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=====================
Developer Information
=====================
This document provides information for developers who maintain or extend
`BTrees`.
Macros
======
`BTrees` are defined using a "template", roughly akin to a C++ template. To
create a new family of `BTrees`, create a source file that defines macros used
to handle differences in key and value types:
Configuration Macros
--------------------
``MASTER_ID``
A string to hold an RCS/CVS Id key to be included in compiled binaries.
``MOD_NAME_PREFIX``
A string (like "IO" or "OO") that provides the prefix used for the module.
This gets used to generate type names and the internal module name string.
``DEFAULT_MAX_BUCKET_SIZE``
An int giving the maximum bucket size (number of key/value pairs). When a
bucket gets larger than this due to an insertion *into a BTREE*, it
splits. Inserting into a bucket directly doesn't split, and functions
that produce a bucket output (e.g., ``union()``) also have no bound on how
large a bucket may get. Someday this will be tunable on `BTree`.
instances.
``DEFAULT_MAX_BTREE_SIZE``
An ``int`` giving the maximum size (number of children) of an internal
btree node. Someday this will be tunable on ``BTree`` instances.
Macros for Keys
---------------
``KEY_TYPE``
The C type declaration for keys (e.g., ``int`` or ``PyObject*``).
``KEY_TYPE_IS_PYOBJECT``
Define if ``KEY_TYPE`` is a ``PyObject*`, else ``undef``.
``KEY_CHECK(K)``
Tests whether the ``PyObject* K`` can be converted to the (``C``) key type
(``KEY_TYPE``). The macro should return a boolean (zero for false,
non-zero for true). When it returns false, its caller should probably set
a ``TypeError`` exception.
``TEST_KEY_SET_OR(V, K, T)``
Like Python's ``cmp()``. Compares K(ey) to T(arget), where ``K``
and ``T`` are ``C`` values of type `KEY_TYPE`. ``V`` is assigned an `int`
value depending on the outcome::
< 0 if K < T
== 0 if K == T
> 0 if K > T
This macro acts like an ``if``, where the following statement is executed
only if a Python exception has been raised because the values could not be
compared.
``DECREF_KEY(K)``
``K`` is a value of ``KEY_TYPE``. If ``KEY_TYPE`` is a flavor of
``PyObject*``, write this to do ``Py_DECREF(K)``. Else (e.g.,
``KEY_TYPE`` is ``int``) make it a nop.
`INCREF_KEY(K)`
``K`` is a value of `KEY_TYPE`. If `KEY_TYPE` is a flavor of
``PyObject*``, write this to do ``Py_INCREF(K)``. Else (e.g., `KEY_TYPE`
is ``int``) make it a nop.
``COPY_KEY(K, E)``
Like ``K=E``. Copy a key from ``E`` to ``K``, both of ``KEY_TYPE``. Note
that this doesn't ``decref K`` or ``incref E`` when ``KEY_TYPE`` is a
``PyObject*``; the caller is responsible for keeping refcounts straight.
``COPY_KEY_TO_OBJECT(O, K)``
Roughly like ``O=K``. ``O`` is a ``PyObject*``, and the macro must build
a Python object form of ``K``, assign it to ``O``, and ensure that ``O``
owns the reference to its new value. It may do this by creating a new
Python object based on ``K`` (e.g., ``PyInt_FromLong(K)`` when
``KEY_TYPE`` is ``int``), or simply by doing ``Py_INCREF(K)`` if
``KEY_TYPE`` is a ``PyObject*``.
``COPY_KEY_FROM_ARG(TARGET, ARG, STATUS)``
Copy an argument to the target without creating a new reference to
``ARG``. ``ARG`` is a ``PyObject*``, and ``TARGET`` is of type
``KEY_TYPE``. If this can't be done (for example, ``KEY_CHECK(ARG)``
returns false), set a Python error and set status to ``0``. If there is
no error, leave status alone.
Macros for Values
-----------------
``VALUE_TYPE``
The C type declaration for values (e.g., ``int`` or ``PyObject*``).
``VALUE_TYPE_IS_PYOBJECT``
Define if ``VALUE_TYPE`` is a ``PyObject*``, else ``undef``.
