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Improve multivariate statistics documentation

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Extended statistics commit 7b504eb282c did not include appropriate
documentation next to where we document regular planner statistics (I
ripped what was submitted before commit and then forgot to put it back),
and while later commit 2686ee1b7ccf added some material, it structurally
depended on what I had ripped out, so the end result wasn't proper.

Fix those problems by shuffling what was added by 2686ee1b7ccf and
including some additional material, so that now chapter 14 "Performance
Tips" now describes the types of multivariate statistics we currently
have, and chapter 68 "How the Planner Uses Statistics" shows some
examples.  The new text should be more in line with previous material,
in (hopefully) the appropriate depth.

While at it, fix a small bug in pg_statistic_ext docs: one column was
listed in the wrong spot.
This commit is contained in:
Alvaro Herrera 2017-04-20 15:43:33 -03:00
parent 8bcb31ad5a
commit 919f6d746e
3 changed files with 323 additions and 146 deletions
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22
doc/src/sgml/catalogs.sgml
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@ -4320,6 +4320,17 @@
<entry>Owner of the statistic</entry>
</row>
<row>
<entry><structfield>stxkeys</structfield></entry>
<entry><type>int2vector</type></entry>
<entry><literal><link linkend="catalog-pg-attribute"><structname>pg_attribute</structname></link>.attnum</literal></entry>
<entry>
This is an array of values that indicate which table columns this
statistic covers. For example a value of <literal>1 3</literal> would
mean that the first and the third table columns make up the statistic key.
</entry>
</row>
<row>
<entry><structfield>stxkind</structfield></entry>
<entry><type>char[]</type></entry>
@ -4332,17 +4343,6 @@
</entry>
</row>
<row>
<entry><structfield>stxkeys</structfield></entry>
<entry><type>int2vector</type></entry>
<entry><literal><link linkend="catalog-pg-attribute"><structname>pg_attribute</structname></link>.attnum</literal></entry>
<entry>
This is an array of values that indicate which table columns this
statistic covers. For example a value of <literal>1 3</literal> would
mean that the first and the third table columns make up the statistic key.
</entry>
</row>
<row>
<entry><structfield>stxndistinct</structfield></entry>
<entry><type>pg_ndistinct</type></entry>

195
doc/src/sgml/perform.sgml
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@ -906,6 +906,8 @@ EXPLAIN ANALYZE SELECT * FROM tenk1 WHERE unique1 &lt; 100 AND unique2 &gt; 9000
<secondary>of the planner</secondary>
</indexterm>
<sect2>
<title>Single-Column Statistics</title>
<para>
As we saw in the previous section, the query planner needs to estimate
the number of rows retrieved by a query in order to make good choices
@ -1043,7 +1045,200 @@ WHERE tablename = 'road';
Further details about the planner's use of statistics can be found in
<xref linkend="planner-stats-details">.
</para>
</sect2>
<sect2 id="planner-stats-extended">
<title>Extended Statistics</title>
<indexterm zone="planner-stats-extended">
<primary>statistics</primary>
<secondary>of the planner</secondary>
</indexterm>
<indexterm>
<primary>correlation</primary>
<secondary>in the query planner</secondary>
</indexterm>
<indexterm>
<primary>pg_statistic_ext</primary>
</indexterm>
<para>
It is common to see slow queries running bad execution plans because
multiple columns used in the query clauses are correlated.
The planner normally assumes that multiple conditions
are independent of each other,
an assumption that does not hold when column values are correlated.
Regular statistics, because of their per-individual-column nature,
do not capture the knowledge of cross-column correlation;
<firstterm>multivariate statistics</firstterm> can be used to instruct
the server to obtain statistics across such a set of columns,
which are later used by the query optimizer
to determine cardinality and selectivity
of clauses involving those columns.
Multivariate statistics are currently the only use of
<firstterm>extended statistics</firstterm>.
</para>
<para>
Extended statistics are created using
<xref linkend="sql-createstatistics">, which see for more details.
