src/backend/access/gist/README

GiST Indexing
=============

This directory contains an implementation of GiST indexing for Postgres.

GiST stands for Generalized Search Tree. It was introduced in the seminal paper
"Generalized Search Trees for Database Systems", 1995, Joseph M. Hellerstein,
Jeffrey F. Naughton, Avi Pfeffer:

    http://www.sai.msu.su/~megera/postgres/gist/papers/gist.ps
    https://dsf.berkeley.edu/papers/sigmod97-gist.pdf

and implemented by J. Hellerstein and P. Aoki in an early version of
PostgreSQL (more details are available from The GiST Indexing Project
at Berkeley at http://gist.cs.berkeley.edu/). As a "university"
project it had a limited number of features and was in rare use.

The current implementation of GiST supports:

  * Variable length keys
  * Composite keys (multi-key)
  * Ordered search (nearest-neighbor search)
  * provides NULL-safe interface to GiST core
  * Concurrency
  * Recovery support via WAL logging
  * Buffering build algorithm

The support for concurrency implemented in PostgreSQL was developed based on
the paper "Access Methods for Next-Generation Database Systems" by
Marcel Kornacker:

    http://www.sai.msu.su/~megera/postgres/gist/papers/concurrency/access-methods-for-next-generation.pdf.gz

Buffering build algorithm for GiST was developed based on the paper "Efficient
Bulk Operations on Dynamic R-trees" by Lars Arge, Klaus Hinrichs, Jan Vahrenhold
and Jeffrey Scott Vitter.

    http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.135.9894&rep=rep1&type=pdf

The original algorithms were modified in several ways:

* They had to be adapted to PostgreSQL conventions. For example, the SEARCH
  algorithm was considerably changed, because in PostgreSQL the search function
  should return one tuple (next), not all tuples at once. Also, it should
  release page locks between calls.
* Since we added support for variable length keys, it's not possible to
  guarantee enough free space for all keys on pages after splitting. User
  defined function picksplit doesn't have information about size of tuples
  (each tuple may contain several keys as in multicolumn index while picksplit
  could work with only one key) and pages.
* We modified original INSERT algorithm for performance reasons. In particular,
  it is now a single-pass algorithm.
* Since the papers were theoretical, some details were omitted and we
  had to find out ourself how to solve some specific problems.

Because of the above reasons, we have revised the interaction of GiST
core and PostgreSQL WAL system. Moreover, we encountered (and solved)
a problem of uncompleted insertions when recovering after crash, which
was not touched in the paper.

Search Algorithm
----------------

The search code maintains a queue of unvisited items, where an "item" is
either a heap tuple known to satisfy the search conditions, or an index
page that is consistent with the search conditions according to inspection
of its parent page's downlink item.  Initially the root page is searched
to find unvisited items in it.  Then we pull items from the queue.  A
heap tuple pointer is just returned immediately; an index page entry
causes that page to be searched, generating more queue entries.

The queue is kept ordered with heap tuple items at the front, then
index page entries, with any newly-added index page entry inserted
before existing index page entries.  This ensures depth-first traversal
of the index, and in particular causes the first few heap tuples to be
returned as soon as possible.  That is helpful in case there is a LIMIT
that requires only a few tuples to be produced.

To implement nearest-neighbor search, the queue entries are augmented
with distance data: heap tuple entries are labeled with exact distance
from the search argument, while index-page entries must be labeled with
the minimum distance that any of their children could have.  Then,
queue entries are retrieved in smallest-distance-first order, with
entries having identical distances managed as stated in the previous
paragraph.

The search algorithm keeps an index page locked only long enough to scan
its entries and queue those that satisfy the search conditions.  Since
insertions can occur concurrently with searches, it is possible for an
index child page to be split between the time we make a queue entry for it
(while visiting its parent page) and the time we actually reach and scan
the child page.  To avoid missing the entries that were moved to the right
sibling, we detect whether a split has occurred by comparing the child
page's NSN (node sequence number, a special-purpose LSN) to the LSN that
the parent had when visited.  If it did, the sibling page is immediately
added to the front of the queue, ensuring that its items will be scanned
in the same order as if they were still on the original child page.

As is usual in Postgres, the search algorithm only guarantees to find index
entries that existed before the scan started; index entries added during
the scan might or might not be visited.  This is okay as long as all
searches use MVCC snapshot rules to reject heap tuples newer than the time
of scan start.  In particular, this means that we need not worry about
cases where a parent page's downlink key is "enlarged" after we look at it.
Any such enlargement would be to add child items that we aren't interested
in returning anyway.


Insert Algorithm
----------------

INSERT guarantees that the GiST tree remains balanced. User defined key method
Penalty is used for choosing a subtree to insert; method PickSplit is used for
the node splitting algorithm; method Union is used for propagating changes
upward to maintain the tree properties.