``TEST_VALUE(X, Y)``
Like Python's ``cmp()``. Compares ``X`` to ``Y``, where ``X`` & ``Y`` are
``C`` values of type ``VALUE_TYPE``. The macro returns an ``int``, with
value::
< 0 if X < Y
== 0 if X == Y
> 0 if X > Y
Bug: There is no provision for determining whether the comparison attempt
failed (set a Python exception).
``DECREF_VALUE(K)``
Like ``DECREF_KEY``, except applied to values of ``VALUE_TYPE``.
``INCREF_VALUE(K)``
Like ``INCREF_KEY``, except applied to values of ``VALUE_TYPE``.
``COPY_VALUE(K, E)``
Like ``COPY_KEY``, except applied to values of ``VALUE_TYPE``.
``COPY_VALUE_TO_OBJECT(O, K)``
Like ``COPY_KEY_TO_OBJECT``, except applied to values of ``VALUE_TYPE``.
``COPY_VALUE_FROM_ARG(TARGET, ARG, STATUS)``
Like ``COPY_KEY_FROM_ARG``, except applied to values of ``VALUE_TYPE``.
``NORMALIZE_VALUE(V, MIN)``
Normalize the value, ``V``, using the parameter ``MIN``. This is almost
certainly a YAGNI. It is a no-op for most types. For integers, ``V`` is
replaced by ``V/MIN`` only if ``MIN > 0``.
Macros for Set Operations
-------------------------
``MERGE_DEFAULT``
A value of ``VALUE_TYPE`` specifying the value to associate with set
elements when sets are merged with mappings via weighed union or weighted
intersection.
``MERGE(O1, w1, O2, w2)``
Performs a weighted merge of two values, ``O1`` and ``O2``, using weights
``w1`` and ``w2``. The result must be of ``VALUE_TYPE``. Note that
weighted unions and weighted intersections are not enabled if this macro
is left undefined.
``MERGE_WEIGHT(O, w)``
Computes a weighted value for ``O``. The result must be of
``VALUE_TYPE``. This is used for "filling out" weighted unions, i.e. to
compute a weighted value for keys that appear in only one of the input
mappings. If left undefined, ``MERGE_WEIGHT`` defaults to::
#define MERGE_WEIGHT(O, w) (O)
``MULTI_INT_UNION``
The value doesn't matter. If defined, `SetOpTemplate.c` compiles code for
a ``multiunion()`` function (compute a union of many input sets at high
speed). This currently makes sense only for structures with integer keys.
BTree Clues
===========
More or less random bits of helpful info.
+ In papers and textbooks, this flavor of BTree is usually called a B+-Tree,
where "+" is a superscript.
+ All keys and all values live in the bucket leaf nodes. Keys in interior
(BTree) nodes merely serve to guide a search efficiently toward the correct
leaf.
+ When a key is deleted, it's physically removed from the bucket it's in, but
this doesn't propagate back up the tree: since keys in interior nodes only
serve to guide searches, it's OK-- and saves time --to leave "stale" keys in
interior nodes.
+ No attempt is made to rebalance the tree after a deletion, unless a bucket
thereby becomes entirely empty. "Classic BTrees" do rebalance, keeping all
buckets at least half full (provided there are enough keys in the entire
tree to fill half a bucket). The tradeoffs are murky. Pathological cases
in the presence of deletion do exist. Pathologies include trees tending
toward only one key per bucket, and buckets at differing depths (all buckets
are at the same depth in a classic BTree).
+ ``DEFAULT_MAX_BUCKET_SIZE`` and ``DEFAULT_MAX_BTREE_SIZE`` are chosen mostly
to "even out" pickle sizes in storage. That's why, e.g., an `IIBTree` has
larger values than an `OOBTree`: pickles store ints more efficiently than
they can store arbitrary Python objects.
+ In a non-empty BTree, every bucket node contains at least one key, and every
BTree node contains at least one child and a non-NULL firstbucket pointer.
However, a BTree node may not contain any keys.
+ An empty BTree consists solely of a BTree node with ``len==0`` and
``firstbucket==NULL``.
