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814 lines
25 KiB
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814 lines
25 KiB
Plaintext
<!-- doc/src/sgml/brin.sgml -->
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<chapter id="BRIN">
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<title>BRIN Indexes</title>
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<indexterm>
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<primary>index</primary>
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<secondary>BRIN</secondary>
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</indexterm>
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<sect1 id="brin-intro">
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<title>Introduction</title>
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<para>
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<acronym>BRIN</acronym> stands for Block Range Index.
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<acronym>BRIN</acronym> is designed for handling very large tables
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in which certain columns have some natural correlation with their
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physical location within the table.
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A <firstterm>block range</> is a group of pages that are physically
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adjacent in the table; for each block range, some summary info is stored
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by the index.
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For example, a table storing a store's sale orders might have
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a date column on which each order was placed, and most of the time
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the entries for earlier orders will appear earlier in the table as well;
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a table storing a ZIP code column might have all codes for a city
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grouped together naturally.
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</para>
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<para>
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<acronym>BRIN</acronym> indexes can satisfy queries via regular bitmap
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index scans, and will return all tuples in all pages within each range if
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the summary info stored by the index is <firstterm>consistent</> with the
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query conditions.
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The query executor is in charge of rechecking these tuples and discarding
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those that do not match the query conditions — in other words, these
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indexes are lossy.
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Because a <acronym>BRIN</acronym> index is very small, scanning the index
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adds little overhead compared to a sequential scan, but may avoid scanning
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large parts of the table that are known not to contain matching tuples.
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</para>
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<para>
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The specific data that a <acronym>BRIN</acronym> index will store,
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as well as the specific queries that the index will be able to satisfy,
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depend on the operator class selected for each column of the index.
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Data types having a linear sort order can have operator classes that
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store the minimum and maximum value within each block range, for instance;
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geometrical types might store the bounding box for all the objects
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in the block range.
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</para>
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<para>
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The size of the block range is determined at index creation time by
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the <literal>pages_per_range</> storage parameter. The number of index
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entries will be equal to the size of the relation in pages divided by
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the selected value for <literal>pages_per_range</>. Therefore, the smaller
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the number, the larger the index becomes (because of the need to
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store more index entries), but at the same time the summary data stored can
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be more precise and more data blocks can be skipped during an index scan.
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</para>
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<sect2 id="brin-operation">
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<title>Index Maintenance</title>
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<para>
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At the time of creation, all existing index pages are scanned and a
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summary index tuple is created for each range, including the
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possibly-incomplete range at the end.
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As new pages are filled with data, page ranges that are already
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summarized will cause the summary information to be updated with data
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from the new tuples.
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When a new page is created that does not fall within the last
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summarized range, that range does not automatically acquire a summary
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tuple; those tuples remain unsummarized until a summarization run is
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invoked later, creating initial summaries.
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This process can be invoked manually using the
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<function>brin_summarize_new_values(regclass)</function> function,
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or automatically when <command>VACUUM</command> processes the table.
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</para>
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</sect2>
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</sect1>
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<sect1 id="brin-builtin-opclasses">
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<title>Built-in Operator Classes</title>
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<para>
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The core <productname>PostgreSQL</productname> distribution
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includes the <acronym>BRIN</acronym> operator classes shown in
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<xref linkend="brin-builtin-opclasses-table">.
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</para>
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<para>
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The <firstterm>minmax</>
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operator classes store the minimum and the maximum values appearing
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in the indexed column within the range. The <firstterm>inclusion</>
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operator classes store a value which includes the values in the indexed
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column within the range.
