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 <sect1 id="xaggr">
  <title>User-Defined Aggregates</title>

  <indexterm zone="xaggr">
   <primary>aggregate function</primary>
   <secondary>user-defined</secondary>
  </indexterm>

  <para>
   Aggregate functions in <productname>PostgreSQL</productname>
   are defined in terms of <firstterm>state values</firstterm>
   and <firstterm>state transition functions</firstterm>.
   That is, an aggregate operates using a state value that is updated
   as each successive input row is processed.
   To define a new aggregate
   function, one selects a data type for the state value,
   an initial value for the state, and a state transition
   function.  The state transition function takes the previous state
   value and the aggregate's input value(s) for the current row, and
   returns a new state value.
   A <firstterm>final function</firstterm>
   can also be specified, in case the desired result of the aggregate
   is different from the data that needs to be kept in the running
   state value.  The final function takes the ending state value
   and returns whatever is wanted as the aggregate result.
   In principle, the transition and final functions are just ordinary
   functions that could also be used outside the context of the
   aggregate.  (In practice, it's often helpful for performance reasons
   to create specialized transition functions that can only work when
   called as part of an aggregate.)
  </para>

  <para>
   Thus, in addition to the argument and result data types seen by a user
   of the aggregate, there is an internal state-value data type that
   might be different from both the argument and result types.
  </para>

  <para>
   If we define an aggregate that does not use a final function,
   we have an aggregate that computes a running function of
   the column values from each row.  <function>sum</function>  is  an
   example  of  this  kind  of aggregate.  <function>sum</function> starts at
   zero and always adds the current  row's  value  to
   its  running  total.  For example, if we want to make a <function>sum</function>
   aggregate to work on a data type for complex numbers,
   we only need the addition function for that data type.
   The aggregate definition would be:

<programlisting>
CREATE AGGREGATE sum (complex)
(
    sfunc = complex_add,
    stype = complex,
    initcond = '(0,0)'
);
</programlisting>

   which we might use like this:

<programlisting>
SELECT sum(a) FROM test_complex;

   sum
-----------
 (34,53.9)
</programlisting>

   (Notice that we are relying on function overloading: there is more than
    one aggregate named <function>sum</function>, but
   <productname>PostgreSQL</productname> can figure out which kind
   of sum applies to a column of type <type>complex</type>.)
  </para>

  <para>
   The above definition of <function>sum</function> will return zero
   (the initial state value) if there are no nonnull input values.
   Perhaps we want to return null in that case instead &mdash; the SQL standard
   expects <function>sum</function> to behave that way.  We can do this simply by
   omitting the <literal>initcond</literal> phrase, so that the initial state
   value is null.  Ordinarily this would mean that the <literal>sfunc</literal>
   would need to check for a null state-value input.  But for
   <function>sum</function> and some other simple aggregates like
   <function>max</function> and <function>min</function>,
   it is sufficient to insert the first nonnull input value into
   the state variable and then start applying the transition function
   at the second nonnull input value.  <productname>PostgreSQL</productname>
   will do that automatically if the initial state value is null and
   the transition function is marked <quote>strict</quote> (i.e., not to be called
   for null inputs).
  </para>

  <para>
   Another bit of default behavior for a <quote>strict</quote> transition function
   is that the previous state value is retained unchanged whenever a
   null input value is encountered.  Thus, null values are ignored.  If you
   need some other behavior for null inputs, do not declare your
   transition function as strict; instead code it to test for null inputs and
   do whatever is needed.
  </para>

  <para>
   <function>avg</function> (average) is a more complex example of an aggregate.
   It requires
   two pieces of running state: the sum of the inputs and the count
   of the number of inputs.  The final result is obtained by dividing
   these quantities.  Average is typically implemented by using an
   array as the state value.  For example,
   the built-in implementation of <function>avg(float8)</function>
   looks like:

<programlisting>
CREATE AGGREGATE avg (float8)
(
    sfunc = float8_accum,
    stype = float8[],
    finalfunc = float8_avg,
    initcond = '{0,0,0}'
);
</programlisting>
  </para>