+ Although a BTree can become unbalanced under a mix of inserts and deletes
(meaning both that there's nothing stronger that can be said about buckets
than that they're not empty, and that buckets can appear at different
depths), a BTree node always has children of the same kind: they're all
buckets, or they're all BTree nodes.
The ``BTREE_SEARCH`` Macro
==========================
For notational ease, consider a fixed BTree node ``x``, and let
::
K(i) mean x->data.key[i]
C(i) mean all the keys reachable from x->data.child[i]
For each ``i`` in ``0`` to ``x->len-1`` inclusive,
::
K(i) <= C(i) < K(i+1)
is a BTree node invariant, where we pretend that ``K(0)`` holds a key smaller
than any possible key, and ``K(x->len)`` holds a key larger than any possible
key. (Note that ``K(x->len)`` doesn't actually exist, and ``K(0)`` is never
used although space for it exists in non-empty BTree nodes.)
When searching for a key ``k``, then, the child pointer we want to follow is
the one at index ``i`` such that ``K(i) <= k < K(i+1)``. There can be at most
one such ``i``, since the ``K(i)`` are strictly increasing. And there is at
least one such ``i`` provided the tree isn't empty (so that ``0 < len``). For
the moment, assume the tree isn't empty (we'll get back to that later).
The macro's chief loop invariant is
::
K(lo) < k < K(hi)
This holds trivially at the start, since ``lo`` is set to ``0``, and ``hi`` to
``x->len``, and we pretend ``K(0)`` is minus infinity and ``K(len)`` is plus
infinity. Inside the loop, if ``K(i) < k`` we set ``lo`` to ``i``, and if
``K(i) > k`` we set ``hi`` to ``i``. These obviously preserve the invariant.
If ``K(i) == k``, the loop breaks and sets the result to ``i``, and since
``K(i) == k`` in that case ``i`` is obviously the correct result.
Other cases depend on how ``i = floor((lo + hi)/2)`` works, exactly. Suppose
``lo + d = hi`` for some ``d >= 0``. Then ``i = floor((lo + lo + d)/2) =
floor(lo + d/2) = lo + floor(d/2)``. So:
a. ``[d == 0] (lo == i == hi)`` if and only if ``(lo == hi)``.
b. ``[d == 1] (lo == i < hi)`` if and only if ``(lo+1 == hi)``.
c. ``[d > 1] (lo < i < hi)`` if and only if ``(lo+1 < hi)``.
If the node is empty ``(x->len == 0)``, then ``lo==i==hi==0`` at the start,
and the loop exits immediately (the first ``i > lo`` test fails), without
entering the body.
Else ``lo < hi`` at the start, and the invariant ``K(lo) < k < K(hi)`` holds.
If ``lo+1 < hi``, we're in case (c): ``i`` is strictly between ``lo`` and
``hi``, so the loop body is entered, and regardless of whether the body sets
the new ``lo`` or the new ``hi`` to ``i``, the new ``lo`` is strictly less
than the new ``hi``, and the difference between the new ``lo`` and new ``hi``
is strictly less than the difference between the old ``lo`` and old ``hi``.
So long as the new ``lo + 1`` remains < the new ``hi``, we stay in this case.
We can't stay in this case forever, though: because ``hi-lo`` decreases on
each trip but remains > ``0``, ``lo+1 == hi`` must eventually become true.
(In fact, it becomes true quickly, in about ``log2(x->len)`` trips; the point
is more that ``lo`` doesn't equal ``hi`` when the loop ends, it has to end
with ``lo+1==hi`` and ``i==lo``).
Then we're in case (b): ``i==lo==hi-1`` then, and the loop exits. The
invariant still holds, with ``lo==i`` and ``hi==lo+1==i+1``::
K(i) < k < K(i+1)
so ``i`` is again the correct answer.
Optimization points:
--------------------
+ Division by 2 is done via shift rather via "/2". These are signed ints, and
almost all C compilers treat signed int division as truncating, and shifting
is not the same as truncation for signed int division. The compiler has no
way to know these values aren't negative, so has to generate longer-winded
code for "/2". But we know these values aren't negative, and exploit it.
