<!-- doc/src/sgml/xindex.sgml -->

<sect1 id="xindex">
 <title>Interfacing Extensions to Indexes</title>

 <indexterm zone="xindex">
  <primary>index</primary>
  <secondary>for user-defined data type</secondary>
 </indexterm>

  <para>
   The procedures described thus far let you define new types, new
   functions, and new operators. However, we cannot yet define an
   index on a column of a new data type.  To do this, we must define an
   <firstterm>operator class</firstterm> for the new data type.  Later in this
   section, we will illustrate this concept in an example: a new
   operator class for the B-tree index method that stores and sorts
   complex numbers in ascending absolute value order.
  </para>

  <para>
   Operator classes can be grouped into <firstterm>operator families</firstterm>
   to show the relationships between semantically compatible classes.
   When only a single data type is involved, an operator class is sufficient,
   so we'll focus on that case first and then return to operator families.
  </para>

 <sect2 id="xindex-opclass">
  <title>Index Methods and Operator Classes</title>

  <para>
   Operator classes are associated with an index access method, such
   as <link linkend="btree">B-Tree</link>
   or <link linkend="gin">GIN</link>. Custom index access method may be
   defined with <xref linkend="sql-create-access-method"/>. See
   <xref linkend="indexam"/> for details.
  </para>

  <para>
   The routines for an index method do not directly know anything
   about the data types that the index method will operate on.
   Instead, an <firstterm>operator
   class</firstterm><indexterm><primary>operator class</primary></indexterm>
   identifies the set of operations that the index method needs to use
   to work with a particular data type.  Operator classes are so
   called because one thing they specify is the set of
   <literal>WHERE</literal>-clause operators that can be used with an index
   (i.e., can be converted into an index-scan qualification).  An
   operator class can also specify some <firstterm>support
   function</firstterm> that are needed by the internal operations of the
   index method, but do not directly correspond to any
   <literal>WHERE</literal>-clause operator that can be used with the index.
  </para>

  <para>
   It is possible to define multiple operator classes for the same
   data type and index method.  By doing this, multiple
   sets of indexing semantics can be defined for a single data type.
   For example, a B-tree index requires a sort ordering to be defined
   for each data type it works on.
   It might be useful for a complex-number data type
   to have one B-tree operator class that sorts the data by complex
   absolute value, another that sorts by real part, and so on.
   Typically, one of the operator classes will be deemed most commonly
   useful and will be marked as the default operator class for that
   data type and index method.
  </para>

  <para>
   The same operator class name
   can be used for several different index methods (for example, both B-tree
   and hash index methods have operator classes named
   <literal>int4_ops</literal>), but each such class is an independent
   entity and must be defined separately.
  </para>
 </sect2>

 <sect2 id="xindex-strategies">
  <title>Index Method Strategies</title>

  <para>
   The operators associated with an operator class are identified by
   <quote>strategy numbers</quote>, which serve to identify the semantics of
   each operator within the context of its operator class.
   For example, B-trees impose a strict ordering on keys, lesser to greater,
   and so operators like <quote>less than</quote> and <quote>greater than or equal
   to</quote> are interesting with respect to a B-tree.
   Because
   <productname>PostgreSQL</productname> allows the user to define operators,
   <productname>PostgreSQL</productname> cannot look at the name of an operator
   (e.g., <literal>&lt;</literal> or <literal>&gt;=</literal>) and tell what kind of
   comparison it is.  Instead, the index method defines a set of
   <quote>strategies</quote>, which can be thought of as generalized operators.
   Each operator class specifies which actual operator corresponds to each
   strategy for a particular data type and interpretation of the index
   semantics.
  </para>

  <para>
   The B-tree index method defines five strategies, shown in <xref
   linkend="xindex-btree-strat-table"/>.
  </para>

   <table tocentry="1" id="xindex-btree-strat-table">
    <title>B-Tree Strategies</title>
    <tgroup cols="2">
     <thead>
      <row>
       <entry>Operation</entry>
       <entry>Strategy Number</entry>
      </row>
     </thead>
     <tbody>
      <row>
       <entry>less than</entry>
       <entry>1</entry>
      </row>
      <row>
       <entry>less than or equal</entry>
       <entry>2</entry>
      </row>
      <row>
       <entry>equal</entry>
       <entry>3</entry>
      </row>
      <row>
       <entry>greater than or equal</entry>
       <entry>4</entry>
      </row>
      <row>
       <entry>greater than</entry>
       <entry>5</entry>
      </row>
     </tbody>
    </tgroup>
   </table>

  <para>
   Hash indexes support only equality comparisons, and so they use only one
   strategy, shown in <xref linkend="xindex-hash-strat-table"/>.
  </para>

   <table tocentry="1" id="xindex-hash-strat-table">
    <title>Hash Strategies</title>
    <tgroup cols="2">
     <thead>
      <row>
       <entry>Operation</entry>
       <entry>Strategy Number</entry>
      </row>
     </thead>
     <tbody>
      <row>
       <entry>equal</entry>
       <entry>1</entry>
      </row>
     </tbody>
    </tgroup>
   </table>

  <para>
   GiST indexes are more flexible: they do not have a fixed set of
   strategies at all.  Instead, the <quote>consistency</quote> support routine
   of each particular GiST operator class interprets the strategy numbers
   however it likes.  As an example, several of the built-in GiST index
   operator classes index two-dimensional geometric objects, providing
   the <quote>R-tree</quote> strategies shown in
   <xref linkend="xindex-rtree-strat-table"/>.  Four of these are true
   two-dimensional tests (overlaps, same, contains, contained by);
   four of them consider only the X direction; and the other four
   provide the same tests in the Y direction.
  </para>