  <note>
   <para>
   <function>float8_accum</function> requires a three-element array, not just
   two elements, because it accumulates the sum of squares as well as
   the sum and count of the inputs.  This is so that it can be used for
   some other aggregates as well as <function>avg</function>.
   </para>
  </note>

  <para>
   Aggregate function calls in SQL allow <literal>DISTINCT</literal>
   and <literal>ORDER BY</literal> options that control which rows are fed
   to the aggregate's transition function and in what order.  These
   options are implemented behind the scenes and are not the concern
   of the aggregate's support functions.
  </para>

  <para>
   For further details see the
   <xref linkend="sql-createaggregate"/>
   command.
  </para>

 <sect2 id="xaggr-moving-aggregates">
  <title>Moving-Aggregate Mode</title>

  <indexterm>
   <primary>moving-aggregate mode</primary>
  </indexterm>

  <indexterm>
   <primary>aggregate function</primary>
   <secondary>moving aggregate</secondary>
  </indexterm>

  <para>
   Aggregate functions can optionally support <firstterm>moving-aggregate
   mode</firstterm>, which allows substantially faster execution of aggregate
   functions within windows with moving frame starting points.
   (See <xref linkend="tutorial-window"/>
   and <xref linkend="syntax-window-functions"/> for information about use of
   aggregate functions as window functions.)
   The basic idea is that in addition to a normal <quote>forward</quote>
   transition function, the aggregate provides an <firstterm>inverse
   transition function</firstterm>, which allows rows to be removed from the
   aggregate's running state value when they exit the window frame.
   For example a <function>sum</function> aggregate, which uses addition as the
   forward transition function, would use subtraction as the inverse
   transition function.  Without an inverse transition function, the window
   function mechanism must recalculate the aggregate from scratch each time
   the frame starting point moves, resulting in run time proportional to the
   number of input rows times the average frame length.  With an inverse
   transition function, the run time is only proportional to the number of
   input rows.
  </para>

  <para>
   The inverse transition function is passed the current state value and the
   aggregate input value(s) for the earliest row included in the current
   state.  It must reconstruct what the state value would have been if the
   given input row had never been aggregated, but only the rows following
   it.  This sometimes requires that the forward transition function keep
   more state than is needed for plain aggregation mode.  Therefore, the
   moving-aggregate mode uses a completely separate implementation from the
   plain mode: it has its own state data type, its own forward transition
   function, and its own final function if needed.  These can be the same as
   the plain mode's data type and functions, if there is no need for extra
   state.
  </para>

  <para>
   As an example, we could extend the <function>sum</function> aggregate given above
   to support moving-aggregate mode like this:

<programlisting>
CREATE AGGREGATE sum (complex)
(
    sfunc = complex_add,
    stype = complex,
    initcond = '(0,0)',
    msfunc = complex_add,
    minvfunc = complex_sub,
    mstype = complex,
    minitcond = '(0,0)'
);
</programlisting>

   The parameters whose names begin with <literal>m</literal> define the
   moving-aggregate implementation.  Except for the inverse transition
   function <literal>minvfunc</literal>, they correspond to the plain-aggregate
   parameters without <literal>m</literal>.
  </para>

  <para>
   The forward transition function for moving-aggregate mode is not allowed
   to return null as the new state value.  If the inverse transition
   function returns null, this is taken as an indication that the inverse
   function cannot reverse the state calculation for this particular input,
   and so the aggregate calculation will be redone from scratch for the
   current frame starting position.  This convention allows moving-aggregate
   mode to be used in situations where there are some infrequent cases that
   are impractical to reverse out of the running state value.  The inverse
   transition function can <quote>punt</quote> on these cases, and yet still come
   out ahead so long as it can work for most cases.  As an example, an
   aggregate working with floating-point numbers might choose to punt when
   a <literal>NaN</literal> (not a number) input has to be removed from the running
   state value.
  </para>

  <para>
   When writing moving-aggregate support functions, it is important to be
   sure that the inverse transition function can reconstruct the correct
   state value exactly.  Otherwise there might be user-visible differences
   in results depending on whether the moving-aggregate mode is used.
   An example of an aggregate for which adding an inverse transition
   function seems easy at first, yet where this requirement cannot be met
   is <function>sum</function> over <type>float4</type> or <type>float8</type> inputs.  A
   naive declaration of <function>sum(<type>float8</type>)</function> could be

