[postgresql] / src / backend / access / nbtree / nbtinsert.c

/*-------------------------------------------------------------------------
 *
 * nbtinsert.c
 *        Item insertion in Lehman and Yao btrees for Postgres.
 *
 * Portions Copyright (c) 1996-2007, PostgreSQL Global Development Group
 * Portions Copyright (c) 1994, Regents of the University of California
 *
 *
 * IDENTIFICATION
 *        $PostgreSQL: pgsql/src/backend/access/nbtree/nbtinsert.c,v 1.163 2007/12/31 04:52:05 tgl Exp $
 *
 *-------------------------------------------------------------------------
 */

#include "postgres.h"

#include "access/heapam.h"
#include "access/nbtree.h"
#include "access/transam.h"
#include "miscadmin.h"
#include "utils/inval.h"


typedef struct
{
        /* context data for _bt_checksplitloc */
        Size            newitemsz;              /* size of new item to be inserted */
        int                     fillfactor;             /* needed when splitting rightmost page */
        bool            is_leaf;                /* T if splitting a leaf page */
        bool            is_rightmost;   /* T if splitting a rightmost page */
        OffsetNumber newitemoff;        /* where the new item is to be inserted */
        int                     leftspace;              /* space available for items on left page */
        int                     rightspace;             /* space available for items on right page */
        int                     olddataitemstotal;              /* space taken by old items */

        bool            have_split;             /* found a valid split? */

        /* these fields valid only if have_split is true */
        bool            newitemonleft;  /* new item on left or right of best split */
        OffsetNumber firstright;        /* best split point */
        int                     best_delta;             /* best size delta so far */
} FindSplitData;


static Buffer _bt_newroot(Relation rel, Buffer lbuf, Buffer rbuf);

static TransactionId _bt_check_unique(Relation rel, IndexTuple itup,
                                 Relation heapRel, Buffer buf, OffsetNumber ioffset,
                                 ScanKey itup_scankey);
static void _bt_findinsertloc(Relation rel,
                                  Buffer *bufptr,
                                  OffsetNumber *offsetptr,
                                  int keysz,
                                  ScanKey scankey,
                                  IndexTuple newtup);
static void _bt_insertonpg(Relation rel, Buffer buf,
                           BTStack stack,
                           IndexTuple itup,
                           OffsetNumber newitemoff,
                           bool split_only_page);
static Buffer _bt_split(Relation rel, Buffer buf, OffsetNumber firstright,
                  OffsetNumber newitemoff, Size newitemsz,
                  IndexTuple newitem, bool newitemonleft);
static OffsetNumber _bt_findsplitloc(Relation rel, Page page,
                                 OffsetNumber newitemoff,
                                 Size newitemsz,
                                 bool *newitemonleft);
static void _bt_checksplitloc(FindSplitData *state,
                                  OffsetNumber firstoldonright, bool newitemonleft,
                                  int dataitemstoleft, Size firstoldonrightsz);
static void _bt_pgaddtup(Relation rel, Page page,
                         Size itemsize, IndexTuple itup,
                         OffsetNumber itup_off, const char *where);
static bool _bt_isequal(TupleDesc itupdesc, Page page, OffsetNumber offnum,
                        int keysz, ScanKey scankey);
static void _bt_vacuum_one_page(Relation rel, Buffer buffer);


/*
 *      _bt_doinsert() -- Handle insertion of a single index tuple in the tree.
 *
 *              This routine is called by the public interface routines, btbuild
 *              and btinsert.  By here, itup is filled in, including the TID.
 */
void
_bt_doinsert(Relation rel, IndexTuple itup,
                         bool index_is_unique, Relation heapRel)
{
        int                     natts = rel->rd_rel->relnatts;
        ScanKey         itup_scankey;
        BTStack         stack;
        Buffer          buf;
        OffsetNumber offset;

        /* we need an insertion scan key to do our search, so build one */
        itup_scankey = _bt_mkscankey(rel, itup);

top:
        /* find the first page containing this key */
        stack = _bt_search(rel, natts, itup_scankey, false, &buf, BT_WRITE);

        offset = InvalidOffsetNumber;