   <table tocentry="1" id="xindex-rtree-strat-table">
    <title>GiST Two-Dimensional <quote>R-tree</quote> Strategies</title>
    <tgroup cols="2">
     <thead>
      <row>
       <entry>Operation</entry>
       <entry>Strategy Number</entry>
      </row>
     </thead>
     <tbody>
      <row>
       <entry>strictly left of</entry>
       <entry>1</entry>
      </row>
      <row>
       <entry>does not extend to right of</entry>
       <entry>2</entry>
      </row>
      <row>
       <entry>overlaps</entry>
       <entry>3</entry>
      </row>
      <row>
       <entry>does not extend to left of</entry>
       <entry>4</entry>
      </row>
      <row>
       <entry>strictly right of</entry>
       <entry>5</entry>
      </row>
      <row>
       <entry>same</entry>
       <entry>6</entry>
      </row>
      <row>
       <entry>contains</entry>
       <entry>7</entry>
      </row>
      <row>
       <entry>contained by</entry>
       <entry>8</entry>
      </row>
      <row>
       <entry>does not extend above</entry>
       <entry>9</entry>
      </row>
      <row>
       <entry>strictly below</entry>
       <entry>10</entry>
      </row>
      <row>
       <entry>strictly above</entry>
       <entry>11</entry>
      </row>
      <row>
       <entry>does not extend below</entry>
       <entry>12</entry>
      </row>
     </tbody>
    </tgroup>
   </table>

  <para>
   SP-GiST indexes are similar to GiST indexes in flexibility: they don't have
   a fixed set of strategies. Instead the support routines of each operator
   class interpret the strategy numbers according to the operator class's
   definition. As an example, the strategy numbers used by the built-in
   operator classes for points are shown in <xref
   linkend="xindex-spgist-point-strat-table"/>.
  </para>

   <table tocentry="1" id="xindex-spgist-point-strat-table">
    <title>SP-GiST Point Strategies</title>
    <tgroup cols="2">
     <thead>
      <row>
       <entry>Operation</entry>
       <entry>Strategy Number</entry>
      </row>
     </thead>
     <tbody>
      <row>
       <entry>strictly left of</entry>
       <entry>1</entry>
      </row>
      <row>
       <entry>strictly right of</entry>
       <entry>5</entry>
      </row>
      <row>
       <entry>same</entry>
       <entry>6</entry>
      </row>
      <row>
       <entry>contained by</entry>
       <entry>8</entry>
      </row>
      <row>
       <entry>strictly below</entry>
       <entry>10</entry>
      </row>
      <row>
       <entry>strictly above</entry>
       <entry>11</entry>
      </row>
     </tbody>
    </tgroup>
   </table>

  <para>
   GIN indexes are similar to GiST and SP-GiST indexes, in that they don't
   have a fixed set of strategies either. Instead the support routines of
   each operator class interpret the strategy numbers according to the
   operator class's definition. As an example, the strategy numbers used by
   the built-in operator class for arrays are shown in
   <xref linkend="xindex-gin-array-strat-table"/>.
  </para>

   <table tocentry="1" id="xindex-gin-array-strat-table">
    <title>GIN Array Strategies</title>
    <tgroup cols="2">
     <thead>
      <row>
       <entry>Operation</entry>
       <entry>Strategy Number</entry>
      </row>
     </thead>
     <tbody>
      <row>
       <entry>overlap</entry>
       <entry>1</entry>
      </row>
      <row>
       <entry>contains</entry>
       <entry>2</entry>
      </row>
      <row>
       <entry>is contained by</entry>
       <entry>3</entry>
      </row>
      <row>
       <entry>equal</entry>
       <entry>4</entry>
      </row>
     </tbody>
    </tgroup>
   </table>

  <para>
   BRIN indexes are similar to GiST, SP-GiST and GIN indexes in that they
   don't have a fixed set of strategies either.  Instead the support routines
   of each operator class interpret the strategy numbers according to the
   operator class's definition. As an example, the strategy numbers used by
   the built-in <literal>Minmax</literal> operator classes are shown in
   <xref linkend="xindex-brin-minmax-strat-table"/>.
  </para>

   <table tocentry="1" id="xindex-brin-minmax-strat-table">
    <title>BRIN Minmax Strategies</title>
    <tgroup cols="2">
     <thead>
      <row>
       <entry>Operation</entry>
       <entry>Strategy Number</entry>
      </row>
     </thead>
     <tbody>
      <row>
       <entry>less than</entry>
       <entry>1</entry>
      </row>
      <row>
       <entry>less than or equal</entry>
       <entry>2</entry>
      </row>
      <row>
       <entry>equal</entry>
       <entry>3</entry>
      </row>
      <row>
       <entry>greater than or equal</entry>
       <entry>4</entry>
      </row>
      <row>
       <entry>greater than</entry>
       <entry>5</entry>
      </row>
     </tbody>
    </tgroup>
   </table>

  <para>
   Notice that all the operators listed above return Boolean values.  In
   practice, all operators defined as index method search operators must
   return type <type>boolean</type>, since they must appear at the top
   level of a <literal>WHERE</literal> clause to be used with an index.
   (Some index access methods also support <firstterm>ordering operators</firstterm>,
   which typically don't return Boolean values; that feature is discussed
   in <xref linkend="xindex-ordering-ops"/>.)
  </para>
 </sect2>

 <sect2 id="xindex-support">
  <title>Index Method Support Routines</title>

  <para>
   Strategies aren't usually enough information for the system to figure
   out how to use an index.  In practice, the index methods require
   additional support routines in order to work. For example, the B-tree
   index method must be able to compare two keys and determine whether one
   is greater than, equal to, or less than the other.  Similarly, the
   hash index method must be able to compute hash codes for key values.
   These operations do not correspond to operators used in qualifications in
   SQL commands;  they are administrative routines used by
   the index methods, internally.
  </para>

  <para>
   Just as with strategies, the operator class identifies which specific
   functions should play each of these roles for a given data type and
   semantic interpretation.  The index method defines the set
   of functions it needs, and the operator class identifies the correct
   functions to use by assigning them to the <quote>support function numbers</quote>
   specified by the index method.
  </para>

  <para>
   Additionally, some opclasses allow users to specify parameters which
   control their behavior.  Each builtin index access method has an optional
   <function>options</function> support function, which defines a set of
   opclass-specific parameters.
  </para>

  <para>
   B-trees require a comparison support function,
   and allow four additional support functions to be
   supplied at the operator class author's option, as shown in <xref
   linkend="xindex-btree-support-table"/>.
   The requirements for these support functions are explained further in
   <xref linkend="btree-support-funcs"/>.
  </para>

