Add SP-GiST (space-partitioned GiST) index access method.

SP-GiST is comparable to GiST in flexibility, but supports non-balanced
partitioned search structures rather than balanced trees.  As described at
PGCon 2011, this new indexing structure can beat GiST in both index build
time and query speed for search problems that it is well matched to.

There are a number of areas that could still use improvement, but at this
point the code seems committable.

Teodor Sigaev and Oleg Bartunov, with considerable revisions by Tom Lane

1 files changed, 706 insertions, 0 deletions

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+<!-- doc/src/sgml/spgist.sgml -->
+
+<chapter id="SPGiST">
+<title>SP-GiST Indexes</title>
+
+   <indexterm>
+    <primary>index</primary>
+    <secondary>SP-GiST</secondary>
+   </indexterm>
+
+<sect1 id="spgist-intro">
+ <title>Introduction</title>
+
+ <para>
+  <acronym>SP-GiST</acronym> is an abbreviation for space-partitioned
+  <acronym>GiST</acronym>.  <acronym>SP-GiST</acronym> supports partitioned
+  search trees, which facilitate development of a wide range of different
+  non-balanced data structures, such as quad-trees, k-d trees, and suffix
+  trees (tries).  The common feature of these structures is that they
+  repeatedly divide the search space into partitions that need not be
+  of equal size.  Searches that are well matched to the partitioning rule
+  can be very fast.
+ </para>
+
+ <para>
+  These popular data structures were originally developed for in-memory
+  usage.  In main memory, they are usually designed as a set of dynamically
+  allocated nodes linked by pointers.  This is not suitable for direct
+  storing on disk, since these chains of pointers can be rather long which
+  would require too many disk accesses.  In contrast, disk-based data
+  structures should have a high fanout to minimize I/O.  The challenge
+  addressed by <acronym>SP-GiST</acronym> is to map search tree nodes to
+  disk pages in such a way that a search need access only a few disk pages,
+  even if it traverses many nodes.
+ </para>
+
+ <para>
+  Like <acronym>GiST</acronym>, <acronym>SP-GiST</acronym> is meant to allow
+  the development of custom data types with the appropriate access methods,
+  by an expert in the domain of the data type, rather than a database expert.
+ </para>
+
+ <para>
+  Some of the information here is derived from Purdue University's
+  SP-GiST Indexing Project
+  <ulink url="http://www.cs.purdue.edu/spgist/">web site</ulink>.
+  The <acronym>SP-GiST</acronym> implementation in
+  <productname>PostgreSQL</productname> is primarily maintained by Teodor
+  Sigaev and Oleg Bartunov, and there is more information on their
+  <!-- URL will be changed -->
+  <ulink url="http://www.sai.msu.su/~megera/wiki/spgist_dev">web site</ulink>.
+ </para>
+
+</sect1>
+
+<sect1 id="spgist-extensibility">
+ <title>Extensibility</title>
+
+ <para>
+  <acronym>SP-GiST</acronym> offers an interface with a high level of
+  abstraction, requiring the access method developer to implement only
+  methods specific to a given data type. The <acronym>SP-GiST</acronym> core
+  is responsible for efficient disk mapping and searching the tree structure.
+  It also takes care of concurrency and logging considerations.
+ </para>
+
+ <para>
+  Leaf tuples of an <acronym>SP-GiST</acronym> tree contain values of the
+  same data type as the indexed column.  Leaf tuples at the root level will
+  always contain the original indexed data value, but leaf tuples at lower
+  levels might contain only a compressed representation, such as a suffix.
+  In that case the operator class support functions must be able to
+  reconstruct the original value using information accumulated from the
+  inner tuples that are passed through to reach the leaf level.
+ </para>
+
+ <para>
+  Inner tuples are more complex, since they are branching points in the
+  search tree.  Each inner tuple contains a set of one or more
+  <firstterm>nodes</>, which represent groups of similar leaf values.
+  A node contains a downlink that leads to either another, lower-level inner
+  tuple, or a short list of leaf tuples that all lie on the same index page.