        /* trade in our read lock for a write lock */
        LockBuffer(buf, BUFFER_LOCK_UNLOCK);
        LockBuffer(buf, BT_WRITE);

        /*
         * If the page was split between the time that we surrendered our read
         * lock and acquired our write lock, then this page may no longer be the
         * right place for the key we want to insert.  In this case, we need to
         * move right in the tree.      See Lehman and Yao for an excruciatingly
         * precise description.
         */
        buf = _bt_moveright(rel, buf, natts, itup_scankey, false, BT_WRITE);

        /*
         * If we're not allowing duplicates, make sure the key isn't already in
         * the index.
         *
         * NOTE: obviously, _bt_check_unique can only detect keys that are already
         * in the index; so it cannot defend against concurrent insertions of the
         * same key.  We protect against that by means of holding a write lock on
         * the target page.  Any other would-be inserter of the same key must
         * acquire a write lock on the same target page, so only one would-be
         * inserter can be making the check at one time.  Furthermore, once we are
         * past the check we hold write locks continuously until we have performed
         * our insertion, so no later inserter can fail to see our insertion.
         * (This requires some care in _bt_insertonpg.)
         *
         * If we must wait for another xact, we release the lock while waiting,
         * and then must start over completely.
         */
        if (index_is_unique)
        {
                TransactionId xwait;

                offset = _bt_binsrch(rel, buf, natts, itup_scankey, false);
                xwait = _bt_check_unique(rel, itup, heapRel, buf, offset, itup_scankey);

                if (TransactionIdIsValid(xwait))
                {
                        /* Have to wait for the other guy ... */
                        _bt_relbuf(rel, buf);
                        XactLockTableWait(xwait);
                        /* start over... */
                        _bt_freestack(stack);
                        goto top;
                }
        }

        /* do the insertion */
        _bt_findinsertloc(rel, &buf, &offset, natts, itup_scankey, itup);
        _bt_insertonpg(rel, buf, stack, itup, offset, false);

        /* be tidy */
        _bt_freestack(stack);
        _bt_freeskey(itup_scankey);
}

/*
 *      _bt_check_unique() -- Check for violation of unique index constraint
 *
 * offset points to the first possible item that could conflict. It can
 * also point to end-of-page, which means that the first tuple to check
 * is the first tuple on the next page.
 *
 * Returns InvalidTransactionId if there is no conflict, else an xact ID
 * we must wait for to see if it commits a conflicting tuple.   If an actual
 * conflict is detected, no return --- just ereport().
 */
static TransactionId
_bt_check_unique(Relation rel, IndexTuple itup, Relation heapRel,
                                 Buffer buf, OffsetNumber offset, ScanKey itup_scankey)
{
        TupleDesc       itupdesc = RelationGetDescr(rel);
        int                     natts = rel->rd_rel->relnatts;
        SnapshotData SnapshotDirty;
        OffsetNumber maxoff;
        Page            page;
        BTPageOpaque opaque;
        Buffer          nbuf = InvalidBuffer;

        InitDirtySnapshot(SnapshotDirty);

        page = BufferGetPage(buf);
        opaque = (BTPageOpaque) PageGetSpecialPointer(page);
        maxoff = PageGetMaxOffsetNumber(page);

        /*
         * Scan over all equal tuples, looking for live conflicts.
         */
        for (;;)
        {
                ItemId          curitemid;
                IndexTuple      curitup;
                BlockNumber nblkno;

                /*
                 * make sure the offset points to an actual item before trying to
                 * examine it...
                 */
                if (offset <= maxoff)
                {
                        curitemid = PageGetItemId(page, offset);

