree

– E.g., of Z-Regions

• [1:9], [10, 18], [19, 28]…


6.2 UB-Trees

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• Z-Value address representation

– Calculated through bit interleaving of the coordinates of the tuple


6.2 UB-Trees

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Tuple = 51, x = 4, y = 5

y

xX = 4 = 100Y = 5 = 101

Z-value = 110010

• Why Z-Values?

– With Z-Values we reduce the dimensionality of the data to one dimension

– Z-Values are then used as keys in B*-trees

• Using B*-Trees results in high node filling degree (at least 50%)

• Logarithmical complexity at search, insert and delete

– Guaranteed maximum node-accesses to locate a key is

– Z-Values are very important for range queries!


6.2 UB-Trees

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• Range queries (RQ) in UB-Trees

– Each query can be specified by 2 coordinates

• qa (the upper left corner of the query rectangle)

• qb (the lower right corner of the query rectangle)

– RQ-algorithm

1. Starts with qa and calculates

its Z-Region

1. Z-Region of qa is [10:18]


6.2 UB-Trees

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• Range queries (RQ) in UB-Trees2. The corresponding page is loaded and filtered with the

query predicate

1. Tupels 15 and 16 fulfill the predicate

3. The next region (inside the query rectangle) on the Z-curve is calculated

1. The next jump point on the Z-curve is 27

4. Repeat steps 2 and 3 until the

end-address of the last filtered

region is bigger than qb


6.2 UB-Trees

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• The critical part of the algorithm is calculating the jump point on the Z-curve which is inside the query rectangle

– If this takes too long it eliminates the advantage obtained through optimized disk access

– How is the jump point optimally calculated?

• From 3 points: qa, qb and the current Z-Region

• By performing bit operations


6.2 UB-Trees

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• Range Queries, UB-Trees and DW

– In DW we have hierarchical organization of dimensions

– No intervals for hierarchical restrictions

– Naive restrictions lead to many point queries instead of one interval on UB-Tree

– This is why we need Multidimensional Hierarchical Clustering (MHC)


6.2 UB-Trees

• With MHC UB-Trees can:

– Artificial encode hierarchies:

• Mapping of hierarchy restrictions to range restrictions

• Mapping is used for physical clustering of the fact table

– Increase computation and space efficiency

• However, modification of query algorithms is necessary


6.2 MHC


6.2 MHC

Item

ProductGroup

Category

Sector

VideoAudio

Camcorder VCR

TR-780 TRV-30 GR-AX 200 GV-500 SLV-E800

Brown Goods White Goods

ALL

...

...

ID 2 11 5 8 21

Item

ProductGroup

Category

Sector

VideoAudio VideoAudio

Camcorder VCR

TR-780 TRV-30TR-780 TRV-30 GR-AX 200 GV-500 SLV-E800

Brown Goods White GoodsBrown Goods White Goods

ALL

...

...

ID 2 11 5 8 21


6.2 MHC

...

Item

ProductGroup

Category

Sector

VideoAudio

Camcorder VCR

TR-780 TRV-30 GR-AX 200 GV-500 SLV-E800

Brown Goods White Goods

ALL

...

...

ID 2 11 5 8 21

0 1

00 01

0 1

Surrogate 00100000

0000 0001 0000 0001 0010

00100001 00110000 00110001 00110010

32 33 48 49 50

...

Item

ProductGroup

Category

Sector

VideoAudio VideoAudio

Camcorder VCR

TR-780 TRV-30TR-780 TRV-30 GR-AX 200 GV-500 SLV-E800

Brown Goods White GoodsBrown Goods White Goods

ALL

...

...