+ The order of _cmp comparisons matters. We're in an interior BTree node, and
are looking at only a tiny fraction of all the keys that exist. So finding
the key exactly in this node is unlikely, and checking ``_cmp == 0`` is a
waste of time to the same extent. It doesn't matter whether we check for
``_cmp < 0`` or ``_cmp > 0`` first, so long as we do both before worrying
about equality.
+ At the start of a routine, it's better to run this macro even if ``x->len``
is ``0`` (check for that afterwards). We just called a function and so
probably drained the pipeline. If the first thing we do then is read up
``self->len`` and check it against ``0``, we just sit there waiting for the
data to get read up, and then another immediate test-and-branch, and for a
very unlikely case (BTree nodes are rarely empty). It's better to get into
the loop right away so the normal case makes progress ASAP.
The ``BUCKET_SEARCH`` Macro
===========================
This has a different job than ``BTREE_SEARCH``: the key ``0`` slot is
legitimate in a bucket, and we want to find the index at which the key
belongs. If the key is larger than the bucket's largest key, a new slot at
index len is where it belongs, else it belongs at the smallest ``i`` with
``keys[i]`` >= the key we're looking for. We also need to know whether or not
the key is present (``BTREE_SEARCH`` didn't care; it only wanted to find the
next node to search).
The mechanics of the search are quite similar, though. The primary
loop invariant changes to (say we're searching for key ``k``)::
K(lo-1) < k < K(hi)
where ``K(i)`` means ``keys[i]``, and we pretend ``K(-1)`` is minus infinity
and ``K(len)`` is plus infinity.
If the bucket is empty, ``lo=hi=i=0`` at the start, the loop body is never
entered, and the macro sets ``INDEX`` to 0 and ``ABSENT`` to true. That's why
``_cmp`` is initialized to 1 (``_cmp`` becomes ``ABSENT``).
Else the bucket is not empty, lo<hi at the start, and the loop body is
entered. The invariant is obviously satisfied then, as ``lo=0`` and
``hi=len``.
If ``K[i]<k``, ``lo`` is set to ``i+1``, preserving that ``K(lo-1) = K[i] <
k``.
If ``K[i]>k``, ``hi`` is set to ``i``, preserving that ``K[hi] = K[i] > k``.
If the loop exits after either of those, ``_cmp != 0``, so ``ABSENT`` becomes
true.
If ``K[i]=k``, the loop breaks, so that ``INDEX`` becomes ``i``, and
``ABSENT`` becomes false (``_cmp=0`` in this case).
The same case analysis for ``BTREE_SEARCH`` on ``lo`` and ``hi`` holds here:
a. ``(lo == i == hi)`` if and only if ``(lo == hi)``.
b. ``(lo == i < hi)`` if and only if ``(lo+1 == hi)``.
c. ``(lo < i < hi)`` if and only if ``(lo+1 < hi)``.
So long as ``lo+1 < hi``, we're in case (c), and either break with equality
(in which case the right results are obviously computed) or narrow the range.
If equality doesn't obtain, the range eventually narrows to cases (a) or (b).
To go from (c) to (a), we must have ``lo+2==hi`` at the start, and
``K[i]=K[lo+1]<k``. Then the new lo gets set to ``i+1 = lo+2 = hi``, and the
loop exits with ``lo=hi=i`` and ``_cmp<0``. This is correct, because we know
that ``k != K(i)`` (loop invariant! we actually know something stronger, that
``k < K(hi)``; since ``i=hi``, this implies ``k != K(i)``).
Else (c) eventually falls into case (b), ``lo+1==hi`` and ``i==lo``. The
invariant tells us ``K(lo-1) < k < K(hi) = K(lo+1)``, so if the key is present
it must be at ``K(lo)``. ``i==lo`` in this case, so we test ``K(lo)`` against
``k``. As always, if equality obtains we do the right thing, else case #b
becomes case (a).
When (b) becomes (a), the last comparison was non-equal, so ``_cmp`` is
non-zero, and the loop exits because ``lo==hi==i`` in case (a). The invariant
then tells us ``K(lo-1) < k < K(lo)``, so the key is in fact not present, it's
correct to exit with ``_cmp`` non-zero, and ``i==lo`` is again the index at
which ``k`` belongs.
