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<tcl>hd_keywords {window functions} {window function}</tcl>

<title>Window Functions</title>
<table_of_contents>

<h1>Introduction to Window Functions</h1>

<p>A window function is an SQL function where the input
values are taken from
a "window" of one or more rows in the results set of a SELECT statement.

<p>Window functions are distinguished from other SQL functions by the
presence of an OVER clause. If a function has an OVER clause,
then it is a window function. If it lacks an OVER clause, then it is an
ordinary aggregate or scalar function. Window functions might also
have a FILTER clause in between the function and the OVER clause.

<p>The syntax for a window function is like this:


<tcl>
RecursiveBubbleDiagram window-function-invocation window-defn frame-spec filter
</tcl>


<p>Unlike ordinary functions, window functions
cannot use the DISTINCT keyword.
Also, Window functions may only appear in the result set and in the
ORDER BY clause of a SELECT statement.

<p>Window functions come in two varieties: 
[aggregate window functions] and
[built-in window functions].  Every aggregate window function
can also work as a ordinary aggregate function, simply by omitting
the OVER and FILTER clauses.  Furthermore, all of the built-in
[aggfunc|aggregate functions] of SQLite can be used as an
aggregate window function by adding an appropriate OVER clause.
Applications can register new aggregate window functions using
the [sqlite3_create_window_function()] interface.
The built-in window functions, however, require special-case
handling in the query planner and hence new window functions
that exhibit the exceptional properties found in the built-in
window functions cannot be added by the application.

<p>Here is an example using the built-in row_number()
window function:

<codeblock>
  CREATE TABLE t0(x INTEGER PRIMARY KEY, y TEXT);
  INSERT INTO t0 VALUES (1, 'aaa'), (2, 'ccc'), (3, 'bbb');

  <i>-- The following SELECT statement returns:</i>
  <i>-- </i>
  <i>--   x | y | row_number</i>
  -----------------------
  <i>--   1 | aaa | 1         </i>
  <i>--   2 | ccc | 3         </i>
  <i>--   3 | bbb | 2         </i>
  <i>-- </i>
  SELECT x, y, row_number() OVER (ORDER BY y) AS row_number FROM t0 ORDER BY x;
</codeblock>

<p>
The row_number() window function
assigns consecutive integers to each
row in order of the "ORDER BY" clause within the
<yynonterm>window-defn</yynonterm> (in this case "ORDER BY y"). Note that 
this does not affect the order in which results are returned from
the overall query.  The order of the final output is
still governed by the ORDER BY clause attached to the SELECT
statement (in this case "ORDER BY x").

<p>Named <yynonterm>window-defn</yynonterm> clauses may also be added to a SELECT 
statement using a WINDOW clause and then referred to by name within window
function invocations. For example, the following SELECT statement contains
two named <yynonterm>window-defs</yynonterm> clauses, "win1" and "win2":

<codeblock>
  SELECT x, y, row_number() OVER <b>win1</b>, rank() OVER <b>win2</b> 
  FROM t0 
  WINDOW <b>win1</b> AS (ORDER BY y RANGE BETWEEN UNBOUNDED PRECEDING AND CURRENT ROW),
         <b>win2</b> AS (PARTITION BY y ORDER BY x)
  ORDER BY x;
</codeblock>

<p>The WINDOW clause, when one is present, comes after any HAVING clause and
before any ORDER BY.

<tcl>hd_fragment aggwinfunc {aggregate window functions}</tcl>
<h1>Aggregate Window Functions</h1>

<p> The examples in this section all assume that the database is populated as
follows:

<codeblock>
  CREATE TABLE t1(a INTEGER PRIMARY KEY, b, c);
  INSERT INTO t1 VALUES   (1, 'A', 'one'  ),
                          (2, 'B', 'two'  ),
                          (3, 'C', 'three'),
                          (4, 'D', 'one'  ),
                          (5, 'E', 'two'  ),
                          (6, 'F', 'three'),
                          (7, 'G', 'one'  );
</codeblock>

<p> An aggregate window function is similar to an 
[aggfunc|ordinary aggregate function], except
adding it to a query does not change the number of rows returned. Instead,
for each row the result of the aggregate window function is as if the
corresponding aggregate were run over all rows in the "window frame"
specified by the OVER clause.


<codeblock>
  <i>-- The following SELECT statement returns:</i>
  <i>-- </i>
  <i>--   a | b | group_concat</i>
  -------------------------
  <i>--   1 | A | A.B         </i>
  <i>--   2 | B | A.B.C       </i>
  <i>--   3 | C | B.C.D       </i>
  <i>--   4 | D | C.D.E       </i>
  <i>--   5 | E | D.E.F       </i>
  <i>--   6 | F | E.F.G       </i>
  <i>--   7 | G | F.G         </i>
  <i>-- </i>
  SELECT a, b, group_concat(b, '.') OVER (
    ORDER BY a ROWS BETWEEN 1 PRECEDING AND 1 FOLLOWING
  ) AS group_concat FROM t1;
</codeblock>

<p> In the example above, the window frame consists of all rows between the
previous row ("1 PRECEDING") and the following row ("1 FOLLOWING"), inclusive,
where rows are sorted according to the ORDER BY clause in the
<yynonterm>window-defn</yynonterm> (in this case "ORDER BY a"). 
For example, the frame for the row with (a=3) consists of rows (2, 'B', 'two'),
(3, 'C', 'three') and (4, 'D', 'one'). The result of group_concat(b, '.') 
for that row is therefore 'B.C.D'.

