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64.1. B-Tree Indexes

64.1. B-Tree Indexes
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64.1.1. Introduction
64.1.2. Behavior of B-Tree Operator Classes
64.1.3. B-Tree Support Functions
64.1.4. Implementation

64.1.1. 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.

64.1.2. 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.

64.1.3. B-Tree Support Functions #

As shown in Table 36.9 , btree defines one required and four optional support functions. The five 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 .

64.1.4. 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.

64.1.4.1. 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 65.6 .

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.

64.1.4.2. 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 UPDATE s 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 UPDATE s 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 UPDATE s. 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 TID s. 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).

64.1.4.3. 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 TID s 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 TID s 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.

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