Data collection is deferred until the next <command>ANALYZE</command>
on the table, after which the stored values can be examined in the
<link linkend="catalog-pg-statistic-ext"><structname>pg_statistic_ext</structname></link>
catalog.
</para>
<para>
The following subsections describe the types of extended statistics
that are currently supported.
</para>
<sect3>
<title>Functional Dependencies</title>
<para>
The simplest type of extended statistics are functional dependencies,
a concept used in definitions of database normal forms.
Put simply, it is said that column <literal>b</> is functionally
dependent on column <literal>a</> if knowledge of the value of
<literal>a</> is sufficient to determine the value of <literal>b</>.
In normalized databases, functional dependencies are allowed only on
primary keys and superkeys. However, many data sets are in practice not
fully normalized for various reasons; intentional denormalization for
performance reasons is a common example.
</para>
<para>
The existance of functional dependencies directly affects the accuracy
of estimates in certain queries.
The reason is that conditions on the dependent columns do not
restrict the result set, but the query planner (lacking functional
dependency knowledge) considers them independent, resulting in
underestimates.
To inform the planner about the functional dependencies, we collect
measurements of dependency during <command>ANALYZE</>. Assessing
the degree of dependency between all sets of columns would be
prohibitively expensive, so the search is limited to potential
dependencies defined using the <literal>dependencies</> option of
extended statistics. It is advisable to create
<literal>dependencies</> statistics if and only if functional
dependencies actually exist, to avoid unnecessary overhead on both
<command>ANALYZE</> and query planning.
</para>
<para>
To inspect functional dependencies on a statistics
<literal>stts</literal>, you may do this:
<programlisting>
CREATE STATISTICS stts WITH (dependencies)
ON (zip, city) FROM zipcodes;
ANALYZE zipcodes;
SELECT stxname, stxkeys, stxdependencies
FROM pg_statistic_ext
WHERE stxname = 'stts';
stxname | stxkeys | stxdependencies
---------+---------+--------------------------------------------
stts | 1 5 | [{1 => 5 : 1.000000}, {5 => 1 : 0.423130}]
(1 row)
</programlisting>
where it can be seen that column 1 (a zip code) fully determines column
5 (city) so the coefficient is 1.0, while city only determines zip code
about 42% of the time, meaning that there are many cities (58%) that are
represented by more than a single ZIP code.
</para>
<para>
When computing the selectivity, the planner inspects all conditions and
attempts to identify which conditions are already implied by other
conditions. The selectivity estimates from any redundant conditions are
ignored from a selectivity point of view. In the example query above,
the selectivity estimates for either of the conditions may be eliminated,
thus improving the overall estimate.
</para>
<sect4>
<title>Limitations of Functional Dependencies</title>
<para>
Functional dependencies are a very simple type of statistics, and
as such have several limitations. The first limitation is that they
only work with simple equality conditions, comparing columns and constant
values. It's not possible to use them to eliminate equality conditions
comparing two columns or a column to an expression, range clauses,
<literal>LIKE</> or any other type of conditions.
</para>
<para>
When eliminating the implied conditions, the planner assumes that the
conditions are compatible. Consider the following example, where
this assumption does not hold:
<programlisting>
EXPLAIN (ANALYZE, TIMING OFF) SELECT * FROM t WHERE a = 1 AND b = 10;
QUERY PLAN
-----------------------------------------------------------------------------
Seq Scan on t (cost=0.00..195.00 rows=100 width=8) (actual rows=0 loops=1)
Filter: ((a = 1) AND (b = 10))
Rows Removed by Filter: 10000
</programlisting>
While there are no rows with such combination of values, the planner
is unable to verify whether the values match &mdash; it only knows that
the columns are functionally dependent.
</para>
<para>
This assumption is related to queries executed on the database; in many
cases, it's actually satisfied (e.g. when the GUI only allows selecting
compatible values). But if that's not the case, functional dependencies
may not be a viable option.
</para>
</sect4>
</sect3>
<sect3>
<title>Multivariate N-Distinct Coefficients</title>
<para>
Single-column statistics store the number of distinct values in each
column. Estimates of the number of distinct values on more than one
column (for example, for <literal>GROUP BY a, b</literal>) are
frequently wrong when the planner only has single-column statistical
data, however, causing it to select bad plans.