+  Each node has a <firstterm>label</> that describes it; for example,
+  in a suffix tree the node label could be the next character of the string
+  value.  Optionally, an inner tuple can have a <firstterm>prefix</> value
+  that describes all its members.  In a suffix tree this could be the common
+  prefix of the represented strings.  The prefix value is not necessarily
+  really a prefix, but can be any data needed by the operator class;
+  for example, in a quad-tree it can store the central point that the four
+  quadrants are measured with respect to.  A quad-tree inner tuple would
+  then also contain four nodes corresponding to the quadrants around this
+  central point.
+ </para>
+
+ <para>
+  Some tree algorithms require knowledge of level (or depth) of the current
+  tuple, so the <acronym>SP-GiST</acronym> core provides the possibility for
+  operator classes to manage level counting while descending the tree.
+  There is also support for incrementally reconstructing the represented
+  value when that is needed.
+ </para>
+
+ <para>
+  There are five user-defined methods that an index operator class for
+  <acronym>SP-GiST</acronym> must provide.  All five follow the convention
+  of accepting two <type>internal</> arguments, the first of which is a
+  pointer to a C struct containing input values for the support method,
+  while the second argument is a pointer to a C struct where output values
+  must be placed.  Four of the methods just return <type>void</>, since
+  all their results appear in the output struct; but
+  <function>leaf_consistent</> additionally returns a <type>boolean</> result.
+  The methods must not modify any fields of their input structs.  In all
+  cases, the output struct is initialized to zeroes before calling the
+  user-defined method.
+ </para>
+
+ <para>
+  The five user-defined methods are:
+ </para>
+
+ <variablelist>
+    <varlistentry>
+     <term><function>config</></term>
+     <listitem>
+      <para>
+       Returns static information about the index implementation, including
+       the datatype OIDs of the prefix and node label data types.
+      </para>
+     <para>
+      The <acronym>SQL</> declaration of the function must look like this:
+<programlisting>
+CREATE FUNCTION my_config(internal, internal) RETURNS void ...
+</programlisting>
+      The first argument is a pointer to a <structname>spgConfigIn</>
+      C struct, containing input data for the function.
+      The second argument is a pointer to a <structname>spgConfigOut</>
+      C struct, which the function must fill with result data.
+<programlisting>
+typedef struct spgConfigIn
+{
+    Oid         attType;        /* Data type to be indexed */
+} spgConfigIn;
+
+typedef struct spgConfigOut
+{
+    Oid         prefixType;     /* Data type of inner-tuple prefixes */
+    Oid         labelType;      /* Data type of inner-tuple node labels */
+    bool        longValuesOK;   /* Opclass can cope with values &gt; 1 page */
+} spgConfigOut;
+</programlisting>
+
+      <structfield>attType</> is passed in order to support polymorphic
+      index operator classes; for ordinary fixed-data-type opclasses, it
+      will always have the same value and so can be ignored.
+     </para>
+
+     <para>
+      For operator classes that do not use prefixes,
+      <structfield>prefixType</> can be set to <literal>VOIDOID</>.
+      Likewise, for operator classes that do not use node labels,
+      <structfield>labelType</> can be set to <literal>VOIDOID</>.
+      <structfield>longValuesOK</> should be set true only when the
+      <structfield>attType</> is of variable length and the operator
+      class is capable of segmenting long values by repeated suffixing
+      (see <xref linkend="spgist-limits">).
+     </para>
+     </listitem>
+    </varlistentry>
+
+    <varlistentry>
+     <term><function>choose</></term>
+     <listitem>
+      <para>
+        Chooses a method for inserting a new value into an inner tuple.
+      </para>
+
+     <para>
+      The <acronym>SQL</> declaration of the function must look like this:
+<programlisting>
+CREATE FUNCTION my_choose(internal, internal) RETURNS void ...
+</programlisting>
+      The first argument is a pointer to a <structname>spgChooseIn</>
+      C struct, containing input data for the function.
+      The second argument is a pointer to a <structname>spgChooseOut</>
+      C struct, which the function must fill with result data.
