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Data Warehousing & Data Mining& 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 indexes
6. Optimization
Bitmap indexesHash Based
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 2
• The basic storage structure is the multidimensional array
– Customized based upon
• The data e.g., sparse or dense
6.1 Multidimensional Storage
• 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
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 3
• 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
6.1 Multidimensional Storage
11 22 53 64 175 Jan (1)• |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
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 4
11
36 47
22 53
78 89
64
911
1116 1217
1012 1313
1518 1619
1414
175
1910
2515
2720
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
6.1 Multidimensional Storage
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|)
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 5
• 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., Pants
6.1 Problems in Array-Storage
11 22 53 64 175 Jan (1)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
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 6
11
36 47
22 53
78 89
64
911
1116 1217
1012 1313
1518 1619
1414
175
1910
2515
2720
Jan (1)
Feb(2)
Feb(3)
Feb(4)
D1
D2
• The problem of dimensions order can be diminished by using caching solutions
– Caching and swapping is performed alsoby the operating system
6.1 Problems in Array-Storage
by the operating system
– MDBMS has to manage its cachessuch that the OS doesn’t perform anydamaging swaps
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 7
• 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
6.1 Problems in Array-Storage
– 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
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 8
• 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
6.1 Problems in Array-Storage
• 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
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 9
• 2 level cube storage
6.1 Problems in Array-Storage
Marketing campaign
Customer
Sparse upper level
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 10
Product
TimeGeo
Dense low 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
6.2 Indexes
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
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 11
• Restriction role
– Besides aggregation, restrictions are the most time-consuming
– Typologies of restrictions based on query types
Range query
6.2 Remember OLAP Queries?
• Range query
– Restricted through intervals in each dimension
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 12
Pro
du
ct (
Art
icle
)
Time (Days)
• Partial range query– Some dimensions are not restricted
– Geometrically described as a sub-space
• Partial match query– Restricts more dimensions on a point,
6.2 Remember OLAP Queries?
Pro
du
ct (
Art
icle
)
Time (Days)
Pro
du
ct (
Art
icle
)
– 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
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 13
Pro
du
ct (
Art
icle
)
Time (Days)
Pro
du
ct (
Art
icle
)
Time (Days)
6.2 Optimization Procedures
R1 R2
MV1
Base-layer of
the data
Layer of
materializationMV2
Logical
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 14
Layer of
partitioning
Layer of
Index struct.
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?
6.2 Index Structures
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
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 15
• Reduce the size of read pages to a
minimum with indexes
6.2 Index Structures
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 16
Pro
du
ct (
Art
icle
)
Time (Days)
Full table scan
Pro
du
ct (
Art
icle
)
Time (Days)
Cluster primary
indexP
rod
uct
(A
rtic
le)
Time (Days)
More secondary
Indexes, bitmap indexes
Pro
du
ct (
Art
icle
)
Time (Days)
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
6.2 Index Structures
• 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
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 17
• Curse of Dimensionality (Richard Bellman)
– The volume increases exponentially with the dimensions number
– More dimensions mean more comparisons will be performed
6.2 Index Structures
performed
• At the present time there is no real scalable indexing
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 18
• 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
6.2 Index Structures
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
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 19
• In the beginning…there were
B-Trees
– Data structures for storing sorted
data with amortized run times for
6.2 Trees
data with amortized run times for
insertion and deletion
– Basic structure of a node
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 20
…
Tree Node
Node Pointers
Key Value Data Pointer
• B-Trees as primary indexes in OLTP
6.2 Trees
2 6 7
1 3 4 5 8 9
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 21
1,
Ad
am
s, $
88
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2,
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$1
9,9
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,00
• 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
6.2 B-Trees
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!
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 22
• Demands
– Can store d-dimensional hypercubes
– Performs point, line, and box queries as fast as possible...
6.2 R-Trees
– ...but also keeps memory usage in check!
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 23
• R-Trees are recommended for lower dimensionality
– Up to 10 dimensions
• More scalable variants:
6.2 R-Trees
• More scalable variants:
– R+-Trees, R*-Trees und X-Trees
– Each up to 20 dimensions
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 24
• 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
belonging to consecutive quarters clustered together
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 25
• R-Trees
– Can organize any-dimensional data
• Representing the data by a minimum bounding rectangle (MBR)
– Each node bounds it’s children
6.2 R-Trees
– 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
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 26
• 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
Root
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 27
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],
6.2 R-Trees
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
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 28
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
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 29
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
6.2 R-Trees - Search
• 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
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 30
• 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
6.2 R-Trees - Search
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“)
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 31
• Search on sales on last 2 quarters of 08, locations e and f
6.2 R-Trees - Search
R1
R3
R5
08 Qtr3
08 Qtr4
09 Qtr1
Time
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 32
R1
R3 R4
R3.1 R3.2 R3.3 R4.1 R4.2 R4.3
R4
R6R2
08 Qtr1
08 Qtr2
08 Qtr3
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
• Node splitting is not trivial
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 33
• 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
6.2 R-Trees - Insert
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.