<p> All of SQLite's [Aggregate Functions|aggregate functions] may
be used as aggregate window functions. It is also possible to 
[user-defined window functions|create user-defined aggregate window functions].


<tcl>hd_fragment ptxn {partition}</tcl>
<h2>The PARTITION BY Clause</h2>

<p> For the purpose of computing window functions, the result set
of a query is divided into one or more "partitions".  A partition consists
of all rows that have the same value for all terms of the PARTITION BY clause
in the <yynonterm>window-defn</yynonterm>.  If there is no PARTITION BY clause,
then the entire result set of the query is a single partition.
Window-function processing is performed separately for each partition.

<p> For example:

<codeblock>
  <i>-- The following SELECT statement returns:</i>
  <i>-- </i>
  <i>--   c     | a | b | group_concat</i>
  ---------------------------------
  <i>--   one   | 1 | A | A.D.G       </i>
  <i>--   one   | 4 | D | D.G         </i>
  <i>--   one   | 7 | G | G           </i>
  <i>--   three | 3 | C | C.F         </i>
  <i>--   three | 6 | F | F           </i>
  <i>--   two   | 2 | B | B.E         </i>
  <i>--   two   | 5 | E | E           </i>
  <i>-- </i>
  SELECT c, a, b, group_concat(b, '.') OVER (
    PARTITION BY c ORDER BY a RANGE BETWEEN CURRENT ROW AND UNBOUNDED FOLLOWING
  ) AS group_concat
  FROM t1 ORDER BY c, a;
</codeblock>

<p> In the query above, the "PARTITION BY c" clause breaks the
result set up into three partitions. The first partition has
three rows with c=='one'.  The second partition has two rows with
c=='three' and the third partition has two rows with c=='two'.

<p> In the example above, all the rows for each partition are
grouped together in the final output.  This is because the PARTITION BY
clause is a prefix of the ORDER BY clause on the overall query.  
But that does not have
to be the case.  A partition can be composed of rows scattered
about haphazardly within the result set.  For example:

<codeblock>
  <i>-- The following SELECT statement returns:</i>
  <i>-- </i>
  <i>--   c     | a | b | group_concat</i>
  ---------------------------------
  <i>--   one   | 1 | A | A.D.G       </i>
  <i>--   two   | 2 | B | B.E         </i>
  <i>--   three | 3 | C | C.F         </i>
  <i>--   one   | 4 | D | D.G         </i>
  <i>--   two   | 5 | E | E           </i>
  <i>--   three | 6 | F | F           </i>
  <i>--   one   | 7 | G | G           </i>
  <i>-- </i>
  SELECT c, a, b, group_concat(b, '.') OVER (
    PARTITION BY c ORDER BY a RANGE BETWEEN CURRENT ROW AND UNBOUNDED FOLLOWING
  ) AS group_concat
  FROM t1 ORDER BY a;
</codeblock>


<tcl>hd_fragment framespec {frame specification} {frames}</tcl>
<h2>Frame Specifications</h2>

<p> The <yynonterm>frame-spec</yynonterm> determines which output rows are
read by an aggregate window function.  The
<yynonterm>frame-spec</yynonterm> consists of four parts:

<ul>
  <li> A frame type - either ROWS, RANGE or GROUPS,
  <li> A starting frame boundary,
  <li> An ending frame boundary,
  <li> An EXCLUDE clause.
</ul>

<p> Here are the syntax details:

<tcl>
RecursiveBubbleDiagram frame-spec
</tcl>


<p>The ending frame boundary can be omitted (if the
BETWEEN and AND keywords that surround the starting frame boundary
are also omitted),
in which case the ending frame boundary defaults to CURRENT ROW.

<p> If the frame type is RANGE or GROUPS, then rows with the same values for
all ORDER BY expressions are considered "peers". Or, if there are no ORDER BY
terms, all rows are peers. Peers are always within the same frame.

<p>The default <yynonterm>frame-spec</yynonterm> is:

<codeblock>
    RANGE BETWEEN UNBOUNDED PRECEDING AND CURRENT ROW EXCLUDE NO OTHERS
</codeblock>

<p>The default means that aggregate window functions read all
rows from the beginning of the partition up to and including the
current row and its peers. This implies that rows that have the same values for
all ORDER BY expressions will also have the same value for the result of the
window function (as the window frame is the same). For example:

<codeblock>
  <i>-- The following SELECT statement returns:</i>
  <i>-- </i>
  <i>--   a | b | c | group_concat</i>
  -----------------------------
  <i>--   1 | A | one   | A.D.G       </i>
  <i>--   2 | B | two   | A.D.G.C.F.B.E</i>
  <i>--   3 | C | three | A.D.G.C.F   </i>
  <i>--   4 | D | one   | A.D.G       </i>
  <i>--   5 | E | two   | A.D.G.C.F.B.E</i>
  <i>--   6 | F | three | A.D.G.C.F   </i>
  <i>--   7 | G | one   | A.D.G       </i>
  <i>-- </i>
  SELECT a, b, c, 
         group_concat(b, '.') OVER (ORDER BY c) AS group_concat 
  FROM t1 ORDER BY a;
</codeblock>

<tcl>hd_fragment frametype {frame type}</tcl>
<h3>Frame Type</h3>

<p>
There are three frame types:  ROWS, GROUPS, and RANGE.
The frame type determines how the starting and ending boundaries
of the frame are measured.

