474 lines
18 KiB
Plaintext
474 lines
18 KiB
Plaintext
<!-- doc/src/sgml/btree.sgml -->
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<chapter id="btree">
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<title>B-Tree Indexes</title>
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<indexterm>
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<primary>index</primary>
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<secondary>B-Tree</secondary>
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</indexterm>
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<sect1 id="btree-intro">
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<title>Introduction</title>
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<para>
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<productname>PostgreSQL</productname> includes an implementation of the
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standard <acronym>btree</acronym> (multi-way balanced tree) index data
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structure. Any data type that can be sorted into a well-defined linear
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order can be indexed by a btree index. The only limitation is that an
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index entry cannot exceed approximately one-third of a page (after TOAST
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compression, if applicable).
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</para>
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<para>
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Because each btree operator class imposes a sort order on its data type,
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btree operator classes (or, really, operator families) have come to be
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used as <productname>PostgreSQL</productname>'s general representation
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and understanding of sorting semantics. Therefore, they've acquired
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some features that go beyond what would be needed just to support btree
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indexes, and parts of the system that are quite distant from the
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btree AM make use of them.
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</para>
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</sect1>
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<sect1 id="btree-behavior">
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<title>Behavior of B-Tree Operator Classes</title>
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<para>
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As shown in <xref linkend="xindex-btree-strat-table"/>, a btree operator
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class must provide five comparison operators,
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<literal><</literal>,
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<literal><=</literal>,
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<literal>=</literal>,
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<literal>>=</literal> and
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<literal>></literal>.
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One might expect that <literal><></literal> should also be part of
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the operator class, but it is not, because it would almost never be
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useful to use a <literal><></literal> WHERE clause in an index
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search. (For some purposes, the planner treats <literal><></literal>
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as associated with a btree operator class; but it finds that operator via
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the <literal>=</literal> operator's negator link, rather than
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from <structname>pg_amop</structname>.)
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</para>
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<para>
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When several data types share near-identical sorting semantics, their
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operator classes can be grouped into an operator family. Doing so is
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advantageous because it allows the planner to make deductions about
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cross-type comparisons. Each operator class within the family should
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contain the single-type operators (and associated support functions)
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for its input data type, while cross-type comparison operators and
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support functions are <quote>loose</quote> in the family. It is
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recommendable that a complete set of cross-type operators be included
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in the family, thus ensuring that the planner can represent any
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comparison conditions that it deduces from transitivity.
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</para>
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<para>
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There are some basic assumptions that a btree operator family must
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satisfy:
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</para>
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<itemizedlist>
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<listitem>
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<para>
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An <literal>=</literal> operator must be an equivalence relation; that
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is, for all non-null values <replaceable>A</replaceable>,
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<replaceable>B</replaceable>, <replaceable>C</replaceable> of the
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data type:
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<itemizedlist>
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<listitem>
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<para>
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<replaceable>A</replaceable> <literal>=</literal>
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<replaceable>A</replaceable> is true
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(<firstterm>reflexive law</firstterm>)
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</para>
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</listitem>
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<listitem>
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<para>
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if <replaceable>A</replaceable> <literal>=</literal>
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<replaceable>B</replaceable>,
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then <replaceable>B</replaceable> <literal>=</literal>
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<replaceable>A</replaceable>
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(<firstterm>symmetric law</firstterm>)
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</para>
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</listitem>
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<listitem>
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<para>
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if <replaceable>A</replaceable> <literal>=</literal>
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<replaceable>B</replaceable> and <replaceable>B</replaceable>
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<literal>=</literal> <replaceable>C</replaceable>,
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then <replaceable>A</replaceable> <literal>=</literal>
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<replaceable>C</replaceable>
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(<firstterm>transitive law</firstterm>)
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</para>
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</listitem>
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</itemizedlist>
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</para>
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</listitem>
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<listitem>
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<para>
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A <literal><</literal> operator must be a strong ordering relation;
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that is, for all non-null values <replaceable>A</replaceable>,
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<replaceable>B</replaceable>, <replaceable>C</replaceable>:
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<itemizedlist>
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<listitem>
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<para>
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<replaceable>A</replaceable> <literal><</literal>
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<replaceable>A</replaceable> is false
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(<firstterm>irreflexive law</firstterm>)
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</para>
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</listitem>
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<listitem>
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<para>
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if <replaceable>A</replaceable> <literal><</literal>
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<replaceable>B</replaceable>
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and <replaceable>B</replaceable> <literal><</literal>
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<replaceable>C</replaceable>,
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then <replaceable>A</replaceable> <literal><</literal>
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<replaceable>C</replaceable>
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(<firstterm>transitive law</firstterm>)
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</para>
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</listitem>
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</itemizedlist>
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</para>
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</listitem>
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<listitem>
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<para>
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Furthermore, the ordering is total; that is, for all non-null
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values <replaceable>A</replaceable>, <replaceable>B</replaceable>:
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<itemizedlist>
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<listitem>
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<para>
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exactly one of <replaceable>A</replaceable> <literal><</literal>
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<replaceable>B</replaceable>, <replaceable>A</replaceable>
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<literal>=</literal> <replaceable>B</replaceable>, and
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<replaceable>B</replaceable> <literal><</literal>
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<replaceable>A</replaceable> is true
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(<firstterm>trichotomy law</firstterm>)
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</para>
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</listitem>
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</itemizedlist>
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(The trichotomy law justifies the definition of the comparison support
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function, of course.)