   <table tocentry="1" id="xindex-btree-support-table">
    <title>B-Tree Support Functions</title>
    <tgroup cols="2">
     <colspec colname="col1" colwidth="3*"/>
     <colspec colname="col2" colwidth="1*"/>
     <thead>
      <row>
       <entry>Function</entry>
       <entry>Support Number</entry>
      </row>
     </thead>
     <tbody>
      <row>
       <entry>
        Compare two keys and return an integer less than zero, zero, or
        greater than zero, indicating whether the first key is less than,
        equal to, or greater than the second
       </entry>
       <entry>1</entry>
      </row>
      <row>
       <entry>
        Return the addresses of C-callable sort support function(s)
        (optional)
       </entry>
       <entry>2</entry>
      </row>
      <row>
       <entry>
        Compare a test value to a base value plus/minus an offset, and return
        true or false according to the comparison result (optional)
       </entry>
       <entry>3</entry>
      </row>
      <row>
       <entry>
        Determine if it is safe for indexes that use the operator
        class to apply the btree deduplication optimization (optional)
       </entry>
       <entry>4</entry>
      </row>
      <row>
       <entry>
        Define options that are specific to this operator class
        (optional)
       </entry>
       <entry>5</entry>
      </row>
      <row>
       <entry>
        Return the addresses of C-callable skip support function(s)
        (optional)
       </entry>
       <entry>6</entry>
      </row>
     </tbody>
    </tgroup>
   </table>

  <para>
   Hash indexes require one support function, and allow two additional ones to
   be supplied at the operator class author's option, as shown in <xref
   linkend="xindex-hash-support-table"/>.
  </para>

   <table tocentry="1" id="xindex-hash-support-table">
    <title>Hash Support Functions</title>
    <tgroup cols="2">
     <colspec colname="col1" colwidth="3*"/>
     <colspec colname="col2" colwidth="1*"/>
     <thead>
      <row>
       <entry>Function</entry>
       <entry>Support Number</entry>
      </row>
     </thead>
     <tbody>
      <row>
       <entry>Compute the 32-bit hash value for a key</entry>
       <entry>1</entry>
      </row>
      <row>
       <entry>
         Compute the 64-bit hash value for a key given a 64-bit salt; if
         the salt is 0, the low 32 bits of the result must match the value
         that would have been computed by function 1
         (optional)
       </entry>
       <entry>2</entry>
      </row>
      <row>
       <entry>
        Define options that are specific to this operator class
        (optional)
       </entry>
       <entry>3</entry>
      </row>
     </tbody>
    </tgroup>
   </table>

  <para>
   GiST indexes have twelve support functions, seven of which are optional,
   as shown in <xref linkend="xindex-gist-support-table"/>.
   (For more information see <xref linkend="gist"/>.)
  </para>

   <table tocentry="1" id="xindex-gist-support-table">
    <title>GiST Support Functions</title>
    <tgroup cols="3">
     <colspec colname="col1" colwidth="2*"/>
     <colspec colname="col2" colwidth="3*"/>
     <colspec colname="col3" colwidth="1*"/>
     <thead>
      <row>
       <entry>Function</entry>
       <entry>Description</entry>
       <entry>Support Number</entry>
      </row>
     </thead>
     <tbody>
      <row>
       <entry><function>consistent</function></entry>
       <entry>determine whether key satisfies the
        query qualifier</entry>
       <entry>1</entry>
      </row>
      <row>
       <entry><function>union</function></entry>
       <entry>compute union of a set of keys</entry>
       <entry>2</entry>
      </row>
      <row>
       <entry><function>compress</function></entry>
       <entry>compute a compressed representation of a key or value
        to be indexed (optional)</entry>
       <entry>3</entry>
      </row>
      <row>
       <entry><function>decompress</function></entry>
       <entry>compute a decompressed representation of a
        compressed key (optional)</entry>
       <entry>4</entry>
      </row>
      <row>
       <entry><function>penalty</function></entry>
       <entry>compute penalty for inserting new key into subtree
       with given subtree's key</entry>
       <entry>5</entry>
      </row>
      <row>
       <entry><function>picksplit</function></entry>
       <entry>determine which entries of a page are to be moved
       to the new page and compute the union keys for resulting pages</entry>
       <entry>6</entry>
      </row>
      <row>
       <entry><function>same</function></entry>
       <entry>compare two keys and return true if they are equal</entry>
       <entry>7</entry>
      </row>
      <row>
       <entry><function>distance</function></entry>
       <entry>determine distance from key to query value (optional)</entry>
       <entry>8</entry>
      </row>
      <row>
       <entry><function>fetch</function></entry>
       <entry>compute original representation of a compressed key for
       index-only scans (optional)</entry>
       <entry>9</entry>
      </row>
      <row>
       <entry><function>options</function></entry>
       <entry>define options that are specific to this operator class
        (optional)</entry>
       <entry>10</entry>
      </row>
      <row>
       <entry><function>sortsupport</function></entry>
       <entry>provide a sort comparator to be used in fast index builds
        (optional)</entry>
       <entry>11</entry>
      </row>
      <row>
       <entry><function>translate_cmptype</function></entry>
       <entry>translate compare types to strategy numbers
        used by the operator class (optional)</entry>
       <entry>12</entry>
      </row>
     </tbody>
    </tgroup>
   </table>

  <para>
   SP-GiST indexes have six support functions, one of which is optional, as
   shown in <xref linkend="xindex-spgist-support-table"/>.
   (For more information see <xref linkend="spgist"/>.)
  </para>