In order to improve n-distinct estimation when multiple columns are
grouped together, the <literal>ndistinct</> option of extended statistics
can be used, which instructs <command>ANALYZE</> to collect n-distinct
estimates for all possible combinations of two or more columns of the set
of columns in the statistics object (the per-column estimates are already
available in <structname>pg_statistic</>).
</para>
<para>
Continuing the above example, the n-distinct coefficients in a ZIP
code table may look like the following:
<programlisting>
CREATE STATISTICS stts2 WITH (ndistinct)
ON (zip, state, city) FROM zipcodes;
ANALYZE zipcodes;
SELECT stxkeys AS k, stxndistinct AS nd
FROM pg_statistic_ext
WHERE stxname = 'stts2';
-[ RECORD 1 ]---------------------------------------------
k | 1 2 5
nd | [{(b 1 2), 33178.000000}, {(b 1 5), 33178.000000},
{(b 2 5), 27435.000000}, {(b 1 2 5), 33178.000000}]
(1 row)
</programlisting>
which indicates that there are three combinations of columns that
have 33178 distinct values: ZIP code and state; ZIP code and city;
and ZIP code, city and state (the fact that they are all equal is
expected given the nature of ZIP-code data). On the other hand,
the combination of city and state only has 27435 distinct values.
</para>
</sect3>
</sect2>
</sect1>
<sect1 id="explicit-joins">

252
doc/src/sgml/planstats.sgml
View File

@ -445,159 +445,141 @@ rows = (outer_cardinality * inner_cardinality) * selectivity
operator-specific selectivity functions are mostly found
in <filename>src/backend/utils/adt/selfuncs.c</filename>.
</para>
</sect1>
<sect2 id="functional-dependencies">
<title>Functional Dependencies</title>
<sect1 id="multivariate-statistics-examples">
<title>Multivariate Statistics Examples</title>
<indexterm>
<primary>row estimation</primary>
<secondary>multivariate</secondary>
</indexterm>
<sect2>
<title>Functional dependencies</title>
<para>
The simplest type of extended statistics are functional dependencies,
used in definitions of database normal forms. When simplified, saying that
<literal>b</> is functionally dependent on <literal>a</> means that
knowledge of value of <literal>a</> is sufficient to determine value of
<literal>b</>.
</para>
<para>
In normalized databases, only functional dependencies on primary keys
and superkeys are allowed. However, in practice, many data sets are not
fully normalized, for example, due to intentional denormalization for
performance reasons.
</para>
<para>
Functional dependencies directly affect accuracy of the estimates, as
conditions on the dependent column(s) do not restrict the result set,
resulting in underestimates.
</para>
<para>
To inform the planner about the functional dependencies, we collect
measurements of dependency during <command>ANALYZE</>. Assessing
dependency between all sets of columns would be prohibitively
expensive, so we limit our search to potential dependencies defined
using the <command>CREATE STATISTICS</> command.
Multivariate correlation can be seen with a very simple data set &mdash; a
table with two columns, both containing the same values:
<programlisting>
CREATE TABLE t (a INT, b INT);
INSERT INTO t SELECT i/100, i/100 FROM generate_series(1,10000) s(i);
CREATE STATISTICS s1 WITH (dependencies) ON (a, b) FROM t;
INSERT INTO t SELECT i % 100, i % 100 FROM generate_series(1, 10000) s(i);
ANALYZE t;
EXPLAIN ANALYZE SELECT * FROM t WHERE a = 1 AND b = 1;
QUERY PLAN
-------------------------------------------------------------------------------------------------
Seq Scan on t (cost=0.00..195.00 rows=100 width=8) (actual time=0.095..3.118 rows=100 loops=1)
</programlisting>
As explained in <xref linkend="planner-stats">, the planner can determine
cardinality of <structname>t</structname> using the number of pages and
rows obtained from <structname>pg_class</structname>:
<programlisting>
SELECT relpages, reltuples FROM pg_class WHERE relname = 't';
relpages | reltuples
----------+-----------
45 | 10000
</programlisting>
The data distribution is very simple; there are only 100 distinct values
in each column, uniformly distributed.