<programlisting>
CREATE AGGREGATE unsafe_sum (float8)
(
    stype = float8,
    sfunc = float8pl,
    mstype = float8,
    msfunc = float8pl,
    minvfunc = float8mi
);
</programlisting>

   This aggregate, however, can give wildly different results than it would
   have without the inverse transition function. For example, consider

<programlisting>
SELECT
  unsafe_sum(x) OVER (ORDER BY n ROWS BETWEEN CURRENT ROW AND 1 FOLLOWING)
FROM (VALUES (1, 1.0e20::float8),
             (2, 1.0::float8)) AS v (n,x);
</programlisting>

   This query returns <literal>0</literal> as its second result, rather than the
   expected answer of <literal>1</literal>.  The cause is the limited precision of
   floating-point values: adding <literal>1</literal> to <literal>1e20</literal> results
   in <literal>1e20</literal> again, and so subtracting <literal>1e20</literal> from that
   yields <literal>0</literal>, not <literal>1</literal>.  Note that this is a limitation
   of floating-point arithmetic in general, not a limitation
   of <productname>PostgreSQL</productname>.
  </para>

 </sect2>

 <sect2 id="xaggr-polymorphic-aggregates">
  <title>Polymorphic and Variadic Aggregates</title>

  <indexterm>
   <primary>aggregate function</primary>
   <secondary>polymorphic</secondary>
  </indexterm>

  <indexterm>
   <primary>aggregate function</primary>
   <secondary>variadic</secondary>
  </indexterm>

  <para>
   Aggregate functions can use polymorphic
   state transition functions or final functions, so that the same functions
   can be used to implement multiple aggregates.
   See <xref linkend="extend-types-polymorphic"/>
   for an explanation of polymorphic functions.
   Going a step further, the aggregate function itself can be specified
   with polymorphic input type(s) and state type, allowing a single
   aggregate definition to serve for multiple input data types.
   Here is an example of a polymorphic aggregate:

<programlisting>
CREATE AGGREGATE array_accum (anycompatible)
(
    sfunc = array_append,
    stype = anycompatiblearray,
    initcond = '{}'
);
</programlisting>

   Here, the actual state type for any given aggregate call is the array type
   having the actual input type as elements.  The behavior of the aggregate
   is to concatenate all the inputs into an array of that type.
   (Note: the built-in aggregate <function>array_agg</function> provides similar
   functionality, with better performance than this definition would have.)
  </para>

  <para>
   Here's the output using two different actual data types as arguments:

<programlisting>
SELECT attrelid::regclass, array_accum(attname)
    FROM pg_attribute
    WHERE attnum &gt; 0 AND attrelid = 'pg_tablespace'::regclass
    GROUP BY attrelid;

   attrelid    |              array_accum
---------------+---------------------------------------
 pg_tablespace | {spcname,spcowner,spcacl,spcoptions}
(1 row)

SELECT attrelid::regclass, array_accum(atttypid::regtype)
    FROM pg_attribute
    WHERE attnum &gt; 0 AND attrelid = 'pg_tablespace'::regclass
    GROUP BY attrelid;

   attrelid    |        array_accum
---------------+---------------------------
 pg_tablespace | {name,oid,aclitem[],text[]}
(1 row)
</programlisting>
  </para>

  <para>
   Ordinarily, an aggregate function with a polymorphic result type has a
   polymorphic state type, as in the above example.  This is necessary
   because otherwise the final function cannot be declared sensibly: it
   would need to have a polymorphic result type but no polymorphic argument
   type, which <command>CREATE FUNCTION</command> will reject on the grounds that
   the result type cannot be deduced from a call.  But sometimes it is
   inconvenient to use a polymorphic state type.  The most common case is
   where the aggregate support functions are to be written in C and the
   state type should be declared as <type>internal</type> because there is
   no SQL-level equivalent for it.  To address this case, it is possible to
   declare the final function as taking extra <quote>dummy</quote> arguments
   that match the input arguments of the aggregate.  Such dummy arguments
   are always passed as null values since no specific value is available when the
   final function is called.  Their only use is to allow a polymorphic
   final function's result type to be connected to the aggregate's input
   type(s).  For example, the definition of the built-in
   aggregate <function>array_agg</function> is equivalent to

