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

<sect1 id="btree">
<title>B-Tree Indexes</title>

   <indexterm>
    <primary>index</primary>
    <secondary>B-Tree</secondary>
   </indexterm>

<sect2 id="btree-intro">
 <title>Introduction</title>

 <para>
  <productname>PostgreSQL</productname> includes an implementation of the
  standard <acronym>btree</acronym> (multi-way balanced tree) index data
  structure.  Any data type that can be sorted into a well-defined linear
  order can be indexed by a btree index.  The only limitation is that an
  index entry cannot exceed approximately one-third of a page (after TOAST
  compression, if applicable).
 </para>

 <para>
  Because each btree operator class imposes a sort order on its data type,
  btree operator classes (or, really, operator families) have come to be
  used as <productname>PostgreSQL</productname>'s general representation
  and understanding of sorting semantics.  Therefore, they've acquired
  some features that go beyond what would be needed just to support btree
  indexes, and parts of the system that are quite distant from the
  btree <acronym>AM</acronym> make use of them.
 </para>

</sect2>

<sect2 id="btree-behavior">
 <title>Behavior of B-Tree Operator Classes</title>

 <para>
  As shown in <xref linkend="xindex-btree-strat-table"/>, a btree operator
  class must provide five comparison operators,
  <literal>&lt;</literal>,
  <literal>&lt;=</literal>,
  <literal>=</literal>,
  <literal>&gt;=</literal> and
  <literal>&gt;</literal>.
  One might expect that <literal>&lt;&gt;</literal> should also be part of
  the operator class, but it is not, because it would almost never be
  useful to use a <literal>&lt;&gt;</literal> WHERE clause in an index
  search.  (For some purposes, the planner treats <literal>&lt;&gt;</literal>
  as associated with a btree operator class; but it finds that operator via
  the <literal>=</literal> operator's negator link, rather than
  from <structname>pg_amop</structname>.)
 </para>

 <para>
  When several data types share near-identical sorting semantics, their
  operator classes can be grouped into an operator family.  Doing so is
  advantageous because it allows the planner to make deductions about
  cross-type comparisons.  Each operator class within the family should
  contain the single-type operators (and associated support functions)
  for its input data type, while cross-type comparison operators and
  support functions are <quote>loose</quote> in the family.  It is
  recommendable that a complete set of cross-type operators be included
  in the family, thus ensuring that the planner can represent any
  comparison conditions that it deduces from transitivity.
 </para>

 <para>
  There are some basic assumptions that a btree operator family must
  satisfy:
 </para>

 <itemizedlist>
  <listitem>
   <para>
    An <literal>=</literal> operator must be an equivalence relation; that
    is, for all non-null values <replaceable>A</replaceable>,
    <replaceable>B</replaceable>, <replaceable>C</replaceable> of the
    data type:

    <itemizedlist>
     <listitem>
      <para>
       <replaceable>A</replaceable> <literal>=</literal>
       <replaceable>A</replaceable> is true
       (<firstterm>reflexive law</firstterm>)
      </para>
     </listitem>
     <listitem>
      <para>
       if <replaceable>A</replaceable> <literal>=</literal>
       <replaceable>B</replaceable>,
       then <replaceable>B</replaceable> <literal>=</literal>
       <replaceable>A</replaceable>
       (<firstterm>symmetric law</firstterm>)
      </para>
     </listitem>
     <listitem>
      <para>
       if <replaceable>A</replaceable> <literal>=</literal>
       <replaceable>B</replaceable> and <replaceable>B</replaceable>
       <literal>=</literal> <replaceable>C</replaceable>,
       then <replaceable>A</replaceable> <literal>=</literal>
       <replaceable>C</replaceable>
       (<firstterm>transitive law</firstterm>)
      </para>
     </listitem>
    </itemizedlist>
   </para>
  </listitem>

  <listitem>
   <para>
    A <literal>&lt;</literal> operator must be a strong ordering relation;
    that is, for all non-null values <replaceable>A</replaceable>,
    <replaceable>B</replaceable>, <replaceable>C</replaceable>:

    <itemizedlist>
     <listitem>
      <para>
       <replaceable>A</replaceable> <literal>&lt;</literal>
       <replaceable>A</replaceable> is false
       (<firstterm>irreflexive law</firstterm>)
      </para>
     </listitem>
     <listitem>
      <para>
       if <replaceable>A</replaceable> <literal>&lt;</literal>
       <replaceable>B</replaceable>
       and <replaceable>B</replaceable> <literal>&lt;</literal>
       <replaceable>C</replaceable>,
       then <replaceable>A</replaceable> <literal>&lt;</literal>
       <replaceable>C</replaceable>
       (<firstterm>transitive law</firstterm>)
      </para>
     </listitem>
    </itemizedlist>
   </para>
  </listitem>