   <table tocentry="1" id="xindex-spgist-support-table">
    <title>SP-GiST Support Functions</title>
    <tgroup cols="3">
     <colspec colname="col1" colwidth="2*"/>
     <colspec colname="col2" colwidth="3*"/>
     <colspec colname="col3" colwidth="1*"/>
     <thead>
      <row>
       <entry>Function</entry>
       <entry>Description</entry>
       <entry>Support Number</entry>
      </row>
     </thead>
     <tbody>
      <row>
       <entry><function>config</function></entry>
       <entry>provide basic information about the operator class</entry>
       <entry>1</entry>
      </row>
      <row>
       <entry><function>choose</function></entry>
       <entry>determine how to insert a new value into an inner tuple</entry>
       <entry>2</entry>
      </row>
      <row>
       <entry><function>picksplit</function></entry>
       <entry>determine how to partition a set of values</entry>
       <entry>3</entry>
      </row>
      <row>
       <entry><function>inner_consistent</function></entry>
       <entry>determine which sub-partitions need to be searched for a
        query</entry>
       <entry>4</entry>
      </row>
      <row>
       <entry><function>leaf_consistent</function></entry>
       <entry>determine whether key satisfies the
        query qualifier</entry>
       <entry>5</entry>
      </row>
      <row>
       <entry><function>options</function></entry>
       <entry>define options that are specific to this operator class
        (optional)</entry>
       <entry>6</entry>
      </row>
     </tbody>
    </tgroup>
   </table>

  <para>
   GIN indexes have seven support functions, four of which are optional,
   as shown in <xref linkend="xindex-gin-support-table"/>.
   (For more information see <xref linkend="gin"/>.)
  </para>

   <table tocentry="1" id="xindex-gin-support-table">
    <title>GIN Support Functions</title>
    <tgroup cols="3">
     <colspec colname="col1" colwidth="2*"/>
     <colspec colname="col2" colwidth="3*"/>
     <colspec colname="col3" colwidth="1*"/>
     <thead>
      <row>
       <entry>Function</entry>
       <entry>Description</entry>
       <entry>Support Number</entry>
      </row>
     </thead>
     <tbody>
      <row>
       <entry><function>compare</function></entry>
       <entry>
        compare two keys and return an integer less than zero, zero,
        or greater than zero, indicating whether the first key is less than,
        equal to, or greater than the second
       </entry>
       <entry>1</entry>
      </row>
      <row>
       <entry><function>extractValue</function></entry>
       <entry>extract keys from a value to be indexed</entry>
       <entry>2</entry>
      </row>
      <row>
       <entry><function>extractQuery</function></entry>
       <entry>extract keys from a query condition</entry>
       <entry>3</entry>
      </row>
      <row>
       <entry><function>consistent</function></entry>
       <entry>
        determine whether value matches query condition (Boolean variant)
        (optional if support function 6 is present)
       </entry>
       <entry>4</entry>
      </row>
      <row>
       <entry><function>comparePartial</function></entry>
       <entry>
        compare partial key from
        query and key from index, and return an integer less than zero, zero,
        or greater than zero, indicating whether GIN should ignore this index
        entry, treat the entry as a match, or stop the index scan (optional)
       </entry>
       <entry>5</entry>
      </row>
      <row>
       <entry><function>triConsistent</function></entry>
       <entry>
        determine whether value matches query condition (ternary variant)
        (optional if support function 4 is present)
       </entry>
       <entry>6</entry>
      </row>
      <row>
       <entry><function>options</function></entry>
       <entry>
        define options that are specific to this operator class
        (optional)
       </entry>
       <entry>7</entry>
      </row>
     </tbody>
    </tgroup>
   </table>

  <para>
   BRIN indexes have five basic support functions, one of which is optional,
   as shown in <xref linkend="xindex-brin-support-table"/>.  Some versions of
   the basic functions require additional support functions to be provided.
   (For more information see <xref linkend="brin-extensibility"/>.)
  </para>

   <table tocentry="1" id="xindex-brin-support-table">
    <title>BRIN Support Functions</title>
    <tgroup cols="3">
     <colspec colname="col1" colwidth="2*"/>
     <colspec colname="col2" colwidth="3*"/>
     <colspec colname="col3" colwidth="1*"/>
     <thead>
      <row>
       <entry>Function</entry>
       <entry>Description</entry>
       <entry>Support Number</entry>
      </row>
     </thead>
     <tbody>
      <row>
       <entry><function>opcInfo</function></entry>
       <entry>
        return internal information describing the indexed columns'
        summary data
       </entry>
       <entry>1</entry>
      </row>
      <row>
       <entry><function>add_value</function></entry>
       <entry>add a new value to an existing summary index tuple</entry>
       <entry>2</entry>
      </row>
      <row>
       <entry><function>consistent</function></entry>
       <entry>determine whether value matches query condition</entry>
       <entry>3</entry>
      </row>
      <row>
       <entry><function>union</function></entry>
       <entry>
        compute union of two summary tuples
       </entry>
       <entry>4</entry>
      </row>
      <row>
       <entry><function>options</function></entry>
       <entry>
        define options that are specific to this operator class
        (optional)
       </entry>
       <entry>5</entry>
      </row>
     </tbody>
    </tgroup>
   </table>

  <para>
   Unlike search operators, support functions return whichever data
   type the particular index method expects; for example in the case
   of the comparison function for B-trees, a signed integer.  The number
   and types of the arguments to each support function are likewise
   dependent on the index method.  For B-tree and hash the comparison and
   hashing support functions take the same input data types as do the
   operators included in the operator class, but this is not the case for
   most GiST, SP-GiST, GIN, and BRIN support functions.
  </para>
 </sect2>

 <sect2 id="xindex-example">
  <title>An Example</title>

  <para>
   Now that we have seen the ideas, here is the promised example of
   creating a new operator class.
   (You can find a working copy of this example in
   <filename>src/tutorial/complex.c</filename> and
   <filename>src/tutorial/complex.sql</filename> in the source
   distribution.)
   The operator class encapsulates
   operators that sort complex numbers in absolute value order, so we
   choose the name <literal>complex_abs_ops</literal>.  First, we need
   a set of operators.  The procedure for defining operators was
   discussed in <xref linkend="xoper"/>.  For an operator class on
   B-trees, the operators we require are:

   <itemizedlist spacing="compact">
    <listitem><simpara>absolute-value less-than (strategy 1)</simpara></listitem>
    <listitem><simpara>absolute-value less-than-or-equal (strategy 2)</simpara></listitem>
    <listitem><simpara>absolute-value equal (strategy 3)</simpara></listitem>
    <listitem><simpara>absolute-value greater-than-or-equal (strategy 4)</simpara></listitem>
    <listitem><simpara>absolute-value greater-than (strategy 5)</simpara></listitem>
   </itemizedlist>
  </para>

