Aggregate functions can optionally support moving-aggregate mode, which allows substantially faster execution of aggregate functions within windows with moving frame starting points. (See Section 3.5 and Section 4.2.8 for information about use of aggregate functions as window functions.) The basic idea is that in addition to a normal “forward” transition function, the aggregate provides an inverse transition function, which allows rows to be removed from the aggregate's running state value when they exit the window frame. For example a sum 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.

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.

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

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

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

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 “punt” 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 NaN (not a number) input has to be removed from the running state value.

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 sum over float4 or float8 inputs. A naive declaration of sum(float8) could be

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

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

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);

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

Aggregate functions can use polymorphic state transition functions or final functions, so that the same functions can be used to implement multiple aggregates. See Section 40.2.5 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:

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

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 array_agg provides similar functionality, with better performance than this definition would have.)

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

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

   attrelid    |        array_accum
---------------+---------------------------
 pg_tablespace | {name,oid,aclitem[],text[]}
(1 row)

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 CREATE FUNCTION 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 internal because there is no SQL-level equivalent for it. To address this case, it is possible to declare the final function as taking extra “dummy” 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 array_agg is equivalent to

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
);

Here, the finalfunc_extra 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 anynonarray argument allows the declaration of array_agg_finalfn to be valid.

An aggregate function can be made to accept a varying number of arguments by declaring its last argument as a VARIADIC array, in much the same fashion as for regular functions; see Section 40.5.6. 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 VARIADIC, but this is not strictly required.

SELECT myaggregate(a ORDER BY a, b, c) FROM ...

SELECT myaggregate(a, b, c ORDER BY a) FROM ...

The aggregates we have been describing so far are “normal” aggregates. PostgreSQL also supports ordered-set aggregates, 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 “direct” 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 percentile_disc is equivalent to:

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
);

This aggregate takes a float8 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:

SELECT percentile_disc(0.5) WITHIN GROUP (ORDER BY income) FROM households;
 percentile_disc
-----------------
           50489

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

Unlike the case for normal aggregates, the sorting of input rows for an ordered-set aggregate is not 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 “tuplesort” 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 “hypothetical” rows into the data to be sorted. While normal aggregates can often be implemented with support functions written in PL/pgSQL 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 internal — 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 READ_ONLY; it must be declared in CREATE AGGREGATE as READ_WRITE, or as SHAREABLE if it's possible for additional final-function calls to make use of the already-sorted state.

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 finalfunc_extra is specified) null values corresponding to the aggregated input(s). As with normal aggregates, finalfunc_extra 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).

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

Optionally, an aggregate function can support partial aggregation. 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.)

To support partial aggregation, the aggregate definition must provide a combine function, 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.

As simple examples, MAX and MIN 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. SUM 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.)

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 initcond, 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.

If the aggregate's state type is declared as internal, 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 NULL it's invalid to simply return the second input, as that value will be in the wrong context and will not have sufficient lifespan.

When the aggregate's state type is declared as internal, it is usually also appropriate for the aggregate definition to provide a serialization function and a deserialization function, 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.

A serialization function must take a single argument of type internal and return a result of type bytea, 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 bytea and internal, and return a result of type internal. (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.

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

A function written in C can detect that it is being called as an aggregate support function by calling AggCheckCallContext, for example:

if (AggCheckCallContext(fcinfo, NULL))

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 int8inc() 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 CREATE AGGREGATE for more detail.)

The second argument of AggCheckCallContext 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 “expanded” objects (see Section 40.13.1) 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 array_append() for an example. (array_append() 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.)

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