+<programlisting>
+typedef struct spgChooseIn
+{
+    Datum       datum;          /* original datum to be indexed */
+    Datum       leafDatum;      /* current datum to be stored at leaf */
+    int         level;          /* current level (counting from zero) */
+
+    /* Data from current inner tuple */
+    bool        allTheSame;     /* tuple is marked all-the-same? */
+    bool        hasPrefix;      /* tuple has a prefix? */
+    Datum       prefixDatum;    /* if so, the prefix value */
+    int         nNodes;         /* number of nodes in the inner tuple */
+    Datum      *nodeLabels;     /* node label values (NULL if none) */
+} spgChooseIn;
+
+typedef enum spgChooseResultType
+{
+    spgMatchNode = 1,           /* descend into existing node */
+    spgAddNode,                 /* add a node to the inner tuple */
+    spgSplitTuple               /* split inner tuple (change its prefix) */
+} spgChooseResultType;
+
+typedef struct spgChooseOut
+{
+    spgChooseResultType resultType;     /* action code, see above */
+    union
+    {
+        struct                  /* results for spgMatchNode */
+        {
+            int         nodeN;      /* descend to this node (index from 0) */
+            int         levelAdd;   /* increment level by this much */
+            Datum       restDatum;  /* new leaf datum */
+        }           matchNode;
+        struct                  /* results for spgAddNode */
+        {
+            Datum       nodeLabel;  /* new node's label */
+            int         nodeN;      /* where to insert it (index from 0) */
+        }           addNode;
+        struct                  /* results for spgSplitTuple */
+        {
+            /* Info to form new inner tuple with one node */
+            bool        prefixHasPrefix;    /* tuple should have a prefix? */
+            Datum       prefixPrefixDatum;  /* if so, its value */
+            Datum       nodeLabel;          /* node's label */
+
+            /* Info to form new lower-level inner tuple with all old nodes */
+            bool        postfixHasPrefix;   /* tuple should have a prefix? */
+            Datum       postfixPrefixDatum; /* if so, its value */
+        }           splitTuple;
+    }           result;
+} spgChooseOut;
+</programlisting>
+
+       <structfield>datum</> is the original datum that was to be inserted
+       into the index.
+       <structfield>leafDatum</> is initially the same as
+       <structfield>datum</>, but can change at lower levels of the tree
+       if the <function>choose</function> or <function>picksplit</function>
+       methods change it.  When the insertion search reaches a leaf page,
+       the current value of <structfield>leafDatum</> is what will be stored
+       in the newly created leaf tuple.
+       <structfield>level</> is the current inner tuple's level, starting at
+       zero for the root level.
+       <structfield>allTheSame</> is true if the current inner tuple is
+       marked as containing multiple equivalent nodes
+       (see <xref linkend="spgist-all-the-same">).
+       <structfield>hasPrefix</> is true if the current inner tuple contains
+       a prefix; if so,
+       <structfield>prefixDatum</> is its value.
+       <structfield>nNodes</> is the number of child nodes contained in the
+       inner tuple, and
+       <structfield>nodeLabels</> is an array of their label values, or
+       NULL if there are no labels.
+      </para>
+
+      <para>
+       The <function>choose</function> function can determine either that
+       the new value matches one of the existing child nodes, or that a new
+       child node must be added, or that the new value is inconsistent with
+       the tuple prefix and so the inner tuple must be split to create a
+       less restrictive prefix.
+      </para>
+
+      <para>
+       If the new value matches one of the existing child nodes,
+       set <structfield>resultType</> to <literal>spgMatchNode</>.
+       Set <structfield>nodeN</> to the index (from zero) of that node in
+       the node array.
+       Set <structfield>levelAdd</> to the increment in
+       <structfield>level</> caused by descending through that node,
+       or leave it as zero if the operator class does not use levels.
+       Set <structfield>restDatum</> to equal <structfield>datum</>
+       if the operator class does not modify datums from one level to the
+       next, or otherwise set it to the modified value to be used as
+       <structfield>leafDatum</> at the next level.
+      </para>
+
+      <para>
+       If a new child node must be added,
+       set <structfield>resultType</> to <literal>spgAddNode</>.
+       Set <structfield>nodeLabel</> to the label to be used for the new
+       node, and set <structfield>nodeN</> to the index (from zero) at which
+       to insert the node in the node array.
+       After the node has been added, the <function>choose</function>
+       function will be called again with the modified inner tuple;
+       that call should result in an <literal>spgMatchNode</> result.
+      </para>
+
+      <para>
+       If the new value is inconsistent with the tuple prefix,
+       set <structfield>resultType</> to <literal>spgSplitTuple</>.
+       This action moves all the existing nodes into a new lower-level
+       inner tuple, and replaces the existing inner tuple with a tuple
+       having a single node that links to the new lower-level inner tuple.
