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

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

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

<sect1 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 AM make use of them.
 </para>

</sect1>

<sect1 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>

</sect1>

<sect1 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 two optional support functions.
 </para>

 <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>

 <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>

 <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>

</sect1>

<sect1 id="btree-implementation">
 <title>Implementation</title>

  <para>
   An introduction to the btree index implementation can be found in
   <filename>src/backend/access/nbtree/README</filename>.
  </para>

</sect1>

</chapter>