                        /*
                         * We can skip items that are marked killed.
                         *
                         * Formerly, we applied _bt_isequal() before checking the kill
                         * flag, so as to fall out of the item loop as soon as possible.
                         * However, in the presence of heavy update activity an index may
                         * contain many killed items with the same key; running
                         * _bt_isequal() on each killed item gets expensive. Furthermore
                         * it is likely that the non-killed version of each key appears
                         * first, so that we didn't actually get to exit any sooner
                         * anyway. So now we just advance over killed items as quickly as
                         * we can. We only apply _bt_isequal() when we get to a non-killed
                         * item or the end of the page.
                         */
                        if (!ItemIdIsDead(curitemid))
                        {
                                ItemPointerData htid;
                                bool            all_dead;

                                /*
                                 * _bt_compare returns 0 for (1,NULL) and (1,NULL) - this's
                                 * how we handling NULLs - and so we must not use _bt_compare
                                 * in real comparison, but only for ordering/finding items on
                                 * pages. - vadim 03/24/97
                                 */
                                if (!_bt_isequal(itupdesc, page, offset, natts, itup_scankey))
                                        break;          /* we're past all the equal tuples */

                                /* okay, we gotta fetch the heap tuple ... */
                                curitup = (IndexTuple) PageGetItem(page, curitemid);
                                htid = curitup->t_tid;

                                /*
                                 * We check the whole HOT-chain to see if there is any tuple
                                 * that satisfies SnapshotDirty.  This is necessary because we
                                 * have just a single index entry for the entire chain.
                                 */
                                if (heap_hot_search(&htid, heapRel, &SnapshotDirty, &all_dead))
                                {
                                        /* it is a duplicate */
                                        TransactionId xwait =
                                        (TransactionIdIsValid(SnapshotDirty.xmin)) ?
                                        SnapshotDirty.xmin : SnapshotDirty.xmax;

                                        /*
                                         * If this tuple is being updated by other transaction
                                         * then we have to wait for its commit/abort.
                                         */
                                        if (TransactionIdIsValid(xwait))
                                        {
                                                if (nbuf != InvalidBuffer)
                                                        _bt_relbuf(rel, nbuf);
                                                /* Tell _bt_doinsert to wait... */
                                                return xwait;
                                        }

                                        /*
                                         * Otherwise we have a definite conflict.  But before
                                         * complaining, look to see if the tuple we want to insert
                                         * is itself now committed dead --- if so, don't complain.
                                         * This is a waste of time in normal scenarios but we must
                                         * do it to support CREATE INDEX CONCURRENTLY.
                                         *
                                         * We must follow HOT-chains here because during
                                         * concurrent index build, we insert the root TID though
                                         * the actual tuple may be somewhere in the HOT-chain.
                                         * While following the chain we might not stop at the
                                         * exact tuple which triggered the insert, but that's OK
                                         * because if we find a live tuple anywhere in this chain,
                                         * we have a unique key conflict.  The other live tuple is
                                         * not part of this chain because it had a different index
                                         * entry.
                                         */
                                        htid = itup->t_tid;
                                        if (heap_hot_search(&htid, heapRel, SnapshotSelf, NULL))
                                        {
                                                /* Normal case --- it's still live */
                                        }
                                        else
                                        {
                                                /*
                                                 * It's been deleted, so no error, and no need to
                                                 * continue searching
                                                 */
                                                break;
                                        }

                                        ereport(ERROR,
                                                        (errcode(ERRCODE_UNIQUE_VIOLATION),
                                                         errmsg("duplicate key value violates unique constraint \"%s\"",
                                                                        RelationGetRelationName(rel))));
                                }
                                else if (all_dead)
                                {
                                        /*
                                         * The conflicting tuple (or whole HOT chain) is dead to
                                         * everyone, so we may as well mark the index entry
                                         * killed.
                                         */
                                        ItemIdMarkDead(curitemid);
                                        opaque->btpo_flags |= BTP_HAS_GARBAGE;
                                        /* be sure to mark the proper buffer dirty... */
                                        if (nbuf != InvalidBuffer)
                                                SetBufferCommitInfoNeedsSave(nbuf);
                                        else
                                                SetBufferCommitInfoNeedsSave(buf);
                                }
                        }
                }