</para>
<para>
The following example shows the result of estimating a <literal>WHERE</>
condition on the <structfield>a</> column:
<programlisting>
EXPLAIN (ANALYZE, TIMING OFF) SELECT * FROM t WHERE a = 1;
QUERY PLAN
-------------------------------------------------------------------------------
Seq Scan on t (cost=0.00..170.00 rows=100 width=8) (actual rows=100 loops=1)
Filter: (a = 1)
Rows Removed by Filter: 9900
</programlisting>
The planner examines the condition and determines the selectivity
of this clause to be 1%. By comparing this estimate and the actual
number of rows, we see that the estimate is very accurate
(in fact exact, as the table is very small). Changing the
<literal>WHERE</> to use the <structfield>b</> column, an identical
plan is generated. Observe what happens if we apply the same
condition on both columns combining them with <literal>AND</>:
<programlisting>
EXPLAIN (ANALYZE, TIMING OFF) SELECT * FROM t WHERE a = 1 AND b = 1;
QUERY PLAN
-----------------------------------------------------------------------------
Seq Scan on t (cost=0.00..195.00 rows=1 width=8) (actual rows=100 loops=1)
Filter: ((a = 1) AND (b = 1))
Rows Removed by Filter: 9900
Planning time: 0.367 ms
Execution time: 3.380 ms
(5 rows)
</programlisting>
The planner is now aware of the functional dependencies and considers
them when computing the selectivity of the second condition. Running
the query without the statistics would lead to quite different estimates.
The planner estimates the selectivity for each condition individually,
arriving to the 1% estimates as above, and then multiplies them, getting
the final 0.01% estimate. The <quote>actual</quote> figures, however,
show that this results in a significant underestimate, as the actual
number of rows matching the conditions (100) is two orders of magnitude
higher than the estimated value.
</para>
<para>
This problem can be fixed by applying functional-dependency
multivariate statistics on the two columns:
<programlisting>
DROP STATISTICS s1;
EXPLAIN ANALYZE SELECT * FROM t WHERE a = 1 AND b = 1;
QUERY PLAN
-----------------------------------------------------------------------------------------------
Seq Scan on t (cost=0.00..195.00 rows=1 width=8) (actual time=0.000..6.379 rows=100 loops=1)
CREATE STATISTICS stts WITH (dependencies) ON (a, b) FROM t;
ANALYZE t;
EXPLAIN (ANALYZE, TIMING OFF) SELECT * FROM t WHERE a = 1 AND b = 1;
QUERY PLAN
-------------------------------------------------------------------------------
Seq Scan on t (cost=0.00..195.00 rows=100 width=8) (actual rows=100 loops=1)
Filter: ((a = 1) AND (b = 1))
Rows Removed by Filter: 9900
Planning time: 0.000 ms
Execution time: 6.379 ms
(5 rows)
</programlisting>
</para>
<para>
If no dependency exists, the collected statistics do not influence the
query plan. The only effect is to slow down <command>ANALYZE</>. Should
partial dependencies exist these will also be stored and applied
during planning.
</para>
<para>
Similarly to per-column statistics, extended statistics are stored in
a system catalog called <structname>pg_statistic_ext</structname>.
To inspect the statistics <literal>s1</literal> defined above,
you may do this:
<programlisting>
SELECT stxname,stxdependencies FROM pg_statistic_ext WHERE stxname = 's1';
stxname | stxdependencies
---------+--------------------------------------------
s1 | [{1 => 2 : 1.000000}, {2 => 1 : 1.000000}]
(1 row)
</programlisting>
This shows that the statistics are defined on table <structname>t</>,
<structfield>attnums</structfield> lists attribute numbers of columns
(references <structname>pg_attribute</structname>). It also shows
the length in bytes of the functional dependencies, as found by
<command>ANALYZE</> when serialized into a <literal>bytea</> column.