Optimization points:
--------------------
+ As for ``BTREE_SEARCH``, shifting of signed ints is cheaper than division.
+ Unlike as for ``BTREE_SEARCH``, there's nothing special about searching an
empty bucket, and the macro computes thoroughly sensible results in that
case.
+ The order of ``_cmp`` comparisons differs from ``BTREE_SEARCH``. When
searching a bucket, it's much more likely (than when searching a BTree node)
that the key is present, so testing ``__cmp==0`` isn't a systematic waste of
cycles. At the extreme, if all searches are successful (key present), on
average this saves one comparison per search, against leaving the
determination of ``_cmp==0`` implicit (as ``BTREE_SEARCH`` does). But even
on successful searches, ``__cmp != 0`` is a more popular outcome than
``__cmp == 0`` across iterations (unless the bucket has only a few keys), so
it's important to check one of the inequality cases first. It turns out
it's better on average to check ``K(i) < key`` (than to check ``K(i) >
key``), because when it pays it narrows the range more (we get a little
boost from setting ``lo=i+1`` in this case; the other case sets ``hi=i``,
which isn't as much of a narrowing).
src/BTrees/Maintainer.txt deleted 100644 → 0
View file @ 5b539583
This document provides information for developers who maintain or
extend BTrees.
Macros
======
BTrees are defined using a "template", roughly akin to a a C++
template. To create a new family of BTrees, create a source file that
defines macros used to handle differences in key and value types:
Configuration Macros
MASTER_ID
A string to hold an RCS/CVS Id key to be included in compiled binaries.
MOD_NAME_PREFIX
A string (like "IO" or "OO") that provides the prefix used for the
module. This gets used to generate type names and the internal module
name string.
DEFAULT_MAX_BUCKET_SIZE
An int giving the maximum bucket size (number of key/value pairs).
When a bucket gets larger than this due to an insertion *into a BTREE*,
it splits. Inserting into a bucket directly doesn't split, and
functions that produce a bucket output (e.g., union()) also have no
bound on how large a bucket may get. Someday this will be tunable
on BTree instances.
DEFAULT_MAX_BTREE_SIZE
An int giving the maximum size (number of children) of an internal
btree node. Someday this will be tunable on BTree instances.
Macros for Keys
KEY_TYPE
The C type declaration for keys (e.g., int or PyObject*).
KEY_TYPE_IS_PYOBJECT
Define if KEY_TYPE is a PyObject*, else undef.
KEY_CHECK(K)
Tests whether the PyObject* K can be converted to the (C) key type
(KEY_TYPE). The macro should return a boolean (zero for false,
non-zero for true). When it returns false, its caller should probably
set a TypeError exception.
TEST_KEY_SET_OR(V, K, T)
Like Python's cmp(). Compares K(ey) to T(arget), where K & T are C
values of type KEY_TYPE. V is assigned an int value depending on
the outcome:
< 0 if K < T
== 0 if K == T
> 0 if K > T
This macro acts like an 'if', where the following statement is
executed only if a Python exception has been raised because the
values could not be compared.
DECREF_KEY(K)
K is a value of KEY_TYPE. If KEY_TYPE is a flavor of PyObject*, write
this to do Py_DECREF(K). Else (e.g., KEY_TYPE is int) make it a nop.
INCREF_KEY(K)
K is a value of KEY_TYPE. If KEY_TYPE is a flavor of PyObject*, write
this to do Py_INCREF(K). Else (e.g., KEY_TYPE is int) make it a nop.
COPY_KEY(K, E)
Like K=E. Copy a key from E to K, both of KEY_TYPE. Note that this
doesn't decref K or incref E when KEY_TYPE is a PyObject*; the caller
is responsible for keeping refcounts straight.
COPY_KEY_TO_OBJECT(O, K)
Roughly like O=K. O is a PyObject*, and the macro must build a Python
object form of K, assign it to O, and ensure that O owns the reference
to its new value. It may do this by creating a new Python object based
on K (e.g., PyInt_FromLong(K) when KEY_TYPE is int), or simply by doing
Py_INCREF(K) if KEY_TYPE is a PyObject*.