<ul>
<li><p><b>ROWS</b>: 
The ROWS frame type means that the starting and ending boundaries
for the frame are determined by counting individual rows relative
to the current row.

<tcl>hd_fragment grouptype {GROUPS frames}</tcl>
<li><p><b>GROUPS</b>:
The GROUPS frame type means that the starting and ending boundaries
are determine by counting "groups" relative to the current group.
A "group" is a set of rows that all have equivalent values for all
all terms of the window ORDER BY clause.  ("Equivalent" means that
the [IS operator] is true when comparing the two values.)
In other words, a group consists of all peers of a row.

<tcl>hd_fragment rangetype {RANGE frames}</tcl>
<li><p><b>RANGE</b>:
The RANGE frame type requires that the ORDER BY clause of the
window have exactly one term.  Call that term "X".  With the
RANGE frame type, the elements of the frame are determined by
computing the value of expression X for all rows in the partition
and framing those rows for which the value of X is within a certain
range of the value of X for the current row.  See the description
in the "[RANGE n PRECEDING|&lt;expr&gt; PRECEDING]" boundary
specification below for details.
</ul>

<p>The ROWS and GROUPS frame types are similar in that they
both determine the extent of a frame by counting relative to
the current row.  The difference is that ROWS counts individual
rows and GROUPS counts peer groups.
The RANGE frame type is different.
The RANGE frame type determines the extent of a frame by
looking for expression values that are within some band of
values relative to the current row.

<tcl>hd_fragment frameboundary {frame boundary}</tcl>
<h3>Frame Boundaries</h3>

<p> There are five ways to describe starting and ending frame boundaries:

<ol>
<li><p> <b>UNBOUNDED PRECEDING</b><br>
     The frame boundary is the first
     row in the [partition].

<li><p> <b>&lt;expr&gt; PRECEDING</b><br>
        &lt;expr&gt; must be a non-negative constant numeric expression.
        The boundary is a row that is &lt;expr&gt; "units" prior to
        the current row.  The meaning of "units" here depends on the
        frame type:
        <ul>
        <li><p> <b>ROWS &rarr;</b>
        The frame boundary is the row that is &lt;expr&gt;
        rows before the current row, or the first row of the
        partition if there are fewer than &lt;expr&gt; rows
        before the current row.  &lt;expr&gt; must be an integer.
        <li><p> <b>GROUPS &rarr;</b>
        A "group" is a set of peer rows - rows that all have
        the same values for every term in the ORDER BY clause.
        The frame boundary is the group that is &lt;expr&gt;
        groups before the group containing the current row, or the
        first group of the partition if there are fewer
        than &lt;expr&gt; groups before the current row.
        For the starting boundary of a frame, the first
        row of the group is used and for the ending boundary
        of a frame, the last row of the group is used.
        &lt;expr&gt; must be an integer.
        <tcl>hd_fragment exprrange {RANGE n PRECEDING} {RANGE n FOLLOWING}</tcl>
        <li><p> <b>RANGE &rarr;</b>
        For this form, the ORDER BY clause of the
        <yynonterm>window-defn</yynonterm> must have a single
        term.  Call that ORDER BY term "X".  Let
        X<sub><small>i</small></sub> be the value of the X
        expression for the i-th row in the partition and let
        X<sub><small>c</small></sub> be the value of X for the
        current row.  Informally, a RANGE bound is the first row
        for which X<sub><small>i</small></sub> is within
        the &lt;expr&gt; of X<sub><small>c</small></sub>.
        More precisely:
        <ol type="a">
        <li> If either X<sub><small>i</small></sub> or
             X<sub><small>c</small></sub> are non-numeric, then
             the boundary is the first row for which the expression
             "X<sub><small>i</small></sub> IS X<sub><small>c</small></sub>"
             is true.
        <li> Else if the ORDER BY is ASC then the boundary
             is the first row for which
             X<sub><small>i</small></sub>&gt;=X<sub><small>c</small></sub>-&lt;expr&gt;.
        <li> Else if the ORDER BY is DESC then the boundary
        is the first row for which
        X<sub><small>i</small></sub>&lt;=X<sub><small>c</small></sub>-&lt;expr&gt;.
        </ol>
        For this form, the &lt;expr&gt; does not have to be an
        integer. It can evaluate to a real number as long as
        it is constant and non-negative.
        </ul>
    The boundary description "0 PRECEDING" always means the same
    thing as "CURRENT ROW".