                /*
                 * Advance to next tuple to continue checking.
                 */
                if (offset < maxoff)
                        offset = OffsetNumberNext(offset);
                else
                {
                        /* If scankey == hikey we gotta check the next page too */
                        if (P_RIGHTMOST(opaque))
                                break;
                        if (!_bt_isequal(itupdesc, page, P_HIKEY,
                                                         natts, itup_scankey))
                                break;
                        /* Advance to next non-dead page --- there must be one */
                        for (;;)
                        {
                                nblkno = opaque->btpo_next;
                                nbuf = _bt_relandgetbuf(rel, nbuf, nblkno, BT_READ);
                                page = BufferGetPage(nbuf);
                                opaque = (BTPageOpaque) PageGetSpecialPointer(page);
                                if (!P_IGNORE(opaque))
                                        break;
                                if (P_RIGHTMOST(opaque))
                                        elog(ERROR, "fell off the end of index \"%s\"",
                                                 RelationGetRelationName(rel));
                        }
                        maxoff = PageGetMaxOffsetNumber(page);
                        offset = P_FIRSTDATAKEY(opaque);
                }
        }

        if (nbuf != InvalidBuffer)
                _bt_relbuf(rel, nbuf);

        return InvalidTransactionId;
}


/*
 *      _bt_findinsertloc() -- Finds an insert location for a tuple
 *
 *              If the new key is equal to one or more existing keys, we can
 *              legitimately place it anywhere in the series of equal keys --- in fact,
 *              if the new key is equal to the page's "high key" we can place it on
 *              the next page.  If it is equal to the high key, and there's not room
 *              to insert the new tuple on the current page without splitting, then
 *              we can move right hoping to find more free space and avoid a split.
 *              (We should not move right indefinitely, however, since that leads to
 *              O(N^2) insertion behavior in the presence of many equal keys.)
 *              Once we have chosen the page to put the key on, we'll insert it before
 *              any existing equal keys because of the way _bt_binsrch() works.
 *
 *              If there's not enough room in the space, we try to make room by
 *              removing any LP_DEAD tuples.
 *
 *              On entry, *buf and *offsetptr point to the first legal position
 *              where the new tuple could be inserted.  The caller should hold an
 *              exclusive lock on *buf.  *offsetptr can also be set to
 *              InvalidOffsetNumber, in which case the function will search for the
 *              right location within the page if needed.  On exit, they point to the
 *              chosen insert location.  If _bt_findinsertloc decides to move right,
 *              the lock and pin on the original page will be released and the new
 *              page returned to the caller is exclusively locked instead.
 *
 *              newtup is the new tuple we're inserting, and scankey is an insertion
 *              type scan key for it.
 */
static void
_bt_findinsertloc(Relation rel,
                                  Buffer *bufptr,
                                  OffsetNumber *offsetptr,
                                  int keysz,
                                  ScanKey scankey,
                                  IndexTuple newtup)
{
        Buffer          buf = *bufptr;
        Page            page = BufferGetPage(buf);
        Size            itemsz;
        BTPageOpaque lpageop;
        bool            movedright,
                                vacuumed;
        OffsetNumber newitemoff;
        OffsetNumber firstlegaloff = *offsetptr;

        lpageop = (BTPageOpaque) PageGetSpecialPointer(page);

        itemsz = IndexTupleDSize(*newtup);
        itemsz = MAXALIGN(itemsz);      /* be safe, PageAddItem will do this but we
                                                                 * need to be consistent */

        /*
         * Check whether the item can fit on a btree page at all. (Eventually, we
         * ought to try to apply TOAST methods if not.) We actually need to be
         * able to fit three items on every page, so restrict any one item to 1/3
         * the per-page available space. Note that at this point, itemsz doesn't
         * include the ItemId.
         */
        if (itemsz > BTMaxItemSize(page))
                ereport(ERROR,
                                (errcode(ERRCODE_PROGRAM_LIMIT_EXCEEDED),
                                 errmsg("index row size %lu exceeds btree maximum, %lu",
                                                (unsigned long) itemsz,
                                                (unsigned long) BTMaxItemSize(page)),
                errhint("Values larger than 1/3 of a buffer page cannot be indexed.\n"
                                "Consider a function index of an MD5 hash of the value, "
                                "or use full text indexing.")));

