B-Tree Indexes
index
B-Tree
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.
Behavior of B-Tree Operator Classes
As shown in , 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.
B-Tree Support Functions
As shown in , btree defines
one required and two optional support functions. The three
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.
inrange
in_range support functions
support functions
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 ). 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.
Implementation
An introduction to the btree index implementation can be found in
src/backend/access/nbtree/README.