Relational Query Optimization

1
Relational Query Optimization

• Chapter 15

2
Query Evaluation

• SQL queries are translated into an extended form
  of relational algebra
• Query Plan Reasoning
• Tree of ops,
• with choice of one among several algorithms for
  each operator
• Query Plan Execution
• Each operator implemented using pull interface
  when operator is pulled for next output tuples,
  it pulls on its inputs and computes them.

3
Overview of Query Evaluation

• Query Plan Optimization
• Ideally Want to find best plan. Practically
  Avoid worst plans!
• Two main issues in query optimization
• For a given query, what plans are considered?
• Algorithm to search plan space for cheapest
  (estimated) plan.
• How is the cost of a plan estimated?
• Cost Models based on I/O estimates

4
Physical Query Plan

• An extended relational algebra tree
• Annotations at each node indicating access
  methods to use for each table
• The implementation methods used for each
  relational operator.
• Costs estimated per operator.

5
Schema for Examples
Sailors (sid integer, sname string, rating
integer, age real) Reserves (sid integer, bid
integer, day dates, rname string)

• Similar to old schema rname added for
  variations.
• Reserves
• Each tuple is 40 bytes long, 100 tuples per
  page, 1000 pages.
• Sailors
• Each tuple is 50 bytes long, 80 tuples per page,
  500 pages.

6
Query Blocks Units of Optimization

• An SQL query is parsed into a collection of query
  blocks.
• An SQL query with no nesting and exactly one
  SELECT,FROM,WHERE, GROUP BY, HAVING clause.
• WHERE is in conjunctive normal form.
• Nested blocks are usually treated as calls to a
  subroutine, made once per outer tuple.
• Optimizing one block at a time.

SELECT S.sname FROM Sailors S WHERE S.age IN
Reference to nested block
SELECT S.sname FROM Sailors S WHERE S.age IN
(SELECT MAX (S2.age) FROM Sailors
S2 GROUP BY S2.rating)
SELECT MAX (S2.age) FROM Sailors S2
GROUP BY S2.rating
Nested block
Outer block
7
Query Block

• For each block, the plans considered are
• All available access methods, for each relation
  in FROM clause.
• All left-deep join trees
• i.e., all ways to join the relations
  one-at-a-time, with the inner relation in the
  FROM clause, considering all relation
  permutations and join methods.

?MAX(S2.age) (Group By S2.rating(?
S2.age(Sailors)))
8
Cost Estimation

• For each plan considered, must estimate cost
• Must estimate cost of each operation in plan
  tree.
• Depends on input cardinalities.
• Depends on algorithm (sequential scan, index
  scan).
• Must also estimate size of result for each
  operation.
• Use information about the input relations.
• For selections and joins, assume independence of
  predicates.
• Must make assumptions about effect of predicates.
• Cost of plan sum of cost of each operator in
  tree.

9
Cost Estimation

• Result size
• product of cardinalities of involved relations
  (FROM)
• product of reduction factors (WHERE).

10
Size Estimation and Reduction Factors
SELECT attribute list FROM relation list WHERE
term1 AND ... AND termk

• Consider a query block
• Maximum tuples in result is product of
  cardinalities of relations in FROM clause.
• Reduction factor (RF) associated with each term
  reflects impact of term in reducing result size.
  
• Result cardinality Max tuples product of
  all RFs.
• Implicit assumption that terms are independent!

11
Reduction Factors
SELECT attribute list FROM relation list WHERE
term1 AND ... AND termk

• Reduction factor (RF) associated with each term
  reflects impact of term in reducing result size
• Term colvalue has RF ?
• Term col1col2 has RF?
• Term colgtvalue has RF?