        /*----------
         * If we will need to split the page to put the item on this page,
         * check whether we can put the tuple somewhere to the right,
         * instead.  Keep scanning right until we
         *              (a) find a page with enough free space,
         *              (b) reach the last page where the tuple can legally go, or
         *              (c) get tired of searching.
         * (c) is not flippant; it is important because if there are many
         * pages' worth of equal keys, it's better to split one of the early
         * pages than to scan all the way to the end of the run of equal keys
         * on every insert.  We implement "get tired" as a random choice,
         * since stopping after scanning a fixed number of pages wouldn't work
         * well (we'd never reach the right-hand side of previously split
         * pages).      Currently the probability of moving right is set at 0.99,
         * which may seem too high to change the behavior much, but it does an
         * excellent job of preventing O(N^2) behavior with many equal keys.
         *----------
         */
        movedright = false;
        vacuumed = false;
        while (PageGetFreeSpace(page) < itemsz)
        {
                Buffer          rbuf;

                /*
                 * before considering moving right, see if we can obtain enough space
                 * by erasing LP_DEAD items
                 */
                if (P_ISLEAF(lpageop) && P_HAS_GARBAGE(lpageop))
                {
                        _bt_vacuum_one_page(rel, buf);

                        /*
                         * remember that we vacuumed this page, because that makes the
                         * hint supplied by the caller invalid
                         */
                        vacuumed = true;

                        if (PageGetFreeSpace(page) >= itemsz)
                                break;                  /* OK, now we have enough space */
                }

                /*
                 * nope, so check conditions (b) and (c) enumerated above
                 */
                if (P_RIGHTMOST(lpageop) ||
                        _bt_compare(rel, keysz, scankey, page, P_HIKEY) != 0 ||
                        random() <= (MAX_RANDOM_VALUE / 100))
                        break;

                /*
                 * step right to next non-dead page
                 *
                 * must write-lock that page before releasing write lock on current
                 * page; else someone else's _bt_check_unique scan could fail to see
                 * our insertion.  write locks on intermediate dead pages won't do
                 * because we don't know when they will get de-linked from the tree.
                 */
                rbuf = InvalidBuffer;

                for (;;)
                {
                        BlockNumber rblkno = lpageop->btpo_next;

                        rbuf = _bt_relandgetbuf(rel, rbuf, rblkno, BT_WRITE);
                        page = BufferGetPage(rbuf);
                        lpageop = (BTPageOpaque) PageGetSpecialPointer(page);
                        if (!P_IGNORE(lpageop))
                                break;
                        if (P_RIGHTMOST(lpageop))
                                elog(ERROR, "fell off the end of index \"%s\"",
                                         RelationGetRelationName(rel));
                }
                _bt_relbuf(rel, buf);
                buf = rbuf;
                movedright = true;
                vacuumed = false;
        }

        /*
         * Now we are on the right page, so find the insert position. If we moved
         * right at all, we know we should insert at the start of the page. If we
         * didn't move right, we can use the firstlegaloff hint if the caller
         * supplied one, unless we vacuumed the page which might have moved tuples
         * around making the hint invalid. If we didn't move right or can't use
         * the hint, find the position by searching.
         */
        if (movedright)
                newitemoff = P_FIRSTDATAKEY(lpageop);
        else if (firstlegaloff != InvalidOffsetNumber && !vacuumed)
                newitemoff = firstlegaloff;
        else
                newitemoff = _bt_binsrch(rel, buf, keysz, scankey, false);