  <para>
   The least error-prone way to define a related set of comparison operators
   is to write the B-tree comparison support function first, and then write the
   other functions as one-line wrappers around the support function.  This
   reduces the odds of getting inconsistent results for corner cases.
   Following this approach, we first write:

<programlisting><![CDATA[
#define Mag(c)  ((c)->x*(c)->x + (c)->y*(c)->y)

static int
complex_abs_cmp_internal(Complex *a, Complex *b)
{
    double      amag = Mag(a),
                bmag = Mag(b);

    if (amag < bmag)
        return -1;
    if (amag > bmag)
        return 1;
    return 0;
}
]]>
</programlisting>

   Now the less-than function looks like:

<programlisting><![CDATA[
PG_FUNCTION_INFO_V1(complex_abs_lt);

Datum
complex_abs_lt(PG_FUNCTION_ARGS)
{
    Complex    *a = (Complex *) PG_GETARG_POINTER(0);
    Complex    *b = (Complex *) PG_GETARG_POINTER(1);

    PG_RETURN_BOOL(complex_abs_cmp_internal(a, b) < 0);
}
]]>
</programlisting>

   The other four functions differ only in how they compare the internal
   function's result to zero.
  </para>

  <para>
   Next we declare the functions and the operators based on the functions
   to SQL:

<programlisting>
CREATE FUNCTION complex_abs_lt(complex, complex) RETURNS bool
    AS '<replaceable>filename</replaceable>', 'complex_abs_lt'
    LANGUAGE C IMMUTABLE STRICT;

CREATE OPERATOR &lt; (
   leftarg = complex, rightarg = complex, procedure = complex_abs_lt,
   commutator = &gt; , negator = &gt;= ,
   restrict = scalarltsel, join = scalarltjoinsel
);
</programlisting>
   It is important to specify the correct commutator and negator operators,
   as well as suitable restriction and join selectivity
   functions, otherwise the optimizer will be unable to make effective
   use of the index.
  </para>

  <para>
   Other things worth noting are happening here:

  <itemizedlist>
   <listitem>
    <para>
     There can only be one operator named, say, <literal>=</literal>
     and taking type <type>complex</type> for both operands.  In this
     case we don't have any other operator <literal>=</literal> for
     <type>complex</type>, but if we were building a practical data
     type we'd probably want <literal>=</literal> to be the ordinary
     equality operation for complex numbers (and not the equality of
     the absolute values).  In that case, we'd need to use some other
     operator name for <function>complex_abs_eq</function>.
    </para>
   </listitem>

   <listitem>
    <para>
     Although <productname>PostgreSQL</productname> can cope with
     functions having the same SQL name as long as they have different
     argument data types, C can only cope with one global function
     having a given name.  So we shouldn't name the C function
     something simple like <filename>abs_eq</filename>.  Usually it's
     a good practice to include the data type name in the C function
     name, so as not to conflict with functions for other data types.
    </para>
   </listitem>

   <listitem>
    <para>
     We could have made the SQL name
     of the function <filename>abs_eq</filename>, relying on
     <productname>PostgreSQL</productname> to distinguish it by
     argument data types from any other SQL function of the same name.
     To keep the example simple, we make the function have the same
     names at the C level and SQL level.
    </para>
   </listitem>
  </itemizedlist>
  </para>

  <para>
   The next step is the registration of the support routine required
   by B-trees.  The example C code that implements this is in the same
   file that contains the operator functions.  This is how we declare
   the function:

<programlisting>
CREATE FUNCTION complex_abs_cmp(complex, complex)
    RETURNS integer
    AS '<replaceable>filename</replaceable>'
    LANGUAGE C IMMUTABLE STRICT;
</programlisting>
  </para>

  <para>
   Now that we have the required operators and support routine,
   we can finally create the operator class:

<programlisting><![CDATA[
CREATE OPERATOR CLASS complex_abs_ops
    DEFAULT FOR TYPE complex USING btree AS
        OPERATOR        1       < ,
        OPERATOR        2       <= ,
        OPERATOR        3       = ,
        OPERATOR        4       >= ,
        OPERATOR        5       > ,
        FUNCTION        1       complex_abs_cmp(complex, complex);
]]>
</programlisting>
  </para>

  <para>
   And we're done!  It should now be possible to create
   and use B-tree indexes on <type>complex</type> columns.
  </para>

  <para>
   We could have written the operator entries more verbosely, as in:
<programlisting>
        OPERATOR        1       &lt; (complex, complex) ,
</programlisting>
   but there is no need to do so when the operators take the same data type
   we are defining the operator class for.
  </para>

  <para>
   The above example assumes that you want to make this new operator class the
   default B-tree operator class for the <type>complex</type> data type.
   If you don't, just leave out the word <literal>DEFAULT</literal>.
  </para>
 </sect2>

 <sect2 id="xindex-opfamily">
  <title>Operator Classes and Operator Families</title>

  <para>
   So far we have implicitly assumed that an operator class deals with
   only one data type.  While there certainly can be only one data type in
   a particular index column, it is often useful to index operations that
   compare an indexed column to a value of a different data type.  Also,
   if there is use for a cross-data-type operator in connection with an
   operator class, it is often the case that the other data type has a
   related operator class of its own.  It is helpful to make the connections
   between related classes explicit, because this can aid the planner in
   optimizing SQL queries (particularly for B-tree operator classes, since
   the planner contains a great deal of knowledge about how to work with them).
  </para>

  <para>
   To handle these needs, <productname>PostgreSQL</productname>
   uses the concept of an <firstterm>operator
   family</firstterm><indexterm><primary>operator family</primary></indexterm>.
   An operator family contains one or more operator classes, and can also
   contain indexable operators and corresponding support functions that
   belong to the family as a whole but not to any single class within the
   family.  We say that such operators and functions are <quote>loose</quote>
   within the family, as opposed to being bound into a specific class.
   Typically each operator class contains single-data-type operators
   while cross-data-type operators are loose in the family.
  </para>

