B-Tree Indexes
index
B-Tree
Introduction
PostgreSQL includes an implementation of the
standard btree (multi-way binary 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 one optional support function.
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. The function must not
return INT_MIN for the A
< B case,
since the value may be negated before being tested for sign. A null
result is disallowed, too.
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.
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.
Implementation
An introduction to the btree index implementation can be found in
src/backend/access/nbtree/README.