<li><p><b>CURRENT ROW</b><br>
        The current row. For RANGE and GROUPS frame types,
        peers of the current row are also included in the frame,
        unless specifically excluded by the EXCLUDE clause.
        This is true regardless of whether CURRENT ROW is used
        as the starting or ending frame boundary.

<li><p><b>&lt;expr&gt; FOLLOWING</b><br>
        This is the same as "&lt;expr&gt; PRECEDING" except that
        the boundary is &lt;expr&gt; units after the current
        rather than before the current row.

<li><p> <b>UNBOUNDED FOLLOWING</b><br>
   The frame boundary is the last
   row in the [partition].

</ol>

<p> The ending frame boundary may not take a form that appears higher in
the above list than the starting frame boundary.

<p> In the following example, the window frame for each row consists of all
rows from the current row to the end of the set, where rows are sorted
according to "ORDER BY a".

<codeblock>
  <i>-- The following SELECT statement returns:</i>
  <i>-- </i>
  <i>--   c     | a | b | group_concat</i>
  ---------------------------------
  <i>--   one   | 1 | A | A.D.G.C.F.B.E</i>
  <i>--   one   | 4 | D | D.G.C.F.B.E </i>
  <i>--   one   | 7 | G | G.C.F.B.E   </i>
  <i>--   three | 3 | C | C.F.B.E     </i>
  <i>--   three | 6 | F | F.B.E       </i>
  <i>--   two   | 2 | B | B.E         </i>
  <i>--   two   | 5 | E | E           </i>
  <i>-- </i>
  SELECT c, a, b, group_concat(b, '.') OVER (
    ORDER BY c, a ROWS BETWEEN CURRENT ROW AND UNBOUNDED FOLLOWING
  ) AS group_concat
  FROM t1 ORDER BY c, a;
</codeblock>

<tcl>hd_fragment wexcls {exclude clause} {EXCLUDE clause}</tcl> 
<h3>The EXCLUDE Clause</h3>

<p> The optional EXCLUDE clause may take any of the following four forms:

<ul>
  <li> <p><b> EXCLUDE NO OTHERS</b>: This is the default. In this case no
  rows are excluded from the window frame as defined by its starting and ending
  frame boundaries.

  <li> <p><b> EXCLUDE CURRENT ROW</b>: In this case the current row is 
  excluded from the window frame.  Peers of the current row remain in
  the frame for the GROUPS and RANGE frame types.

  <li> <p><b> EXCLUDE GROUP</b>: In this case the current row and all other
  rows that are peers of the current row are excluded from the frame. When
  processing an EXCLUDE clause, all rows with the same ORDER BY values, or all
  rows in the partition if there is no ORDER BY clause, are considered peers,
  even if the frame type is ROWS.

  <li> <p><b> EXCLUDE TIES</b>: In this case the current row is part of the
  frame, but peers of the current row are excluded.
</ul>

<p> The following example demonstrates the effect of the various
forms of the EXCLUDE clause:

<codeblock>
  <i>-- The following SELECT statement returns:</i>
  <i>-- </i>
  <i>--   c    | a | b | no_others     | current_row | grp       | ties</i>
  <i>--  one   | 1 | A | A.D.G         | D.G         |           | A</i>
  <i>--  one   | 4 | D | A.D.G         | A.G         |           | D</i>
  <i>--  one   | 7 | G | A.D.G         | A.D         |           | G</i>
  <i>--  three | 3 | C | A.D.G.C.F     | A.D.G.F     | A.D.G     | A.D.G.C</i>
  <i>--  three | 6 | F | A.D.G.C.F     | A.D.G.C     | A.D.G     | A.D.G.F</i>
  <i>--  two   | 2 | B | A.D.G.C.F.B.E | A.D.G.C.F.E | A.D.G.C.F | A.D.G.C.F.B</i>
  <i>--  two   | 5 | E | A.D.G.C.F.B.E | A.D.G.C.F.B | A.D.G.C.F | A.D.G.C.F.E</i>
  <i>-- </i>
  SELECT c, a, b,
    group_concat(b, '.') OVER (
      ORDER BY c GROUPS BETWEEN UNBOUNDED PRECEDING AND CURRENT ROW EXCLUDE NO OTHERS
    ) AS no_others,
    group_concat(b, '.') OVER (
      ORDER BY c GROUPS BETWEEN UNBOUNDED PRECEDING AND CURRENT ROW EXCLUDE CURRENT ROW
    ) AS current_row,
    group_concat(b, '.') OVER (
      ORDER BY c GROUPS BETWEEN UNBOUNDED PRECEDING AND CURRENT ROW EXCLUDE GROUP
    ) AS grp,
    group_concat(b, '.') OVER (
      ORDER BY c GROUPS BETWEEN UNBOUNDED PRECEDING AND CURRENT ROW EXCLUDE TIES
    ) AS ties
  FROM t1 ORDER BY c, a;
</codeblock>

<h2>The FILTER Clause</h2>

<tcl>
RecursiveBubbleDiagram filter
</tcl>

<p>If a FILTER clause is provided, then only rows for which the <i>expr</i> is
true are included in the window frame. The aggregate window still returns a
value for every row, but those for which the FILTER expression evaluates to
other than true are not included in the window frame for any row. For example:

<codeblock>
  <i>-- The following SELECT statement returns:</i>
  <i>-- </i>
  <i>--   c     | a | b | group_concat</i>
  ---------------------------------
  <i>--   one   | 1 | A | A           </i>
  <i>--   two   | 2 | B | A           </i>
  <i>--   three | 3 | C | A.C         </i>
  <i>--   one   | 4 | D | A.C.D       </i>
  <i>--   two   | 5 | E | A.C.D       </i>
  <i>--   three | 6 | F | A.C.D.F     </i>
  <i>--   one   | 7 | G | A.C.D.F.G   </i>
  <i>-- </i>
  SELECT c, a, b, group_concat(b, '.') FILTER (WHERE c!='two') OVER (
    ORDER BY a
  ) AS group_concat
  FROM t1 ORDER BY a;
</codeblock>


<tcl>hd_fragment builtins {built-in window functions} {built-ins}</tcl>
<h1>Built-in Window Functions</h1>

<p> As well as aggregate window functions, SQLite features a set of built-in
window functions based on 
<a href=https://www.postgresql.org/docs/10/static/functions-window.html>
those supported by PostgreSQL</a>.

<p> Built-in window functions honor any PARTITION BY clause in the same way
as aggregate window functions - each selected row is assigned to a partition 
and each partition is processed separately. The ways in which any ORDER BY
clause affects each built-in window function is described below. Some of
the window functions (rank(), dense_rank(), percent_rank() and ntile()) use
the concept of "peer groups" (rows within the same partition that have the
same values for all ORDER BY expressions). In these cases, it does not matter
whether the <yynonterm>frame-spec</yynonterm> specifies ROWS, GROUPS, or RANGE.
For the purposes of built-in window function processing, rows with the same values
for all ORDER BY expressions are considered peers regardless of the frame type.

<p> Most built-in window functions ignore the
<yynonterm>frame-spec</yynonterm>, the exceptions being first_value(),
last_value() and nth_value(). It is a syntax error to specify a FILTER 
clause as part of a built-in window function invocation.

<tcl>hd_fragment biwinfunc {builtin window functions}</tcl>
<p> SQLite supports the following 11 built-in window functions:

<dl>
  <dt><p><b>row_number()</b>
  <dd><p> The number of the row within the current partition. Rows are
       numbered starting from 1 in the order defined by the ORDER BY clause in
       the window definition, or in arbitrary order otherwise.
  <dt><p><b>rank()</b>
  <dd><p> The row_number() of the first peer in each group - the rank of the
       current row with gaps. If there is no ORDER BY clause, then all rows
       are considered peers and this function always returns 1.
  <dt><p><b>dense_rank()</b>
  <dd><p> The number of the current row's peer group within its partition - the
       rank of the current row without gaps. Partitions are numbered starting
       from 1 in the order defined by the ORDER BY clause in the window
       definition. If there is no ORDER BY clause, then all rows are
       considered peers and this function always returns 1.
  <dt><p><b>percent_rank()</b>
  <dd><p> Despite the name, this function always returns a value between 0.0
       and 1.0 equal to (<i>rank</i> - 1)/(<i>partition-rows</i> - 1), where
       <i>rank</i> is the value returned by built-in window function rank()
       and <i>partition-rows</i> is the total number of rows in the
       partition. If the partition contains only one row, this function
       returns 0.0.
  <dt><p><b>cume_dist()</b>
  <dd><p> The cumulative distribution. Calculated as
       <i>row-number</i>/<i>partition-rows</i>, where <i>row-number</i> is
       the value returned by row_number() for the last peer in the group
       and <i>partition-rows</i> the number of rows in the partition.
  <dt><p><b>ntile(N)</b>
  <dd><p> Argument <i>N</i> is handled as an integer. This function divides the
       partition into N groups as evenly as possible and assigns an integer
       between 1 and <i>N</i> to each group, in the order defined by the ORDER
       BY clause, or in arbitrary order otherwise. If necessary, larger groups
       occur first. This function returns the integer value assigned to the
       group that the current row is a part of.

  <dt><p><b>lag(expr)<br>lag(expr, offset)<br>lag(expr, offset, default)</b>
  <dd><p> The first form of the lag() function returns the result of evaluating
       expression <i>expr</i> against the previous row in the partition. Or, if
       there is no previous row (because the current row is the first), NULL.

      <p> If the <i>offset</i> argument is provided, then it must be a
       non-negative integer. In this case the value returned is the result
       of evaluating <i>expr</i> against the row <i>offset</i> rows before the
       current row within the partition. If <i>offset</i> is 0, then
       <i>expr</i> is evaluated against the current row. If there is no row
       <i>offset</i> rows before the current row, NULL is returned.

      <p> If <i>default</i> is also provided, then it is returned instead of
      NULL if the row identified by <i>offset</i> does not exist.
       
  <dt><p><b>lead(expr)<br>lead(expr, offset)<br>lead(expr, offset, default)</b>
  <dd><p> The first form of the lead() function returns the result of evaluating
       expression <i>expr</i> against the next row in the partition. Or, if
       there is no next row (because the current row is the last), NULL.