  <para>
   All the operators and functions in an operator family must have compatible
   semantics, where the compatibility requirements are set by the index
   method.  You might therefore wonder why bother to single out particular
   subsets of the family as operator classes; and indeed for many purposes
   the class divisions are irrelevant and the family is the only interesting
   grouping.  The reason for defining operator classes is that they specify
   how much of the family is needed to support any particular index.
   If there is an index using an operator class, then that operator class
   cannot be dropped without dropping the index &mdash; but other parts of
   the operator family, namely other operator classes and loose operators,
   could be dropped.  Thus, an operator class should be specified to contain
   the minimum set of operators and functions that are reasonably needed
   to work with an index on a specific data type, and then related but
   non-essential operators can be added as loose members of the operator
   family.
  </para>

  <para>
   As an example, <productname>PostgreSQL</productname> has a built-in
   B-tree operator family <literal>integer_ops</literal>, which includes operator
   classes <literal>int8_ops</literal>, <literal>int4_ops</literal>, and
   <literal>int2_ops</literal> for indexes on <type>bigint</type> (<type>int8</type>),
   <type>integer</type> (<type>int4</type>), and <type>smallint</type> (<type>int2</type>)
   columns respectively.  The family also contains cross-data-type comparison
   operators allowing any two of these types to be compared, so that an index
   on one of these types can be searched using a comparison value of another
   type.  The family could be duplicated by these definitions:

<programlisting><![CDATA[
CREATE OPERATOR FAMILY integer_ops USING btree;

CREATE OPERATOR CLASS int8_ops
DEFAULT FOR TYPE int8 USING btree FAMILY integer_ops AS
  -- standard int8 comparisons
  OPERATOR 1 < ,
  OPERATOR 2 <= ,
  OPERATOR 3 = ,
  OPERATOR 4 >= ,
  OPERATOR 5 > ,
  FUNCTION 1 btint8cmp(int8, int8) ,
  FUNCTION 2 btint8sortsupport(internal) ,
  FUNCTION 3 in_range(int8, int8, int8, boolean, boolean) ,
  FUNCTION 4 btequalimage(oid) ,
  FUNCTION 6 btint8skipsupport(internal) ;

CREATE OPERATOR CLASS int4_ops
DEFAULT FOR TYPE int4 USING btree FAMILY integer_ops AS
  -- standard int4 comparisons
  OPERATOR 1 < ,
  OPERATOR 2 <= ,
  OPERATOR 3 = ,
  OPERATOR 4 >= ,
  OPERATOR 5 > ,
  FUNCTION 1 btint4cmp(int4, int4) ,
  FUNCTION 2 btint4sortsupport(internal) ,
  FUNCTION 3 in_range(int4, int4, int4, boolean, boolean) ,
  FUNCTION 4 btequalimage(oid) ,
  FUNCTION 6 btint4skipsupport(internal) ;

CREATE OPERATOR CLASS int2_ops
DEFAULT FOR TYPE int2 USING btree FAMILY integer_ops AS
  -- standard int2 comparisons
  OPERATOR 1 < ,
  OPERATOR 2 <= ,
  OPERATOR 3 = ,
  OPERATOR 4 >= ,
  OPERATOR 5 > ,
  FUNCTION 1 btint2cmp(int2, int2) ,
  FUNCTION 2 btint2sortsupport(internal) ,
  FUNCTION 3 in_range(int2, int2, int2, boolean, boolean) ,
  FUNCTION 4 btequalimage(oid) ,
  FUNCTION 6 btint2skipsupport(internal) ;

ALTER OPERATOR FAMILY integer_ops USING btree ADD
  -- cross-type comparisons int8 vs int2
  OPERATOR 1 < (int8, int2) ,
  OPERATOR 2 <= (int8, int2) ,
  OPERATOR 3 = (int8, int2) ,
  OPERATOR 4 >= (int8, int2) ,
  OPERATOR 5 > (int8, int2) ,
  FUNCTION 1 btint82cmp(int8, int2) ,

  -- cross-type comparisons int8 vs int4
  OPERATOR 1 < (int8, int4) ,
  OPERATOR 2 <= (int8, int4) ,
  OPERATOR 3 = (int8, int4) ,
  OPERATOR 4 >= (int8, int4) ,
  OPERATOR 5 > (int8, int4) ,
  FUNCTION 1 btint84cmp(int8, int4) ,

  -- cross-type comparisons int4 vs int2
  OPERATOR 1 < (int4, int2) ,
  OPERATOR 2 <= (int4, int2) ,
  OPERATOR 3 = (int4, int2) ,
  OPERATOR 4 >= (int4, int2) ,
  OPERATOR 5 > (int4, int2) ,
  FUNCTION 1 btint42cmp(int4, int2) ,

  -- cross-type comparisons int4 vs int8
  OPERATOR 1 < (int4, int8) ,
  OPERATOR 2 <= (int4, int8) ,
  OPERATOR 3 = (int4, int8) ,
  OPERATOR 4 >= (int4, int8) ,
  OPERATOR 5 > (int4, int8) ,
  FUNCTION 1 btint48cmp(int4, int8) ,

  -- cross-type comparisons int2 vs int8
  OPERATOR 1 < (int2, int8) ,
  OPERATOR 2 <= (int2, int8) ,
  OPERATOR 3 = (int2, int8) ,
  OPERATOR 4 >= (int2, int8) ,
  OPERATOR 5 > (int2, int8) ,
  FUNCTION 1 btint28cmp(int2, int8) ,

  -- cross-type comparisons int2 vs int4
  OPERATOR 1 < (int2, int4) ,
  OPERATOR 2 <= (int2, int4) ,
  OPERATOR 3 = (int2, int4) ,
  OPERATOR 4 >= (int2, int4) ,
  OPERATOR 5 > (int2, int4) ,
  FUNCTION 1 btint24cmp(int2, int4) ,

  -- cross-type in_range functions
  FUNCTION 3 in_range(int4, int4, int8, boolean, boolean) ,
  FUNCTION 3 in_range(int4, int4, int2, boolean, boolean) ,
  FUNCTION 3 in_range(int2, int2, int8, boolean, boolean) ,
  FUNCTION 3 in_range(int2, int2, int4, boolean, boolean) ;
]]>
</programlisting>