<programlisting>
CREATE FUNCTION array_agg_transfn(internal, anynonarray)
  RETURNS internal ...;
CREATE FUNCTION array_agg_finalfn(internal, anynonarray)
  RETURNS anyarray ...;

CREATE AGGREGATE array_agg (anynonarray)
(
    sfunc = array_agg_transfn,
    stype = internal,
    finalfunc = array_agg_finalfn,
    finalfunc_extra
);
</programlisting>

   Here, the <literal>finalfunc_extra</literal> option specifies that the final
   function receives, in addition to the state value, extra dummy
   argument(s) corresponding to the aggregate's input argument(s).
   The extra <type>anynonarray</type> argument allows the declaration
   of <function>array_agg_finalfn</function> to be valid.
  </para>

  <para>
   An aggregate function can be made to accept a varying number of arguments
   by declaring its last argument as a <literal>VARIADIC</literal> array, in much
   the same fashion as for regular functions; see
   <xref linkend="xfunc-sql-variadic-functions"/>.  The aggregate's transition
   function(s) must have the same array type as their last argument.  The
   transition function(s) typically would also be marked <literal>VARIADIC</literal>,
   but this is not strictly required.
  </para>

  <note>
   <para>
    Variadic aggregates are easily misused in connection with
    the <literal>ORDER BY</literal> option (see <xref linkend="syntax-aggregates"/>),
    since the parser cannot tell whether the wrong number of actual arguments
    have been given in such a combination.  Keep in mind that everything to
    the right of <literal>ORDER BY</literal> is a sort key, not an argument to the
    aggregate.  For example, in
<programlisting>
SELECT myaggregate(a ORDER BY a, b, c) FROM ...
</programlisting>
    the parser will see this as a single aggregate function argument and
    three sort keys.  However, the user might have intended
<programlisting>
SELECT myaggregate(a, b, c ORDER BY a) FROM ...
</programlisting>
    If <literal>myaggregate</literal> is variadic, both these calls could be
    perfectly valid.
   </para>

   <para>
    For the same reason, it's wise to think twice before creating aggregate
    functions with the same names and different numbers of regular arguments.
   </para>
  </note>

 </sect2>

 <sect2 id="xaggr-ordered-set-aggregates">
  <title>Ordered-Set Aggregates</title>

  <indexterm>
   <primary>aggregate function</primary>
   <secondary>ordered set</secondary>
  </indexterm>

  <para>
   The aggregates we have been describing so far are <quote>normal</quote>
   aggregates.  <productname>PostgreSQL</productname> also
   supports <firstterm>ordered-set aggregates</firstterm>, which differ from
   normal aggregates in two key ways.  First, in addition to ordinary
   aggregated arguments that are evaluated once per input row, an
   ordered-set aggregate can have <quote>direct</quote> arguments that are
   evaluated only once per aggregation operation.  Second, the syntax
   for the ordinary aggregated arguments specifies a sort ordering
   for them explicitly.  An ordered-set aggregate is usually
   used to implement a computation that depends on a specific row
   ordering, for instance rank or percentile, so that the sort ordering
   is a required aspect of any call.  For example, the built-in
   definition of <function>percentile_disc</function> is equivalent to:

<programlisting>
CREATE FUNCTION ordered_set_transition(internal, anyelement)
  RETURNS internal ...;
CREATE FUNCTION percentile_disc_final(internal, float8, anyelement)
  RETURNS anyelement ...;

CREATE AGGREGATE percentile_disc (float8 ORDER BY anyelement)
(
    sfunc = ordered_set_transition,
    stype = internal,
    finalfunc = percentile_disc_final,
    finalfunc_extra
);
</programlisting>

   This aggregate takes a <type>float8</type> direct argument (the percentile
   fraction) and an aggregated input that can be of any sortable data type.
   It could be used to obtain a median household income like this:

<programlisting>
SELECT percentile_disc(0.5) WITHIN GROUP (ORDER BY income) FROM households;
 percentile_disc
-----------------
           50489
</programlisting>

   Here, <literal>0.5</literal> is a direct argument; it would make no sense
   for the percentile fraction to be a value varying across rows.
  </para>

  <para>
   Unlike the case for normal aggregates, the sorting of input rows for
   an ordered-set aggregate is <emphasis>not</emphasis> done behind the scenes,
   but is the responsibility of the aggregate's support functions.
   The typical implementation approach is to keep a reference to
   a <quote>tuplesort</quote> object in the aggregate's state value, feed the
   incoming rows into that object, and then complete the sorting and
   read out the data in the final function.  This design allows the
   final function to perform special operations such as injecting
   additional <quote>hypothetical</quote> rows into the data to be sorted.
   While normal aggregates can often be implemented with support
   functions written in <application>PL/pgSQL</application> or another
   PL language, ordered-set aggregates generally have to be written in
   C, since their state values aren't definable as any SQL data type.
   (In the above example, notice that the state value is declared as
   type <type>internal</type> &mdash; this is typical.)
   Also, because the final function performs the sort, it is not possible
   to continue adding input rows by executing the transition function again
   later.  This means the final function is not <literal>READ_ONLY</literal>;
   it must be declared in <link linkend="sql-createaggregate"><command>CREATE AGGREGATE</command></link>
   as <literal>READ_WRITE</literal>, or as <literal>SHAREABLE</literal> if
   it's possible for additional final-function calls to make use of the
   already-sorted state.
  </para>

  <para>
   The state transition function for an ordered-set aggregate receives
   the current state value plus the aggregated input values for
   each row, and returns the updated state value.  This is the
   same definition as for normal aggregates, but note that the direct
   arguments (if any) are not provided.  The final function receives
   the last state value, the values of the direct arguments if any,
   and (if <literal>finalfunc_extra</literal> is specified) null values
   corresponding to the aggregated input(s).  As with normal
   aggregates, <literal>finalfunc_extra</literal> is only really useful if the
   aggregate is polymorphic; then the extra dummy argument(s) are needed
   to connect the final function's result type to the aggregate's input
   type(s).
  </para>

  <para>
   Currently, ordered-set aggregates cannot be used as window functions,
   and therefore there is no need for them to support moving-aggregate mode.
  </para>

 </sect2>

  <sect2 id="xaggr-partial-aggregates">
  <title>Partial Aggregation</title>

  <indexterm>
   <primary>aggregate function</primary>
   <secondary>partial aggregation</secondary>
  </indexterm>

  <para>
   Optionally, an aggregate function can support <firstterm>partial
   aggregation</firstterm>.  The idea of partial aggregation is to run the aggregate's
   state transition function over different subsets of the input data
   independently, and then to combine the state values resulting from those
   subsets to produce the same state value that would have resulted from
   scanning all the input in a single operation.  This mode can be used for
   parallel aggregation by having different worker processes scan different
   portions of a table.  Each worker produces a partial state value, and at
   the end those state values are combined to produce a final state value.
   (In the future this mode might also be used for purposes such as combining
   aggregations over local and remote tables; but that is not implemented
   yet.)
  </para>

  <para>
   To support partial aggregation, the aggregate definition must provide
   a <firstterm>combine function</firstterm>, which takes two values of the
   aggregate's state type (representing the results of aggregating over two
   subsets of the input rows) and produces a new value of the state type,
   representing what the state would have been after aggregating over the
   combination of those sets of rows.  It is unspecified what the relative
   order of the input rows from the two sets would have been.  This means
   that it's usually impossible to define a useful combine function for
   aggregates that are sensitive to input row order.
  </para>

  <para>
   As simple examples, <literal>MAX</literal> and <literal>MIN</literal> aggregates can be
   made to support partial aggregation by specifying the combine function as
   the same greater-of-two or lesser-of-two comparison function that is used
   as their transition function.  <literal>SUM</literal> aggregates just need an
   addition function as combine function.  (Again, this is the same as their
   transition function, unless the state value is wider than the input data
   type.)
  </para>

