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 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.)

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}' );

`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 38.2.5 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, 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 38.5.5. 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.

Variadic aggregates are easily misused in connection with
the `ORDER BY`

option (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 of `ORDER BY`

is a sort key, not an argument to
the aggregate. For example, in

SELECT 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 `myaggregate`

is 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.)
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 38.12.1)
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.