@ -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-dependenci es">
<title>Functional Dependenci es</title>
<sect1 id="multivariate-statistics-exampl es">
<title>Multivariate Statistics Exampl es</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>
<indexterm>
<primary>row estimation</primary>
<secondary>multivariate</secondary>
</indexterm>
<sect2>
<title>Functional dependencies</title>
<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>
Multivariate correlation can be seen with a very simple data set — a
table with two columns, both containing the same values:
<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.
<programlisting>
CREATE TABLE t (a INT, b INT);
INSERT INTO t SELECT i % 100, i % 100 FROM generate_series(1, 10000) s(i);
ANALYZE t;
</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>
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.
The following example shows the result of estimating a <literal>WHERE</>
condition on the <structfield>a</> column:
<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;
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)
Filter: ((a = 1) AND (b = 1))
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
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 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>
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)
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: 0.000 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.
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>
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:
This problem can be fixed by applying functional-dependency
multivariate statistics on the two columns:
<programlisting>
SELECT stxname,stxdependencies FROM pg_statistic_ext WHERE stxname = 's1';
stxname | stxdependencies
---------+--------------------------------------------
s1 | [{1 => 2 : 1.000000}, {2 => 1 : 1.000000}]
(1 row)
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
</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>
</sect2>
<sect2>
<title>Multivariate N-Distinct coefficients</title>
<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.
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
-> 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
-> 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
-> Seq Scan on t (cost=0.00..145.00 rows=10000 width=8) (actual rows=10000 loops=1)
</programlisting>
</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>
</chapter>
Alvaro Herrera
9 years ago