To insert a tuple, we first have to find a suitable leaf page to insert to.
The algorithm walks down the tree, starting from the root, along the path
of smallest Penalty. At each step:

1. Has this page been split since we looked at the parent? If so, it's
possible that we should be inserting to the other half instead, so retreat
back to the parent.
2. If this is a leaf node, we've found our target node.
3. Otherwise use Penalty to pick a new target subtree.
4. Check the key representing the target subtree. If it doesn't already cover
the key we're inserting, replace it with the Union of the old downlink key
and the key being inserted. (Actually, we always call Union, and just skip
the replacement if the Unioned key is the same as the existing key)
5. Replacing the key in step 4 might cause the page to be split. In that case,
propagate the change upwards and restart the algorithm from the first parent
that didn't need to be split.
6. Walk down to the target subtree, and goto 1.

This differs from the insertion algorithm in the original paper. In the
original paper, you first walk down the tree until you reach a leaf page, and
then you adjust the downlink in the parent, and propagate the adjustment up,
all the way up to the root in the worst case. But we adjust the downlinks to
cover the new key already when we walk down, so that when we reach the leaf
page, we don't need to update the parents anymore, except to insert the
downlinks if we have to split the page. This makes crash recovery simpler:
after inserting a key to the page, the tree is immediately self-consistent
without having to update the parents. Even if we split a page and crash before
inserting the downlink to the parent, the tree is self-consistent because the
right half of the split is accessible via the rightlink of the left page
(which replaced the original page).

Note that the algorithm can walk up and down the tree before reaching a leaf
page, if internal pages need to split while adjusting the downlinks for the
new key. Eventually, you should reach the bottom, and proceed with the
insertion of the new tuple.

Once we've found the target page to insert to, we check if there's room
for the new tuple. If there is, the tuple is inserted, and we're done.
If it doesn't fit, however, the page needs to be split. Note that it is
possible that a page needs to be split into more than two pages, if keys have
different lengths or more than one key is being inserted at a time (which can
happen when inserting downlinks for a page split that resulted in more than
two pages at the lower level). After splitting a page, the parent page needs
to be updated. The downlink for the new page needs to be inserted, and the
downlink for the old page, which became the left half of the split, needs to
be updated to only cover those tuples that stayed on the left page. Inserting
the downlink in the parent can again lead to a page split, recursing up to the
root page in the worst case.

gistplacetopage is the workhorse function that performs one step of the
insertion. If the tuple fits, it inserts it to the given page, otherwise
it splits the page, and constructs the new downlink tuples for the split
pages. The caller must then call gistplacetopage() on the parent page to
insert the downlink tuples. The parent page that holds the downlink to
the child might have migrated as a result of concurrent splits of the
parent, gistFindCorrectParent() is used to find the parent page.

Splitting the root page works slightly differently. At root split,
gistplacetopage() allocates the new child pages and replaces the old root
page with the new root containing downlinks to the new children, all in one
operation.


findPath is a subroutine of findParent, used when the correct parent page
can't be found by following the rightlinks at the parent level:

findPath( stack item )
	push stack, [root, 0, 0] // page, LSN, parent
	while( stack )
		ptr = top of stack
		latch( ptr->page, S-mode )
		if ( ptr->parent->page->lsn < ptr->page->nsn )
			push stack, [ ptr->page->rightlink, 0, ptr->parent ]
		end
		for( each tuple on page )
			if ( tuple->pagepointer == item->page )
				return stack
			else
				add to stack at the end [tuple->pagepointer,0, ptr]
			end
		end
		unlatch( ptr->page )
		pop stack
	end


gistFindCorrectParent is used to re-find the parent of a page during
insertion. It might have migrated to the right since we traversed down the
tree because of page splits.

findParent( stack item )
	parent = item->parent
	if ( parent->page->lsn != parent->lsn )
		while(true)
			search parent tuple on parent->page, if found the return
			rightlink = parent->page->rightlink
			unlatch( parent->page )
			if ( rightlink is incorrect )
				break loop
			end
			parent->page = rightlink
			latch( parent->page, X-mode )
		end
		newstack = findPath( item->parent )
		replace part of stack to new one
		latch( parent->page, X-mode )
		return findParent( item )
	end

pageSplit function decides how to distribute keys to the new pages after
page split:

pageSplit(page, allkeys)
	(lkeys, rkeys) = pickSplit( allkeys )
	if ( page is root )
		lpage = new page
	else
		lpage = page
	rpage = new page
	if ( no space left on rpage )
		newkeys = pageSplit( rpage, rkeys )
	else
		push newkeys, union(rkeys)
	end
	if ( no space left on lpage )
		push newkeys, pageSplit( lpage, lkeys )
	else
		push newkeys, union(lkeys)
	end
	return newkeys



Concurrency control
-------------------
As a rule of thumb, if you need to hold a lock on multiple pages at the
same time, the locks should be acquired in the following order: child page
before parent, and left-to-right at the same level. Always acquiring the
locks in the same order avoids deadlocks.

