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65.1. B-Tree Indexes #

   65.1.1. Introduction
   65.1.2. Behavior of B-Tree Operator Classes
   65.1.3. B-Tree Support Functions
   65.1.4. Implementation

65.1.1. Introduction #

   PostgreSQL includes an implementation of the standard btree (multi-way
   balanced tree) index data structure. Any data type that can be sorted
   into a well-defined linear order can be indexed by a btree index. The
   only limitation is that an index entry cannot exceed approximately
   one-third of a page (after TOAST compression, if applicable).

   Because each btree operator class imposes a sort order on its data
   type, btree operator classes (or, really, operator families) have come
   to be used as PostgreSQL's general representation and understanding of
   sorting semantics. Therefore, they've acquired some features that go
   beyond what would be needed just to support btree indexes, and parts of
   the system that are quite distant from the btree AM make use of them.

65.1.2. Behavior of B-Tree Operator Classes #

   As shown in Table 36.3, a btree operator class must provide five
   comparison operators, <, <=, =, >= and >. One might expect that <>
   should also be part of the operator class, but it is not, because it
   would almost never be useful to use a <> WHERE clause in an index
   search. (For some purposes, the planner treats <> as associated with a
   btree operator class; but it finds that operator via the = operator's
   negator link, rather than from pg_amop.)

   When several data types share near-identical sorting semantics, their
   operator classes can be grouped into an operator family. Doing so is
   advantageous because it allows the planner to make deductions about
   cross-type comparisons. Each operator class within the family should
   contain the single-type operators (and associated support functions)
   for its input data type, while cross-type comparison operators and
   support functions are “loose” in the family. It is recommendable that a
   complete set of cross-type operators be included in the family, thus
   ensuring that the planner can represent any comparison conditions that
   it deduces from transitivity.

   There are some basic assumptions that a btree operator family must
   satisfy:
     * An = operator must be an equivalence relation; that is, for all
       non-null values A, B, C of the data type:
          + A = A is true (reflexive law)
          + if A = B, then B = A (symmetric law)
          + if A = B and B = C, then A = C (transitive law)
     * A < operator must be a strong ordering relation; that is, for all
       non-null values A, B, C:
          + A < A is false (irreflexive law)
          + if A < B and B < C, then A < C (transitive law)
     * Furthermore, the ordering is total; that is, for all non-null
       values A, B:
          + exactly one of A < B, A = B, and B < A is true (trichotomy
            law)
       (The trichotomy law justifies the definition of the comparison
       support function, of course.)

   The other three operators are defined in terms of = and < in the
   obvious way, and must act consistently with them.

   For an operator family supporting multiple data types, the above laws
   must hold when A, B, C are taken from any data types in the family. The
   transitive laws are the trickiest to ensure, as in cross-type
   situations they represent statements that the behaviors of two or three
   different operators are consistent. As an example, it would not work to
   put float8 and numeric into the same operator family, at least not with
   the current semantics that numeric values are converted to float8 for
   comparison to a float8. Because of the limited accuracy of float8, this
   means there are distinct numeric values that will compare equal to the
   same float8 value, and thus the transitive law would fail.

   Another requirement for a multiple-data-type family is that any
   implicit or binary-coercion casts that are defined between data types
   included in the operator family must not change the associated sort
   ordering.

   It should be fairly clear why a btree index requires these laws to hold
   within a single data type: without them there is no ordering to arrange
   the keys with. Also, index searches using a comparison key of a
   different data type require comparisons to behave sanely across two
   data types. The extensions to three or more data types within a family
   are not strictly required by the btree index mechanism itself, but the
   planner relies on them for optimization purposes.

65.1.3. B-Tree Support Functions #

   As shown in Table 36.9, btree defines one required and five optional
   support functions. The six user-defined methods are:

   order
          For each combination of data types that a btree operator family
          provides comparison operators for, it must provide a comparison
          support function, registered in pg_amproc with support function
          number 1 and amproclefttype/amprocrighttype equal to the left
          and right data types for the comparison (i.e., the same data
          types that the matching operators are registered with in
          pg_amop). The comparison function must take two non-null values
          A and B and return an int32 value that is < 0, 0, or > 0 when A
          < B, A = B, or A > B, respectively. A null result is disallowed:
          all values of the data type must be comparable. See
          src/backend/access/nbtree/nbtcompare.c for examples.

