Doc: move info for btree opclass implementors into main documentation.

Up to now, useful info for writing a new btree opclass has been buried
in the backend's nbtree/README file.  Let's move it into the SGML docs,
in preparation for extending it with info about "in_range" functions
in the upcoming window RANGE patch.

To do this, I chose to create a new chapter for btree indexes in Part VII
(Internals), parallel to the chapters that exist for the newer index AMs.
This is a pretty short chapter as-is.  At some point somebody might care
to flesh it out with more detail about btree internals, but that is
beyond the scope of my ambition for today.

Discussion: https://postgr.es/m/[email protected]

1 files changed, 267 insertions, 0 deletions

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+<!-- 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>