COPY_KEY_FROM_ARG(TARGET, ARG, STATUS)
Copy an argument to the target without creating a new reference to ARG.
ARG is a PyObject*, and TARGET is of type KEY_TYPE. If this can't be
done (for example, KEY_CHECK(ARG) returns false), set a Python error
and set status to 0. If there is no error, leave status alone.
Macros for Values
VALUE_TYPE
The C type declaration for values (e.g., int or PyObject*).
VALUE_TYPE_IS_PYOBJECT
Define if VALUE_TYPE is a PyObject*, else undef.
TEST_VALUE(X, Y)
Like Python's cmp(). Compares X to Y, where X & Y are C values of
type VALUE_TYPE. The macro returns an int, with value
< 0 if X < Y
== 0 if X == Y
> 0 if X > Y
Bug: There is no provision for determining whether the comparison
attempt failed (set a Python exception).
DECREF_VALUE(K)
Like DECREF_KEY, except applied to values of VALUE_TYPE.
INCREF_VALUE(K)
Like INCREF_KEY, except applied to values of VALUE_TYPE.
COPY_VALUE(K, E)
Like COPY_KEY, except applied to values of VALUE_TYPE.
COPY_VALUE_TO_OBJECT(O, K)
Like COPY_KEY_TO_OBJECT, except applied to values of VALUE_TYPE.
COPY_VALUE_FROM_ARG(TARGET, ARG, STATUS)
Like COPY_KEY_FROM_ARG, except applied to values of VALUE_TYPE.
NORMALIZE_VALUE(V, MIN)
Normalize the value, V, using the parameter MIN. This is almost
certainly a YAGNI. It is a no op for most types. For integers, V is
replaced by V/MIN only if MIN > 0.
Macros for Set Operations
MERGE_DEFAULT
A value of VALUE_TYPE specifying the value to associate with set
elements when sets are merged with mappings via weighed union or
weighted intersection.
MERGE(O1, w1, O2, w2)
Performs a weighted merge of two values, O1 and O2, using weights w1
and w2. The result must be of VALUE_TYPE. Note that weighted unions
and weighted intersections are not enabled if this macro is left
undefined.
MERGE_WEIGHT(O, w)
Computes a weighted value for O. The result must be of VALUE_TYPE.
This is used for "filling out" weighted unions, i.e. to compute a
weighted value for keys that appear in only one of the input
mappings. If left undefined, MERGE_WEIGHT defaults to
#define MERGE_WEIGHT(O, w) (O)
MULTI_INT_UNION
The value doesn't matter. If defined, SetOpTemplate.c compiles
code for a multiunion() function (compute a union of many input sets
at high speed). This currently makes sense only for structures with
integer keys.
BTree Clues
===========
More or less random bits of helpful info.
+ In papers and textbooks, this flavor of BTree is usually called
a B+-Tree, where "+" is a superscript.
+ All keys and all values live in the bucket leaf nodes. Keys in
interior (BTree) nodes merely serve to guide a search efficiently
toward the correct leaf.
+ When a key is deleted, it's physically removed from the bucket
it's in, but this doesn't propagate back up the tree: since keys
in interior nodes only serve to guide searches, it's OK-- and
saves time --to leave "stale" keys in interior nodes.
+ No attempt is made to rebalance the tree after a deletion, unless
a bucket thereby becomes entirely empty. "Classic BTrees" do
rebalance, keeping all buckets at least half full (provided there
are enough keys in the entire tree to fill half a bucket). The
tradeoffs are murky. Pathological cases in the presence of
deletion do exist. Pathologies include trees tending toward only
one key per bucket, and buckets at differing depths (all buckets
are at the same depth in a classic BTree).
+ DEFAULT_MAX_BUCKET_SIZE and DEFAULT_MAX_BTREE_SIZE are chosen
mostly to "even out" pickle sizes in storage. That's why, e.g.,
an IIBTree has larger values than an OOBTree: pickles store ints
more efficiently than they can store arbitrary Python objects.
+ In a non-empty BTree, every bucket node contains at least one key,
and every BTree node contains at least one child and a non-NULL
firstbucket pointer. However, a BTree node may not contain any keys.
