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

<sect1 id="cube" xreflabel="cube">
 <title>cube</title>

 <indexterm zone="cube">
  <primary>cube (extension)</primary>
 </indexterm>

 <para>
  This module implements a data type <type>cube</> for
  representing multidimensional cubes.
 </para>

 <sect2>
  <title>Syntax</title>

  <para>
   <xref linkend="cube-repr-table"> shows the valid external
   representations for the <type>cube</>
   type.  <replaceable>x</>, <replaceable>y</>, etc. denote
   floating-point numbers.
  </para>

  <table id="cube-repr-table">
   <title>Cube External Representations</title>
   <tgroup cols="2">
    <tbody>
     <row>
      <entry><literal><replaceable>x</></literal></entry>
      <entry>A one-dimensional point
       (or, zero-length one-dimensional interval)
      </entry>
     </row>
     <row>
      <entry><literal>(<replaceable>x</>)</literal></entry>
      <entry>Same as above</entry>
     </row>
     <row>
      <entry><literal><replaceable>x1</>,<replaceable>x2</>,...,<replaceable>xn</></literal></entry>
      <entry>A point in n-dimensional space, represented internally as a
      zero-volume cube
      </entry>
     </row>
     <row>
      <entry><literal>(<replaceable>x1</>,<replaceable>x2</>,...,<replaceable>xn</>)</literal></entry>
      <entry>Same as above</entry>
     </row>
     <row>
      <entry><literal>(<replaceable>x</>),(<replaceable>y</>)</literal></entry>
      <entry>A one-dimensional interval starting at <replaceable>x</> and ending at <replaceable>y</> or vice versa; the
       order does not matter
      </entry>
     </row>
     <row>
      <entry><literal>[(<replaceable>x</>),(<replaceable>y</>)]</literal></entry>
      <entry>Same as above</entry>
     </row>
     <row>
      <entry><literal>(<replaceable>x1</>,...,<replaceable>xn</>),(<replaceable>y1</>,...,<replaceable>yn</>)</literal></entry>
      <entry>An n-dimensional cube represented by a pair of its diagonally
       opposite corners
      </entry>
     </row>
     <row>
      <entry><literal>[(<replaceable>x1</>,...,<replaceable>xn</>),(<replaceable>y1</>,...,<replaceable>yn</>)]</literal></entry>
      <entry>Same as above</entry>
     </row>
    </tbody>
   </tgroup>
  </table>

  <para>
   It does not matter which order the opposite corners of a cube are
   entered in.  The <type>cube</> functions
   automatically swap values if needed to create a uniform
   <quote>lower left &mdash; upper right</> internal representation.
   When corners coincide cube stores only one corner along with a
   special flag in order to reduce size wasted.
  </para>

  <para>
   White space is ignored, so <literal>[(<replaceable>x</>),(<replaceable>y</>)]</literal> is the same as
   <literal>[ ( <replaceable>x</> ), ( <replaceable>y</> ) ]</literal>.
  </para>
 </sect2>

 <sect2>
  <title>Precision</title>

  <para>
   Values are stored internally as 64-bit floating point numbers. This means
   that numbers with more than about 16 significant digits will be truncated.
  </para>
 </sect2>

 <sect2>
  <title>Usage</title>

  <para>
   The <filename>cube</> module includes a GiST index operator class for
   <type>cube</> values.
   The operators supported by the GiST operator class are shown in <xref linkend="cube-gist-operators">.
  </para>

  <table id="cube-gist-operators">
   <title>Cube GiST Operators</title>
   <tgroup cols="2">
    <thead>
     <row>
      <entry>Operator</entry>
      <entry>Description</entry>
     </row>
    </thead>

    <tbody>
     <row>
      <entry><literal>a = b</></entry>
      <entry>The cubes a and b are identical.</entry>
     </row>

     <row>
      <entry><literal>a &amp;&amp; b</></entry>
      <entry>The cubes a and b overlap.</entry>
     </row>

