Data Warehouse
Indexing & Optimization
PA220: Database systems for data analytics
Contents
• Approaches to indexing
• Data partitioning
• Joins
• Materialized views
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Why Indexes?
• Consider a 100 GB table; at 100 MB/s read speed we need 17 minutes
for a full table scan
• Query for the number of Bosch S500 washing machines sold in
Germany last month
• Applying restrictions (product, location) the selectivity would be strongly
improved
• If we have 30 locations, 10,000 products and 24 months in the DW,
the selectivity value is 1/30 * 1/ 10,000 * 1/24 = 0,000 000 14
• So… we read 100 GB for 1,4KB of data
• The problem is: filter data in a fact table as much as possible
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Why Indexes?
• Reduce the size of read pages to a minimum with indexes
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Index Types
• Tree structures
• B+ tree, R trees, …
• Hash based
• Dynamic hash table
• Special
• Bitmap index
• Block-Range INdex (in Pg)
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Multidimensional Data
• B+ Tree
• classic structure – very efficient in updates
• supports point and range queries
• limited to 1D data
• UB-Tree
• uses B* tree and
• Z-curve to linearize n-dim data
• R-Tree
• wrapping by n-dim rectangles
• R+, R*, X-Tree
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UB-Trees
• Convert n-dim data to a single dimension by the Z-curve and
• Index by B* tree
• The Z-curve provides for good performance
for range queries!
• Consecutive values on the Z-curve index similar data
• Similarity by means of neighborhood
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UB-Trees
• Z-Value address representation
• Calculate the z-values such that
neighboring data is clustered together
• Calculated through bit interleaving of
the coordinates of the tuple
• To localize a value with coordinates
one must perform de-interleaving
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For Z-value 51, we have
the offset 50.
50 in binary is 110010
Z-value = 110010
Y = 101 = 5 X = 100 = 4
row at x=4, y=5
We have Z-regions –
describes one block
in storage.
E.g. [1-9], [10-18].
UB-Trees – Range Query
• 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)
• Range Query Algorithm
1. Starts with qa and calculates its Z-Region
• Z-Region of qa is [10:18]
2. The corresponding page is loaded and filtered with the query predicate
• E.g., value 10 has after de-interleaving x=1 and y=2,
which is outside the query rectangle
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Q: x[2;5], y[3;6]
UB-Trees – Range Query
• Range Query Algorithm (cont.)
3. After qa, all values on the Z-curve are
de-interleaved and checked by their
coordinates
• The data is only accessed from the disk.
• The next jump point on the Z-curve is 27.
4. Repeat Steps 2 and 3 until the decoded
end-address of the last filtered region is bigger than qb
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Calculating the jump point mostly involves:
• Performing bit operations and comparisons
• 3 points: qa, qb and the current Z-Value
R-Trees
• Like B-trees
• Data objects stored in leaf nodes
• Nodes represented by minimum bounding rectangles
• High-balanced structure
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Query S:
7 out of 12 nodes are checked.
R-Trees Querying
• Many MBR overlaps deteriorate query performance
• All nodes get visited in the worst case.
• Key is insertion/split optimization
• Minimize volume by MBR → overlaps.
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R+ Tree
• Eliminates overlaps by replication of objects in leaves
• Improves performance of point queries
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Bitmap Index
• Good for data which has a small number of distinct values
• E.g., gender data, clothing sizes
• Similar performance as B+ tree for read-only data
• also, when all values are distinct
• A bitmap index for an attribute is:
• A collection of bitmaps (bit-vectors)
• The number of bit-vectors represents the number of distinct values of an attr. in the relation
• Bitmap (bit vector/array) is an array data structure that stores individual bits
• Compressed by Run-length encoding
• The length of each bit-vector is the cardinality of the relation
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Bitmap Index
• Example
• Records are allocated permanent numbers.
• There is a mapping between record numbers and record addresses.
• Deletion
• in the fact table → tombstones
• in the index → bit is cleared
• Insertion → bit-vectors extended
• Update → del & ins
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Shop dim Sales fact Bitmap on Shop of Sales
3
2
1
Bitmap Index – Queries
• Combine OR/AND values
• OR/AND bit ops on vectors
• E.g., Saturn | P&C
• Combine different indexes on the same table
• Bitmap indexes should be used when the selectivity value is low.
• Not very good for range queries on values.