+ An empty BTree consists solely of a BTree node with len==0 and
firstbucket==NULL.
+ Although a BTree can become unbalanced under a mix of inserts and
deletes (meaning both that there's nothing stronger that can be
said about buckets than that they're not empty, and that buckets
can appear at different depths), a BTree node always has children
of the same kind: they're all buckets, or they're all BTree nodes.
The BTREE_SEARCH Macro
======================
For notational ease, consider a fixed BTree node x, and let
K(i) mean x->data.key[i]
C(i) mean all the keys reachable from x->data.child[i]
For each i in 0 to x->len-1 inclusive,
K(i) <= C(i) < K(i+1)
is a BTree node invariant, where we pretend that K(0) holds a key
smaller than any possible key, and K(x->len) holds a key larger
than any possible key. (Note that K(x->len) doesn't actually exist,
and K(0) is never used although space for it exists in non-empty
BTree nodes.)
When searching for a key k, then, the child pointer we want to follow
is the one at index i such that K(i) <= k < K(i+1). There can be
at most one such i, since the K(i) are strictly increasing. And there
is at least one such i provided the tree isn't empty (so that 0 < len).
For the moment, assume the tree isn't empty (we'll get back to that
later).
The macro's chief loop invariant is
K(lo) < k < K(hi)
This holds trivially at the start, since lo is set to 0, and hi to
x->len, and we pretend K(0) is minus infinity and K(len) is plus
infinity. Inside the loop, if K(i) < k we set lo to i, and if
K(i) > k we set hi to i. These obviously preserve the invariant.
If K(i) == k, the loop breaks and sets the result to i, and since
K(i) == k in that case i is obviously the correct result.
Other cases depend on how i = floor((lo + hi)/2) works, exactly.
Suppose lo + d = hi for some d >= 0. Then i = floor((lo + lo + d)/2) =
floor(lo + d/2) = lo + floor(d/2). So:
a. [d == 0] (lo == i == hi) if and only if (lo == hi).
b. [d == 1] (lo == i < hi) if and only if (lo+1 == hi).
c. [d > 1] (lo < i < hi) if and only if (lo+1 < hi).
If the node is empty (x->len == 0), then lo==i==hi==0 at the start,
and the loop exits immediately (the first "i > lo" test fails),
without entering the body.
Else lo < hi at the start, and the invariant K(lo) < k < K(hi) holds.
If lo+1 < hi, we're in case #c: i is strictly between lo and hi,
so the loop body is entered, and regardless of whether the body sets
the new lo or the new hi to i, the new lo is strictly less than the
new hi, and the difference between the new lo and new hi is strictly
less than the difference between the old lo and old hi. So long as
the new lo + 1 remains < the new hi, we stay in this case. We can't
stay in this case forever, though: because hi-lo decreases on each
trip but remains > 0, lo+1 == hi must eventually become true. (In
fact, it becomes true quickly, in about log2(x->len) trips; the
point is more that lo doesn't equal hi when the loop ends, it has to
end with lo+1==hi and i==lo).
Then we're in case #b: i==lo==hi-1 then, and the loop exits. The
invariant still holds, with lo==i and hi==lo+1==i+1:
K(i) < k < K(i+1)
so i is again the correct answer.
Optimization points:
+ Division by 2 is done via shift rather via "/2". These are
signed ints, and almost all C compilers treat signed int division
as truncating, and shifting is not the same as truncation for
signed int division. The compiler has no way to know these values
aren't negative, so has to generate longer-winded code for "/2".
But we know these values aren't negative, and exploit it.
+ The order of _cmp comparisons matters. We're in an interior
BTree node, and are looking at only a tiny fraction of all the
keys that exist. So finding the key exactly in this node is
unlikely, and checking _cmp == 0 is a waste of time to the same
extent. It doesn't matter whether we check for _cmp < 0 or
_cmp > 0 first, so long as we do both before worrying about
equality.