+       Set <structfield>prefixHasPrefix</> to indicate whether the new
+       upper tuple should have a prefix, and if so set
+       <structfield>prefixPrefixDatum</> to the prefix value.  This new
+       prefix value must be sufficiently less restrictive than the original
+       to accept the new value to be indexed, and it should be no longer
+       than the original prefix.
+       Set <structfield>nodeLabel</> to the label to be used for the
+       node that will point to the new lower-level inner tuple.
+       Set <structfield>postfixHasPrefix</> to indicate whether the new
+       lower-level inner tuple should have a prefix, and if so set
+       <structfield>postfixPrefixDatum</> to the prefix value.  The
+       combination of these two prefixes and the additional label must
+       have the same meaning as the original prefix, because there is
+       no opportunity to alter the node labels that are moved to the new
+       lower-level tuple, nor to change any child index entries.
+       After the node has been split, the <function>choose</function>
+       function will be called again with the replacement inner tuple.
+       That call will usually result in an <literal>spgAddNode</> result,
+       since presumably the node label added in the split step will not
+       match the new value; so after that, there will be a third call
+       that finally returns <literal>spgMatchNode</> and allows the
+       insertion to descend to the leaf level.
+      </para>
+     </listitem>
+    </varlistentry>
+
+    <varlistentry>
+     <term><function>picksplit</></term>
+     <listitem>
+      <para>
+       Decides how to create a new inner tuple over a set of leaf tuples.
+      </para>
+
+      <para>
+        The <acronym>SQL</> declaration of the function must look like this:
+<programlisting>
+CREATE FUNCTION my_picksplit(internal, internal) RETURNS void ...
+</programlisting>
+      The first argument is a pointer to a <structname>spgPickSplitIn</>
+      C struct, containing input data for the function.
+      The second argument is a pointer to a <structname>spgPickSplitOut</>
+      C struct, which the function must fill with result data.
+<programlisting>
+typedef struct spgPickSplitIn
+{
+    int         nTuples;        /* number of leaf tuples */
+    Datum      *datums;         /* their datums (array of length nTuples) */
+    int         level;          /* current level (counting from zero) */
+} spgPickSplitIn;
+
+typedef struct spgPickSplitOut
+{
+    bool        hasPrefix;      /* new inner tuple should have a prefix? */
+    Datum       prefixDatum;    /* if so, its value */
+
+    int         nNodes;         /* number of nodes for new inner tuple */
+    Datum      *nodeLabels;     /* their labels (or NULL for no labels) */
+
+    int        *mapTuplesToNodes;   /* node index for each leaf tuple */
+    Datum      *leafTupleDatums;    /* datum to store in each new leaf tuple */
+} spgPickSplitOut;
+</programlisting>
+
+       <structfield>nTuples</> is the number of leaf tuples provided.
+       <structfield>datums</> is an array of their datum values.
+       <structfield>level</> is the current level that all the leaf tuples
+       share, which will become the level of the new inner tuple.
+      </para>
+
+      <para>
+       Set <structfield>hasPrefix</> to indicate whether the new inner
+       tuple should have a prefix, and if so set
+       <structfield>prefixDatum</> to the prefix value.
+       Set <structfield>nNodes</> to indicate the number of nodes that
+       the new inner tuple will contain, and
+       set <structfield>nodeLabels</> to an array of their label values.
+       (If the nodes do not require labels, set <structfield>nodeLabels</>
+       to NULL; see <xref linkend="spgist-null-labels"> for details.)
+       Set <structfield>mapTuplesToNodes</> to an array that gives the index
+       (from zero) of the node that each leaf tuple should be assigned to.
+       Set <structfield>leafTupleDatums</> to an array of the values to
+       be stored in the new leaf tuples (these will be the same as the
+       input <structfield>datums</> if the operator class does not modify
+       datums from one level to the next).
+       Note that the <function>picksplit</> function is
+       responsible for palloc'ing the
+       <structfield>nodeLabels</>, <structfield>mapTuplesToNodes</> and
+       <structfield>leafTupleDatums</> arrays.