  <listitem>
   <para>
    Furthermore, the ordering is total; that is, for all non-null
    values <replaceable>A</replaceable>, <replaceable>B</replaceable>:

    <itemizedlist>
     <listitem>
      <para>
       exactly one of <replaceable>A</replaceable> <literal>&lt;</literal>
       <replaceable>B</replaceable>, <replaceable>A</replaceable>
       <literal>=</literal> <replaceable>B</replaceable>, and
       <replaceable>B</replaceable> <literal>&lt;</literal>
       <replaceable>A</replaceable> is true
       (<firstterm>trichotomy law</firstterm>)
      </para>
     </listitem>
    </itemizedlist>

    (The trichotomy law justifies the definition of the comparison support
    function, of course.)
   </para>
  </listitem>
 </itemizedlist>

 <para>
  The other three operators are defined in terms of <literal>=</literal>
  and <literal>&lt;</literal> in the obvious way, and must act consistently
  with them.
 </para>

 <para>
  For an operator family supporting multiple data types, the above laws must
  hold when <replaceable>A</replaceable>, <replaceable>B</replaceable>,
  <replaceable>C</replaceable> are taken from any data types in the family.
  The transitive laws are the trickiest to ensure, as in cross-type
  situations they represent statements that the behaviors of two or three
  different operators are consistent.
  As an example, it would not work to put <type>float8</type>
  and <type>numeric</type> into the same operator family, at least not with
  the current semantics that <type>numeric</type> values are converted
  to <type>float8</type> for comparison to a <type>float8</type>.  Because
  of the limited accuracy of <type>float8</type>, this means there are
  distinct <type>numeric</type> values that will compare equal to the
  same <type>float8</type> value, and thus the transitive law would fail.
 </para>

 <para>
  Another requirement for a multiple-data-type family is that any implicit
  or binary-coercion casts that are defined between data types included in
  the operator family must not change the associated sort ordering.
 </para>

 <para>
  It should be fairly clear why a btree index requires these laws to hold
  within a single data type: without them there is no ordering to arrange
  the keys with.  Also, index searches using a comparison key of a
  different data type require comparisons to behave sanely across two
  data types.  The extensions to three or more data types within a family
  are not strictly required by the btree index mechanism itself, but the
  planner relies on them for optimization purposes.
 </para>

</sect2>

<sect2 id="btree-support-funcs">
 <title>B-Tree Support Functions</title>

 <para>
  As shown in <xref linkend="xindex-btree-support-table"/>, btree defines
  one required and five optional support functions.  The six
  user-defined methods are:
 </para>
 <variablelist>
  <varlistentry>
   <term><function>order</function></term>
   <listitem>
    <para>
     For each combination of data types that a btree operator family
     provides comparison operators for, it must provide a comparison
     support function, registered in
     <structname>pg_amproc</structname> with support function number 1
     and
     <structfield>amproclefttype</structfield>/<structfield>amprocrighttype</structfield>
     equal to the left and right data types for the comparison (i.e.,
     the same data types that the matching operators are registered
     with in <structname>pg_amop</structname>).  The comparison
     function must take two non-null values
     <replaceable>A</replaceable> and <replaceable>B</replaceable> and
     return an <type>int32</type> value that is
     <literal>&lt;</literal> <literal>0</literal>,
     <literal>0</literal>, or <literal>&gt;</literal>
     <literal>0</literal> when <replaceable>A</replaceable>
     <literal>&lt;</literal> <replaceable>B</replaceable>,
     <replaceable>A</replaceable> <literal>=</literal>
     <replaceable>B</replaceable>, or <replaceable>A</replaceable>
     <literal>&gt;</literal> <replaceable>B</replaceable>,
     respectively.  A null result is disallowed: all values of the
     data type must be comparable.  See
     <filename>src/backend/access/nbtree/nbtcompare.c</filename> for
     examples.
    </para>