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 34
• ChooseLeaf(E)– Initialize
• Set N to be the root node
– Leaf check• If N is a leaf, return N
6.2 R-Trees
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
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 35
• Node splitting
– A full node contains M entries
– Divide the collection of M+1 entries between
2 nodes.
6.2 R-Trees - Insert
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
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 36
• Node splitting
6.2 R-Trees - Insert
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 37
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
6.2 R-Trees - Insert
• Number of possibilities ~ 2M-1
• For M ~ 50 Number of possibilities ~ 600 Trillion
– Which is even more than theObama administration spentwith the crisis!!!
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 38
• Quadratic-cost algorithm
– A heuristic to find a small-area split
– Cost is quadratic in M and linear in the number of dimensions
6.2 R-Trees - Insert
– 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
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 39
• 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
6.2 R-Trees - Insert
• 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)
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 40
• 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
6.2 R-Trees - Insert
» the rectangle with the highest minimum coordinates
» and the rectangle with the lowest maximum coordinates
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 41
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
6.2 R-Trees - Insert
Dx = (Ex – Ax)/max(x)
– Order the next entries so that the volume growth is the smallest from one step to another
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 42
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
• Quadratic vs. Linear cost algorithm
– Quadratic:
• Choose two objects that create as much empty space as possible
– Linear:
6.2 R-Trees - Insert
– 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!
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 43
• 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
6.2 R-Trees - Delete
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
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 44
• 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
and inserted
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 45
• Let
– M = Maximum number of entries in a node.
– m <= M/2
– N = Number of records
• Properties
6.2 R-Trees
• 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
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 46
– 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
6.2 R-Trees
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
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 47
• Advantages
– Efficient for non-point queries
– No downward cascading splits
– Guaranteed utilization
6.2 R-Trees
Guaranteed utilization
• Disadvantages
– Dimension dependent fan-out
– Overlapping regions - search performance problem
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 48
• R+-Trees enhances retrieval performance by avoiding visiting multiple pathswhen searching for point queries
– No overlap for MBRs at the samelevel (internal nodes)
6.2 R-Tree Variations
level (internal nodes)
– Specific object’s entry might beduplicated
– Insertions might lead to a series of update operations in a chain-reaction
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 49
• 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
6.2 R-Tree Variations
An object ID may be stored in more than one leaf node
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 50
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
6.2 R-Tree Variations
and 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
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 51
• 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:
6.2 R-Tree Variations
– 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
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 52
• 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
• Z-values are used as keys in B*-Tree
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 53
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
multidimensional range queries
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 54
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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]
6.2 UB-Trees
– 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]…
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 55
1
3 4
2 5
7 8
6
9
11 12
10 13
15 16
14
17
19 20
18 21
23 24
22
25
27 28
26 29
31 32
30
33
35 36
34 37
39 40
38
41
43 44
42 45
47 48
46
49
51 52
50 53
55 56
54
57
59 60
58 61
63 64
62
• Z-Value address representation
– Calculated through bit interleaving of the coordinates of the tuple
6.2 UB-Trees
Tuple = 51, x = 4, y = 5
xX = 4 = 100Y = 5 = 101
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 56
1
3 4
2 5
7 8
6
9
11 12
10 13
15 16
14
17
19 20
18 21
23 24
22
25
27 28
26 29
31 32
30
33
35 36
34 37
39 40
38
41
43 44
42 45
47 48
46
49
51 52
50 53
55 56
54
57
59 60
58 61
63 64
62
0
1
2
3
4
5
6
7
0 1 2 3 4 5 6 7
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
6.2 UB-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!
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 57
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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
6.2 UB-Trees
– RQ-algorithm
1. Starts with qa and calculates
its Z-Region
1. Z-Region of qa is [10:18]
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 58
1
3 4
2 5
7 8
6
9
11 12
10 13
15 16
14
17
19 20
18 21
23 24
22
25
27 28
26 29
31 32
30
33
35 36
34 37
39 40
38
41
43 44
42 45
47 48
46
49
51 52
50 53
55 56
54
57
59 60
58 61
63 64
62
• 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-
6.2 UB-Trees
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
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 59
1
3 4
2 5
7 8
6
9
11 12
10 13
15 16
14
17
19 20
18 21
23 24
22
25
27 28
26 29
31 32
30
33
35 36
34 37
39 40
38
41
43 44
42 45
47 48
46
49
51 52
50 53
55 56
54
57
59 60
58 61
63 64
62
• 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
6.2 UB-Trees
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
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 60
• Range Queries, UB-Trees and DW
– In DW we have hierarchical organization of dimensions
– No intervals for hierarchical restrictions
6.2 UB-Trees
– Naive restrictions lead to many point queries instead of one interval on UB-Tree
– This is why we need Multidimensional Hierarchical Clustering (MHC)
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 61
• 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
6.2 MHC
– Increase computation and space efficiency
• However, modification of query algorithms is necessary
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 62
6.2 MHC
Category
Sector
VideoAudio
Brown Goods White Goods
ALL
...