  <para>
   The combine function is treated much like a transition function that
   happens to take a value of the state type, not of the underlying input
   type, as its second argument.  In particular, the rules for dealing
   with null values and strict functions are similar.  Also, if the aggregate
   definition specifies a non-null <literal>initcond</literal>, keep in mind that
   that will be used not only as the initial state for each partial
   aggregation run, but also as the initial state for the combine function,
   which will be called to combine each partial result into that state.
  </para>

  <para>
   If the aggregate's state type is declared as <type>internal</type>, it is
   the combine function's responsibility that its result is allocated in
   the correct memory context for aggregate state values.  This means in
   particular that when the first input is <literal>NULL</literal> it's invalid
   to simply return the second input, as that value will be in the wrong
   context and will not have sufficient lifespan.
  </para>

  <para>
   When the aggregate's state type is declared as <type>internal</type>, it is
   usually also appropriate for the aggregate definition to provide a
   <firstterm>serialization function</firstterm> and a <firstterm>deserialization
   function</firstterm>, which allow such a state value to be copied from one process
   to another.  Without these functions, parallel aggregation cannot be
   performed, and future applications such as local/remote aggregation will
   probably not work either.
  </para>

  <para>
   A serialization function must take a single argument of
   type <type>internal</type> and return a result of type <type>bytea</type>, which
   represents the state value packaged up into a flat blob of bytes.
   Conversely, a deserialization function reverses that conversion.  It must
   take two arguments of types <type>bytea</type> and <type>internal</type>, and
   return a result of type <type>internal</type>.  (The second argument is unused
   and is always zero, but it is required for type-safety reasons.)  The
   result of the deserialization function should simply be allocated in the
   current memory context, as unlike the combine function's result, it is not
   long-lived.
  </para>

  <para>
   Worth noting also is that for an aggregate to be executed in parallel,
   the aggregate itself must be marked <literal>PARALLEL SAFE</literal>.  The
   parallel-safety markings on its support functions are not consulted.
  </para>

 </sect2>

 <sect2 id="xaggr-support-functions">
  <title>Support Functions for Aggregates</title>

  <indexterm>
   <primary>aggregate function</primary>
   <secondary>support functions for</secondary>
  </indexterm>

  <para>
   A function written in C can detect that it is being called as an
   aggregate support function by calling
   <function>AggCheckCallContext</function>, for example:
<programlisting>
if (AggCheckCallContext(fcinfo, NULL))
</programlisting>
   One reason for checking this is that when it is true, the first input
   must be a temporary state value and can therefore safely be modified
   in-place rather than allocating a new copy.
   See <function>int8inc()</function> for an example.
   (While aggregate transition functions are always allowed to modify
   the transition value in-place, aggregate final functions are generally
   discouraged from doing so; if they do so, the behavior must be declared
   when creating the aggregate.  See <xref linkend="sql-createaggregate"/>
   for more detail.)
  </para>

  <para>
   The second argument of <function>AggCheckCallContext</function> can be used to
   retrieve the memory context in which aggregate state values are being kept.
   This is useful for transition functions that wish to use <quote>expanded</quote>
   objects (see <xref linkend="xtypes-toast"/>) as their state values.
   On first call, the transition function should return an expanded object
   whose memory context is a child of the aggregate state context, and then
   keep returning the same expanded object on subsequent calls.  See
   <function>array_append()</function> for an example.  (<function>array_append()</function>
   is not the transition function of any built-in aggregate, but it is written
   to behave efficiently when used as transition function of a custom
   aggregate.)
  </para>

  <para>
   Another support routine available to aggregate functions written in C
   is <function>AggGetAggref</function>, which returns the <literal>Aggref</literal>
   parse node that defines the aggregate call.  This is mainly useful
   for ordered-set aggregates, which can inspect the substructure of
   the <literal>Aggref</literal> node to find out what sort ordering they are
   supposed to implement.  Examples can be found
   in <filename>orderedsetaggs.c</filename> in the <productname>PostgreSQL</productname>
   source code.
  </para>

 </sect2>

 </sect1>