    <para>
     If the compared values are of a collatable data type, the
     appropriate collation OID will be passed to the comparison
     support function, using the standard
     <function>PG_GET_COLLATION()</function> mechanism.
    </para>
   </listitem>
  </varlistentry>
  <varlistentry>
   <term><function>sortsupport</function></term>
   <listitem>
    <para>
     Optionally, a btree operator family may provide <firstterm>sort
      support</firstterm> function(s), registered under support
     function number 2.  These functions allow implementing
     comparisons for sorting purposes in a more efficient way than
     naively calling the comparison support function.  The APIs
     involved in this are defined in
     <filename>src/include/utils/sortsupport.h</filename>.
    </para>
   </listitem>
  </varlistentry>
  <varlistentry>
   <term><function>in_range</function></term>
   <listitem>
    <indexterm>
     <primary>in_range support functions</primary>
    </indexterm>

    <indexterm>
     <primary>support functions</primary>
     <secondary>in_range</secondary>
    </indexterm>
    <para>
     Optionally, a btree operator family may provide
     <firstterm>in_range</firstterm> support function(s), registered
     under support function number 3.  These are not used during btree
     index operations; rather, they extend the semantics of the
     operator family so that it can support window clauses containing
     the <literal>RANGE</literal> <replaceable>offset</replaceable>
     <literal>PRECEDING</literal> and <literal>RANGE</literal>
     <replaceable>offset</replaceable> <literal>FOLLOWING</literal>
     frame bound types (see <xref
      linkend="syntax-window-functions"/>).  Fundamentally, the extra
     information provided is how to add or subtract an
     <replaceable>offset</replaceable> value in a way that is
     compatible with the family's data ordering.
    </para>

    <para>
     An <function>in_range</function> function must have the signature
<synopsis>
in_range(<replaceable>val</replaceable> type1, <replaceable>base</replaceable> type1, <replaceable>offset</replaceable> type2, <replaceable>sub</replaceable> bool, <replaceable>less</replaceable> bool)
returns bool
</synopsis>
     <replaceable>val</replaceable> and
     <replaceable>base</replaceable> must be of the same type, which
     is one of the types supported by the operator family (i.e., a
     type for which it provides an ordering).  However,
     <replaceable>offset</replaceable> could be of a different type,
     which might be one otherwise unsupported by the family.  An
     example is that the built-in <literal>time_ops</literal> family
     provides an <function>in_range</function> function that has
     <replaceable>offset</replaceable> of type <type>interval</type>.
     A family can provide <function>in_range</function> functions for
     any of its supported types and one or more
     <replaceable>offset</replaceable> types.  Each
     <function>in_range</function> function should be entered in
     <structname>pg_amproc</structname> with
     <structfield>amproclefttype</structfield> equal to
     <type>type1</type> and <structfield>amprocrighttype</structfield>
     equal to <type>type2</type>.
    </para>

    <para>
     The essential semantics of an <function>in_range</function>
     function depend on the two Boolean flag parameters.  It should
     add or subtract <replaceable>base</replaceable> and
     <replaceable>offset</replaceable>, then compare
     <replaceable>val</replaceable> to the result, as follows:
     <itemizedlist>
      <listitem>
       <para>
        if <literal>!</literal><replaceable>sub</replaceable> and
        <literal>!</literal><replaceable>less</replaceable>, return
        <replaceable>val</replaceable> <literal>&gt;=</literal>
        (<replaceable>base</replaceable> <literal>+</literal>
        <replaceable>offset</replaceable>)
       </para>
      </listitem>
      <listitem>
       <para>
        if <literal>!</literal><replaceable>sub</replaceable> and
        <replaceable>less</replaceable>, return
        <replaceable>val</replaceable> <literal>&lt;=</literal>
        (<replaceable>base</replaceable> <literal>+</literal>
        <replaceable>offset</replaceable>)
       </para>
      </listitem>
      <listitem>
       <para>
        if <replaceable>sub</replaceable> and
        <literal>!</literal><replaceable>less</replaceable>, return
        <replaceable>val</replaceable> <literal>&gt;=</literal>
        (<replaceable>base</replaceable> <literal>-</literal>
        <replaceable>offset</replaceable>)
       </para>
      </listitem>
      <listitem>
       <para>
        if <replaceable>sub</replaceable> and
        <replaceable>less</replaceable>, return
        <replaceable>val</replaceable> <literal>&lt;=</literal>
        (<replaceable>base</replaceable> <literal>-</literal>
        <replaceable>offset</replaceable>)
       </para>
      </listitem>
     </itemizedlist>
     Before doing so, the function should check the sign of
     <replaceable>offset</replaceable>: if it is less than zero, raise
     error
     <literal>ERRCODE_INVALID_PRECEDING_OR_FOLLOWING_SIZE</literal>
     (22013) with error text like <quote>invalid preceding or
      following size in window function</quote>.  (This is required by
     the SQL standard, although nonstandard operator families might
     perhaps choose to ignore this restriction, since there seems to
     be little semantic necessity for it.) This requirement is
     delegated to the <function>in_range</function> function so that
     the core code needn't understand what <quote>less than
      zero</quote> means for a particular data type.
    </para>