+      </para>
+
+      <para>
+       If more than one leaf tuple is supplied, it is expected that the
+       <function>picksplit</> function will classify them into more than
+       one node; otherwise it is not possible to split the leaf tuples
+       across multiple pages, which is the ultimate purpose of this
+       operation.  Therefore, if the <function>picksplit</> function
+       ends up placing all the leaf tuples in the same node, the core
+       SP-GiST code will override that decision and generate an inner
+       tuple in which the leaf tuples are assigned at random to several
+       identically-labeled nodes.  Such a tuple is marked
+       <literal>allTheSame</> to signify that this has happened.  The
+       <function>choose</> and <function>inner_consistent</> functions
+       must take suitable care with such inner tuples.
+       See <xref linkend="spgist-all-the-same"> for more information.
+      </para>
+
+      <para>
+       <function>picksplit</> can be applied to a single leaf tuple only
+       in the case that the <function>config</> function set
+       <structfield>longValuesOK</> to true and a larger-than-a-page input
+       value has been supplied.  In this case the point of the operation is
+       to strip off a prefix and produce a new, shorter leaf datum value.
+       The call will be repeated until a leaf datum short enough to fit on
+       a page has been produced.  See <xref linkend="spgist-limits"> for
+       more information.
+      </para>
+     </listitem>
+    </varlistentry>
+
+    <varlistentry>
+     <term><function>inner_consistent</></term>
+     <listitem>
+      <para>
+       Returns set of nodes (branches) to follow during tree search.
+      </para>
+
+      <para>
+       The <acronym>SQL</> declaration of the function must look like this:
+<programlisting>
+CREATE FUNCTION my_inner_consistent(internal, internal) RETURNS void ...
+</programlisting>
+      The first argument is a pointer to a <structname>spgInnerConsistentIn</>
+      C struct, containing input data for the function.
+      The second argument is a pointer to a <structname>spgInnerConsistentOut</>
+      C struct, which the function must fill with result data.
+
+<programlisting>
+typedef struct spgInnerConsistentIn
+{
+    StrategyNumber strategy;    /* operator strategy number */
+    Datum       query;          /* operator's RHS value */
+
+    Datum       reconstructedValue;     /* value reconstructed at parent */
+    int         level;          /* current level (counting from zero) */
+
+    /* Data from current inner tuple */
+    bool        allTheSame;     /* tuple is marked all-the-same? */
+    bool        hasPrefix;      /* tuple has a prefix? */
+    Datum       prefixDatum;    /* if so, the prefix value */
+    int         nNodes;         /* number of nodes in the inner tuple */
+    Datum      *nodeLabels;     /* node label values (NULL if none) */
+} spgInnerConsistentIn;
+
+typedef struct spgInnerConsistentOut
+{
+    int         nNodes;         /* number of child nodes to be visited */
+    int        *nodeNumbers;    /* their indexes in the node array */
+    int        *levelAdds;      /* increment level by this much for each */
+    Datum      *reconstructedValues;    /* associated reconstructed values */
+} spgInnerConsistentOut;
+</programlisting>
+
+       <structfield>strategy</> and
+       <structfield>query</> describe the index search condition.
+       <structfield>reconstructedValue</> is the value reconstructed for the
+       parent tuple; it is <literal>(Datum) 0</> at the root level or if the
+       <function>inner_consistent</> function did not provide a value at the
+       parent level.
+       <structfield>level</> is the current inner tuple's level, starting at
+       zero for the root level.
+       <structfield>allTheSame</> is true if the current inner tuple is
+       marked <quote>all-the-same</>; in this case all the nodes have the
+       same label (if any) and so either all or none of them match the query
+       (see <xref linkend="spgist-all-the-same">).
+       <structfield>hasPrefix</> is true if the current inner tuple contains
+       a prefix; if so,
+       <structfield>prefixDatum</> is its value.
+       <structfield>nNodes</> is the number of child nodes contained in the
+       inner tuple, and
+       <structfield>nodeLabels</> is an array of their label values, or
+       NULL if the nodes do not have labels.
+      </para>
+
+      <para>
+       <structfield>nNodes</> must be set to the number of child nodes that
+       need to be visited by the search, and
+       <structfield>nodeNumbers</> must be set to an array of their indexes.
+       If the operator class keeps track of levels, set
+       <structfield>levelAdds</> to an array of the level increments
+       required when descending to each node to be visited.  (Often these
+       increments will be the same for all the nodes, but that's not
+       necessarily so, so an array is used.)