+ At the start of a routine, it's better to run this macro even
if x->len is 0 (check for that afterwards). We just called a
function and so probably drained the pipeline. If the first thing
we do then is read up self->len and check it against 0, we just
sit there waiting for the data to get read up, and then another
immediate test-and-branch, and for a very unlikely case (BTree
nodes are rarely empty). It's better to get into the loop right
away so the normal case makes progress ASAP.
The BUCKET_SEARCH Macro
=======================
This has a different job than BTREE_SEARCH: the key 0 slot is
legitimate in a bucket, and we want to find the index at which the
key belongs. If the key is larger than the bucket's largest key, a
new slot at index len is where it belongs, else it belongs at the
smallest i with keys[i] >= the key we're looking for. We also need
to know whether or not the key is present (BTREE_SEARCH didn't care;
it only wanted to find the next node to search).
The mechanics of the search are quite similar, though. The primary
loop invariant changes to (say we're searching for key k):
K(lo-1) < k < K(hi)
where K(i) means keys[i], and we pretend K(-1) is minus infinity and
K(len) is plus infinity.
If the bucket is empty, lo=hi=i=0 at the start, the loop body is never
entered, and the macro sets INDEX to 0 and ABSENT to true. That's why
_cmp is initialized to 1 (_cmp becomes ABSENT).
Else the bucket is not empty, lo<hi at the start, and the loop body
is entered. The invariant is obviously satisfied then, as lo=0 and
hi=len.
If K[i]<k, lo is set to i+1, preserving that K(lo-1) = K[i] < k.
If K[i]>k, hi is set to i, preserving that K[hi] = K[i] > k.
If the loop exits after either of those, _cmp != 0, so ABSENT becomes
true.
If K[i]=k, the loop breaks, so that INDEX becomes i, and ABSENT
becomes false (_cmp=0 in this case).
The same case analysis for BTREE_SEARCH on lo and hi holds here:
a. (lo == i == hi) if and only if (lo == hi).
b. (lo == i < hi) if and only if (lo+1 == hi).
c. (lo < i < hi) if and only if (lo+1 < hi).
So long as lo+1 < hi, we're in case #c, and either break with
equality (in which case the right results are obviously computed) or
narrow the range. If equality doesn't obtain, the range eventually
narrows to cases #a or #b.
To go from #c to #a, we must have lo+2==hi at the start, and
K[i]=K[lo+1]<k. Then the new lo gets set to i+1 = lo+2 = hi, and the
loop exits with lo=hi=i and _cmp<0. This is correct, because we
know that k != K(i) (loop invariant! we actually know something
stronger, that k < K(hi); since i=hi, this implies k != K(i)).
Else #c eventually falls into case #b, lo+1==hi and i==lo. The
invariant tells us K(lo-1) < k < K(hi) = K(lo+1), so if the key
is present it must be at K(lo). i==lo in this case, so we test
K(lo) against k. As always, if equality obtains we do the right
thing, else case #b becomes case #a.
When #b becomes #a, the last comparison was non-equal, so _cmp is
non-zero, and the loop exits because lo==hi==i in case #a. The
invariant then tells us K(lo-1) < k < K(lo), so the key is in fact
not present, it's correct to exit with _cmp non-zero, and i==lo is
again the index at which k belongs.
Optimization points:
+ As for BTREE_SEARCH, shifting of signed ints is cheaper than
division.
+ Unlike as for BTREE_SEARCH, there's nothing special about searching
an empty bucket, and the macro computes thoroughly sensible results
in that case.
+ The order of _cmp comparisons differs from BTREE_SEARCH. When
searching a bucket, it's much more likely (than when searching a
BTree node) that the key is present, so testing __cmp==0 isn't a
systematic waste of cycles. At the extreme, if all searches are
successful (key present), on average this saves one comparison per
search, against leaving the determination of _cmp==0 implicit (as
BTREE_SEARCH does). But even on successful searches, __cmp != 0 is
a more popular outcome than __cmp == 0 across iterations (unless
the bucket has only a few keys), so it's important to check one
of the inequality cases first. It turns out it's better on average
to check K(i) < key (than to check K(i) > key), because when it
pays it narrows the range more (we get a little boost from setting
lo=i+1 in this case; the other case sets hi=i, which isn't as much
of a narrowing).
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