          If the compared values are of a collatable data type, the
          appropriate collation OID will be passed to the comparison
          support function, using the standard PG_GET_COLLATION()
          mechanism.

   sortsupport
          Optionally, a btree operator family may provide sort support
          function(s), registered under support function number 2. These
          functions allow implementing comparisons for sorting purposes in
          a more efficient way than naively calling the comparison support
          function. The APIs involved in this are defined in
          src/include/utils/sortsupport.h.

   in_range
          Optionally, a btree operator family may provide in_range support
          function(s), registered under support function number 3. These
          are not used during btree index operations; rather, they extend
          the semantics of the operator family so that it can support
          window clauses containing the RANGE offset PRECEDING and RANGE
          offset FOLLOWING frame bound types (see Section 4.2.8).
          Fundamentally, the extra information provided is how to add or
          subtract an offset value in a way that is compatible with the
          family's data ordering.

          An in_range function must have the signature

in_range(val type1, base type1, offset type2, sub bool, less bool)
returns bool

          val and base must be of the same type, which is one of the types
          supported by the operator family (i.e., a type for which it
          provides an ordering). However, offset could be of a different
          type, which might be one otherwise unsupported by the family. An
          example is that the built-in time_ops family provides an
          in_range function that has offset of type interval. A family can
          provide in_range functions for any of its supported types and
          one or more offset types. Each in_range function should be
          entered in pg_amproc with amproclefttype equal to type1 and
          amprocrighttype equal to type2.

          The essential semantics of an in_range function depend on the
          two Boolean flag parameters. It should add or subtract base and
          offset, then compare val to the result, as follows:

          + if !sub and !less, return val >= (base + offset)
          + if !sub and less, return val <= (base + offset)
          + if sub and !less, return val >= (base - offset)
          + if sub and less, return val <= (base - offset)

          Before doing so, the function should check the sign of offset:
          if it is less than zero, raise error
          ERRCODE_INVALID_PRECEDING_OR_FOLLOWING_SIZE (22013) with error
          text like “invalid preceding or following size in window
          function”. (This is required by the SQL standard, although
          nonstandard operator families might perhaps choose to ignore
          this restriction, since there seems to be little semantic
          necessity for it.) This requirement is delegated to the in_range
          function so that the core code needn't understand what “less
          than zero” means for a particular data type.

          An additional expectation is that in_range functions should, if
          practical, avoid throwing an error if base + offset or base -
          offset would overflow. The correct comparison result can be
          determined even if that value would be out of the data type's
          range. Note that if the data type includes concepts such as
          “infinity” or “NaN”, extra care may be needed to ensure that
          in_range's results agree with the normal sort order of the
          operator family.

          The results of the in_range function must be consistent with the
          sort ordering imposed by the operator family. To be precise,
          given any fixed values of offset and sub, then:

          + If in_range with less = true is true for some val1 and base,
            it must be true for every val2 <= val1 with the same base.
          + If in_range with less = true is false for some val1 and base,
            it must be false for every val2 >= val1 with the same base.
          + If in_range with less = true is true for some val and base1,
            it must be true for every base2 >= base1 with the same val.
          + If in_range with less = true is false for some val and base1,
            it must be false for every base2 <= base1 with the same val.

          Analogous statements with inverted conditions hold when less =
          false.

          If the type being ordered (type1) is collatable, the appropriate
          collation OID will be passed to the in_range function, using the
          standard PG_GET_COLLATION() mechanism.

          in_range functions need not handle NULL inputs, and typically
          will be marked strict.

   equalimage
          Optionally, a btree operator family may provide equalimage
          (“equality implies image equality”) support functions,
          registered under support function number 4. These functions
          allow the core code to determine when it is safe to apply the
          btree deduplication optimization. Currently, equalimage
          functions are only called when building or rebuilding an index.

          An equalimage function must have the signature

equalimage(opcintype oid) returns bool

          The return value is static information about an operator class
          and collation. Returning true indicates that the order function
          for the operator class is guaranteed to only return 0
          (“arguments are equal”) when its A and B arguments are also
          interchangeable without any loss of semantic information. Not
          registering an equalimage function or returning false indicates
          that this condition cannot be assumed to hold.

          The opcintype argument is the pg_type.oid of the data type that
          the operator class indexes. This is a convenience that allows
          reuse of the same underlying equalimage function across operator
          classes. If opcintype is a collatable data type, the appropriate
          collation OID will be passed to the equalimage function, using
          the standard PG_GET_COLLATION() mechanism.