    <para>
     An additional expectation is that <function>in_range</function>
     functions should, if practical, avoid throwing an error if
     <replaceable>base</replaceable> <literal>+</literal>
     <replaceable>offset</replaceable> or
     <replaceable>base</replaceable> <literal>-</literal>
     <replaceable>offset</replaceable> would overflow.  The correct
     comparison result can be determined even if that value would be
     out of the data type's range.  Note that if the data type
     includes concepts such as <quote>infinity</quote> or
     <quote>NaN</quote>, extra care may be needed to ensure that
     <function>in_range</function>'s results agree with the normal
     sort order of the operator family.
    </para>

    <para>
     The results of the <function>in_range</function> function must be
     consistent with the sort ordering imposed by the operator family.
     To be precise, given any fixed values of
     <replaceable>offset</replaceable> and
     <replaceable>sub</replaceable>, then:
     <itemizedlist>
      <listitem>
       <para>
        If <function>in_range</function> with
        <replaceable>less</replaceable> = true is true for some
        <replaceable>val1</replaceable> and
        <replaceable>base</replaceable>, it must be true for every
        <replaceable>val2</replaceable> <literal>&lt;=</literal>
        <replaceable>val1</replaceable> with the same
        <replaceable>base</replaceable>.
       </para>
      </listitem>
      <listitem>
       <para>
        If <function>in_range</function> with
        <replaceable>less</replaceable> = true is false for some
        <replaceable>val1</replaceable> and
        <replaceable>base</replaceable>, it must be false for every
        <replaceable>val2</replaceable> <literal>&gt;=</literal>
        <replaceable>val1</replaceable> with the same
        <replaceable>base</replaceable>.
       </para>
      </listitem>
      <listitem>
       <para>
        If <function>in_range</function> with
        <replaceable>less</replaceable> = true is true for some
        <replaceable>val</replaceable> and
        <replaceable>base1</replaceable>, it must be true for every
        <replaceable>base2</replaceable> <literal>&gt;=</literal>
        <replaceable>base1</replaceable> with the same
        <replaceable>val</replaceable>.
       </para>
      </listitem>
      <listitem>
       <para>
        If <function>in_range</function> with
        <replaceable>less</replaceable> = true is false for some
        <replaceable>val</replaceable> and
        <replaceable>base1</replaceable>, it must be false for every
        <replaceable>base2</replaceable> <literal>&lt;=</literal>
        <replaceable>base1</replaceable> with the same
        <replaceable>val</replaceable>.
       </para>
      </listitem>
     </itemizedlist>
     Analogous statements with inverted conditions hold when
     <replaceable>less</replaceable> = false.
    </para>

    <para>
     If the type being ordered (<type>type1</type>) is collatable, the
     appropriate collation OID will be passed to the
     <function>in_range</function> function, using the standard
     PG_GET_COLLATION() mechanism.
    </para>