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</para>
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<table id="brin-builtin-opclasses-table">
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<title>Built-in <acronym>BRIN</acronym> Operator Classes</title>
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<tgroup cols="3">
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<thead>
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<row>
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<entry>Name</entry>
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<entry>Indexed Data Type</entry>
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<entry>Indexable Operators</entry>
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</row>
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</thead>
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<tbody>
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<row>
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<entry><literal>abstime_minmax_ops</literal></entry>
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<entry><type>abstime</type></entry>
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<entry>
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<literal><</literal>
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<literal><=</literal>
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<literal>=</literal>
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<literal>>=</literal>
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<literal>></literal>
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</entry>
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</row>
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<row>
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<entry><literal>int8_minmax_ops</literal></entry>
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<entry><type>bigint</type></entry>
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<entry>
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<literal><</literal>
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<literal><=</literal>
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<literal>=</literal>
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<literal>>=</literal>
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<literal>></literal>
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</entry>
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</row>
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<row>
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<entry><literal>bit_minmax_ops</literal></entry>
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<entry><type>bit</type></entry>
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<entry>
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<literal><</literal>
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<literal><=</literal>
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<literal>=</literal>
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<literal>>=</literal>
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<literal>></literal>
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</entry>
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</row>
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<row>
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<entry><literal>varbit_minmax_ops</literal></entry>
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<entry><type>bit varying</type></entry>
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<entry>
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<literal><</literal>
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<literal><=</literal>
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<literal>=</literal>
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<literal>>=</literal>
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<literal>></literal>
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</entry>
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</row>
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<row>
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<entry><literal>box_inclusion_ops</></entry>
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<entry><type>box</type></entry>
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<entry>
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<literal><<</>
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<literal>&<</>
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<literal>&&</>
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<literal>&></>
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<literal>>></>
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<literal>~=</>
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<literal>@></>
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<literal><@</>
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<literal>&<|</>
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<literal><<|</>
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<literal>|>></literal>
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<literal>|&></>
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</entry>
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</row>
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<row>
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<entry><literal>bytea_minmax_ops</literal></entry>
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<entry><type>bytea</type></entry>
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<entry>
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<literal><</literal>
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<literal><=</literal>
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<literal>=</literal>
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<literal>>=</literal>
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<literal>></literal>
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</entry>
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</row>
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<row>
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<entry><literal>bpchar_minmax_ops</literal></entry>
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<entry><type>character</type></entry>
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<entry>
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<literal><</literal>
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<literal><=</literal>
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<literal>=</literal>
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<literal>>=</literal>
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<literal>></literal>
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</entry>
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</row>
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<row>
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<entry><literal>char_minmax_ops</literal></entry>
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<entry><type>"char"</type></entry>
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<entry>
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<literal><</literal>
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<literal><=</literal>
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<literal>=</literal>
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<literal>>=</literal>
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<literal>></literal>
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</entry>
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</row>
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<row>
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<entry><literal>date_minmax_ops</literal></entry>
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<entry><type>date</type></entry>
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<entry>
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<literal><</literal>
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<literal><=</literal>
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<literal>=</literal>
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<literal>>=</literal>
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<literal>></literal>
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</entry>
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</row>
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<row>
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<entry><literal>float8_minmax_ops</literal></entry>
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<entry><type>double precision</type></entry>
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<entry>
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<literal><</literal>
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<literal><=</literal>
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<literal>=</literal>
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<literal>>=</literal>
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<literal>></literal>
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</entry>
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</row>
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<row>
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<entry><literal>inet_minmax_ops</literal></entry>
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<entry><type>inet</type></entry>
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<entry>
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<literal><</literal>
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<literal><=</literal>
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<literal>=</literal>
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<literal>>=</literal>
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<literal>></literal>
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</entry>
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</row>
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<row>
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<entry><literal>network_inclusion_ops</literal></entry>
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<entry><type>inet</type></entry>
