<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>

<sect2>

<title>Single-Column Statistics</title>

</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>