postgresql/doc/src/sgml/btree.sgml

<!-- 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 binary 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 one optional support function.
 </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.  The function must not
  return <literal>INT_MIN</literal> for the <replaceable>A</replaceable>
  <literal>&lt;</literal> <replaceable>B</replaceable> case,
  since the value may be negated before being tested for sign.  A null
  result is disallowed, too.
  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>

</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>