+       If value reconstruction is needed, set
+       <structfield>reconstructedValues</> to an array of the values
+       reconstructed for each child node to be visited; otherwise, leave
+       <structfield>reconstructedValues</> as NULL.
+       Note that the <function>inner_consistent</> function is
+       responsible for palloc'ing the
+       <structfield>nodeNumbers</>, <structfield>levelAdds</> and
+       <structfield>reconstructedValues</> arrays.
+      </para>
+     </listitem>
+    </varlistentry>
+
+    <varlistentry>
+     <term><function>leaf_consistent</></term>
+     <listitem>
+      <para>
+       Returns true if a leaf tuple satisfies a query.
+      </para>
+
+      <para>
+       The <acronym>SQL</> declaration of the function must look like this:
+<programlisting>
+CREATE FUNCTION my_leaf_consistent(internal, internal) RETURNS bool ...
+</programlisting>
+      The first argument is a pointer to a <structname>spgLeafConsistentIn</>
+      C struct, containing input data for the function.
+      The second argument is a pointer to a <structname>spgLeafConsistentOut</>
+      C struct, which the function must fill with result data.
+<programlisting>
+typedef struct spgLeafConsistentIn
+{
+    StrategyNumber strategy;    /* operator strategy number */
+    Datum       query;          /* operator's RHS value */
+
+    Datum       reconstructedValue;     /* value reconstructed at parent */
+    int         level;          /* current level (counting from zero) */
+
+    Datum       leafDatum;      /* datum in leaf tuple */
+} spgLeafConsistentIn;
+
+typedef struct spgLeafConsistentOut
+{
+    bool        recheck;        /* set true if operator must be rechecked */
+} spgLeafConsistentOut;
+</programlisting>
+
+       <structfield>strategy</> and
+       <structfield>query</> define the index search condition.
+       <structfield>reconstructedValue</> is the value reconstructed for the
+       parent tuple; it is <literal>(Datum) 0</> at the root level or if the
+       <function>inner_consistent</> function did not provide a value at the
+       parent level.
+       <structfield>level</> is the current leaf tuple's level, starting at
+       zero for the root level.
+       <structfield>leafDatum</> is the key value stored in the current
+       leaf tuple.
+      </para>
+
+      <para>
+       The function must return <literal>true</> if the leaf tuple matches the
+       query, or <literal>false</> if not.  In the <literal>true</> case,
+       <structfield>recheck</> may be set to <literal>true</> if the match
+       is uncertain and so the operator must be re-applied to the actual heap
+       tuple to verify the match.
+      </para>
+     </listitem>
+    </varlistentry>
+   </variablelist>
+
+  <para>
+   All the SP-GiST support methods are normally called in a short-lived
+   memory context; that is, <varname>CurrentMemoryContext</> will be reset
+   after processing of each tuple.  It is therefore not very important to
+   worry about pfree'ing everything you palloc.  (The <function>config</>
+   method is an exception: it should try to avoid leaking memory.  But
+   usually the <function>config</> method need do nothing but assign
+   constants into the passed parameter struct.)
+  </para>
+
+  <para>
+   If the indexed column is of a collatable data type, the index collation
+   will be passed to all the support methods, using the standard
+   <function>PG_GET_COLLATION()</> mechanism.
+  </para>
+
+</sect1>
+
+<sect1 id="spgist-implementation">
+ <title>Implementation</title>
+
+  <para>
+   This section covers implementation details and other tricks that are
+   useful for implementors of <acronym>SP-GiST</acronym> operator classes to
+   know.
+  </para>
+
+ <sect2 id="spgist-limits">
+  <title>SP-GiST Limits</title>
+
+  <para>
+   Individual leaf tuples and inner tuples must fit on a single index page
+   (8KB by default).  Therefore, when indexing values of variable-length
+   data types, long values can only be supported by methods such as suffix
+   trees, in which each level of the tree includes a prefix that is short
+   enough to fit on a page, and the final leaf level includes a suffix also
+   short enough to fit on a page.  The operator class should set
+   <structfield>longValuesOK</> to TRUE only if it is prepared to arrange for
+   this to happen.  Otherwise, the <acronym>SP-GiST</acronym> core will
+   reject any request to index a value that is too large to fit
+   on an index page.
+  </para>
+
+  <para>
+   Likewise, it is the operator class's responsibility that inner tuples
+   do not grow too large to fit on an index page; this limits the number
+   of child nodes that can be used in one inner tuple, as well as the
+   maximum size of a prefix value.