   Notice that this definition <quote>overloads</quote> the operator strategy and
   support function numbers: each number occurs multiple times within the
   family.  This is allowed so long as each instance of a
   particular number has distinct input data types.  The instances that have
   both input types equal to an operator class's input type are the
   primary operators and support functions for that operator class,
   and in most cases should be declared as part of the operator class rather
   than as loose members of the family.
  </para>

  <para>
   In a B-tree operator family, all the operators in the family must sort
   compatibly, as is specified in detail in <xref linkend="btree-behavior"/>.
   For each
   operator in the family there must be a support function having the same
   two input data types as the operator.  It is recommended that a family be
   complete, i.e., for each combination of data types, all operators are
   included.  Each operator class should include just the non-cross-type
   operators and support function for its data type.
  </para>

  <para>
   To build a multiple-data-type hash operator family, compatible hash
   support functions must be created for each data type supported by the
   family.  Here compatibility means that the functions are guaranteed to
   return the same hash code for any two values that are considered equal
   by the family's equality operators, even when the values are of different
   types.  This is usually difficult to accomplish when the types have
   different physical representations, but it can be done in some cases.
   Furthermore, casting a value from one data type represented in the operator
   family to another data type also represented in the operator family via
   an implicit or binary coercion cast must not change the computed hash value.
   Notice that there is only one support function per data type, not one
   per equality operator.  It is recommended that a family be complete, i.e.,
   provide an equality operator for each combination of data types.
   Each operator class should include just the non-cross-type equality
   operator and the support function for its data type.
  </para>

  <para>
   GiST, SP-GiST, and GIN indexes do not have any explicit notion of
   cross-data-type operations.  The set of operators supported is just
   whatever the primary support functions for a given operator class can
   handle.
  </para>

  <para>
   In BRIN, the requirements depends on the framework that provides the
   operator classes.  For operator classes based on <literal>minmax</literal>,
   the behavior required is the same as for B-tree operator families:
   all the operators in the family must sort compatibly, and casts must
   not change the associated sort ordering.
  </para>

  <note>
   <para>
    Prior to <productname>PostgreSQL</productname> 8.3, there was no concept
    of operator families, and so any cross-data-type operators intended to be
    used with an index had to be bound directly into the index's operator
    class.  While this approach still works, it is deprecated because it
    makes an index's dependencies too broad, and because the planner can
    handle cross-data-type comparisons more effectively when both data types
    have operators in the same operator family.
   </para>
  </note>
 </sect2>

 <sect2 id="xindex-opclass-dependencies">
  <title>System Dependencies on Operator Classes</title>

   <indexterm>
    <primary>ordering operator</primary>
   </indexterm>

  <para>
   <productname>PostgreSQL</productname> uses operator classes to infer the
   properties of operators in more ways than just whether they can be used
   with indexes.  Therefore, you might want to create operator classes
   even if you have no intention of indexing any columns of your data type.
  </para>

  <para>
   In particular, there are SQL features such as <literal>ORDER BY</literal> and
   <literal>DISTINCT</literal> that require comparison and sorting of values.
   To implement these features on a user-defined data type,
   <productname>PostgreSQL</productname> looks for the default B-tree operator
   class for the data type.  The <quote>equals</quote> member of this operator
   class defines the system's notion of equality of values for
   <literal>GROUP BY</literal> and <literal>DISTINCT</literal>, and the sort ordering
   imposed by the operator class defines the default <literal>ORDER BY</literal>
   ordering.
  </para>

  <para>
   If there is no default B-tree operator class for a data type, the system
   will look for a default hash operator class.  But since that kind of
   operator class only provides equality, it is only able to support grouping
   not sorting.
  </para>

  <para>
   When there is no default operator class for a data type, you will get
   errors like <quote>could not identify an ordering operator</quote> if you
   try to use these SQL features with the data type.
  </para>

   <note>
    <para>
     In <productname>PostgreSQL</productname> versions before 7.4,
     sorting and grouping operations would implicitly use operators named
     <literal>=</literal>, <literal>&lt;</literal>, and <literal>&gt;</literal>.  The new
     behavior of relying on default operator classes avoids having to make
     any assumption about the behavior of operators with particular names.
    </para>
   </note>

  <para>
   Sorting by a non-default B-tree operator class is possible by specifying
   the class's less-than operator in a <literal>USING</literal> option,
   for example
<programlisting>
SELECT * FROM mytable ORDER BY somecol USING ~&lt;~;
</programlisting>
   Alternatively, specifying the class's greater-than operator
   in <literal>USING</literal> selects a descending-order sort.
  </para>

  <para>
   Comparison of arrays of a user-defined type also relies on the semantics
   defined by the type's default B-tree operator class.  If there is no
   default B-tree operator class, but there is a default hash operator class,
   then array equality is supported, but not ordering comparisons.
  </para>

  <para>
   Another SQL feature that requires even more data-type-specific knowledge
   is the <literal>RANGE</literal> <replaceable>offset</replaceable>
   <literal>PRECEDING</literal>/<literal>FOLLOWING</literal> framing option
   for window functions (see <xref linkend="syntax-window-functions"/>).
   For a query such as
<programlisting>
SELECT sum(x) OVER (ORDER BY x RANGE BETWEEN 5 PRECEDING AND 10 FOLLOWING)
  FROM mytable;
</programlisting>
   it is not sufficient to know how to order by <literal>x</literal>;
   the database must also understand how to <quote>subtract 5</quote> or
   <quote>add 10</quote> to the current row's value of <literal>x</literal>
   to identify the bounds of the current window frame.  Comparing the
   resulting bounds to other rows' values of <literal>x</literal> is
   possible using the comparison operators provided by the B-tree operator
   class that defines the <literal>ORDER BY</literal> ordering &mdash; but
   addition and subtraction operators are not part of the operator class, so
   which ones should be used?  Hard-wiring that choice would be undesirable,
   because different sort orders (different B-tree operator classes) might
   need different behavior.  Therefore, a B-tree operator class can specify
   an <firstterm>in_range</firstterm> support function that encapsulates the
   addition and subtraction behaviors that make sense for its sort order.
   It can even provide more than one in_range support function, in case
   there is more than one data type that makes sense to use as the offset
   in <literal>RANGE</literal> clauses.
   If the B-tree operator class associated with the window's <literal>ORDER
   BY</literal> clause does not have a matching in_range support function,
   the <literal>RANGE</literal> <replaceable>offset</replaceable>
   <literal>PRECEDING</literal>/<literal>FOLLOWING</literal>
   option is not supported.
  </para>