12
Cost of a Plan (cont')

• Reduction Factor.
• Column value. - Given index I on column,
  assume uniform distribution. 1/Nkeys(I). -
  Otherwise, fixed reduction factor 1/10
• Column1 column2 - Given indexes I1 and I2 on
  column1 and column2, assuming each key value in
  I1 (smaller one) has a matching value in I2.
  1/MAX(Nkeys(I1), Nkeys(I2)). - One index I,
  1/Nkeys(I) - otherwise, 1/10
• Column gt values - Given an index I on column,
  arithmetic type. High(I) value / High(I)
  Low(I). - Not arithmetic type, or no index. A
  fraction Less than half is arbitrarily chosen.
• Column IN (list of values) - reduction factor
  for (column value) number of items in the
  list.
• Assumptions for reduction factors
• Uniform distribution of values
• Independent distribution of values in different
  columns.

13
Cost of a Plan (cont)

• Improved Statistics Histograms
• Uniform distribution is not accurate since real
  data is not uniformly distributed.
• Histogram data structure maintained by DBMS
  to approximate data distribution.
• Equiwidth divide range of column values into
  subranges (buckets). Assuming distribution
  within histogram bucket is uniform.
• Equidepth number of tuples within each bucket is
  equal (almost).
• Compressed equidepth maintain separate counts
  for small number of very frequent values, and
  maintain equidepth histogram to cover remaining
  values.

5.0
5.0
5.0
9.0
Equiwidth
Equidepth
2.67
2.25
2.5
1.33
1.0
1.75
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Bucket 1 2 3 4 5 Count 8 4 15 3
15
Bucket 1 2 3 4 5 Count 9 10 10 7
9
14
Relational Algebra Equivalences

• Allow to choose different join orders and to
  push selections and projections ahead of joins.
• Selections
  (Cascade)

Combine several selections into one selection.
Evaluate the conjunctive condition to each of
the tuple.
Separate conjunctions into smaller selection
operations, Able to combine as Join.
(Commute)

• Projections

(Cascade)
Where ai ? ai1 for i 1n-1.
15
Relational Algebra Equivalences
(Associative)

• Joins

R (S T) (R S) T
(Commute)
(R S) (S R)
R (S T) (T R) S

• Show that

• When joining several relations, we are free to
  join the relations in any order we choose.

16
Selects, Projects and Joins

• Case1 ?a (?c(R)) ? ?c (?a(R))
• A projection commutes with a selection that only
  uses attributes retained by the projection.
• XCase2 R c S ? ?c (R?S)
• Combine a selection with a cross-product to form
  a join.
• Case3 i.e., (R S) (R) S
  
• A selection on just attributes of R commutes with
  Join.

17
Selects, Projects and Joins

• Case1 A projection commutes with a selection
  that only uses attributes retained by the
  projection.
• ?a (?c(R)) ? ?c (?a(R))
• Case2 Combine a selection with a cross-product
  to form a join. R c S ? ?c (R?S)
• Case3 A selection on just attributes of R
  commutes with Join. i.e., (R S)
  (R) S

18
Selects, Projects and Joins

• In general, part of the selection condition can
  be pushed ahead of the Join.
• ?c1?c2 ?c3 (R S) ? ?c1(?c2 (R) ?c3 (S))
• Project ?a (R?S) ? ?a1 (R) ? ?a2 (S) ?a (R
  c S) ? ?a1 (R) c ?a2 (S) (Where c appears
  in a)

19
Enumeration of Alternative Plans

• There are two main cases
• Single-relation plans
• Multiple-relation plans

20
Single Relation Plans

• Queries over a single relation combination of
  selects, projects, and aggregate operations
• Main decision which access path to use in
  retrieving tuples.
• Most selective access path (file scan / index) if
  only single operator considered.
• Different operations are essentially carried out
  together.
• e.g., if an index is used for a selection,
  projection is done for each retrieved tuple, and
  the resulting tuples are pipelined into the
  aggregate computation.