• → Range-encoded Bitmap Index
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3
2
1
100010
001000
----------
101010
Multi-component Bitmap Index
• Encoding using a different numeration system
• E.g., <div,mod> classes
• Idea:
• transform values into more dimensions and project
• intersection of projections gives the original value
• E.g., the month attribute has values between 0 and 11.
• Encode by X = 3*Z+Y
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1 0
A_,1 A_,0A_,2A_,3A1,_ A0,_A2,_
Multi-component Bitmap Index
• If we have 100 (0..99) different days to index we can use a multicomponent
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
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Range-encoded Bitmap Index
• Requires a logical ordering of values
• Idea:
• set the bit in all bit-vectors of the values following this current one
• range queries will check just 2 bit-vectors
• Disadvantage:
• a point query requires reading 2 vectors (= NOT previous AND current)
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Range-encoded Bitmap Index
• Query: Persons born between March and August
• So, persons which didn’t exist in February but existed in August.
• Just 2 vectors read: ((NOT A1) AND A7)
• Normal bitmap would require 6 vectors to read.
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Summary of Indexes
• B-Trees are not fit for multidimensional data
• UB-trees can be applicable
• R-Trees may not scale to many dimensions
• Bitmap indexes are typically only a fraction of the size of the indexed
data in the table
• Bitmap indexes reduce response time for large classes of ad hoc
queries
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Data Partitioning
• Breaking data into “non-overlapping” parts
• Horizontal vs. vertical
• May correspond to granularity of a dimension
• Improves:
• Business query performance,
• i.e., minimize the amount of data to scan
• Data availability,
• e.g., back-up/restores can run at the partition level
• Database administration,
• e.g., archiving data, recreating indexes, loading tables
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Data Partitioning
• Approaches:
• Logical partitioning by
• Date, Line of business, Geography, Organizational unit, Combinations of these factors, …
• Physical partitioning
• Makes data available to different processing nodes
• Possible parallelization on multiple disks/machines
• Implementation:
• Application level
• Database system
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Data Partitioning
• Horizontal – splitting out the rows of a table into multiple tables
• Vertical – splitting out the columns of a table into multiple tables
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Partitioning key
Horizontal Partitioning
• Distributes records into disjoint tables
• Typically, “view” over union of the table is available
• Types:
• range – a range of values per table
• list – enumeration of values per table
• hash – result of a hash function determines the table
• In DWs typically:
• Usually, generated reports can identify the partitioning key.
• Time dimension – weeks, months or age of data
• Another dim if it does not change much – branch, region
• Table size – requires some meta-data to constraint the contents
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Vertical Partitioning
• Involves creating tables with fewer columns and using additional
tables to store the remaining columns
• Usually called row splitting
• Row splitting creates one-to-one relationships between the partitions
• Different physical storage might be used
• E.g., storing infrequently used or very wide columns on a different device
• In DWs typically:
• move seldom used columns from a highly-used table to another
• create a view that merges them
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Vertical Partitioning
• Mini-dimension with outrigger is a solution
• Many dimension attributes are used very frequently as browsing constraints
• In big dimensions these constraints can be hard to find among the lesser used ones
• Logical groups of often used constraints can be separated into small dimensions
• which are very well indexed and easily accessible for browsing
• E.g., demography dimension
• Notice the foreign key in
customer
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Summary of Partitioning
• Advantages
• Records used together are grouped together
• Each partition can be optimized for performance
• Security, recovery
• Partitions stored on different disks reduces contention
• Take advantage of parallel processing capability
• Disadvantages
• Slow retrieval across partitions (expensive joins when vertical partitioning)
• Complexity
• Recommendations
• A table is larger than 2GB (from Oracle)
• A table has more than 100 million rows (practice)
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Join Optimization
• Often queries over several partitions are needed
• This results in joins over the data
• Though joins are generally expensive operations, the overall cost of the query
may strongly differ with the chosen evaluation plan for the joins
• Joins are commutative and associative
• R ⋈ S ≡ S ⋈ R
• R ⋈ (S ⋈ T) ≡ (S ⋈ R) ⋈ T
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Join Optimization
• This allows to evaluate individual joins in any order
• Results in join trees
• Different join trees may show very different evaluation performance
• Join trees have different shapes
• Within a shape, there are different relation assignments possible
• Number of possible join trees grows rapidly (n!)