        *bufptr = buf;
        *offsetptr = newitemoff;
}

/*----------
 *      _bt_insertonpg() -- Insert a tuple on a particular page in the index.
 *
 *              This recursive procedure does the following things:
 *
 *                      +  if necessary, splits the target page (making sure that the
 *                         split is equitable as far as post-insert free space goes).
 *                      +  inserts the tuple.
 *                      +  if the page was split, pops the parent stack, and finds the
 *                         right place to insert the new child pointer (by walking
 *                         right using information stored in the parent stack).
 *                      +  invokes itself with the appropriate tuple for the right
 *                         child page on the parent.
 *                      +  updates the metapage if a true root or fast root is split.
 *
 *              On entry, we must have the right buffer in which to do the
 *              insertion, and the buffer must be pinned and write-locked.      On return,
 *              we will have dropped both the pin and the lock on the buffer.
 *
 *              The locking interactions in this code are critical.  You should
 *              grok Lehman and Yao's paper before making any changes.  In addition,
 *              you need to understand how we disambiguate duplicate keys in this
 *              implementation, in order to be able to find our location using
 *              L&Y "move right" operations.  Since we may insert duplicate user
 *              keys, and since these dups may propagate up the tree, we use the
 *              'afteritem' parameter to position ourselves correctly for the
 *              insertion on internal pages.
 *----------
 */
static void
_bt_insertonpg(Relation rel,
                           Buffer buf,
                           BTStack stack,
                           IndexTuple itup,
                           OffsetNumber newitemoff,
                           bool split_only_page)
{
        Page            page;
        BTPageOpaque lpageop;
        OffsetNumber firstright = InvalidOffsetNumber;
        Size            itemsz;

        page = BufferGetPage(buf);
        lpageop = (BTPageOpaque) PageGetSpecialPointer(page);

        itemsz = IndexTupleDSize(*itup);
        itemsz = MAXALIGN(itemsz);      /* be safe, PageAddItem will do this but we
                                                                 * need to be consistent */

        /*
         * Do we need to split the page to fit the item on it?
         *
         * Note: PageGetFreeSpace() subtracts sizeof(ItemIdData) from its result,
         * so this comparison is correct even though we appear to be accounting
         * only for the item and not for its line pointer.
         */
        if (PageGetFreeSpace(page) < itemsz)
        {
                bool            is_root = P_ISROOT(lpageop);
                bool            is_only = P_LEFTMOST(lpageop) && P_RIGHTMOST(lpageop);
                bool            newitemonleft;
                Buffer          rbuf;

                /* Choose the split point */
                firstright = _bt_findsplitloc(rel, page,
                                                                          newitemoff, itemsz,
                                                                          &newitemonleft);

                /* split the buffer into left and right halves */
                rbuf = _bt_split(rel, buf, firstright,
                                                 newitemoff, itemsz, itup, newitemonleft);

                /*----------
                 * By here,
                 *
                 *              +  our target page has been split;
                 *              +  the original tuple has been inserted;
                 *              +  we have write locks on both the old (left half)
                 *                 and new (right half) buffers, after the split; and
                 *              +  we know the key we want to insert into the parent
                 *                 (it's the "high key" on the left child page).
                 *
                 * We're ready to do the parent insertion.  We need to hold onto the
                 * locks for the child pages until we locate the parent, but we can
                 * release them before doing the actual insertion (see Lehman and Yao
                 * for the reasoning).
                 *----------
                 */
                _bt_insert_parent(rel, buf, rbuf, stack, is_root, is_only);
        }
        else
        {
                Buffer          metabuf = InvalidBuffer;
                Page            metapg = NULL;
                BTMetaPageData *metad = NULL;
                OffsetNumber itup_off;
                BlockNumber itup_blkno;

                itup_off = newitemoff;
                itup_blkno = BufferGetBlockNumber(buf);

                /*
                 * If we are doing this insert because we split a page that was the
                 * only one on its tree level, but was not the root, it may have been
                 * the "fast root".  We need to ensure that the fast root link points
                 * at or above the current page.  We can safely acquire a lock on the
                 * metapage here --- see comments for _bt_newroot().
                 */
                if (split_only_page)
                {
                        Assert(!P_ISLEAF(lpageop));