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<entry>
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<literal>&&</>
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<literal>>>=</>
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<literal><<=</literal>
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<literal>=</literal>
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<literal>>></>
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<literal><<</literal>
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</entry>
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</row>
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<row>
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<entry><literal>int4_minmax_ops</literal></entry>
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<entry><type>integer</type></entry>
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<entry>
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<literal><</literal>
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<literal><=</literal>
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<literal>=</literal>
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<literal>>=</literal>
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<literal>></literal>
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</entry>
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</row>
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<row>
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<entry><literal>interval_minmax_ops</literal></entry>
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<entry><type>interval</type></entry>
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<entry>
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<literal><</literal>
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<literal><=</literal>
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<literal>=</literal>
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<literal>>=</literal>
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<literal>></literal>
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</entry>
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</row>
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<row>
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<entry><literal>macaddr_minmax_ops</literal></entry>
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<entry><type>macaddr</type></entry>
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<entry>
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<literal><</literal>
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<literal><=</literal>
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<literal>=</literal>
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<literal>>=</literal>
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<literal>></literal>
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</entry>
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</row>
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<row>
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<entry><literal>name_minmax_ops</literal></entry>
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<entry><type>name</type></entry>
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<entry>
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<literal><</literal>
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<literal><=</literal>
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<literal>=</literal>
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<literal>>=</literal>
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<literal>></literal>
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</entry>
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</row>
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<row>
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<entry><literal>numeric_minmax_ops</literal></entry>
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<entry><type>numeric</type></entry>
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<entry>
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<literal><</literal>
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<literal><=</literal>
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<literal>=</literal>
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<literal>>=</literal>
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<literal>></literal>
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</entry>
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</row>
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<row>
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<entry><literal>pg_lsn_minmax_ops</literal></entry>
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<entry><type>pg_lsn</type></entry>
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<entry>
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<literal><</literal>
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<literal><=</literal>
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<literal>=</literal>
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<literal>>=</literal>
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<literal>></literal>
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</entry>
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</row>
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<row>
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<entry><literal>oid_minmax_ops</literal></entry>
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<entry><type>oid</type></entry>
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<entry>
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<literal><</literal>
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<literal><=</literal>
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<literal>=</literal>
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<literal>>=</literal>
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<literal>></literal>
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</entry>
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</row>
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<row>
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<entry><literal>range_inclusion_ops</></entry>
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<entry><type>any range type</type></entry>
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<entry>
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<literal><<</>
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<literal>&<</>
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<literal>&&</>
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<literal>&></>
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<literal>>></>
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<literal>@></>
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<literal><@</>
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<literal>-|-</>
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<literal>=</>
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<literal><</literal>
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<literal><=</literal>
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<literal>=</literal>
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<literal>></literal>
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<literal>>=</literal>
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</entry>
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</row>
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<row>
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<entry><literal>float4_minmax_ops</literal></entry>
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<entry><type>real</type></entry>
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<entry>
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<literal><</literal>
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<literal><=</literal>
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<literal>=</literal>
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<literal>>=</literal>
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<literal>></literal>
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</entry>
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</row>
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<row>
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<entry><literal>reltime_minmax_ops</literal></entry>
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<entry><type>reltime</type></entry>
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<entry>
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<literal><</literal>
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<literal><=</literal>
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<literal>=</literal>
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<literal>>=</literal>
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<literal>></literal>
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</entry>
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</row>
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<row>
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<entry><literal>int2_minmax_ops</literal></entry>
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<entry><type>smallint</type></entry>
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<entry>
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<literal><</literal>
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<literal><=</literal>
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<literal>=</literal>
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<literal>>=</literal>
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<literal>></literal>
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</entry>
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</row>
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<row>
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<entry><literal>text_minmax_ops</literal></entry>
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<entry><type>text</type></entry>
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<entry>