• DBMS’ optimizer considers
• statistics to minimize result size
• all possibilities → impossible for large n
• heuristics to pick promising ones
• when the number of relations is high (e.g., >6)
• e.g., genetic algorithms
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Join Selection Heuristics
• Join relations that relate by an attribute/condition
• avoiding cross joins
• Minimize the result size (A is the common attr.)
•
𝑇 𝑅 ∗𝑇(𝑆)
max(𝑉 𝑅,𝐴 ,𝑉 𝑆,𝐴 )
• Availability of indexes and selectivity of
other conditions
• User tuning
• Hints in Oracle
• Set the parameter join_collapse_limit in PostgreSQL
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Join Selection Heuristics in DWs
• OLTP’s heuristics are not suitable in DWs
• E.g., join Sales with Geo in the following case:
• Sales has 10 mil records, in Germany there are 10 stores,
in January 2016 there were products sold in 20 days, and
the Electronics group has 50 products
• If 20 % of our sales were performed in Germany,
• the selectivity value is high.
• so, an index would not help that much
• The intermediate result would still comprise 2 mil records
• Cross join is recommended
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Join Selection Heuristics in DWs
• The cross join of the dimension tables is recommended
• Geo dimension – 10 stores in Germany
• Time dimension – 20 days in Jan 2016
• Product dimension – 50 products in Electronics
• 10m facts in Sales
• 10*20*50 = 10,000 records after performing the cross product
• But can also be expensive!
• Allows
• a single pass over the Sales
• using an index on the most selective attribute yet07.10.2021 PA220 DB for Analytics 33
Join Selection Heuristics in DWs
• If cross join is too large, intersect partial joins
• applicable when all dimension FKs are indexed
• in fact, it is a semi-join (no record duplication can take place)
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Summary of Joins
• Prefer a cross-join on dimensions first
• If not all dimension FKs are indexed
• Intersect semi-joins otherwise
• Avoid standard DBMS’s plans
• But check the plan first ☺
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Materialized Views
• Views whose tuples are stored in the database are said to be
materialized
• They provide fast access, like a (very high-level) cache
• Need to maintain the view’s contents as the underlying tables change
• Ideally, we want incremental view maintenance algorithms
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Materialized Views
• How can we use MV in DW?
• E.g., we have queries requiring us to join the Sales table with another table
and aggregate the result
• SELECT P.Categ, SUM(S.Qty) FROM Product P, Sales S WHERE P.ProdID=S.ProdID
GROUP BY P.Categ
• SELECT G.Store, SUM(S.Qty) FROM Geo G, Sales S WHERE G.GeoID=S.GeoID
GROUP BY G.Store
• …
• There are more solutions to speed up such queries
• Pre-compute the two joins involved (product with sales and geo with sales)
• Pre-compute each query in its entirety
• Or use a common and already materialized view
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Materialized Views
• Having the following view materialized
• CREATE MATERIALIZED VIEW Totalsales(ProdID, GeoID, total) AS
SELECT S.ProdID, S.GeoID, SUM(S.Qty) FROM Sales S
GROUP BY S.ProdID, S.GeoID
• We can use it in our queries
• SELECT P.Categ, SUM(T.Total) FROM Product P, Totalsales T
WHERE P.ProdID=T.ProdID GROUP BY P.Categ
• SELECT G.Store, SUM(T.Total) FROM Geo G, Totalsales T
WHERE G.GeoID=T.GeoID GROUP BY G.Store
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Materialized Views
• MV issues
• Choice of materialized views
• What views should we materialize, and what indexes should we build on the precomputed
results?
• Utilization
• Given a query and a set of materialized views, can we use the materialized views to
answer the query?
• Maintenance
• How frequently should we refresh materialized views to make them consistent with the
underlying tables?
• And how can we do this incrementally?
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Materialized Views: Utilization
• Utilization must be transparent
• Queries are internally rewritten to use the available MVs by the query
rewriter
• The query rewriter performs integration of the MV based on the query
execution graph
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Materialized Views: Utilization
• E.g., mono-block query (perfect match)
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Materialized Views: Integration
• Correctness:
• A query Q` represents a valid replacement of query Q by utilizing the
materialized view M, if Q and Q` always deliver the same result.
• Implementation requires the following conditions:
• The selection condition in M cannot be more restrictive than the one in Q.
• The projection from Q must be a subset of the projection from M.
• It must be possible to derive the aggregation functions in Q from ones in M.
• Additional selection conditions in Q must be possible also on M.