     <row>
      <entry><literal>a @&gt; b</></entry>
      <entry>The cube a contains the cube b.</entry>
     </row>

     <row>
      <entry><literal>a &lt;@ b</></entry>
      <entry>The cube a is contained in the cube b.</entry>
     </row>

     <row>
      <entry><literal>a -&gt; n</></entry>
      <entry>Get n-th coordinate of cube.</entry>
     </row>

     <row>
      <entry><literal>a ~&gt; n</></entry>
      <entry>
        Get n-th coordinate in 'normalized' cube representation. Noramlization
        means coordinate rearrangement to form (lower left, upper right).
      </entry>
     </row>
    </tbody>
   </tgroup>
  </table>

  <para>
   (Before PostgreSQL 8.2, the containment operators <literal>@&gt;</> and <literal>&lt;@</> were
   respectively called <literal>@</> and <literal>~</>.  These names are still available, but are
   deprecated and will eventually be retired.  Notice that the old names
   are reversed from the convention formerly followed by the core geometric
   data types!)
  </para>

  <para>
   GiST index can be used to retrieve nearest neighbours via several metric
   operators. As always any of them can be used as ordinary function.
  </para>

  <table id="cube-gistknn-operators">
   <title>Cube GiST-kNN Operators</title>
   <tgroup cols="2">
    <thead>
     <row>
      <entry>Operator</entry>
      <entry>Description</entry>
     </row>
    </thead>
    <tbody>
     <row>
      <entry><literal>a &lt;-&gt; b</></entry>
      <entry>Euclidean distance between a and b</entry>
     </row>

     <row>
      <entry><literal>a &lt;#&gt; b</></entry>
      <entry>Taxicab (L-1 metric) distance between a and b</entry>
     </row>

     <row>
      <entry><literal>a &lt;=&gt; b</></entry>
      <entry>Chebyshev (L-inf metric) distance between a and b</entry>
     </row>
    </tbody>
   </tgroup>
  </table>

  <para>
   Selection of nearing neigbours can be done in the following way:
  </para>
<programlisting>
SELECT c FROM test 
ORDER BY cube(array[0.5,0.5,0.5])<->c 
LIMIT 1;
</programlisting>


  <para>
   Also kNN framework allows us to cheat with metrics in order to get results
   sorted by selected coodinate directly from the index without extra sorting
   step. That technique significantly faster on small values of LIMIT, however
   with bigger values of LIMIT planner will switch automatically to standart
   index scan and sort.
   That behavior can be achieved using coordinate operator
   (cube c)~&gt;(int offset).
  </para>
<programlisting>
=> select cube(array[0.41,0.42,0.43])~>2 as coord;
 coord 
-------
  0.42
(1 row)
</programlisting>

  <para>
   So using that operator as kNN metric we can obtain cubes sorted by it's
   coordinate.
  </para>
  <para>
   To get cubes ordered by first coordinate of lower left corner ascending
   one can use the following query:
  </para>
<programlisting>
SELECT c FROM test ORDER BY c~>1 LIMIT 5;
</programlisting>
  <para>
   And to get cubes descending by first coordinate of upper right corner
   of 2d-cube:
  </para>
<programlisting>
SELECT c FROM test ORDER BY c~>3 DESC LIMIT 5;
</programlisting>



  <para>
   The standard B-tree operators are also provided, for example

   <informaltable>
    <tgroup cols="2">
     <thead>
      <row>
       <entry>Operator</entry>
       <entry>Description</entry>
      </row>
     </thead>

     <tbody>
      <row>
       <entry><literal>[a, b] &lt; [c, d]</literal></entry>
       <entry>Less than</entry>
      </row>

      <row>
       <entry><literal>[a, b] &gt; [c, d]</literal></entry>
       <entry>Greater than</entry>
      </row>
     </tbody>
    </tgroup>
   </informaltable>

   These operators do not make a lot of sense for any practical
   purpose but sorting. These operators first compare (a) to (c),
   and if these are equal, compare (b) to (d). That results in
   reasonably good sorting in most cases, which is useful if
   you want to use ORDER BY with this type.
  </para>