    <para>
     <function>in_range</function> functions need not handle NULL
     inputs, and typically will be marked strict.
    </para>
   </listitem>
  </varlistentry>
  <varlistentry>
   <term><function>equalimage</function></term>
   <listitem>
    <para>
     Optionally, a btree operator family may provide
     <function>equalimage</function> (<quote>equality implies image
      equality</quote>) support functions, registered under support
     function number 4.  These functions allow the core code to
     determine when it is safe to apply the btree deduplication
     optimization.  Currently, <function>equalimage</function>
     functions are only called when building or rebuilding an index.
    </para>
    <para>
     An <function>equalimage</function> function must have the
     signature
<synopsis>
equalimage(<replaceable>opcintype</replaceable> <type>oid</type>) returns bool
</synopsis>
     The return value is static information about an operator class
     and collation.  Returning <literal>true</literal> indicates that
     the <function>order</function> function for the operator class is
     guaranteed to only return <literal>0</literal> (<quote>arguments
      are equal</quote>) when its <replaceable>A</replaceable> and
     <replaceable>B</replaceable> arguments are also interchangeable
     without any loss of semantic information.  Not registering an
     <function>equalimage</function> function or returning
     <literal>false</literal> indicates that this condition cannot be
     assumed to hold.
    </para>
    <para>
     The <replaceable>opcintype</replaceable> argument is the
     <literal><structname>pg_type</structname>.oid</literal> of the
     data type that the operator class indexes.  This is a convenience
     that allows reuse of the same underlying
     <function>equalimage</function> function across operator classes.
     If <replaceable>opcintype</replaceable> is a collatable data
     type, the appropriate collation OID will be passed to the
     <function>equalimage</function> function, using the standard
     <function>PG_GET_COLLATION()</function> mechanism.
    </para>
    <para>
     As far as the operator class is concerned, returning
     <literal>true</literal> indicates that deduplication is safe (or
     safe for the collation whose OID was passed to its
     <function>equalimage</function> function).  However, the core
     code will only deem deduplication safe for an index when
     <emphasis>every</emphasis> indexed column uses an operator class
     that registers an <function>equalimage</function> function, and
     each function actually returns <literal>true</literal> when
     called.
    </para>
    <para>
     Image equality is <emphasis>almost</emphasis> the same condition
     as simple bitwise equality.  There is one subtle difference: When
     indexing a varlena data type, the on-disk representation of two
     image equal datums may not be bitwise equal due to inconsistent
     application of <acronym>TOAST</acronym> compression on input.
     Formally, when an operator class's
     <function>equalimage</function> function returns
     <literal>true</literal>, it is safe to assume that the
     <literal>datum_image_eq()</literal> C function will always agree
     with the operator class's <function>order</function> function
     (provided that the same collation OID is passed to both the
     <function>equalimage</function> and <function>order</function>
     functions).
    </para>
    <para>
     The core code is fundamentally unable to deduce anything about
     the <quote>equality implies image equality</quote> status of an
     operator class within a multiple-data-type family based on
     details from other operator classes in the same family.  Also, it
     is not sensible for an operator family to register a cross-type
     <function>equalimage</function> function, and attempting to do so
     will result in an error.  This is because <quote>equality implies
      image equality</quote> status does not just depend on
     sorting/equality semantics, which are more or less defined at the
     operator family level.  In general, the semantics that one
     particular data type implements must be considered separately.
    </para>
    <para>
     The convention followed by the operator classes included with the
     core <productname>PostgreSQL</productname> distribution is to
     register a stock, generic <function>equalimage</function>
     function.  Most operator classes register
     <function>btequalimage()</function>, which indicates that
     deduplication is safe unconditionally.  Operator classes for
     collatable data types such as <type>text</type> register
     <function>btvarstrequalimage()</function>, which indicates that
     deduplication is safe with deterministic collations.  Best
     practice for third-party extensions is to register their own
     custom function to retain control.
    </para>
   </listitem>
  </varlistentry>
  <varlistentry>
   <term><function>options</function></term>
   <listitem>
    <para>
     Optionally, a B-tree operator family may provide
     <function>options</function> (<quote>operator class specific
     options</quote>) support functions, registered under support
     function number 5.  These functions define a set of user-visible
     parameters that control operator class behavior.
    </para>
    <para>
     An <function>options</function> support function must have the
     signature
<synopsis>
options(<replaceable>relopts</replaceable> <type>local_relopts *</type>) returns void
</synopsis>
     The function is passed a pointer to a <structname>local_relopts</structname>
     struct, which needs to be filled with a set of operator class
     specific options.  The options can be accessed from other support
     functions using the <literal>PG_HAS_OPCLASS_OPTIONS()</literal> and
     <literal>PG_GET_OPCLASS_OPTIONS()</literal> macros.
    </para>
    <para>
     Currently, no B-Tree operator class has an <function>options</function>
     support function.  B-tree doesn't allow flexible representation of keys
     like GiST, SP-GiST, GIN and BRIN do.  So, <function>options</function>
     probably doesn't have much application in the current B-tree index
     access method.  Nevertheless, this support function was added to B-tree
     for uniformity, and will probably find uses during further
     evolution of B-tree in <productname>PostgreSQL</productname>.
    </para>
   </listitem>
  </varlistentry>
  <varlistentry>
   <term><function>skipsupport</function></term>
   <listitem>
    <para>
     Optionally, a btree operator family may provide a <firstterm>skip
      support</firstterm> function, registered under support function number 6.
     These functions give the B-tree code a way to iterate through every
     possible value that can be represented by an operator class's underlying
     input type, in key space order.  This is used by the core code when it
     applies the skip scan optimization.  The APIs involved in this are
     defined in <filename>src/include/utils/skipsupport.h</filename>.
    </para>
    <para>
     Operator classes that do not provide a skip support function are still
     eligible to use skip scan.  The core code can still use its fallback
     strategy, though that might be suboptimal for some discrete types.  It
     usually doesn't make sense (and may not even be feasible) for operator
     classes on continuous types to provide a skip support function.
    </para>
    <para>
     It is not sensible for an operator family to register a cross-type
     <function>skipsupport</function> function, and attempting to do so will
     result in an error.  This is because determining the next indexable value
     must happen by incrementing a value copied from an index tuple.  The
     values generated must all be of the same underlying data type (the
     <quote>skipped</quote> index column's opclass input type).
    </para>
   </listitem>
  </varlistentry>
 </variablelist>