21
Single Relation Plans without Index
? s.rating, COUNT()(HAVING COUNT
DISTINCT(S.sname) gt 2(GROUP Bys.rating(
?s.rating, S.sname ( ? s.ratinggt5 ? S.age20
(Sailors)))))
SELECT S.rating, COUNT() FROM Sailors S WHERE
S.rating gt 5 AND S.age 20 GROUP BY
S.rating HAVING COUNT DISTINCT (S.sname) gt 2

• A file scan to retrieve tuples and apply
  selections and projections.
• Writing out tuples after the selections and
  projections.
• Sorting these tuples to implement GROUP BY
  clause.
• Note GROUP BY and HAVING are done on-the-fly.
• e.g., Cost Cost1(scan) cost2 (writing
  ltS.rating, S.snamegt pairs) cost3
  (sorting as per the GROUP BY clause).
  Sailors 500 pages cost1 500, cost3
  3Npages 60 (assuming two passes)
• cost2 500 ratio of tuple size RFs
  20 ratio of tuple size pair size / tuple
  size 0.8
• RF(s.ratinggt5) 0.5 RF(S.age20) 0.1

22
Single-Relation Plans with Index

• Single-index access path
• When several indexes match the selection
  conditions, choose the most selective access
  path.
• Multiple-index access path
• Several indexes using Alternatives (2) or (3) for
  data entries match the selection condition.
• Retrieve rids using them individually.
• Intersect result set. Sort by page id.
• Sorted-index access path
• Grouping attributes is a prefix of a tree index.
• Using index to retrieve tuples in the order
  required by GROUP BY clause.
• Index-only access path
• All attributes mentioned in the query are
  included in the search key for some dense index
  on the relation in the FROM clause.
• Index-only scan to compute the answer.

23
Single-Relation Plans with Index

• Index I on primary key matches selection
• Cost is Height(I)1 for a B tree, about 1.2 for
  hash index.
• Clustered index I matching one or more selects
• (NPages(I)NPages(R)) product of RFs of
  matching selects.
• Non-clustered index I matching one or more
  selects
• (NPages(I)NTuples(R)) product of RFs of
  matching selects.
• Sequential scan of file
• NPages(R).
• Note Typically, no duplicate elimination on
  projections! (Exception Done on answers if user
  says DISTINCT.)

24
Example
SELECT S.sid FROM Sailors S WHERE S.rating8

• Sailors 500 pages, 80 tuples/page
• Case2 index on rating.
• Clustered index (1/NKeys(I))
  (NPages(I)NPages(S)) (1/10) (50500) pages
  Unclustered index (1/NKeys(I))
  (NPages(I)NTuples(S)) (1/10) (5040000)
  pages
• Case3 index on sid.
• Would have to retrieve all tuples/pages. With a
  clustered index, the cost is 50500, with
  unclustered index, 5040000.
• Case4 file scan
• We retrieve all file pages (500).

25
Queries Over Multiple Relations

• Fundamental decision in System R only left-deep
  join trees are considered.
• As the number of joins increases, the number of
  alternative plans grows rapidly we need to
  restrict the search space.
• Left-deep trees allow us to generate all fully
  pipelined plans.
• Intermediate results not written to temporary
  files.
• Not all left-deep trees are fully pipelined
  (e.g., SM join).

26
Enumeration of Left-Deep Plans

• Left-deep plans differ only in the order of
  relations, the access method for each relation,
  and the join method for each join.
• Enumerated using N passes (if N relations
  joined)
• Pass 1 Find best 1-relation plan for each
  relation.
• Pass 2 Find best way to join result of each
  1-relation plan (as outer) to another relation.
  (All 2-relation plans.)
• Pass N Find best way to join result of a
  (N-1)-relation plan (as outer) to the Nth
  relation. (All N-relation plans.)
• For each subset of relations, retain only
• Cheapest plan overall, plus
• Cheapest plan for each interesting order of
  tuples.

B
A
C
D
Pass 1
Pass 2
Pass 3
27
Enumeration of Left-Deep Plans (contd.)

• Pass1 1 relation plan.
• Identify selection terms in the WHERE clause that
  mention only attributes of A. (perform first
  access of A, before Join)
• Identify attributes of A not mentioned in SELECT
  or WHERE clause (Project out when first access
  of A, before Join)
• Cheapest plan keeping order.
• E.g., a file scan (as cheapest overall plan for
  fetching all tuples) and a B tree index (as the
  cheapest plan for fetching all tuples in the
  search key order).