Category
Sector
VideoAudio VideoAudio
Brown Goods White GoodsBrown Goods White Goods
ALL
...
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 63
Item
Product
GroupCamcorder VCR
TR-780 TRV-30 GR-AX 200 GV-500 SLV-E800
...
ID 2 11 5 8 21
Item
Product
GroupCamcorder VCR
TR-780 TRV-30TR-780 TRV-30 GR-AX 200 GV-500 SLV-E800
...
ID 2 11 5 8 21
6.2 MHC
...Category
Sector
VideoAudio
Brown Goods White Goods
ALL
...
...
0 1
00 01...Category
Sector
VideoAudio VideoAudio
Brown Goods White GoodsBrown Goods White Goods
ALL
...
...
0 1
00 01
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 64
Item
ProductGroup
Camcorder VCR
TR-780 TRV-30 GR-AX 200 GV-500 SLV-E800
...
ID 2 11 5 8 21
0 1
Surrogate 00100000
0000 0001 0000 0001 0010
00100001 00110000 00110001 00110010
32 33 48 49 50
Item
ProductGroup
Camcorder VCR
TR-780 TRV-30TR-780 TRV-30 GR-AX 200 GV-500 SLV-E800
...
ID 2 11 5 8 21
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
6.2 Bitmap Indexes
Nr Shop Sum
1 Saturn 150
2 Real 65
3 P&C 160
4 Real 45and Sum
– A bitmap index for attribute
Shop looks like this
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 65
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
6.2 Bitmap Indexes
– 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
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 66
• 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
6.2 Bitmap Indexes
– 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)
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 67
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
6.2 Bitmap Indexes
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)
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 68
• Handling modification
– Assume record numbers are
not changed
– Deletion
6.2 Bitmap Indexes
Nr Shop Sum
1 Saturn 150
2 Real 65
3 P&C 160
4 Real 45
5 Saturn 350Deletion
• Tombstone replaces deleted record
• Corresponding bit is set to 0
• E.g. delete the 5th record
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 69
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 value
6.2 Bitmap Indexes
Nr Shop Sum
1 Saturn 150
2 Real 65
3 P&C 160
4 Real 45
5 Saturn 350
– If new record contains a new valueof the attribute, add one bit-vector
• E.g., insert new record with REWE as shop
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 70
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
6.2 Bitmap IndexesNr Shop Sum
1 Saturn 150
2 REWE 65
3 P&C 160
4 Real 45
5 Saturn 350
6 Real 80the 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
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 71
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
6.2 Bitmap IndexesNr Shop Sum
1 Saturn 150
2 Real 65
3 P&C 160
4 Real 45
5 Saturn 350
6 Real 80Saturn OR P&C = Result
– Bitmap indexes should be used when selectivity is high
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 72
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
• 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
6.2 Bitmap indexes
– 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
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 73
• 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
6.2 Bitmap indexes
x = 4 *y+z, where 0 ≤ y ≤2, and 0 ≤z ≤3, called <3,4> basis encoding
• 9 = 4*2+1
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 74
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>
6.2 Multi-component bitmap indexes
<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
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 75
• Range-encoded bitmap indexes– Idea: set the bits of all bitmap vectors to 1 if they are
higher or equal to the given value
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
– 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)
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 76
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
((NOT A ) AND A )
6.2 Range-encoded bitmap indexes
• For point queries, 2 vector reads are however necessary!
– E.g., persons born in March: ((NOT A1) AND A2)
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 77
• Combine multi-component bitmap indexes with range-encoding bitmap indexes and we have multi-component-range-encoding bitmap indexes
6.2 Bitmap indexes flavors
• 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
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 78
• 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
6.2 Grid Files
– 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)
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 79
6.2 Grid Files
XY
0 6 25 32
A
F
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 80
K
O
Z
• Properties
– Supports multi-dimensional data, but not high number of dimension
– Every key is treated as primary key
6.2 Grid Files
– 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
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 81
• 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)
6.2 Grid Files: Queries
(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
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 82
• E.g., Find 5<X<9 AND “Mat”<Y<“Robot”
6.2 Grid Files: Range queries
XY
0 6 25 32
A
F
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 83
F
K
O
Z
• Determine the bucket into which insertion must occur
– If space in bucket, insert
– Else, split bucket
6.2 Grid Files: Insert
– 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!!!
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 84
6.2 Grid Files: Insert
a) b)
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 85
a) b)
c) d)
6.2 Grid Files: Insert
d) e)
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 86
d) e)
f) g)
• Delete the data node, and
– Merge data pages/blocks ifpossible
– Merge directory pages ifpossible
6.2 Grid Files: Delete
a)possible
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 87
a)
b) c)
6.2 Grid Files: Delete
c) d)
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 88
c) d)
e) f)
• Optimization
– Partitioning
– Joins
– Materialized Views
Next lecture
Data Warehousing & OLAP – Wolf-Tilo Balke – Institut für Informationssysteme – TU Braunschweig 89