ID 2 11 5 8 21

0 1

00 01

0 1

Surrogate 00100000

0000 0001 0000 0001 0010

00100001 00110000 00110001 00110010

32 33 48 49 50

• Bitmap Indexes

– Lets assume a relation Expenses

with three attributes: Nr, Shop

and Sum

– A bitmap index for attribute

Shop looks like this


6.2 Bitmap Indexes

Nr Shop Sum

1 Saturn 150

2 Real 65

3 P&C 160

4 Real 45

5 Saturn 350

6 Real 80

Value Vector

P&C 001000

Real 010101

Saturn 100010

• A bitmap index for an attribute of relation is:– A collection of bit-vectors

– The number of bit-vectors represents the number of distinct values of the attribute in the relation

– The length of each bit-vector is called the cardinality of the relation

– The bit-vector for value v has 1 in position i, if the ith

record has v in attribute A, and it has 0 otherwise

• Records are allocated permanent numbers

• There is a mapping between record numbers and record addresses


6.2 Bitmap Indexes

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• Advantages

– Very efficient when used for partial match queries

– They offer the advantage of buckets

• In our example each index vector is a bucket

– E.g., the Saturn bitmap vector is a bucket

of 2, telling us that records having value

Saturn in attribute Shop are first and

5th record in the table

– They can also help answer range queries

– Efficient hardware support for bitmap operations (AND, OR, XOR, NOT)


6.2 Bitmap Indexes

Value Vector

P&C 001000

Real 010101

Saturn 100010

• Assume:– There are n records in the table

– Attribute A has m distinct values in the table

• The size of a bitmap index on attribute A is m*n• Significant number of 0’s is m is big, and of 1’s if m

is small– Opportunity to compress

• Run Length Encoding (RLE)

• Gzip (Lempel-Ziv, LZ)

• Byte-Aligned Bitmap Compression (BBC): variable byte length encoding (Oracle patent)


6.2 Bitmap Indexes

• Handling modification

– Assume record numbers are

not changed

– Deletion

• Tombstone replaces deleted record

• Corresponding bit is set to 0

• E.g. delete the 5th record


6.2 Bitmap Indexes

Nr Shop Sum

1 Saturn 150

2 Real 65

3 P&C 160

4 Real 45

5 Saturn 350

6 Real 80

Value Vector

P&C 001000

Real 010101

Saturn 100010

Value Vector

P&C 001000

Real 010101

Saturn 100000

Before After

• Insertion record is assigned thenext record number

– A bit of value 0 or 1 is appendedto each bit vector

– If new record contains a new valueof the attribute, add one bit-vector

• E.g., insert new record with REWE as shop


6.2 Bitmap Indexes

Nr Shop Sum

1 Saturn 150

2 Real 65

3 P&C 160

4 Real 45

5 Saturn 350

6 Real 80

7 REWE 23

Value Vector

P&C 001000

Real 010101

Saturn 100010

Value Vector

P&C 0010000

Real 0101010

Saturn 1000100

REWE 0000001

BeforeAfter

• Modification

– Change the bit corresponding tothe old value of the modified record to 0

– Change the bit corresponding tothe new value of the modified record to 1

– If the new value is a new value of A, then insert a new bit-vector: e.g., replace Shop for record 2 to REWE


6.2 Bitmap IndexesNr Shop Sum

1 Saturn 150

2 REWE 65

3 P&C 160

4 Real 45

5 Saturn 350

6 Real 80

Value Vector

P&C 001000

Real 010101

Saturn 100010

Value Vector

P&C 001000

Real 000101

Saturn 100010

REWE 010000

BeforeAfter

• Select

– Basic AND, OR bit operations:

• E.g., select the sums we have spent inSaturn and P&C

– Bitmap indexes should be used when selectivity is high


6.2 Bitmap IndexesNr Shop Sum

1 Saturn 150

2 Real 65

3 P&C 160

4 Real 45

5 Saturn 350

6 Real 80

Value Vector

P&C 001000

Real 010101

Saturn 100010

Saturn OR P&C = Result

1 0 1

0 0 0

0 1 1

0 0 0

1 0 1

0 0 0

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• Advantages– Operations are efficient and easy to implement (directly

supported by hardware)

• Disadvantages– For each new value of an attribute a new bitmap-vector is

introduced• If we bitmap index an attribute like birthday (only day) we have

365 vectors: 365/8 bits ≈ 46 Bytes for a record, just for that

• Solution to such problems is multi-component bitmaps

– Not fit for range queries where many bitmap vectors have to be read

• Solution: range-encoded bitmap indexes


6.2 Bitmap indexes

• Multi-component bitmap indexes

– Encoding using a different numeration system

• E.g., for the month attribute, between 0 and 11 values can be encoded as x = 4 *y+z, where 0 ≤ y ≤2, and 0 ≤z ≤3, called <3,4> basis encoding

