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Aggregate functions in PostgreSQL are defined in terms of *state values* and *state
transition functions*. 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 *final function* 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 last 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.)

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.

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. `sum`

is
an example of this kind of aggregate. `sum`

starts at zero and always adds the current
row's value to its running total. For example, if we want to make
a `sum`

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:

CREATE AGGREGATE sum (complex) ( sfunc = complex_add, stype = complex, initcond = '(0,0)' );

which we might use like this:

SELECT sum(a) FROM test_complex; sum ----------- (34,53.9)

(Notice that we are relying on function overloading: there is
more than one aggregate named `sum`

,
but PostgreSQL can figure out
which kind of sum applies to a column of type `complex`.)

The above definition of `sum`

will
return zero (the initial state value) if there are no nonnull
input values. Perhaps we want to return null in that case instead
— the SQL standard expects `sum`

to
behave that way. We can do this simply by omitting the `initcond` phrase, so that the initial state value
is null. Ordinarily this would mean that the `sfunc` would need to check for a null state-value
input. But for `sum`

and some other
simple aggregates like `max`

and
`min`

, 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. PostgreSQL will do that
automatically if the initial state value is null and the
transition function is marked "strict"
(i.e., not to be called for null inputs).

Another bit of default behavior for a "strict" 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.

`avg`

(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 `avg(float8)`

looks like:

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

Note:`float8_accum`

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`avg`

.

Aggregate function calls in SQL allow `DISTINCT` and `ORDER BY`
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.

For further details see the CREATE AGGREGATE command.

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(`

could be`float8`)

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 35.2.5 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 (anyelement) ( sfunc = array_append, stype = anyarray, 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, anyelement) RETURNS internal ...; CREATE FUNCTION array_agg_finalfn(internal, anyelement) RETURNS anyarray ...; CREATE AGGREGATE array_agg (anyelement) ( 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 `anyelement` 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
35.4.5. 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.

Note:Variadic aggregates are easily misused in connection with theORDER BYoption (see Section 4.2.7), 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 ofORDER BYis a sort key, not an argument to the aggregate. For example, inSELECT myaggregate(a ORDER BY a, b, c) FROM ...the parser will see this as a single aggregate function argument and three sort keys. However, the user might have intended

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

myaggregateis variadic, both these calls could be perfectly valid.For the same reason, it's wise to think twice before creating aggregate functions with the same names and different numbers of regular arguments.

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.)

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.

A function written in C can detect that it is being called
as an aggregate transition or final function by calling
`AggCheckCallContext`

, for
example:

if (AggCheckCallContext(fcinfo, NULL))

One reason for checking this is that when it is true for a
transition function, 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. (This is the only case where it is safe for a
function to modify a pass-by-reference input. In particular,
final functions for normal aggregates must not modify their
inputs in any case, because in some cases they will be
re-executed on the same final state value.)

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.