</sect2>

<sect2 id="btree-implementation">
 <title>Implementation</title>

 <para>
  This section covers B-Tree index implementation details that may be
  of use to advanced users.  See
  <filename>src/backend/access/nbtree/README</filename> in the source
  distribution for a much more detailed, internals-focused description
  of the B-Tree implementation.
 </para>
 <sect3 id="btree-structure">
  <title>B-Tree Structure</title>
  <para>
   <productname>PostgreSQL</productname> B-Tree indexes are
   multi-level tree structures, where each level of the tree can be
   used as a doubly-linked list of pages.  A single metapage is stored
   in a fixed position at the start of the first segment file of the
   index.  All other pages are either leaf pages or internal pages.
   Leaf pages are the pages on the lowest level of the tree.  All
   other levels consist of internal pages.  Each leaf page contains
   tuples that point to table rows.  Each internal page contains
   tuples that point to the next level down in the tree.  Typically,
   over 99% of all pages are leaf pages.  Both internal pages and leaf
   pages use the standard page format described in <xref
    linkend="storage-page-layout"/>.
  </para>
  <para>
   New leaf pages are added to a B-Tree index when an existing leaf
   page cannot fit an incoming tuple.  A <firstterm>page
    split</firstterm> operation makes room for items that originally
   belonged on the overflowing page by moving a portion of the items
   to a new page.  Page splits must also insert a new
   <firstterm>downlink</firstterm> to the new page in the parent page,
   which may cause the parent to split in turn.  Page splits
   <quote>cascade upwards</quote> in a recursive fashion.  When the
   root page finally cannot fit a new downlink, a <firstterm>root page
    split</firstterm> operation takes place.  This adds a new level to
   the tree structure by creating a new root page that is one level
   above the original root page.
  </para>
 </sect3>