      <p> If the <i>offset</i> argument is provided, then it must be a
       non-negative integer. In this case the value returned is the result
       of evaluating <i>expr</i> against the row <i>offset</i> rows after the
       current row within the partition. If <i>offset</i> is 0, then
       <i>expr</i> is evaluated against the current row. If there is no row
       <i>offset</i> rows after the current row, NULL is returned.

      <p> If <i>default</i> is also provided, then it is returned instead of
      NULL if the row identified by <i>offset</i> does not exist.
  <dt><p><b>first_value(expr)</b>
  <dd><p> This built-in window function calculates the window frame for each
       row in the same way as an aggregate window function. It returns the
       value of <i>expr</i> evaluated against the first row in the window frame
       for each row.
  <dt><p><b>last_value(expr)</b>
  <dd><p> This built-in window function calculates the window frame for each
       row in the same way as an aggregate window function. It returns the
       value of <i>expr</i> evaluated against the last row in the window frame
       for each row.
  <dt><p><b>nth_value(expr, N)</b>
  <dd><p> This built-in window function calculates the window frame for each
       row in the same way as an aggregate window function. It returns the
       value of <i>expr</i> evaluated against the row <i>N</i> of the window
       frame. Rows are numbered within the window frame starting from 1 in
       the order defined by the ORDER BY clause if one is present, or in
       arbitrary order otherwise. If there is no <i>N</i>th row in the
       partition, then NULL is returned.
  <dd>
</dl>

<p>The examples in this section all assume the following data:

<codeblock>
  CREATE TABLE t2(a, b);
  INSERT INTO t2 VALUES('a', 'one'), 
                       ('a', 'two'), 
                       ('a', 'three'), 
                       ('b', 'four'), 
                       ('c', 'five'), 
                       ('c', 'six');
</codeblock>

<p>The following example illustrates the behaviour of the five ranking
functions - row_number(), rank(), dense_rank(), percent_rank() and 
cume_dist().

<codeblock>
  <i>-- The following SELECT statement returns:</i>
  <i>-- </i>
  <i>--   a | row_number | rank | dense_rank | percent_rank | cume_dist</i>
  ------------------------------------------------------------------
  <i>--   a |          1 |    1 |          1 |          0.0 |       0.5</i>
  <i>--   a |          2 |    1 |          1 |          0.0 |       0.5</i>
  <i>--   a |          3 |    1 |          1 |          0.0 |       0.5</i>
  <i>--   b |          4 |    4 |          2 |          0.6 |       0.66</i>
  <i>--   c |          5 |    5 |          3 |          0.8 |       1.0</i>
  <i>--   c |          6 |    5 |          3 |          0.8 |       1.0</i>
  <i>-- </i>
  SELECT a                        AS a,
         row_number() OVER win    AS row_number,
         rank() OVER win          AS rank,
         dense_rank() OVER win    AS dense_rank,
         percent_rank() OVER win  AS percent_rank,
         cume_dist() OVER win     AS cume_dist
  FROM t2
  WINDOW win AS (ORDER BY a);
</codeblock>

<p>The example below uses ntile() to divide the six rows into two groups (the
ntile(2) call) and into four groups (the ntile(4) call). For ntile(2), there
are three rows assigned to each group. For ntile(4), there are two groups of
two and two groups of one. The larger groups of two appear first.

<codeblock>
  <i>-- The following SELECT statement returns:</i>
  <i>-- </i>
  <i>--   a | b     | ntile_2 | ntile_4</i>
  ----------------------------------
  <i>--   a | one   |       1 |       1</i>
  <i>--   a | two   |       1 |       1</i>
  <i>--   a | three |       1 |       2</i>
  <i>--   b | four  |       2 |       2</i>
  <i>--   c | five  |       2 |       3</i>
  <i>--   c | six   |       2 |       4</i>
  <i>-- </i>
  SELECT a                        AS a,
         b                        AS b,
         ntile(2) OVER win        AS ntile_2,
         ntile(4) OVER win        AS ntile_4
  FROM t2
  WINDOW win AS (ORDER BY a);
</codeblock>

<p> The next example demonstrates lag(), lead(), first_value(), last_value()
and nth_value(). The <yynonterm>frame-spec</yynonterm> is ignored by 
both lag() and lead(), but respected by first_value(), last_value()
and nth_value().

<codeblock>
  <i>-- The following SELECT statement returns:</i>
  <i>-- </i>
  <i>--   b | lead | lag  | first_value | last_value | nth_value_3</i>
  -------------------------------------------------------------
  <i>--   A | C    | NULL | A           | A          | NULL       </i>
  <i>--   B | D    | A    | A           | B          | NULL       </i>
  <i>--   C | E    | B    | A           | C          | C          </i>
  <i>--   D | F    | C    | A           | D          | C          </i>
  <i>--   E | G    | D    | A           | E          | C          </i>
  <i>--   F | n/a  | E    | A           | F          | C          </i>
  <i>--   G | n/a  | F    | A           | G          | C          </i>
  <i>-- </i>
  SELECT b                          AS b,
         lead(b, 2, 'n/a') OVER win AS lead,
         lag(b) OVER win            AS lag,
         first_value(b) OVER win    AS first_value,
         last_value(b) OVER win     AS last_value,
         nth_value(b, 3) OVER win   AS nth_value_3
  FROM t1
  WINDOW win AS (ORDER BY b ROWS BETWEEN UNBOUNDED PRECEDING AND CURRENT ROW)
</codeblock>