  <para>
   Another important point is that an equality operator that
   appears in a hash operator family is a candidate for hash joins,
   hash aggregation, and related optimizations.  The hash operator family
   is essential here since it identifies the hash function(s) to use.
  </para>
 </sect2>

 <sect2 id="xindex-ordering-ops">
  <title>Ordering Operators</title>

  <para>
   Some index access methods (currently, only GiST and SP-GiST) support the concept of
   <firstterm>ordering operators</firstterm>.  What we have been discussing so far
   are <firstterm>search operators</firstterm>.  A search operator is one for which
   the index can be searched to find all rows satisfying
   <literal>WHERE</literal>
   <replaceable>indexed_column</replaceable>
   <replaceable>operator</replaceable>
   <replaceable>constant</replaceable>.
   Note that nothing is promised about the order in which the matching rows
   will be returned.  In contrast, an ordering operator does not restrict the
   set of rows that can be returned, but instead determines their order.
   An ordering operator is one for which the index can be scanned to return
   rows in the order represented by
   <literal>ORDER BY</literal>
   <replaceable>indexed_column</replaceable>
   <replaceable>operator</replaceable>
   <replaceable>constant</replaceable>.
   The reason for defining ordering operators that way is that it supports
   nearest-neighbor searches, if the operator is one that measures distance.
   For example, a query like
<programlisting><![CDATA[
SELECT * FROM places ORDER BY location <-> point '(101,456)' LIMIT 10;
]]>
</programlisting>
   finds the ten places closest to a given target point.  A GiST index
   on the location column can do this efficiently because
   <literal>&lt;-&gt;</literal> is an ordering operator.
  </para>

  <para>
   While search operators have to return Boolean results, ordering operators
   usually return some other type, such as float or numeric for distances.
   This type is normally not the same as the data type being indexed.
   To avoid hard-wiring assumptions about the behavior of different data
   types, the definition of an ordering operator is required to name
   a B-tree operator family that specifies the sort ordering of the result
   data type.  As was stated in the previous section, B-tree operator families
   define <productname>PostgreSQL</productname>'s notion of ordering, so
   this is a natural representation.  Since the point <literal>&lt;-&gt;</literal>
   operator returns <type>float8</type>, it could be specified in an operator
   class creation command like this:
<programlisting><![CDATA[
OPERATOR 15    <-> (point, point) FOR ORDER BY float_ops
]]>
</programlisting>
   where <literal>float_ops</literal> is the built-in operator family that includes
   operations on <type>float8</type>.  This declaration states that the index
   is able to return rows in order of increasing values of the
   <literal>&lt;-&gt;</literal> operator.
  </para>
 </sect2>

 <sect2 id="xindex-opclass-features">
  <title>Special Features of Operator Classes</title>

  <para>
   There are two special features of operator classes that we have
   not discussed yet, mainly because they are not useful
   with the most commonly used index methods.
  </para>

  <para>
   Normally, declaring an operator as a member of an operator class
   (or family) means that the index method can retrieve exactly the set of rows
   that satisfy a <literal>WHERE</literal> condition using the operator.  For example:
<programlisting>
SELECT * FROM table WHERE integer_column &lt; 4;
</programlisting>
   can be satisfied exactly by a B-tree index on the integer column.
   But there are cases where an index is useful as an inexact guide to
   the matching rows.  For example, if a GiST index stores only bounding boxes
   for geometric objects, then it cannot exactly satisfy a <literal>WHERE</literal>
   condition that tests overlap between nonrectangular objects such as
   polygons.  Yet we could use the index to find objects whose bounding
   box overlaps the bounding box of the target object, and then do the
   exact overlap test only on the objects found by the index.  If this
   scenario applies, the index is said to be <quote>lossy</quote> for the
   operator.  Lossy index searches are implemented by having the index
   method return a <firstterm>recheck</firstterm> flag when a row might or might
   not really satisfy the query condition.  The core system will then
   test the original query condition on the retrieved row to see whether
   it should be returned as a valid match.  This approach works if
   the index is guaranteed to return all the required rows, plus perhaps
   some additional rows, which can be eliminated by performing the original
   operator invocation.  The index methods that support lossy searches
   (currently, GiST, SP-GiST and GIN) allow the support functions of individual
   operator classes to set the recheck flag, and so this is essentially an
   operator-class feature.
  </para>

  <para>
   Consider again the situation where we are storing in the index only
   the bounding box of a complex object such as a polygon.  In this
   case there's not much value in storing the whole polygon in the index
   entry &mdash; we might as well store just a simpler object of type
   <type>box</type>.  This situation is expressed by the <literal>STORAGE</literal>
   option in <command>CREATE OPERATOR CLASS</command>: we'd write something like:

<programlisting>
CREATE OPERATOR CLASS polygon_ops
    DEFAULT FOR TYPE polygon USING gist AS
        ...
        STORAGE box;
</programlisting>

   At present, only the GiST, SP-GiST, GIN and BRIN index methods support a
   <literal>STORAGE</literal> type that's different from the column data type.
   The GiST <function>compress</function> and <function>decompress</function> support
   routines must deal with data-type conversion when <literal>STORAGE</literal>
   is used.  SP-GiST likewise requires a <function>compress</function>
   support function to convert to the storage type, when that is different;
   if an SP-GiST opclass also supports retrieving data, the reverse
   conversion must be handled by the <function>consistent</function> function.
   In GIN, the <literal>STORAGE</literal> type identifies the type of
   the <quote>key</quote> values, which normally is different from the type
   of the indexed column &mdash; for example, an operator class for
   integer-array columns might have keys that are just integers.  The
   GIN <function>extractValue</function> and <function>extractQuery</function> support
   routines are responsible for extracting keys from indexed values.
   BRIN is similar to GIN: the <literal>STORAGE</literal> type identifies the
   type of the stored summary values, and operator classes' support
   procedures are responsible for interpreting the summary values
   correctly.
  </para>
 </sect2>

</sect1>