          As far as the operator class is concerned, returning true
          indicates that deduplication is safe (or safe for the collation
          whose OID was passed to its equalimage function). However, the
          core code will only deem deduplication safe for an index when
          every indexed column uses an operator class that registers an
          equalimage function, and each function actually returns true
          when called.

          Image equality is almost the same condition as simple bitwise
          equality. There is one subtle difference: When indexing a
          varlena data type, the on-disk representation of two image equal
          datums may not be bitwise equal due to inconsistent application
          of TOAST compression on input. Formally, when an operator
          class's equalimage function returns true, it is safe to assume
          that the datum_image_eq() C function will always agree with the
          operator class's order function (provided that the same
          collation OID is passed to both the equalimage and order
          functions).

          The core code is fundamentally unable to deduce anything about
          the “equality implies image equality” status of an operator
          class within a multiple-data-type family based on details from
          other operator classes in the same family. Also, it is not
          sensible for an operator family to register a cross-type
          equalimage function, and attempting to do so will result in an
          error. This is because “equality implies image equality” status
          does not just depend on sorting/equality semantics, which are
          more or less defined at the operator family level. In general,
          the semantics that one particular data type implements must be
          considered separately.

          The convention followed by the operator classes included with
          the core PostgreSQL distribution is to register a stock, generic
          equalimage function. Most operator classes register
          btequalimage(), which indicates that deduplication is safe
          unconditionally. Operator classes for collatable data types such
          as text register btvarstrequalimage(), which indicates that
          deduplication is safe with deterministic collations. Best
          practice for third-party extensions is to register their own
          custom function to retain control.

   options
          Optionally, a B-tree operator family may provide options
          (“operator class specific options”) support functions,
          registered under support function number 5. These functions
          define a set of user-visible parameters that control operator
          class behavior.

          An options support function must have the signature

options(relopts local_relopts *) returns void

          The function is passed a pointer to a local_relopts struct,
          which needs to be filled with a set of operator class specific
          options. The options can be accessed from other support
          functions using the PG_HAS_OPCLASS_OPTIONS() and
          PG_GET_OPCLASS_OPTIONS() macros.

          Currently, no B-Tree operator class has an options support
          function. B-tree doesn't allow flexible representation of keys
          like GiST, SP-GiST, GIN and BRIN do. So, options probably
          doesn't have much application in the current B-tree index access
          method. Nevertheless, this support function was added to B-tree
          for uniformity, and will probably find uses during further
          evolution of B-tree in PostgreSQL.

   skipsupport
          Optionally, a btree operator family may provide a skip support
          function, registered under support function number 6. These
          functions give the B-tree code a way to iterate through every
          possible value that can be represented by an operator class's
          underlying input type, in key space order. This is used by the
          core code when it applies the skip scan optimization. The APIs
          involved in this are defined in src/include/utils/skipsupport.h.

          Operator classes that do not provide a skip support function are
          still eligible to use skip scan. The core code can still use its
          fallback strategy, though that might be suboptimal for some
          discrete types. It usually doesn't make sense (and may not even
          be feasible) for operator classes on continuous types to provide
          a skip support function.

          It is not sensible for an operator family to register a
          cross-type skipsupport function, and attempting to do so will
          result in an error. This is because determining the next
          indexable value must happen by incrementing a value copied from
          an index tuple. The values generated must all be of the same
          underlying data type (the “skipped” index column's opclass input
          type).

65.1.4. Implementation #

   This section covers B-Tree index implementation details that may be of
   use to advanced users. See src/backend/access/nbtree/README in the
   source distribution for a much more detailed, internals-focused
   description of the B-Tree implementation.

65.1.4.1. B-Tree Structure #

   PostgreSQL B-Tree indexes are multi-level tree structures, where each
   level of the tree can be used as a doubly-linked list of pages. A
   single metapage is stored in a fixed position at the start of the first
   segment file of the index. All other pages are either leaf pages or
   internal pages. Leaf pages are the pages on the lowest level of the
   tree. All other levels consist of internal pages. Each leaf page
   contains tuples that point to table rows. Each internal page contains
   tuples that point to the next level down in the tree. Typically, over
   99% of all pages are leaf pages. Both internal pages and leaf pages use
   the standard page format described in Section 66.6.