</para>
<para>
When computing the selectivity, the planner inspects all conditions and
attempts to identify which conditions are already implied by other
conditions. The selectivity estimates from any redundant conditions are
ignored from a selectivity point of view. In the example query above,
the selectivity estimates for either of the conditions may be eliminated,
thus improving the overall estimate.
</para>
<sect3 id="functional-dependencies-limitations">
<title>Limitations of functional dependencies</title>
<para>
Functional dependencies are a very simple type of statistics, and
as such have several limitations. The first limitation is that they
only work with simple equality conditions, comparing columns and constant
values. It's not possible to use them to eliminate equality conditions
comparing two columns or a column to an expression, range clauses,
<literal>LIKE</> or any other type of conditions.
</para>
<para>
When eliminating the implied conditions, the planner assumes that the
conditions are compatible. Consider the following example, violating
this assumption:
<programlisting>
EXPLAIN ANALYZE SELECT * FROM t WHERE a = 1 AND b = 10;
QUERY PLAN
-----------------------------------------------------------------------------------------------
Seq Scan on t (cost=0.00..195.00 rows=100 width=8) (actual time=2.992..2.992 rows=0 loops=1)
Filter: ((a = 1) AND (b = 10))
Rows Removed by Filter: 10000
Planning time: 0.232 ms
Execution time: 3.033 ms
(5 rows)
</programlisting>
While there are no rows with such combination of values, the planner
is unable to verify whether the values match - it only knows that
the columns are functionally dependent.
</para>
<para>
This assumption is more about queries executed on the database - in many
cases, it's actually satisfied (e.g. when the GUI only allows selecting
compatible values). But if that's not the case, functional dependencies
may not be a viable option.
</para>
<para>
For additional information about functional dependencies, see
<filename>src/backend/statistics/README.dependencies</>.
</para>
</sect3>
</sect2>
</sect1>
<sect2>
<title>Multivariate N-Distinct coefficients</title>
<para>
A similar problem occurs with estimation of the cardinality of distinct
elements, used to determine the number of groups that would be generated
by a <command>GROUP BY</command> clause. When <command>GROUP BY</command>
lists a single column, the n-distinct estimate (which can be seen as the
number of rows returned by the aggregate execution node) is very accurate:
<programlisting>
EXPLAIN (ANALYZE, TIMING OFF) SELECT COUNT(*) FROM t GROUP BY a;
QUERY PLAN
-----------------------------------------------------------------------------------------
HashAggregate (cost=195.00..196.00 rows=100 width=12) (actual rows=100 loops=1)
Group Key: a
-&gt; Seq Scan on t (cost=0.00..145.00 rows=10000 width=4) (actual rows=10000 loops=1)
</programlisting>
But without multivariate statistics, the estimate for the number of
groups in a query with two columns in <command>GROUP BY</command>, as
in the following example, is off by an order of magnitude:
<programlisting>
EXPLAIN (ANALYZE, TIMING OFF) SELECT COUNT(*) FROM t GROUP BY a, b;
QUERY PLAN
--------------------------------------------------------------------------------------------
HashAggregate (cost=220.00..230.00 rows=1000 width=16) (actual rows=100 loops=1)
Group Key: a, b
-&gt; Seq Scan on t (cost=0.00..145.00 rows=10000 width=8) (actual rows=10000 loops=1)
</programlisting>
By dropping the existing statistics and re-creating it to include n-distinct
calculation, the estimate is much improved:
<programlisting>
DROP STATISTICS stts;
CREATE STATISTICS stts WITH (dependencies, ndistinct) ON (a, b) FROM t;
ANALYZE t;
EXPLAIN (ANALYZE, TIMING OFF) SELECT COUNT(*) FROM t GROUP BY a, b;
QUERY PLAN
--------------------------------------------------------------------------------------------
HashAggregate (cost=220.00..221.00 rows=100 width=16) (actual rows=100 loops=1)
Group Key: a, b
-&gt; Seq Scan on t (cost=0.00..145.00 rows=10000 width=8) (actual rows=10000 loops=1)
</programlisting>
</para>
</sect2>
</sect1>
</chapter>