<tcl>hd_fragment wchaining {window chaining}</tcl>
<h1>Window Chaining</h1>

<p>
Window chaining is a shorthand that allows one window to be defined in terms 
of another. Specifically, the shorthand allows the new window to implicitly
copy the PARTITION BY and optionally ORDER BY clauses of the base window. For
example, in the following:

<codeblock>
  SELECT group_concat(b, '.') OVER (
    win ROWS BETWEEN UNBOUNDED PRECEDING AND CURRENT ROW
  )
  FROM t1
  WINDOW win AS (PARTITION BY a ORDER BY c)
</codeblock>

<p>
the window used by the group_concat() function is equivalent
to "PARTITION BY a ORDER BY c ROWS BETWEEN UNBOUNDED PRECEDING
AND CURRENT ROW". In order to use window
chaining, all of the following must be true:

<ul>
  <li><p>The new window definition must not include a PARTITION BY clause. The
       PARTITION BY clause, if there is one, must be supplied by the base
       window specification.

  <li><p>If the base window has an ORDER BY clause, it is copied into the new
       window. In this case the new window must not specify an ORDER BY clause.
       If the base window has no ORDER BY clause, one may be specified as part
       of the new window definition.
  
  <li><p>The base window may not specify a frame specification.  The frame
      specification can only be given in the new window specification.
</ul>

<p>The two fragments of SQL below are similar, but not entirely equivalent, as
the latter will fail if the definition of window "win" contains a frame
specification.

<codeblock>
  SELECT group_concat(b, '.') OVER win ...
  SELECT group_concat(b, '.') OVER (win) ...
</codeblock>

<tcl>hd_fragment udfwinfunc {user-defined window functions} \
 {application-defined window functions}</tcl>
<h1>User-Defined Aggregate Window Functions</h1>

<p> User-defined aggregate window functions may be created using the
[sqlite3_create_window_function]() API. Implementing an aggregate window
function is very similar to an ordinary aggregate function. Any user-defined
aggregate window function may also be used as an ordinary aggregate. To 
implement a user-defined aggregate window function the application must
supply four callback functions:

<table striped=1>
<tr><th>Callback <th>Description
<tr><td>xStep <td>
      This method is required by both window aggregate and legacy aggregate
      function implementations. It is invoked to add a row to the current
      window. The function arguments, if any, corresponding to the row being
      added are passed to the implementation of xStep.

<tr><td>xFinal <td>
      This method is required by both window aggregate and legacy aggregate
      function implementations. It is invoked to return the current value
      of the aggregate (determined by the contents of the current window),
      and to free any resources allocated by earlier calls to xStep.

<tr><td>xValue <td>
      This method is only required window aggregate functions, not legacy
      aggregate function implementations. It is invoked to return the current
      value of the aggregate. Unlike xFinal, the implementation should not
      delete any context.

<tr><td>xInverse <td>
      This method is only required window aggregate functions, not legacy
      aggregate function implementations. It is invoked to remove a row
      from the current window. The function arguments, if any, correspond
      to the row being removed.
</table>

<p> The C code below implements a simple window aggregate function named
sumint(). This works in the same way as the built-in sum() function, except
that it throws an exception if passed an argument that is not an integer
value.

<codeblock>
<tcl>

proc C_Code {text} {
  hd_puts "<pre>\n"
  set iLine 0
  foreach zLine [split [string trim $text "\n"] "\n"] {
    regsub {^  } $zLine {} zLine 
    set zSubspec {<i>&</i>}
    regsub {(/\*|\*\*|\*/).*} $zLine $zSubspec zLine 
    if {[regexp {(.*)(sqlite3[a-z0-9_]*)(.*)} $zLine -> one two three]} {
      hd_puts $one
      hd_resolve "\[$two\]"
      hd_puts $three
    } else {
      hd_puts $zLine
    }
    hd_puts "\n"
  }
  hd_puts "</pre>\n"
}

C_Code {
  /*
  ** xStep for sumint().
  **
  ** Add the value of the argument to the aggregate context (an integer).
  */
  static void sumintStep(
    sqlite3_context *ctx, 
    int nArg, 
    sqlite3_value *apArg[]
  ){
    sqlite3_int64 *pInt;
  
    assert( nArg==1 );
    if( sqlite3_value_type(apArg[0])!=SQLITE_INTEGER ){
      sqlite3_result_error(ctx, "invalid argument", -1);
      return;
    }
    pInt = (sqlite3_int64*)sqlite3_aggregate_context(ctx, sizeof(sqlite3_int64));
    if( pInt ){
      *pInt += sqlite3_value_int64(apArg[0]);
    }
  }
  