The search algorithm only looks at and locks one page at a time. Consequently
there's a race condition between a search and a page split. A page split
happens in two phases: 1. The page is split 2. The downlink is inserted to the
parent. If a search looks at the parent page between those steps, before the
downlink is inserted, it will still find the new right half by following the
rightlink on the left half. But it must not follow the rightlink if it saw the
downlink in the parent, or the page will be visited twice!

A split initially marks the left page with the F_FOLLOW_RIGHT flag. If a scan
sees that flag set, it knows that the right page is missing the downlink, and
should be visited too. When split inserts the downlink to the parent, it
clears the F_FOLLOW_RIGHT flag in the child, and sets the NSN field in the
child page header to match the LSN of the insertion on the parent. If the
F_FOLLOW_RIGHT flag is not set, a scan compares the NSN on the child and the
LSN it saw in the parent. If NSN < LSN, the scan looked at the parent page
before the downlink was inserted, so it should follow the rightlink. Otherwise
the scan saw the downlink in the parent page, and will/did follow that as
usual.

A scan can't normally see a page with the F_FOLLOW_RIGHT flag set, because
a page split keeps the child pages locked until the downlink has been inserted
to the parent and the flag cleared again. But if a crash happens in the middle
of a page split, before the downlinks are inserted into the parent, that will
leave a page with F_FOLLOW_RIGHT in the tree. Scans handle that just fine,
but we'll eventually want to fix that for performance reasons. And more
importantly, dealing with pages with missing downlink pointers in the parent
would complicate the insertion algorithm. So when an insertion sees a page
with F_FOLLOW_RIGHT set, it immediately tries to bring the split that
crashed in the middle to completion by adding the downlink in the parent.

Buffering build algorithm
-------------------------

In the buffering index build algorithm, some or all internal nodes have a
buffer attached to them. When a tuple is inserted at the top, the descend down
the tree is stopped as soon as a buffer is reached, and the tuple is pushed to
the buffer. When a buffer gets too full, all the tuples in it are flushed to
the lower level, where they again hit lower level buffers or leaf pages. This
makes the insertions happen in more of a breadth-first than depth-first order,
which greatly reduces the amount of random I/O required.

In the algorithm, levels are numbered so that leaf pages have level zero,
and internal node levels count up from 1. This numbering ensures that a page's
level number never changes, even when the root page is split.

Level                    Tree

3                         *
                      /       \
2                *                 *
              /  |  \           /  |  \
1          *     *     *     *     *     *
          / \   / \   / \   / \   / \   / \
0        o   o o   o o   o o   o o   o o   o

* - internal page
o - leaf page

Internal pages that belong to certain levels have buffers associated with
them. Leaf pages never have buffers. Which levels have buffers is controlled
by "level step" parameter: level numbers that are multiples of level_step
have buffers, while others do not. For example, if level_step = 2, then
pages on levels 2, 4, 6, ... have buffers. If level_step = 1 then every
internal page has a buffer.

Level        Tree (level_step = 1)                Tree (level_step = 2)

3                      *                                     *
                   /       \                             /       \
2             *(b)              *(b)                *(b)              *(b)
           /  |  \           /  |  \             /  |  \           /  |  \
1       *(b)  *(b)  *(b)  *(b)  *(b)  *(b)    *     *     *     *     *     *
       / \   / \   / \   / \   / \   / \     / \   / \   / \   / \   / \   / \
0     o   o o   o o   o o   o o   o o   o   o   o o   o o   o o   o o   o o   o

(b) - buffer

Logically, a buffer is just bunch of tuples. Physically, it is divided in
pages, backed by a temporary file. Each buffer can be in one of two states:
a) Last page of the buffer is kept in main memory. A node buffer is
automatically switched to this state when a new index tuple is added to it,
or a tuple is removed from it.
b) All pages of the buffer are swapped out to disk. When a buffer becomes too
full, and we start to flush it, all other buffers are switched to this state.

When an index tuple is inserted, its initial processing can end in one of the
following points:
1) Leaf page, if the depth of the index <= level_step, meaning that
   none of the internal pages have buffers associated with them.
2) Buffer of topmost level page that has buffers.

New index tuples are processed until one of the buffers in the topmost
buffered level becomes half-full. When a buffer becomes half-full, it's added
to the emptying queue, and will be emptied before a new tuple is processed.

Buffer emptying process means that index tuples from the buffer are moved
into buffers at a lower level, or leaf pages. First, all the other buffers are
swapped to disk to free up the memory. Then tuples are popped from the buffer
one by one, and cascaded down the tree to the next buffer or leaf page below
the buffered node.