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<literal><</literal>
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<literal><=</literal>
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<literal>=</literal>
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<literal>>=</literal>
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<literal>></literal>
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</entry>
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</row>
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<row>
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<entry><literal>tid_minmax_ops</literal></entry>
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<entry><type>tid</type></entry>
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<entry>
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<literal><</literal>
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<literal><=</literal>
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<literal>=</literal>
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<literal>>=</literal>
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<literal>></literal>
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</entry>
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</row>
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<row>
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<entry><literal>timestamp_minmax_ops</literal></entry>
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<entry><type>timestamp without time zone</type></entry>
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<entry>
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<literal><</literal>
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<literal><=</literal>
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<literal>=</literal>
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<literal>>=</literal>
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<literal>></literal>
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</entry>
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</row>
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<row>
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<entry><literal>timestamptz_minmax_ops</literal></entry>
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<entry><type>timestamp with time zone</type></entry>
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<entry>
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<literal><</literal>
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<literal><=</literal>
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<literal>=</literal>
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<literal>>=</literal>
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<literal>></literal>
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</entry>
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</row>
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<row>
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<entry><literal>time_minmax_ops</literal></entry>
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<entry><type>time without time zone</type></entry>
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<entry>
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<literal><</literal>
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<literal><=</literal>
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<literal>=</literal>
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<literal>>=</literal>
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<literal>></literal>
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</entry>
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</row>
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<row>
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<entry><literal>timetz_minmax_ops</literal></entry>
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<entry><type>time with time zone</type></entry>
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<entry>
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<literal><</literal>
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<literal><=</literal>
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<literal>=</literal>
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<literal>>=</literal>
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<literal>></literal>
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</entry>
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</row>
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<row>
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<entry><literal>uuid_minmax_ops</literal></entry>
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<entry><type>uuid</type></entry>
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<entry>
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<literal><</literal>
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<literal><=</literal>
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<literal>=</literal>
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<literal>>=</literal>
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<literal>></literal>
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</entry>
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</row>
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</tbody>
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</tgroup>
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</table>
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</sect1>
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|
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<sect1 id="brin-extensibility">
|
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<title>Extensibility</title>
|
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|
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<para>
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The <acronym>BRIN</acronym> interface has a high level of abstraction,
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requiring the access method implementer only to implement the semantics
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of the data type being accessed. The <acronym>BRIN</acronym> layer
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itself takes care of concurrency, logging and searching the index structure.
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</para>
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<para>
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All it takes to get a <acronym>BRIN</acronym> access method working is to
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implement a few user-defined methods, which define the behavior of
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summary values stored in the index and the way they interact with
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scan keys.
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In short, <acronym>BRIN</acronym> combines
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extensibility with generality, code reuse, and a clean interface.
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</para>
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<para>
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There are four methods that an operator class for <acronym>BRIN</acronym>
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must provide:
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<variablelist>
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<varlistentry>
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<term><function>BrinOpcInfo *opcInfo(Oid type_oid)</></term>
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<listitem>
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<para>
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Returns internal information about the indexed columns' summary data.
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The return value must point to a palloc'd <structname>BrinOpcInfo</>,
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which has this definition:
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<programlisting>
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typedef struct BrinOpcInfo
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{
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/* Number of columns stored in an index column of this opclass */
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uint16 oi_nstored;
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/* Opaque pointer for the opclass' private use */
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void *oi_opaque;
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/* Type cache entries of the stored columns */
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TypeCacheEntry *oi_typcache[FLEXIBLE_ARRAY_MEMBER];
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} BrinOpcInfo;
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</programlisting>
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<structname>BrinOpcInfo</>.<structfield>oi_opaque</> can be used by the
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operator class routines to pass information between support procedures
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during an index scan.
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</para>
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</listitem>
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</varlistentry>
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|
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<varlistentry>
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<term><function>bool consistent(BrinDesc *bdesc, BrinValues *column,
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ScanKey key)</function></term>
|
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<listitem>
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<para>
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Returns whether the ScanKey is consistent with the given indexed
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values for a range.
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The attribute number to use is passed as part of the scan key.