 <sect3 id="btree-deletion">
  <title>Bottom-up Index Deletion</title>
  <para>
   B-Tree indexes are not directly aware that under MVCC, there might
   be multiple extant versions of the same logical table row; to an
   index, each tuple is an independent object that needs its own index
   entry.  <quote>Version churn</quote> tuples may sometimes
   accumulate and adversely affect query latency and throughput.  This
   typically occurs with <command>UPDATE</command>-heavy workloads
   where most individual updates cannot apply the
   <link linkend="storage-hot"><acronym>HOT</acronym> optimization.</link>
   Changing the value of only
   one column covered by one index during an <command>UPDATE</command>
   <emphasis>always</emphasis> necessitates a new set of index tuples
   &mdash; one for <emphasis>each and every</emphasis> index on the
   table.  Note in particular that this includes indexes that were not
   <quote>logically modified</quote> by the <command>UPDATE</command>.
   All indexes will need a successor physical index tuple that points
   to the latest version in the table.  Each new tuple within each
   index will generally need to coexist with the original
   <quote>updated</quote> tuple for a short period of time (typically
   until shortly after the <command>UPDATE</command> transaction
   commits).
  </para>
  <para>
   B-Tree indexes incrementally delete version churn index tuples by
   performing <firstterm>bottom-up index deletion</firstterm> passes.
   Each deletion pass is triggered in reaction to an anticipated
   <quote>version churn page split</quote>.  This only happens with
   indexes that are not logically modified by
   <command>UPDATE</command> statements, where concentrated build up
   of obsolete versions in particular pages would occur otherwise.  A
   page split will usually be avoided, though it's possible that
   certain implementation-level heuristics will fail to identify and
   delete even one garbage index tuple (in which case a page split or
   deduplication pass resolves the issue of an incoming new tuple not
   fitting on a leaf page).  The worst-case number of versions that
   any index scan must traverse (for any single logical row) is an
   important contributor to overall system responsiveness and
   throughput.  A bottom-up index deletion pass targets suspected
   garbage tuples in a single leaf page based on
   <emphasis>qualitative</emphasis> distinctions involving logical
   rows and versions.  This contrasts with the <quote>top-down</quote>
   index cleanup performed by autovacuum workers, which is triggered
   when certain <emphasis>quantitative</emphasis> table-level
   thresholds are exceeded (see <xref linkend="autovacuum"/>).
  </para>
  <note>
   <para>
    Not all deletion operations that are performed within B-Tree
    indexes are bottom-up deletion operations.  There is a distinct
    category of index tuple deletion: <firstterm>simple index tuple
     deletion</firstterm>.  This is a deferred maintenance operation
    that deletes index tuples that are known to be safe to delete
    (those whose item identifier's <literal>LP_DEAD</literal> bit is
    already set).  Like bottom-up index deletion, simple index
    deletion takes place at the point that a page split is anticipated
    as a way of avoiding the split.
   </para>
   <para>
    Simple deletion is opportunistic in the sense that it can only
    take place when recent index scans set the
    <literal>LP_DEAD</literal> bits of affected items in passing.
    Prior to <productname>PostgreSQL</productname> 14, the only
    category of B-Tree deletion was simple deletion.  The main
    differences between it and bottom-up deletion are that only the
    former is opportunistically driven by the activity of passing
    index scans, while only the latter specifically targets version
    churn from <command>UPDATE</command>s that do not logically modify
    indexed columns.
   </para>
  </note>
  <para>
   Bottom-up index deletion performs the vast majority of all garbage
   index tuple cleanup for particular indexes with certain workloads.
   This is expected with any B-Tree index that is subject to
   significant version churn from <command>UPDATE</command>s that
   rarely or never logically modify the columns that the index covers.
   The average and worst-case number of versions per logical row can
   be kept low purely through targeted incremental deletion passes.
   It's quite possible that the on-disk size of certain indexes will
   never increase by even one single page/block despite
   <emphasis>constant</emphasis> version churn from
   <command>UPDATE</command>s.  Even then, an exhaustive <quote>clean
    sweep</quote> by a <command>VACUUM</command> operation (typically
   run in an autovacuum worker process) will eventually be required as
   a part of <emphasis>collective</emphasis> cleanup of the table and
   each of its indexes.
  </para>
  <para>
   Unlike <command>VACUUM</command>, bottom-up index deletion does not
   provide any strong guarantees about how old the oldest garbage
   index tuple may be.  No index can be permitted to retain
   <quote>floating garbage</quote> index tuples that became dead prior
   to a conservative cutoff point shared by the table and all of its
   indexes collectively.  This fundamental table-level invariant makes
   it safe to recycle table <acronym>TID</acronym>s.  This is how it
   is possible for distinct logical rows to reuse the same table
   <acronym>TID</acronym> over time (though this can never happen with
   two logical rows whose lifetimes span the same
   <command>VACUUM</command> cycle).
  </para>
 </sect3>