Emptying a buffer has the interesting dynamic property that any intermediate
pages between the buffer being emptied, and the next buffered or leaf level
below it, become cached. If there are no more buffers below the node, the leaf
pages where the tuples finally land on get cached too. If there are, the last
buffer page of each buffer below is kept in memory. This is illustrated in
the figures below:

   Buffer being emptied to
     lower-level buffers               Buffer being emptied to leaf pages

               +(fb)                                 +(fb)
            /     \                                /     \
        +             +                        +             +
      /   \         /   \                    /   \         /   \
    *(ab)   *(ab) *(ab)   *(ab)            x       x     x       x

+    - cached internal page
x    - cached leaf page
*    - non-cached internal page
(fb) - buffer being emptied
(ab) - buffers being appended to, with last page in memory

In the beginning of the index build, the level-step is chosen so that all those
pages involved in emptying one buffer fit in cache, so after each of those
pages have been accessed once and cached, emptying a buffer doesn't involve
any more I/O. This locality is where the speedup of the buffering algorithm
comes from.

Emptying one buffer can fill up one or more of the lower-level buffers,
triggering emptying of them as well. Whenever a buffer becomes too full, it's
added to the emptying queue, and will be emptied after the current buffer has
been processed.

To keep the size of each buffer limited even in the worst case, buffer emptying
is scheduled as soon as a buffer becomes half-full, and emptying it continues
until 1/2 of the nominal buffer size worth of tuples has been emptied. This
guarantees that when buffer emptying begins, all the lower-level buffers
are at most half-full. In the worst case that all the tuples are cascaded down
to the same lower-level buffer, that buffer therefore has enough space to
accommodate all the tuples emptied from the upper-level buffer. There is no
hard size limit in any of the data structures used, though, so this only needs
to be approximate; small overfilling of some buffers doesn't matter.

If an internal page that has a buffer associated with it is split, the buffer
needs to be split too. All tuples in the buffer are scanned through and
relocated to the correct sibling buffers, using the penalty function to decide
which buffer each tuple should go to.

After all tuples from the heap have been processed, there are still some index
tuples in the buffers. At this point, final buffer emptying starts. All buffers
are emptied in top-down order. This is slightly complicated by the fact that
new buffers can be allocated during the emptying, due to page splits. However,
the new buffers will always be siblings of buffers that haven't been fully
emptied yet; tuples never move upwards in the tree. The final emptying loops
through buffers at a given level until all buffers at that level have been
emptied, and then moves down to the next level.

Bulk delete algorithm (VACUUM)
------------------------------

VACUUM works in two stages:

In the first stage, we scan the whole index in physical order. To make sure
that we don't miss any dead tuples because a concurrent page split moved them,
we check the F_FOLLOW_RIGHT flags and NSN on each page, to detect if the
page has been concurrently split. If a concurrent page split is detected, and
one half of the page was moved to a position that we already scanned, we
"jump backwards" to scan the page again. This is the same mechanism that
B-tree VACUUM uses, but because we already have NSNs on pages, to detect page
splits during searches, we don't need a "vacuum cycle ID" concept for that
like B-tree does.

While we scan all the pages, we also make note of any completely empty leaf
pages. We will try to unlink them from the tree after the scan. We also record
the block numbers of all internal pages; they are needed to locate parents of
the empty pages while unlinking them.

We try to unlink any empty leaf pages from the tree, so that their space can
be reused. In order to delete an empty page, its downlink must be removed from
the parent. We scan all the internal pages, whose block numbers we memorized
in the first stage, and look for downlinks to pages that we have memorized as
being empty. Whenever we find one, we acquire a lock on the parent and child
page, re-check that the child page is still empty. Then, we remove the
downlink and mark the child as deleted, and release the locks.

The insertion algorithm would get confused, if an internal page was completely
empty. So we never delete the last child of an internal page, even if it's
empty. Currently, we only support deleting leaf pages.

This page deletion algorithm works on a best-effort basis. It might fail to
find a downlink, if a concurrent page split moved it after the first stage.
In that case, we won't be able to remove all empty pages. That's OK, it's
not expected to happen very often, and hopefully the next VACUUM will clean
it up.

When we have deleted a page, it's possible that an in-progress search will
still descend on the page, if it saw the downlink before we removed it. The
search will see that it is deleted, and ignore it, but as long as that can
happen, we cannot reuse the page. To "wait out" any in-progress searches, when
a page is deleted, it's labeled with the current next-transaction counter
value. The page is not recycled, until that XID is no longer visible to
anyone. That's much more conservative than necessary, but let's keep it
simple.


Authors:
	Teodor Sigaev	<teodor@sigaev.ru>
	Oleg Bartunov	<oleg@sai.msu.su>