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</para>
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</listitem>
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</itemizedlist>
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<para>
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The other three operators are defined in terms of <literal>=</literal>
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and <literal><</literal> in the obvious way, and must act consistently
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with them.
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</para>
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<para>
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For an operator family supporting multiple data types, the above laws must
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hold when <replaceable>A</replaceable>, <replaceable>B</replaceable>,
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<replaceable>C</replaceable> are taken from any data types in the family.
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The transitive laws are the trickiest to ensure, as in cross-type
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situations they represent statements that the behaviors of two or three
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different operators are consistent.
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As an example, it would not work to put <type>float8</type>
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and <type>numeric</type> into the same operator family, at least not with
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the current semantics that <type>numeric</type> values are converted
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to <type>float8</type> for comparison to a <type>float8</type>. Because
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of the limited accuracy of <type>float8</type>, this means there are
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distinct <type>numeric</type> values that will compare equal to the
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same <type>float8</type> value, and thus the transitive law would fail.
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</para>
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<para>
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Another requirement for a multiple-data-type family is that any implicit
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or binary-coercion casts that are defined between data types included in
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the operator family must not change the associated sort ordering.
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</para>
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<para>
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It should be fairly clear why a btree index requires these laws to hold
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within a single data type: without them there is no ordering to arrange
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the keys with. Also, index searches using a comparison key of a
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different data type require comparisons to behave sanely across two
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data types. The extensions to three or more data types within a family
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are not strictly required by the btree index mechanism itself, but the
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planner relies on them for optimization purposes.
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</para>
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</sect1>
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<sect1 id="btree-support-funcs">
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<title>B-Tree Support Functions</title>
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<para>
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As shown in <xref linkend="xindex-btree-support-table"/>, btree defines
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one required and two optional support functions. The three
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user-defined methods are:
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</para>
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<variablelist>
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<varlistentry>
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<term><function>order</function></term>
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<listitem>
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<para>
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For each combination of data types that a btree operator family
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provides comparison operators for, it must provide a comparison
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support function, registered in
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<structname>pg_amproc</structname> with support function number 1
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and
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<structfield>amproclefttype</structfield>/<structfield>amprocrighttype</structfield>
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equal to the left and right data types for the comparison (i.e.,
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the same data types that the matching operators are registered
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with in <structname>pg_amop</structname>). The comparison
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function must take two non-null values
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<replaceable>A</replaceable> and <replaceable>B</replaceable> and
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return an <type>int32</type> value that is
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<literal><</literal> <literal>0</literal>,
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<literal>0</literal>, or <literal>></literal>
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<literal>0</literal> when <replaceable>A</replaceable>
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<literal><</literal> <replaceable>B</replaceable>,
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<replaceable>A</replaceable> <literal>=</literal>
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<replaceable>B</replaceable>, or <replaceable>A</replaceable>
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<literal>></literal> <replaceable>B</replaceable>,
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respectively. A null result is disallowed: all values of the
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data type must be comparable. See
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<filename>src/backend/access/nbtree/nbtcompare.c</filename> for
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examples.