28
Enumeration of Left-Deep Plans (contd.)

• Pass 2 All 2-relational Plans.
• Each of the single-relation plan from Pass 1 as
  the outer relation, and every other relation as
  the inner relation.
• Examine WHERE clauses
• Selections involving only attributes of inner
  relation (apply before Join).
• Selections defining the Join.
• Selections involving attributes of other
  relations (apply after Join).
• Actually, only selections that are really applied
  before the join are those that match the chosen
  access paths for A and B.
• Depending on Join algorithm chosen, the cost may
  include the materializing the outer relation.

29
Enumeration of Plans (Contd.)

• ORDER BY, GROUP BY, aggregates etc. handled as a
  final step, using either an interestingly
  ordered plan or an additional sorting operator.
• An N-1 way plan is not combined with an
  additional relation unless there is a join
  condition between them, unless all predicates in
  WHERE have been used up.
• i.e., avoid Cartesian products if possible.
• In spite of pruning plan space, this approach is
  still exponential in the of tables.

30
Example
Sailors B tree on rating Hash on
sid Reserves B tree on bid

• Pass1
• Sailors
• Choice1 B tree matches ratinggt5,
  probably cheapest.
• if selection is expected to retrieve a lot of
  tuples, and index is unclustered, file scan may
  be cheaper.
• Decision B tree plan kept (because tuples are
  in rating order).
• Reserves B tree on bid matches bid500
  cheapest.

• Pass 2
• each plan retained from Pass 1 as the outer.
• how to join it with the (only) other relation.
  e.g., Reserves as outer Hash index can be
  used to get Sailors tuples
• that satisfy sid outer tuples sid value.

31
Nested Queries
SELECT S.sname FROM Sailors S WHERE EXISTS
(SELECT FROM Reserves R WHERE
R.bid103 AND R.sidS.sid)

• Nested block is optimized independently, with the
  outer tuple considered as providing a selection
  condition.
• Outer block is optimized with the cost of
  calling nested block computation taken into
  account.
• Implicit ordering of these blocks means that some
  good strategies are not considered.
• The non-nested version of the query is typically
  optimized better.

Nested block to optimize SELECT FROM
Reserves R WHERE R.bid103 AND S.sid
outer value
Equivalent non-nested query SELECT S.sname FROM
Sailors S, Reserves R WHERE S.sidR.sid AND
R.bid103
32
Highlights of System R Optimizer

• Impact
• Most widely used currently works well for lt 10
  joins.
• Cost estimation
• Statistics, maintained in system catalogs, used
  to estimate cost of operations and result sizes.
• Considers combination of CPU and I/O costs.
• Plan Space
• Only the space of left-deep plans is considered.
• Left-deep plans allow output of each operator to
  be pipelined into the next operator without
  storing it in a temporary relation.
• Focus on optimizing SQLs without nesting. No
  duplication in projections. (unless DISTINCT
  used)
• Cartesian products avoided.

33
Other Approaches to Query Optimization

• Exclusive search is not suitable for large number
  of Joins.
• Rule-based optimizers a set of rules to guide
  the generation of candidate plans.
• Randomized plan generation probabilistic
  algorithms to explore a large space of plans.

34
Summary

• Query optimization is an important task in a
  relational DBMS.
• Must understand optimization in order to
  understand the performance impact of a given
  database design (relations, indexes) on a
  workload (set of queries).
• Two parts to optimizing a query
• Consider a set of alternative plans.
• Must prune search space typically, left-deep
  plans only.
• Must estimate cost of each plan that is
  considered.
• Must estimate size of result and cost for each
  plan node.
• Key issues Statistics, indexes, operator
  implementations.

35
Summary

• Single-relation queries
• All access paths considered, cheapest is chosen.
• Issues Selections that match index, whether
  index key has all needed fields and/or provides
  tuples in a desired order.
• Multiple-relation queries
• All single-relation plans are first enumerated.
• Selections/projections considered as early as
  possible.
• Next, for each 1-relation plan, all ways of
  joining another relation (as inner) are
  considered.
• Next, for each 2-relation plan that is
  retained, all ways of joining another relation
  (as inner) are considered, etc.
• At each level, for each subset of relations, only
  best plan for each interesting order of tuples is
  retained.