  <para>
   <xref linkend="cube-functions-table"> shows the available functions.
  </para>

  <table id="cube-functions-table">
   <title>Cube Functions</title>
   <tgroup cols="2">
    <tbody>
     <row>
      <entry><literal>cube(float8) returns cube</literal></entry>
      <entry>Makes a one dimensional cube with both coordinates the same.
       <literal>cube(1) == '(1)'</literal>
      </entry>
     </row>

     <row>
      <entry><literal>cube(float8, float8) returns cube</literal></entry>
      <entry>Makes a one dimensional cube.
       <literal>cube(1,2) == '(1),(2)'</literal>
      </entry>
     </row>

     <row>
      <entry><literal>cube(float8[]) returns cube</literal></entry>
      <entry>Makes a zero-volume cube using the coordinates
       defined by the array.
       <literal>cube(ARRAY[1,2]) == '(1,2)'</literal>
      </entry>
     </row>

     <row>
      <entry><literal>cube(float8[], float8[]) returns cube</literal></entry>
      <entry>Makes a cube with upper right and lower left
       coordinates as defined by the two arrays, which must be of the
       same length.
       <literal>cube('{1,2}'::float[], '{3,4}'::float[]) == '(1,2),(3,4)'
       </literal>
      </entry>
     </row>

     <row>
      <entry><literal>cube(cube, float8) returns cube</literal></entry>
      <entry>Makes a new cube by adding a dimension on to an
       existing cube with the same values for both parts of the new coordinate.
       This is useful for building cubes piece by piece from calculated values.
       <literal>cube('(1)',2) == '(1,2),(1,2)'</literal>
      </entry>
     </row>

     <row>
      <entry><literal>cube(cube, float8, float8) returns cube</literal></entry>
      <entry>Makes a new cube by adding a dimension on to an
       existing cube. This is useful for building cubes piece by piece from
       calculated values. <literal>cube('(1,2)',3,4) == '(1,3),(2,4)'</literal>
      </entry>
     </row>

     <row>
      <entry><literal>cube_dim(cube) returns int</literal></entry>
      <entry>Returns the number of dimensions of the cube
      </entry>
     </row>

     <row>
      <entry><literal>cube_ll_coord(cube, int) returns double </literal></entry>
      <entry>Returns the n'th coordinate value for the lower left
       corner of a cube
      </entry>
     </row>

    <row>
      <entry><literal>cube_ur_coord(cube, int) returns double
      </literal></entry>
      <entry>Returns the n'th coordinate value for the
       upper right corner of a cube
      </entry>
     </row>

     <row>
      <entry><literal>cube_is_point(cube) returns bool</literal></entry>
      <entry>Returns true if a cube is a point, that is,
       the two defining corners are the same.</entry>
     </row>

     <row>
      <entry><literal>cube_distance(cube, cube) returns double</literal></entry>
      <entry>Returns the distance between two cubes. If both
       cubes are points, this is the normal distance function.
      </entry>
     </row>

     <row>
      <entry><literal>cube_subset(cube, int[]) returns cube
      </literal></entry>
      <entry>Makes a new cube from an existing cube, using a list of
       dimension indexes from an array. Can be used to find both the LL and UR
       coordinates of a single dimension, e.g.
       <literal>cube_subset(cube('(1,3,5),(6,7,8)'), ARRAY[2]) = '(3),(7)'</>.
       Or can be used to drop dimensions, or reorder them as desired, e.g.
       <literal>cube_subset(cube('(1,3,5),(6,7,8)'), ARRAY[3,2,1,1]) = '(5, 3,
       1, 1),(8, 7, 6, 6)'</>.
      </entry>
     </row>

     <row>
      <entry><literal>cube_union(cube, cube) returns cube</literal></entry>
      <entry>Produces the union of two cubes
      </entry>
     </row>