 <sect3 id="btree-deduplication">
  <title>Deduplication</title>
  <para>
   A duplicate is a leaf page tuple (a tuple that points to a table
   row) where <emphasis>all</emphasis> indexed key columns have values
   that match corresponding column values from at least one other leaf
   page tuple in the same index.  Duplicate tuples are quite common in
   practice.  B-Tree indexes can use a special, space-efficient
   representation for duplicates when an optional technique is
   enabled: <firstterm>deduplication</firstterm>.
  </para>
  <para>
   Deduplication works by periodically merging groups of duplicate
   tuples together, forming a single <firstterm>posting list</firstterm> tuple for each
   group.  The column key value(s) only appear once in this
   representation.  This is followed by a sorted array of
   <acronym>TID</acronym>s that point to rows in the table.  This
   significantly reduces the storage size of indexes where each value
   (or each distinct combination of column values) appears several
   times on average.  The latency of queries can be reduced
   significantly.  Overall query throughput may increase
   significantly.  The overhead of routine index vacuuming may also be
   reduced significantly.
  </para>
  <note>
   <para>
    B-Tree deduplication is just as effective with
    <quote>duplicates</quote> that contain a NULL value, even though
    NULL values are never equal to each other according to the
    <literal>=</literal> member of any B-Tree operator class.  As far
    as any part of the implementation that understands the on-disk
    B-Tree structure is concerned, NULL is just another value from the
    domain of indexed values.
   </para>
  </note>
  <para>
   The deduplication process occurs lazily, when a new item is
   inserted that cannot fit on an existing leaf page, though only when
   index tuple deletion could not free sufficient space for the new
   item (typically deletion is briefly considered and then skipped
   over).  Unlike GIN posting list tuples, B-Tree posting list tuples
   do not need to expand every time a new duplicate is inserted; they
   are merely an alternative physical representation of the original
   logical contents of the leaf page.  This design prioritizes
   consistent performance with mixed read-write workloads.  Most
   client applications will at least see a moderate performance
   benefit from using deduplication.  Deduplication is enabled by
   default.
  </para>
  <para>
   <command>CREATE INDEX</command> and <command>REINDEX</command>
   apply deduplication to create posting list tuples, though the
   strategy they use is slightly different.  Each group of duplicate
   ordinary tuples encountered in the sorted input taken from the
   table is merged into a posting list tuple
   <emphasis>before</emphasis> being added to the current pending leaf
   page.  Individual posting list tuples are packed with as many
   <acronym>TID</acronym>s as possible.  Leaf pages are written out in
   the usual way, without any separate deduplication pass.  This
   strategy is well-suited to <command>CREATE INDEX</command> and
   <command>REINDEX</command> because they are once-off batch
   operations.
  </para>
  <para>
   Write-heavy workloads that don't benefit from deduplication due to
   having few or no duplicate values in indexes will incur a small,
   fixed performance penalty (unless deduplication is explicitly
   disabled).  The <literal>deduplicate_items</literal> storage
   parameter can be used to disable deduplication within individual
   indexes.  There is never any performance penalty with read-only
   workloads, since reading posting list tuples is at least as
   efficient as reading the standard tuple representation.  Disabling
   deduplication isn't usually helpful.
  </para>
  <para>
   It is sometimes possible for unique indexes (as well as unique
   constraints) to use deduplication.  This allows leaf pages to
   temporarily <quote>absorb</quote> extra version churn duplicates.
   Deduplication in unique indexes augments bottom-up index deletion,
   especially in cases where a long-running transaction holds a
   snapshot that blocks garbage collection.  The goal is to buy time
   for the bottom-up index deletion strategy to become effective
   again.  Delaying page splits until a single long-running
   transaction naturally goes away can allow a bottom-up deletion pass
   to succeed where an earlier deletion pass failed.
  </para>
  <tip>
   <para>
    A special heuristic is applied to determine whether a
    deduplication pass in a unique index should take place.  It can
    often skip straight to splitting a leaf page, avoiding a
    performance penalty from wasting cycles on unhelpful deduplication
    passes.  If you're concerned about the overhead of deduplication,
    consider setting <literal>deduplicate_items = off</literal>
    selectively.  Leaving deduplication enabled in unique indexes has
    little downside.
   </para>
  </tip>
  <para>
   Deduplication cannot be used in all cases due to
   implementation-level restrictions.  Deduplication safety is
   determined when <command>CREATE INDEX</command> or
   <command>REINDEX</command> is run.
  </para>
  <para>
   Note that deduplication is deemed unsafe and cannot be used in the
   following cases involving semantically significant differences
   among equal datums:
  </para>
  <para>
   <itemizedlist>
    <listitem>
     <para>
      <type>text</type>, <type>varchar</type>, and <type>char</type>
      cannot use deduplication when a
      <emphasis>nondeterministic</emphasis> collation is used.  Case
      and accent differences must be preserved among equal datums.
     </para>
    </listitem>

    <listitem>
     <para>
      <type>numeric</type> cannot use deduplication.  Numeric display
      scale must be preserved among equal datums.
     </para>
    </listitem>

    <listitem>
     <para>
      <type>jsonb</type> cannot use deduplication, since the
      <type>jsonb</type> B-Tree operator class uses
      <type>numeric</type> internally.
     </para>
    </listitem>

    <listitem>
     <para>
      <type>float4</type> and <type>float8</type> cannot use
      deduplication.  These types have distinct representations for
      <literal>-0</literal> and <literal>0</literal>, which are
      nevertheless considered equal.  This difference must be
      preserved.
     </para>
    </listitem>
   </itemizedlist>
  </para>
  <para>
   There is one further implementation-level restriction that may be
   lifted in a future version of
   <productname>PostgreSQL</productname>:
  </para>
  <para>
   <itemizedlist>
    <listitem>
     <para>
      Container types (such as composite types, arrays, or range
      types) cannot use deduplication.
     </para>
    </listitem>
   </itemizedlist>
  </para>
  <para>
   There is one further implementation-level restriction that applies
   regardless of the operator class or collation used:
  </para>
  <para>
   <itemizedlist>
    <listitem>
     <para>
      <literal>INCLUDE</literal> indexes can never use deduplication.
     </para>
    </listitem>
   </itemizedlist>
  </para>

 </sect3>
</sect2>

</sect1>