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</para>
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<para>
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If the compared values are of a collatable data type, the
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appropriate collation OID will be passed to the comparison
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support function, using the standard
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<function>PG_GET_COLLATION()</function> mechanism.
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</para>
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</listitem>
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</varlistentry>
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<varlistentry>
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<term><function>sortsupport</function></term>
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<listitem>
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<para>
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Optionally, a btree operator family may provide <firstterm>sort
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support</firstterm> function(s), registered under support
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function number 2. These functions allow implementing
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comparisons for sorting purposes in a more efficient way than
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naively calling the comparison support function. The APIs
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involved in this are defined in
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<filename>src/include/utils/sortsupport.h</filename>.
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</para>
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</listitem>
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</varlistentry>
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<varlistentry>
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<term><function>inrange</function></term>
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<listitem>
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<indexterm>
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<primary>in_range support functions</primary>
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</indexterm>
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<indexterm>
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<primary>support functions</primary>
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<secondary>in_range</secondary>
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</indexterm>
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<para>
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Optionally, a btree operator family may provide
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<firstterm>in_range</firstterm> support function(s), registered
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under support function number 3. These are not used during btree
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index operations; rather, they extend the semantics of the
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operator family so that it can support window clauses containing
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the <literal>RANGE</literal> <replaceable>offset</replaceable>
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<literal>PRECEDING</literal> and <literal>RANGE</literal>
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<replaceable>offset</replaceable> <literal>FOLLOWING</literal>
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frame bound types (see <xref
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linkend="syntax-window-functions"/>). Fundamentally, the extra
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information provided is how to add or subtract an
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<replaceable>offset</replaceable> value in a way that is
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compatible with the family's data ordering.
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</para>
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<para>
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An <function>in_range</function> function must have the signature
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<synopsis>
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in_range(<replaceable>val</replaceable> type1, <replaceable>base</replaceable> type1, <replaceable>offset</replaceable> type2, <replaceable>sub</replaceable> bool, <replaceable>less</replaceable> bool)
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returns bool
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</synopsis>
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<replaceable>val</replaceable> and
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<replaceable>base</replaceable> must be of the same type, which
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is one of the types supported by the operator family (i.e., a
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type for which it provides an ordering). However,
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<replaceable>offset</replaceable> could be of a different type,
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which might be one otherwise unsupported by the family. An
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example is that the built-in <literal>time_ops</literal> family
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provides an <function>in_range</function> function that has
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<replaceable>offset</replaceable> of type <type>interval</type>.
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A family can provide <function>in_range</function> functions for
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any of its supported types and one or more
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<replaceable>offset</replaceable> types. Each
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<function>in_range</function> function should be entered in
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<structname>pg_amproc</structname> with
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<structfield>amproclefttype</structfield> equal to
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<type>type1</type> and <structfield>amprocrighttype</structfield>
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equal to <type>type2</type>.
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</para>
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<para>
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The essential semantics of an <function>in_range</function>
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function depend on the two Boolean flag parameters. It should
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add or subtract <replaceable>base</replaceable> and
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<replaceable>offset</replaceable>, then compare
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<replaceable>val</replaceable> to the result, as follows:
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<itemizedlist>
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<listitem>
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<para>
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if <literal>!</literal><replaceable>sub</replaceable> and
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<literal>!</literal><replaceable>less</replaceable>, return
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<replaceable>val</replaceable> <literal>>=</literal>
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(<replaceable>base</replaceable> <literal>+</literal>
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<replaceable>offset</replaceable>)
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</para>
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</listitem>
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<listitem>
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<para>
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if <literal>!</literal><replaceable>sub</replaceable> and
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<replaceable>less</replaceable>, return
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<replaceable>val</replaceable> <literal><=</literal>
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(<replaceable>base</replaceable> <literal>+</literal>
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<replaceable>offset</replaceable>)
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</para>
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</listitem>
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<listitem>
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<para>
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if <replaceable>sub</replaceable> and
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<literal>!</literal><replaceable>less</replaceable>, return
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<replaceable>val</replaceable> <literal>>=</literal>
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(<replaceable>base</replaceable> <literal>-</literal>
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<replaceable>offset</replaceable>)
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</para>
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</listitem>
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<listitem>
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<para>
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if <replaceable>sub</replaceable> and
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<replaceable>less</replaceable>, return
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<replaceable>val</replaceable> <literal><=</literal>
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(<replaceable>base</replaceable> <literal>-</literal>
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<replaceable>offset</replaceable>)
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</para>
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</listitem>
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</itemizedlist>
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Before doing so, the function should check the sign of
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<replaceable>offset</replaceable>: if it is less than zero, raise
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error
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<literal>ERRCODE_INVALID_PRECEDING_OR_FOLLOWING_SIZE</literal>
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(22013) with error text like <quote>invalid preceding or
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following size in window function</quote>. (This is required by
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the SQL standard, although nonstandard operator families might
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perhaps choose to ignore this restriction, since there seems to
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be little semantic necessity for it.) This requirement is
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delegated to the <function>in_range</function> function so that
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the core code needn't understand what <quote>less than
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zero</quote> means for a particular data type.