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</para>
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</listitem>
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</varlistentry>
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<varlistentry>
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<term><function>bool addValue(BrinDesc *bdesc, BrinValues *column,
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Datum newval, bool isnull)</function></term>
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<listitem>
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<para>
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Given an index tuple and an indexed value, modifies the indicated
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attribute of the tuple so that it additionally represents the new value.
|
|
If any modification was done to the tuple, <literal>true</literal> is
|
|
returned.
|
|
</para>
|
|
</listitem>
|
|
</varlistentry>
|
|
|
|
<varlistentry>
|
|
<term><function>bool unionTuples(BrinDesc *bdesc, BrinValues *a,
|
|
BrinValues *b)</function></term>
|
|
<listitem>
|
|
<para>
|
|
Consolidates two index tuples. Given two index tuples, modifies the
|
|
indicated attribute of the first of them so that it represents both tuples.
|
|
The second tuple is not modified.
|
|
</para>
|
|
</listitem>
|
|
</varlistentry>
|
|
</variablelist>
|
|
|
|
The core distribution includes support for two types of operator classes:
|
|
minmax and inclusion. Operator class definitions using them are shipped for
|
|
in-core data types as appropriate. Additional operator classes can be
|
|
defined by the user for other data types using equivalent definitions,
|
|
without having to write any source code; appropriate catalog entries being
|
|
declared is enough. Note that assumptions about the semantics of operator
|
|
strategies are embedded in the support procedures' source code.
|
|
</para>
|
|
|
|
<para>
|
|
Operator classes that implement completely different semantics are also
|
|
possible, provided implementations of the four main support procedures
|
|
described above are written. Note that backwards compatibility across major
|
|
releases is not guaranteed: for example, additional support procedures might
|
|
be required in later releases.
|
|
</para>
|
|
|
|
<para>
|
|
To write an operator class for a data type that implements a totally
|
|
ordered set, it is possible to use the minmax support procedures
|
|
alongside the corresponding operators, as shown in
|
|
<xref linkend="brin-extensibility-minmax-table">.
|
|
All operator class members (procedures and operators) are mandatory.
|
|
</para>
|
|
|
|
<table id="brin-extensibility-minmax-table">
|
|
<title>Procedure and Support Numbers for Minmax Operator Classes</title>
|
|
<tgroup cols="2">
|
|
<thead>
|
|
<row>
|
|
<entry>Operator class member</entry>
|
|
<entry>Object</entry>
|
|
</row>
|
|
</thead>
|
|
<tbody>
|
|
<row>
|
|
<entry>Support Procedure 1</entry>
|
|
<entry>internal function <function>brin_minmax_opcinfo()</function></entry>
|
|
</row>
|
|
<row>
|
|
<entry>Support Procedure 2</entry>
|
|
<entry>internal function <function>brin_minmax_add_value()</function></entry>
|
|
</row>
|
|
<row>
|
|
<entry>Support Procedure 3</entry>
|
|
<entry>internal function <function>brin_minmax_consistent()</function></entry>
|
|
</row>
|
|
<row>
|
|
<entry>Support Procedure 4</entry>
|
|
<entry>internal function <function>brin_minmax_union()</function></entry>
|
|
</row>
|
|
<row>
|
|
<entry>Operator Strategy 1</entry>
|
|
<entry>operator less-than</entry>
|
|
</row>
|
|
<row>
|
|
<entry>Operator Strategy 2</entry>
|
|
<entry>operator less-than-or-equal-to</entry>
|
|
</row>
|
|
<row>
|
|
<entry>Operator Strategy 3</entry>
|
|
<entry>operator equal-to</entry>
|
|
</row>
|
|
<row>
|
|
<entry>Operator Strategy 4</entry>
|
|
<entry>operator greater-than-or-equal-to</entry>
|
|
</row>
|
|
<row>
|
|
<entry>Operator Strategy 5</entry>
|
|
<entry>operator greater-than</entry>
|
|
</row>
|
|
</tbody>
|
|
</tgroup>
|
|
</table>
|
|
|
|
<para>
|
|
To write an operator class for a complex data type which has values
|
|
included within another type, it's possible to use the inclusion support
|
|
procedures alongside the corresponding operators, as shown
|
|
in <xref linkend="brin-extensibility-inclusion-table">. It requires
|
|
only a single additional function, which can be written in any language.