  /*
  ** xInverse for sumint().
  **
  ** This does the opposite of xStep() - subtracts the value of the argument
  ** from the current context value. The error checking can be omitted from
  ** this function, as it is only ever called after xStep() (so the aggregate
  ** context has already been allocated) and with a value that has already
  ** been passed to xStep() without error (so it must be an integer).
  */
  static void sumintInverse(
    sqlite3_context *ctx, 
    int nArg, 
    sqlite3_value *apArg[]
  ){
    sqlite3_int64 *pInt;
    assert( sqlite3_value_type(apArg[0])==SQLITE_INTEGER );
    pInt = (sqlite3_int64*)sqlite3_aggregate_context(ctx, sizeof(sqlite3_int64));
    *pInt -= sqlite3_value_int64(apArg[0]);
  }
  
  /*
  ** xFinal for sumint().
  **
  ** Return the current value of the aggregate window function. Because
  ** this implementation does not allocate any resources beyond the buffer
  ** returned by sqlite3_aggregate_context, which is automatically freed
  ** by the system, there are no resources to free. And so this method is
  ** identical to xValue().
  */
  static void sumintFinal(sqlite3_context *ctx){
    sqlite3_int64 res = 0;
    sqlite3_int64 *pInt;
    pInt = (sqlite3_int64*)sqlite3_aggregate_context(ctx, 0);
    if( pInt ) res = *pInt;
    sqlite3_result_int64(ctx, res);
  }
  
  /*
  ** xValue for sumint().
  **
  ** Return the current value of the aggregate window function. Because
  */
  static void sumintValue(sqlite3_context *ctx){
    sqlite3_int64 res = 0;
    sqlite3_int64 *pInt;
    pInt = (sqlite3_int64*)sqlite3_aggregate_context(ctx, 0);
    if( pInt ) res = *pInt;
    sqlite3_result_int64(ctx, res);
  }

  /*
  ** Register sumint() window aggregate with database handle db. 
  */
  int register_sumint(sqlite3 *db){
    return sqlite3_create_window_function(db, "sumint", 1, SQLITE_UTF8, 0,
        sumintStep, sumintFinal, sumintValue, sumintInverse, 0
    );
  }
}
</tcl>
</codeblock>

<p> The following example uses the sumint() function implemented by the above
C code. For each row, the window consists of the preceding row (if any), the current row and the following row (again, if any):

<codeblock>
  CREATE TABLE t3(x, y);
  INSERT INTO t3 VALUES('a', 4),
                       ('b', 5),
                       ('c', 3),
                       ('d', 8),
                       ('e', 1);

  <i>-- Assuming the database is populated using the above script, the </i>
  <i>-- following SELECT statement returns:</i>
  <i>-- </i>
  <i>--   x | sum_y</i>
  --------------
  <i>--   a | 9    </i>
  <i>--   b | 12   </i>
  <i>--   c | 16   </i>
  <i>--   d | 12   </i>
  <i>--   e | 9    </i>
  <i>-- </i>
  SELECT x, sumint(y) OVER (
    ORDER BY x ROWS BETWEEN 1 PRECEDING AND 1 FOLLOWING
  ) AS sum_y
  FROM t3 ORDER BY x;
</codeblock>

<p>In processing the query above, SQLite invokes the sumint callbacks as
follows:

<p>
<ol>
  <li> <b>xStep(4)</b> - add "4" to the current window.
  <li> <b>xStep(5)</b> - add "5" to the current window.
  <li> <b>xValue()</b> - invoke xValue() to obtain the value of sumint() for
       the row with (x='a'). The window currently consists of values 4 and 5, 
       and so the result is 9.
  <li> <b>xStep(3)</b> - add "3" to the current window.
  <li> <b>xValue()</b> - invoke xValue() to obtain the value of sumint() for
       the row with (x='b'). The window currently consists of values 4, 5 and
       3, and so the result is 12.
  <li> <b>xInverse(4)</b> - remove "4" from the window. 
  <li> <b>xStep(8)</b> - add "8" to the current window. The window now consists
       of values 5, 3 and 8.
  <li> <b>xValue()</b> - invoked to obtain the value for the row with (x='c').
       In this case, 16.
  <li> <b>xInverse(5)</b> - remove value "5" from the window.
  <li> <b>xStep(1)</b> - add value "1" to the window.
  <li> <b>xValue()</b> - invoked to obtain the value for row (x='d').
  <li> <b>xInverse(3)</b> - remove value "3" from the window. The window now
       contains values 8 and 1 only.
  <li> <b>xValue()</b> - invoked to obtain the value for row (x='d'). 9.
</ol>

<h1>History</h1>

<p>Window function support was first added to SQLite with release
[version 3.25.0] ([dateof:3.25.0]). The SQLite developers used
the <a href=http://www.postgresql.org>PostgreSQL</a> window function
documentation as their primary reference for how window functions
ought to behave.  Many test cases have been run against PostgreSQL
to ensure that window functions operate the same way in both
SQLite and PostgreSQL.

<p>In SQLite [version 3.28.0] ([dateof:3.28.0]),
windows function support was extended to include the EXCLUDE clause,
GROUPS frame types, window chaining, and support for
"&lt;expr&gt; PRECEDING" and "&lt;expr&gt; FOLLOWING" boundaries
in RANGE frames.