   New leaf pages are added to a B-Tree index when an existing leaf page
   cannot fit an incoming tuple. A page split operation makes room for
   items that originally belonged on the overflowing page by moving a
   portion of the items to a new page. Page splits must also insert a new
   downlink to the new page in the parent page, which may cause the parent
   to split in turn. Page splits “cascade upwards” in a recursive fashion.
   When the root page finally cannot fit a new downlink, a root page split
   operation takes place. This adds a new level to the tree structure by
   creating a new root page that is one level above the original root
   page.

65.1.4.2. Bottom-up Index Deletion #

   B-Tree indexes are not directly aware that under MVCC, there might be
   multiple extant versions of the same logical table row; to an index,
   each tuple is an independent object that needs its own index entry.
   “Version churn” tuples may sometimes accumulate and adversely affect
   query latency and throughput. This typically occurs with UPDATE-heavy
   workloads where most individual updates cannot apply the HOT
   optimization. Changing the value of only one column covered by one
   index during an UPDATE always necessitates a new set of index tuples —
   one for each and every index on the table. Note in particular that this
   includes indexes that were not “logically modified” by the UPDATE. All
   indexes will need a successor physical index tuple that points to the
   latest version in the table. Each new tuple within each index will
   generally need to coexist with the original “updated” tuple for a short
   period of time (typically until shortly after the UPDATE transaction
   commits).

   B-Tree indexes incrementally delete version churn index tuples by
   performing bottom-up index deletion passes. Each deletion pass is
   triggered in reaction to an anticipated “version churn page split”.
   This only happens with indexes that are not logically modified by
   UPDATE statements, where concentrated build up of obsolete versions in
   particular pages would occur otherwise. A page split will usually be
   avoided, though it's possible that certain implementation-level
   heuristics will fail to identify and delete even one garbage index
   tuple (in which case a page split or deduplication pass resolves the
   issue of an incoming new tuple not fitting on a leaf page). The
   worst-case number of versions that any index scan must traverse (for
   any single logical row) is an important contributor to overall system
   responsiveness and throughput. A bottom-up index deletion pass targets
   suspected garbage tuples in a single leaf page based on qualitative
   distinctions involving logical rows and versions. This contrasts with
   the “top-down” index cleanup performed by autovacuum workers, which is
   triggered when certain quantitative table-level thresholds are exceeded
   (see Section 24.1.6).

Note

   Not all deletion operations that are performed within B-Tree indexes
   are bottom-up deletion operations. There is a distinct category of
   index tuple deletion: simple index tuple deletion. This is a deferred
   maintenance operation that deletes index tuples that are known to be
   safe to delete (those whose item identifier's LP_DEAD bit is already
   set). Like bottom-up index deletion, simple index deletion takes place
   at the point that a page split is anticipated as a way of avoiding the
   split.

   Simple deletion is opportunistic in the sense that it can only take
   place when recent index scans set the LP_DEAD bits of affected items in
   passing. Prior to PostgreSQL 14, the only category of B-Tree deletion
   was simple deletion. The main differences between it and bottom-up
   deletion are that only the former is opportunistically driven by the
   activity of passing index scans, while only the latter specifically
   targets version churn from UPDATEs that do not logically modify indexed
   columns.

   Bottom-up index deletion performs the vast majority of all garbage
   index tuple cleanup for particular indexes with certain workloads. This
   is expected with any B-Tree index that is subject to significant
   version churn from UPDATEs that rarely or never logically modify the
   columns that the index covers. The average and worst-case number of
   versions per logical row can be kept low purely through targeted
   incremental deletion passes. It's quite possible that the on-disk size
   of certain indexes will never increase by even one single page/block
   despite constant version churn from UPDATEs. Even then, an exhaustive
   “clean sweep” by a VACUUM operation (typically run in an autovacuum
   worker process) will eventually be required as a part of collective
   cleanup of the table and each of its indexes.

   Unlike VACUUM, bottom-up index deletion does not provide any strong
   guarantees about how old the oldest garbage index tuple may be. No
   index can be permitted to retain “floating garbage” index tuples that
   became dead prior to a conservative cutoff point shared by the table
   and all of its indexes collectively. This fundamental table-level
   invariant makes it safe to recycle table TIDs. This is how it is
   possible for distinct logical rows to reuse the same table TID over
   time (though this can never happen with two logical rows whose
   lifetimes span the same VACUUM cycle).