|
|
More functions can be defined for additional functionality. All operators
|
|
are optional. Some operators require other operators, as shown as
|
|
dependencies on the table.
|
|
</para>
|
|
|
|
<table id="brin-extensibility-inclusion-table">
|
|
<title>Procedure and Support Numbers for Inclusion Operator Classes</title>
|
|
<tgroup cols="3">
|
|
<thead>
|
|
<row>
|
|
<entry>Operator class member</entry>
|
|
<entry>Object</entry>
|
|
<entry>Dependency</entry>
|
|
</row>
|
|
</thead>
|
|
<tbody>
|
|
<row>
|
|
<entry>Support Procedure 1</entry>
|
|
<entry>internal function <function>brin_inclusion_opcinfo()</function></entry>
|
|
<entry></entry>
|
|
</row>
|
|
<row>
|
|
<entry>Support Procedure 2</entry>
|
|
<entry>internal function <function>brin_inclusion_add_value()</function></entry>
|
|
<entry></entry>
|
|
</row>
|
|
<row>
|
|
<entry>Support Procedure 3</entry>
|
|
<entry>internal function <function>brin_inclusion_consistent()</function></entry>
|
|
<entry></entry>
|
|
</row>
|
|
<row>
|
|
<entry>Support Procedure 4</entry>
|
|
<entry>internal function <function>brin_inclusion_union()</function></entry>
|
|
<entry></entry>
|
|
</row>
|
|
<row>
|
|
<entry>Support Procedure 11</entry>
|
|
<entry>function to merge two elements</entry>
|
|
<entry></entry>
|
|
</row>
|
|
<row>
|
|
<entry>Support Procedure 12</entry>
|
|
<entry>optional function to check whether two elements are mergeable</entry>
|
|
<entry></entry>
|
|
</row>
|
|
<row>
|
|
<entry>Support Procedure 13</entry>
|
|
<entry>optional function to check if an element is contained within another</entry>
|
|
<entry></entry>
|
|
</row>
|
|
<row>
|
|
<entry>Support Procedure 14</entry>
|
|
<entry>optional function to check whether an element is empty</entry>
|
|
<entry></entry>
|
|
</row>
|
|
<row>
|
|
<entry>Operator Strategy 1</entry>
|
|
<entry>operator left-of</entry>
|
|
<entry>Operator Strategy 4</entry>
|
|
</row>
|
|
<row>
|
|
<entry>Operator Strategy 2</entry>
|
|
<entry>operator does-not-extend-to-the-right-of</entry>
|
|
<entry>Operator Strategy 5</entry>
|
|
</row>
|
|
<row>
|
|
<entry>Operator Strategy 3</entry>
|
|
<entry>operator overlaps</entry>
|
|
<entry></entry>
|
|
</row>
|
|
<row>
|
|
<entry>Operator Strategy 4</entry>
|
|
<entry>operator does-not-extend-to-the-left-of</entry>
|
|
<entry>Operator Strategy 1</entry>
|
|
</row>
|
|
<row>
|
|
<entry>Operator Strategy 5</entry>
|
|
<entry>operator right-of</entry>
|
|
<entry>Operator Strategy 2</entry>
|
|
</row>
|
|
<row>
|
|
<entry>Operator Strategy 6, 18</entry>
|
|
<entry>operator same-as-or-equal-to</entry>
|
|
<entry>Operator Strategy 7</entry>
|
|
</row>
|
|
<row>
|
|
<entry>Operator Strategy 7, 13, 16, 24, 25</entry>
|
|
<entry>operator contains-or-equal-to</entry>
|
|
<entry></entry>
|
|
</row>
|
|
<row>
|
|
<entry>Operator Strategy 8, 14, 26, 27</entry>
|
|
<entry>operator is-contained-by-or-equal-to</entry>
|
|
<entry>Operator Strategy 3</entry>
|
|
</row>
|
|
<row>
|
|
<entry>Operator Strategy 9</entry>
|
|
<entry>operator does-not-extend-above</entry>
|
|
<entry>Operator Strategy 11</entry>
|
|
</row>
|
|
<row>
|
|
<entry>Operator Strategy 10</entry>
|
|