+  </para>
+
+  <para>
+   Another limitation is that when an inner tuple's node points to a set
+   of leaf tuples, those tuples must all be in the same index page.
+   (This is a design decision to reduce seeking and save space in the
+   links that chain such tuples together.)  If the set of leaf tuples
+   grows too large for a page, a split is performed and an intermediate
+   inner tuple is inserted.  For this to fix the problem, the new inner
+   tuple <emphasis>must</> divide the set of leaf values into more than one
+   node group.  If the operator class's <function>picksplit</> function
+   fails to do that, the <acronym>SP-GiST</acronym> core resorts to
+   extraordinary measures described in <xref linkend="spgist-all-the-same">.
+  </para>
+ </sect2>
+
+ <sect2 id="spgist-null-labels">
+  <title>SP-GiST Without Node Labels</title>
+
+  <para>
+   Some tree algorithms use a fixed set of nodes for each inner tuple;
+   for example, in a quad-tree there are always exactly four nodes
+   corresponding to the four quadrants around the inner tuple's centroid
+   point.  In such a case the code typically works with the nodes by
+   number, and there is no need for explicit node labels.  To suppress
+   node labels (and thereby save some space), the <function>picksplit</>
+   function can return NULL for the <structfield>nodeLabels</> array.
+   This will in turn result in <structfield>nodeLabels</> being NULL during
+   subsequent calls to <function>choose</> and <function>inner_consistent</>.
+   In principle, node labels could be used for some inner tuples and omitted
+   for others in the same index.
+  </para>
+
+  <para>
+   When working with an inner tuple having unlabeled nodes, it is an error
+   for <function>choose</> to return <literal>spgAddNode</>, since the set
+   of nodes is supposed to be fixed in such cases.  Also, there is no
+   provision for generating an unlabeled node in <literal>spgSplitTuple</>
+   actions, since it is expected that an <literal>spgAddNode</> action will
+   be needed as well.
+  </para>
+ </sect2>
+
+ <sect2 id="spgist-all-the-same">
+  <title><quote>All-the-same</> Inner Tuples</title>
+
+  <para>
+   The <acronym>SP-GiST</acronym> core can override the results of the
+   operator class's <function>picksplit</> function when
+   <function>picksplit</> fails to divide the supplied leaf values into
+   at least two node categories.  When this happens, the new inner tuple
+   is created with multiple nodes that each have the same label (if any)
+   that <function>picksplit</> gave to the one node it did use, and the
+   leaf values are divided at random among these equivalent nodes.
+   The <literal>allTheSame</> flag is set on the inner tuple to warn the
+   <function>choose</> and <function>inner_consistent</> functions that the
+   tuple does not have the node set that they might otherwise expect.
+  </para>
+
+  <para>
+   When dealing with an <literal>allTheSame</> tuple, a <function>choose</>
+   result of <literal>spgMatchNode</> is interpreted to mean that the new
+   value can be assigned to any of the equivalent nodes; the core code will
+   ignore the supplied  <structfield>nodeN</> value and descend into one
+   of the nodes at random (so as to keep the tree balanced).  It is an
+   error for <function>choose</> to return <literal>spgAddNode</>, since
+   that would make the nodes not all equivalent; the
+   <literal>spgSplitTuple</> action must be used if the value to be inserted
+   doesn't match the existing nodes.
+  </para>
+
+  <para>
+   When dealing with an <literal>allTheSame</> tuple, the
+   <function>inner_consistent</> function should return either all or none
+   of the nodes as targets for continuing the index search, since they are
+   all equivalent.  This may or may not require any special-case code,
+   depending on how much the <function>inner_consistent</> function normally
+   assumes about the meaning of the nodes.
+  </para>
+ </sect2>
+
+</sect1>
+
+<sect1 id="spgist-examples">
+ <title>Examples</title>
+
+ <para>
+  The <productname>PostgreSQL</productname> source distribution includes
+  several examples of index operator classes for
+  <acronym>SP-GiST</acronym>.  The core system currently provides suffix
+  trees over text columns and two types of trees over points: quad-tree and
+  k-d tree.  Look into <filename>src/backend/access/spgist/</> to see the
+  code.
+ </para>
+
+</sect1>
+
+</chapter>