                        metabuf = _bt_getbuf(rel, BTREE_METAPAGE, BT_WRITE);
                        metapg = BufferGetPage(metabuf);
                        metad = BTPageGetMeta(metapg);

                        if (metad->btm_fastlevel >= lpageop->btpo.level)
                        {
                                /* no update wanted */
                                _bt_relbuf(rel, metabuf);
                                metabuf = InvalidBuffer;
                        }
                }

                /* Do the update.  No ereport(ERROR) until changes are logged */
                START_CRIT_SECTION();

                _bt_pgaddtup(rel, page, itemsz, itup, newitemoff, "page");

                MarkBufferDirty(buf);

                if (BufferIsValid(metabuf))
                {
                        metad->btm_fastroot = itup_blkno;
                        metad->btm_fastlevel = lpageop->btpo.level;
                        MarkBufferDirty(metabuf);
                }

                /* XLOG stuff */
                if (!rel->rd_istemp)
                {
                        xl_btree_insert xlrec;
                        BlockNumber xldownlink;
                        xl_btree_metadata xlmeta;
                        uint8           xlinfo;
                        XLogRecPtr      recptr;
                        XLogRecData rdata[4];
                        XLogRecData *nextrdata;
                        IndexTupleData trunctuple;

                        xlrec.target.node = rel->rd_node;
                        ItemPointerSet(&(xlrec.target.tid), itup_blkno, itup_off);

                        rdata[0].data = (char *) &xlrec;
                        rdata[0].len = SizeOfBtreeInsert;
                        rdata[0].buffer = InvalidBuffer;
                        rdata[0].next = nextrdata = &(rdata[1]);

                        if (P_ISLEAF(lpageop))
                                xlinfo = XLOG_BTREE_INSERT_LEAF;
                        else
                        {
                                xldownlink = ItemPointerGetBlockNumber(&(itup->t_tid));
                                Assert(ItemPointerGetOffsetNumber(&(itup->t_tid)) == P_HIKEY);

                                nextrdata->data = (char *) &xldownlink;
                                nextrdata->len = sizeof(BlockNumber);
                                nextrdata->buffer = InvalidBuffer;
                                nextrdata->next = nextrdata + 1;
                                nextrdata++;

                                xlinfo = XLOG_BTREE_INSERT_UPPER;
                        }

                        if (BufferIsValid(metabuf))
                        {
                                xlmeta.root = metad->btm_root;
                                xlmeta.level = metad->btm_level;
                                xlmeta.fastroot = metad->btm_fastroot;
                                xlmeta.fastlevel = metad->btm_fastlevel;

                                nextrdata->data = (char *) &xlmeta;
                                nextrdata->len = sizeof(xl_btree_metadata);
                                nextrdata->buffer = InvalidBuffer;
                                nextrdata->next = nextrdata + 1;
                                nextrdata++;

                                xlinfo = XLOG_BTREE_INSERT_META;
                        }

                        /* Read comments in _bt_pgaddtup */
                        if (!P_ISLEAF(lpageop) && newitemoff == P_FIRSTDATAKEY(lpageop))
                        {
                                trunctuple = *itup;
                                trunctuple.t_info = sizeof(IndexTupleData);
                                nextrdata->data = (char *) &trunctuple;
                                nextrdata->len = sizeof(IndexTupleData);
                        }
                        else
                        {
                                nextrdata->data = (char *) itup;
                                nextrdata->len = IndexTupleDSize(*itup);
                        }
                        nextrdata->buffer = buf;
                        nextrdata->buffer_std = true;
                        nextrdata->next = NULL;

                        recptr = XLogInsert(RM_BTREE_ID, xlinfo, rdata);

                        if (BufferIsValid(metabuf))
                        {
                                PageSetLSN(metapg, recptr);
                                PageSetTLI(metapg, ThisTimeLineID);
                        }

                        PageSetLSN(page, recptr);
                        PageSetTLI(page, ThisTimeLineID);
                }