<entry>operator is-below</entry>
|
|
<entry>Operator Strategy 12</entry>
|
|
</row>
|
|
<row>
|
|
<entry>Operator Strategy 11</entry>
|
|
<entry>operator is-above</entry>
|
|
<entry>Operator Strategy 9</entry>
|
|
</row>
|
|
<row>
|
|
<entry>Operator Strategy 12</entry>
|
|
<entry>operator does-not-extend-below</entry>
|
|
<entry>Operator Strategy 10</entry>
|
|
</row>
|
|
<row>
|
|
<entry>Operator Strategy 20</entry>
|
|
<entry>operator less-than</entry>
|
|
<entry>Operator Strategy 5</entry>
|
|
</row>
|
|
<row>
|
|
<entry>Operator Strategy 21</entry>
|
|
<entry>operator less-than-or-equal-to</entry>
|
|
<entry>Operator Strategy 5</entry>
|
|
</row>
|
|
<row>
|
|
<entry>Operator Strategy 22</entry>
|
|
<entry>operator greater-than</entry>
|
|
<entry>Operator Strategy 1</entry>
|
|
</row>
|
|
<row>
|
|
<entry>Operator Strategy 23</entry>
|
|
<entry>operator greater-than-or-equal-to</entry>
|
|
<entry>Operator Strategy 1</entry>
|
|
</row>
|
|
</tbody>
|
|
</tgroup>
|
|
</table>
|
|
|
|
<para>
|
|
Support procedure numbers 1-10 are reserved for the BRIN internal
|
|
functions, so the SQL level functions start with number 11. Support
|
|
function number 11 is the main function required to build the index.
|
|
It should accept two arguments with the same data type as the operator class,
|
|
and return the union of them. The inclusion operator class can store union
|
|
values with different data types if it is defined with the
|
|
<literal>STORAGE</> parameter. The return value of the union
|
|
function should match the <literal>STORAGE</> data type.
|
|
</para>
|
|
|
|
<para>
|
|
Support procedure numbers 12 and 14 are provided to support
|
|
irregularities of built-in data types. Procedure number 12
|
|
is used to support network addresses from different families which
|
|
are not mergeable. Procedure number 14 is used to support
|
|
empty ranges. Procedure number 13 is an optional but
|
|
recommended one, which allows the new value to be checked before
|
|
it is passed to the union function. As the BRIN framework can shortcut
|
|
some operations when the union is not changed, using this
|
|
function can improve index performance.
|
|
</para>
|
|
|
|
<para>
|
|
Both minmax and inclusion operator classes support cross-data-type
|
|
operators, though with these the dependencies become more complicated.
|
|
The minmax operator class requires a full set of operators to be
|
|
defined with both arguments having the same data type. It allows
|
|
additional data types to be supported by defining extra sets
|
|
of operators. Inclusion operator class operator strategies are dependent
|
|
on another operator strategy as shown in
|
|
<xref linkend="brin-extensibility-inclusion-table">, or the same
|
|
operator strategy as themselves. They require the dependency
|
|
operator to be defined with the <literal>STORAGE</> data type as the
|
|
left-hand-side argument and the other supported data type to be the
|
|
right-hand-side argument of the supported operator. See
|
|
<literal>float4_minmax_ops</> as an example of minmax, and
|
|
<literal>box_inclusion_ops</> as an example of inclusion.
|
|
</para>
|
|
</sect1>
|
|
</chapter>
|