65.1.4.3. Deduplication #

   A duplicate is a leaf page tuple (a tuple that points to a table row)
   where all indexed key columns have values that match corresponding
   column values from at least one other leaf page tuple in the same
   index. Duplicate tuples are quite common in practice. B-Tree indexes
   can use a special, space-efficient representation for duplicates when
   an optional technique is enabled: deduplication.

   Deduplication works by periodically merging groups of duplicate tuples
   together, forming a single posting list tuple for each group. The
   column key value(s) only appear once in this representation. This is
   followed by a sorted array of TIDs that point to rows in the table.
   This significantly reduces the storage size of indexes where each value
   (or each distinct combination of column values) appears several times
   on average. The latency of queries can be reduced significantly.
   Overall query throughput may increase significantly. The overhead of
   routine index vacuuming may also be reduced significantly.

Note

   B-Tree deduplication is just as effective with “duplicates” that
   contain a NULL value, even though NULL values are never equal to each
   other according to the = member of any B-Tree operator class. As far as
   any part of the implementation that understands the on-disk B-Tree
   structure is concerned, NULL is just another value from the domain of
   indexed values.

   The deduplication process occurs lazily, when a new item is inserted
   that cannot fit on an existing leaf page, though only when index tuple
   deletion could not free sufficient space for the new item (typically
   deletion is briefly considered and then skipped over). Unlike GIN
   posting list tuples, B-Tree posting list tuples do not need to expand
   every time a new duplicate is inserted; they are merely an alternative
   physical representation of the original logical contents of the leaf
   page. This design prioritizes consistent performance with mixed
   read-write workloads. Most client applications will at least see a
   moderate performance benefit from using deduplication. Deduplication is
   enabled by default.

   CREATE INDEX and REINDEX apply deduplication to create posting list
   tuples, though the strategy they use is slightly different. Each group
   of duplicate ordinary tuples encountered in the sorted input taken from
   the table is merged into a posting list tuple before being added to the
   current pending leaf page. Individual posting list tuples are packed
   with as many TIDs as possible. Leaf pages are written out in the usual
   way, without any separate deduplication pass. This strategy is
   well-suited to CREATE INDEX and REINDEX because they are once-off batch
   operations.

   Write-heavy workloads that don't benefit from deduplication due to
   having few or no duplicate values in indexes will incur a small, fixed
   performance penalty (unless deduplication is explicitly disabled). The
   deduplicate_items storage parameter can be used to disable
   deduplication within individual indexes. There is never any performance
   penalty with read-only workloads, since reading posting list tuples is
   at least as efficient as reading the standard tuple representation.
   Disabling deduplication isn't usually helpful.

   It is sometimes possible for unique indexes (as well as unique
   constraints) to use deduplication. This allows leaf pages to
   temporarily “absorb” extra version churn duplicates. Deduplication in
   unique indexes augments bottom-up index deletion, especially in cases
   where a long-running transaction holds a snapshot that blocks garbage
   collection. The goal is to buy time for the bottom-up index deletion
   strategy to become effective again. Delaying page splits until a single
   long-running transaction naturally goes away can allow a bottom-up
   deletion pass to succeed where an earlier deletion pass failed.

Tip

   A special heuristic is applied to determine whether a deduplication
   pass in a unique index should take place. It can often skip straight to
   splitting a leaf page, avoiding a performance penalty from wasting
   cycles on unhelpful deduplication passes. If you're concerned about the
   overhead of deduplication, consider setting deduplicate_items = off
   selectively. Leaving deduplication enabled in unique indexes has little
   downside.

   Deduplication cannot be used in all cases due to implementation-level
   restrictions. Deduplication safety is determined when CREATE INDEX or
   REINDEX is run.

   Note that deduplication is deemed unsafe and cannot be used in the
   following cases involving semantically significant differences among
   equal datums:

     * text, varchar, and char cannot use deduplication when a
       nondeterministic collation is used. Case and accent differences
       must be preserved among equal datums.
     * numeric cannot use deduplication. Numeric display scale must be
       preserved among equal datums.
     * jsonb cannot use deduplication, since the jsonb B-Tree operator
       class uses numeric internally.
     * float4 and float8 cannot use deduplication. These types have
       distinct representations for -0 and 0, which are nevertheless
       considered equal. This difference must be preserved.

   There is one further implementation-level restriction that may be
   lifted in a future version of PostgreSQL:

     * Container types (such as composite types, arrays, or range types)
       cannot use deduplication.

   There is one further implementation-level restriction that applies
   regardless of the operator class or collation used:

     * INCLUDE indexes can never use deduplication.