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Materialized Views: Integration
• A way to integrate a more restrictive view:
• Split the query Q in two parts, Qa and Qb, such that
• σ(Qa) = (σ(Q) ⋀ σ(M)) and
• σ(Qb) = (σ(Q) ⋀ ¬σ(M))
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Materialized Views & DWs
• Often store aggregated results
• For a set of “n” group-by attributes, there are 2n possible combinations
• Too much to materialize all
• What to materialize?
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Materialized Views & DWs
• Choosing the views to materialize
• Static choice:
• The choice is performed at a certain time point by the DB administrator (not very often)
or by an algorithm
• The set of MVs remains unmodified until the next refresh
• The chosen MVs correspond to older queries
• Dynamical choice:
• The MV set adapts itself according to new queries
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Views to Materialize
• Static choice
• Choose which views to materialize, in concordance with the “benefit” they bring
• The benefit is computed based on a cost function
• The cost function involves
• Query costs
• Statistical approximations of the frequency of the query
• Actualization/maintenance costs
• Classical knapsack problem – a limit on MV storage and the cost of each MV
• Greedy algorithm
• Input: the lattice of cuboids, the expected cardinality of each node, and the maximum storage
size available to save MVs
• It calculates the nodes from the lattice which bring the highest benefit according to the cost
function, until there is no more space to store MVs
• Output: the list of lattice nodes to be materialized
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Views to Materialize
• Disadvantages of static choice
• OLAP applications are interactive
• Usually, the user runs a series of queries to explain a behavior he has observed, which
happened for the first time
• So now the query set comprises hard to predict, ad-hoc queries
• Even if the query pattern is observed after a while, it is unknown for how
much time it will remain used
• Queries are always changing
• Often modification to the data leads to high update effort
• There are, however, also for OLAP applications, some often repeating
queries that should in any case be statically materialized
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Views to Materialize
• Dynamic choice
• Monitor the queries being executed over time
• Maintain a materialized view processing plan (MVPP) by incorporating most
frequently executed queries
• Modify MVPP incrementally by executing MVPP generation algorithm
• as a background process
• Decide on the views to be materialized
• Reorganize the existing views
• It works on the same principle as caching, but with semantic
knowledge
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Views to Materialize
• Dynamic choice
• Updates of cached MV:
• In each step, the cost of MV in the cache as well as of the query is calculated
• All MVs as well as the query result are sorted according to their costs
• The cache is then filled with MV in the order of their costs, from high to low
• This way it can happen that one or more old MVs are replaced with the current query
• Factors consider in the cost function:
• Time of the last access
• Frequency of query
• Size of the materialized view
• The costs a new calculation or actualization would produce for a MV
• Number of queries which were answered with the MV
• Number of queries which could be answered with this MV
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Maintenance of Materialized Views
• Keeping a materialized view up-to-date with the underlying data
• How do we refresh a view when an underlying table is refreshed?
• When should we refresh a view in response to a change in the underlying table?
• Approaches:
• Re-computation – re-calculated from the scratch
• Incremental – updated by new data, not easy to implement
• Immediate – as part of the transaction that modifies the underlying data tables
• Advantage: materialized view is always consistent
• Disadvantage: updates are slowed down
• Deferred – some time later, in a separate transaction
• Advantage: can scale to maintain many views without slowing updates
• Disadvantage: view briefly becomes inconsistent
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Maintenance of Materialized Views
• Incremental maintenance
• Changes to database relations are used to compute changes to the
materialized view, which is then updated
• Considering that we have a materialized view V, and that the basis relations
suffer modifications through inserts, updates or deletes, we can calculate V`
as follows
• V` = (V - Δ-) ∪ Δ+, where Δ- and Δ+ represent deleted and inserted tuples, respectively
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Maintenance of Materialized Views
• Deferred update options:
• Lazy
• delay refresh until next query on view, then refresh before answering the query
• Periodic (Snapshot)
• refresh periodically – queries are possibly answered using outdated version of view
tuples
• widely used in DWs
• Event-based
• e.g., refresh after a fixed number of updates to underlying data tables
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Summary
• Bitmap indexes are universal, space efficient
• R*-trees, X-trees for multidimensional data
• Partitioning
• Records used together should be stored together
• Mini-dimension
• Joins
• Computing cross join on dimension table is an option
• Materialized views can replace parts of a query
• Select what to materialize (not everything) statically or dynamically
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