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</para>
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<para>
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An additional expectation is that <function>in_range</function>
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functions should, if practical, avoid throwing an error if
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<replaceable>base</replaceable> <literal>+</literal>
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<replaceable>offset</replaceable> or
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<replaceable>base</replaceable> <literal>-</literal>
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<replaceable>offset</replaceable> would overflow. The correct
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comparison result can be determined even if that value would be
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out of the data type's range. Note that if the data type
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includes concepts such as <quote>infinity</quote> or
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<quote>NaN</quote>, extra care may be needed to ensure that
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<function>in_range</function>'s results agree with the normal
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sort order of the operator family.
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</para>
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<para>
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The results of the <function>in_range</function> function must be
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consistent with the sort ordering imposed by the operator family.
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To be precise, given any fixed values of
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<replaceable>offset</replaceable> and
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<replaceable>sub</replaceable>, then:
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<itemizedlist>
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<listitem>
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<para>
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If <function>in_range</function> with
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<replaceable>less</replaceable> = true is true for some
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<replaceable>val1</replaceable> and
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<replaceable>base</replaceable>, it must be true for every
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<replaceable>val2</replaceable> <literal><=</literal>
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<replaceable>val1</replaceable> with the same
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<replaceable>base</replaceable>.
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</para>
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</listitem>
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<listitem>
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<para>
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If <function>in_range</function> with
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<replaceable>less</replaceable> = true is false for some
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<replaceable>val1</replaceable> and
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<replaceable>base</replaceable>, it must be false for every
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<replaceable>val2</replaceable> <literal>>=</literal>
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<replaceable>val1</replaceable> with the same
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<replaceable>base</replaceable>.
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</para>
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</listitem>
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<listitem>
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<para>
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If <function>in_range</function> with
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<replaceable>less</replaceable> = true is true for some
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<replaceable>val</replaceable> and
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<replaceable>base1</replaceable>, it must be true for every
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<replaceable>base2</replaceable> <literal>>=</literal>
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<replaceable>base1</replaceable> with the same
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<replaceable>val</replaceable>.
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</para>
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</listitem>
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<listitem>
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<para>
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If <function>in_range</function> with
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<replaceable>less</replaceable> = true is false for some
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<replaceable>val</replaceable> and
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<replaceable>base1</replaceable>, it must be false for every
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<replaceable>base2</replaceable> <literal><=</literal>
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<replaceable>base1</replaceable> with the same
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<replaceable>val</replaceable>.
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</para>
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</listitem>
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</itemizedlist>
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Analogous statements with inverted conditions hold when
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<replaceable>less</replaceable> = false.
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</para>
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<para>
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If the type being ordered (<type>type1</type>) is collatable, the
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appropriate collation OID will be passed to the
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<function>in_range</function> function, using the standard
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PG_GET_COLLATION() mechanism.
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</para>
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<para>
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<function>in_range</function> functions need not handle NULL
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inputs, and typically will be marked strict.
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</para>
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</listitem>
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</varlistentry>
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</variablelist>
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</sect1>
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<sect1 id="btree-implementation">
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<title>Implementation</title>
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<para>
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An introduction to the btree index implementation can be found in
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<filename>src/backend/access/nbtree/README</filename>.
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</para>
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</sect1>
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</chapter>
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