• 9 = 4*2+1


6.2 Bitmap indexes

Month Dec Nov Oct Sep Aug Jul Jun Mai Apr Mar Feb Jan

M A11 A10 A9 A8 A7 A6 A5 A4 A3 A2 A1 A0

9 0 0 1 0 0 0 0 0 0 0 0 0

X Y Z

M A2,1 A1,1 A0,1 A3,0 A2,0 A1,0 A0,0

9 1 0 0 0 0 1 0

• Advantage of multi-component bitmap indexes

– If we have 100 (0..99) different Days to index we can use a multi-component bitmap index with basis of <10,10>

– The storage is reduced from 100 to 20 bitmap-vectors (10 for y and 10 for z)

– The read-access for a point (1 day out of 100) query needs however 2 read operations instead of just 1


6.2 Multi-component bitmap indexes

• Range-encoded bitmap indexes– Idea: set the bits of all bitmap vectors to 1 if they are

higher or equal to the given value

– Query people born between March and August• For normal encoded bitmap indexes read 6 vectors, for range-

encoded indexes, we can solve the query with just 2 vectors read: ((NOT A2) AND A7)


6.2 Bitmap indexes

Month Dec Nov Oct Sep Aug Jul Jun Mai Apr Mar Feb Jan

M A11 A10 A9 A8 A7 A6 A5 A4 A3 A2 A1 A0

0 1 1 1 1 1 1 1 1 1 1 1 1

3 1 1 1 1 1 1 1 1 1 0 0 0

5 1 1 1 1 1 1 1 0 0 0 0 0

11 1 0 0 0 0 0 0 0 0 0 0 0

• If the query is limited only on one side, (e.g., persons born in or after March), 1vector is enough (NOT A1)

• For point queries, 2 vector reads are however necessary!

– E.g., persons born in March: ((NOT A1) AND A2)


6.2 Range-encoded bitmap indexes

• Combine multi-component bitmap indexes with range-encoding bitmap indexes and we have multi-component-range-encoding bitmap indexes

• Interval-encoded bitmap indexes

– Each bitmap-vector represents an interval

– It also needs to read at most 2 vectors, but the storage is half, compared to range-encoded bitmap indexes


6.2 Bitmap indexes flavors

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14

• Partitions the range of key values for each key into several buckets

• Dynamic structure using a grid directory

– Grid array: a 2 dimensional array with pointers to buckets (this array can be large, disk resident) G(0,…, nx-1, 0, …, ny-1)

– Linear scales: two 1 dimensional arrays that used to access the grid array (main memory) X(0, …, nx-1), Y(0, …, ny-1)


6.2 Grid Files


6.2 Grid Files

XY

0 6 25 32

A

F

K

O

Z

• Properties

– Supports multi-dimensional data, but not high number of dimension

– Every key is treated as primary key

– The index structure adapts itself dynamically to maintain storage efficiency

– Guarantee two disk accesses for point queries

– Values of key must be in linearly-ordered domain


6.2 Grid Files

• 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


6.2 Grid Files: Queries

• E.g., Find 5<X<9 AND “Mat”<Y<“Robot”


6.2 Grid Files: Range queries

XY

0 6 25 32

A

F

K

O

Z

• Determine the bucket into which insertion must occur

– If space in bucket, insert

– Else, split bucket

– If bucket split causes a cell directory to split do so and adjust linear scales

• Insertion of these new entries potentially requires a complete reorganization of the cell directory… expensive!!!


6.2 Grid Files: Insert

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15



a) b)

c) d)



d) e)

f) g)

• Delete the data node, and

– Merge data pages/blocks ifpossible

– Merge directory pages ifpossible


6.2 Grid Files: Delete

a)

b) c)


6.2 Grid Files: Delete

c) d)

e) f)

• Optimization

– Partitioning

– Joins

– Materialized Views


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