     <row>
      <entry><literal>cube_inter(cube, cube) returns cube</literal></entry>
      <entry>Produces the intersection of two cubes
      </entry>
     </row>

     <row>
      <entry><literal>cube_enlarge(cube c, double r, int n) returns cube</literal></entry>
      <entry>Increases the size of a cube by a specified radius in at least
       n dimensions. If the radius is negative the cube is shrunk instead. This
       is useful for creating bounding boxes around a point for searching for
       nearby points. All defined dimensions are changed by the radius r.
       LL coordinates are decreased by r and UR coordinates are increased by r.
       If a LL coordinate is increased to larger than the corresponding UR
       coordinate (this can only happen when r &lt; 0) than both coordinates
       are set to their average.  If n is greater than the number of defined
       dimensions and the cube is being increased (r &gt;= 0) then 0 is used
       as the base for the extra coordinates.
      </entry>
     </row>
    </tbody>
   </tgroup>
  </table>
 </sect2>

 <sect2>
  <title>Defaults</title>

  <para>
   I believe this union:
  </para>
<programlisting>
select cube_union('(0,5,2),(2,3,1)', '0');
cube_union
-------------------
(0, 0, 0),(2, 5, 2)
(1 row)
</programlisting>

   <para>
    does not contradict common sense, neither does the intersection
   </para>

<programlisting>
select cube_inter('(0,-1),(1,1)', '(-2),(2)');
cube_inter
-------------
(0, 0),(1, 0)
(1 row)
</programlisting>

   <para>
    In all binary operations on differently-dimensioned cubes, I assume the
    lower-dimensional one to be a Cartesian projection, i. e., having zeroes
    in place of coordinates omitted in the string representation. The above
    examples are equivalent to:
   </para>

<programlisting>
cube_union('(0,5,2),(2,3,1)','(0,0,0),(0,0,0)');
cube_inter('(0,-1),(1,1)','(-2,0),(2,0)');
</programlisting>

   <para>
    The following containment predicate uses the point syntax,
    while in fact the second argument is internally represented by a box.
    This syntax makes it unnecessary to define a separate point type
    and functions for (box,point) predicates.
   </para>

<programlisting>
select cube_contains('(0,0),(1,1)', '0.5,0.5');
cube_contains
--------------
t
(1 row)
</programlisting>
 </sect2>

 <sect2>
  <title>Notes</title>

  <para>
   For examples of usage, see the regression test <filename>sql/cube.sql</>.
  </para>

  <para>
   To make it harder for people to break things, there
   is a limit of 100 on the number of dimensions of cubes. This is set
   in <filename>cubedata.h</> if you need something bigger.
  </para>
 </sect2>

 <sect2>
  <title>Credits</title>

  <para>
   Original author: Gene Selkov, Jr. <email>selkovjr@mcs.anl.gov</email>,
   Mathematics and Computer Science Division, Argonne National Laboratory.
  </para>

  <para>
   My thanks are primarily to Prof. Joe Hellerstein
   (<ulink url="http://db.cs.berkeley.edu/jmh/"></ulink>) for elucidating the
   gist of the GiST (<ulink url="http://gist.cs.berkeley.edu/"></ulink>), and
   to his former student Andy Dong for his example written for Illustra.
   I am also grateful to all Postgres developers, present and past, for
   enabling myself to create my own world and live undisturbed in it. And I
   would like to acknowledge my gratitude to Argonne Lab and to the
   U.S. Department of Energy for the years of faithful support of my database
   research.
  </para>

  <para>
   Minor updates to this package were made by Bruno Wolff III
   <email>bruno@wolff.to</email> in August/September of 2002. These include
   changing the precision from single precision to double precision and adding
   some new functions.
  </para>

  <para>
   Additional updates were made by Joshua Reich <email>josh@root.net</email> in
   July 2006. These include <literal>cube(float8[], float8[])</literal> and
   cleaning up the code to use the V1 call protocol instead of the deprecated
   V0 protocol.
  </para>
 </sect2>

</sect1>