                END_CRIT_SECTION();

                /* release buffers; send out relcache inval if metapage changed */
                if (BufferIsValid(metabuf))
                {
                        CacheInvalidateRelcache(rel);
                        _bt_relbuf(rel, metabuf);
                }

                _bt_relbuf(rel, buf);
        }
}

/*
 *      _bt_split() -- split a page in the btree.
 *
 *              On entry, buf is the page to split, and is pinned and write-locked.
 *              firstright is the item index of the first item to be moved to the
 *              new right page.  newitemoff etc. tell us about the new item that
 *              must be inserted along with the data from the old page.
 *
 *              Returns the new right sibling of buf, pinned and write-locked.
 *              The pin and lock on buf are maintained.
 */
static Buffer
_bt_split(Relation rel, Buffer buf, OffsetNumber firstright,
                  OffsetNumber newitemoff, Size newitemsz, IndexTuple newitem,
                  bool newitemonleft)
{
        Buffer          rbuf;
        Page            origpage;
        Page            leftpage,
                                rightpage;
        BTPageOpaque ropaque,
                                lopaque,
                                oopaque;
        Buffer          sbuf = InvalidBuffer;
        Page            spage = NULL;
        BTPageOpaque sopaque = NULL;
        Size            itemsz;
        ItemId          itemid;
        IndexTuple      item;
        OffsetNumber leftoff,
                                rightoff;
        OffsetNumber maxoff;
        OffsetNumber i;
        bool            isroot;

        rbuf = _bt_getbuf(rel, P_NEW, BT_WRITE);
        origpage = BufferGetPage(buf);
        leftpage = PageGetTempPage(origpage, sizeof(BTPageOpaqueData));
        rightpage = BufferGetPage(rbuf);

        _bt_pageinit(leftpage, BufferGetPageSize(buf));
        /* rightpage was already initialized by _bt_getbuf */

        /*
         * Copy the original page's LSN and TLI into leftpage, which will become
         * the updated version of the page.  We need this because XLogInsert will
         * examine these fields and possibly dump them in a page image.
         */
        PageSetLSN(leftpage, PageGetLSN(origpage));
        PageSetTLI(leftpage, PageGetTLI(origpage));

        /* init btree private data */
        oopaque = (BTPageOpaque) PageGetSpecialPointer(origpage);
        lopaque = (BTPageOpaque) PageGetSpecialPointer(leftpage);
        ropaque = (BTPageOpaque) PageGetSpecialPointer(rightpage);

        isroot = P_ISROOT(oopaque);

        /* if we're splitting this page, it won't be the root when we're done */
        /* also, clear the SPLIT_END and HAS_GARBAGE flags in both pages */
        lopaque->btpo_flags = oopaque->btpo_flags;
        lopaque->btpo_flags &= ~(BTP_ROOT | BTP_SPLIT_END | BTP_HAS_GARBAGE);
        ropaque->btpo_flags = lopaque->btpo_flags;
        lopaque->btpo_prev = oopaque->btpo_prev;
        lopaque->btpo_next = BufferGetBlockNumber(rbuf);
        ropaque->btpo_prev = BufferGetBlockNumber(buf);
        ropaque->btpo_next = oopaque->btpo_next;
        lopaque->btpo.level = ropaque->btpo.level = oopaque->btpo.level;
        /* Since we already have write-lock on both pages, ok to read cycleid */
        lopaque->btpo_cycleid = _bt_vacuum_cycleid(rel);
        ropaque->btpo_cycleid = lopaque->btpo_cycleid;

        /*
         * If the page we're splitting is not the rightmost page at its level in
         * the tree, then the first entry on the page is the high key for the
         * page.  We need to copy that to the right half.  Otherwise (meaning the
         * rightmost page case), all the items on the right half will be user
         * data.
         */
        rightoff = P_HIKEY;

        if (!P_RIGHTMOST(oopaque))
        {
                itemid = PageGetItemId(origpage, P_HIKEY);
                itemsz = ItemIdGetLength(itemid);
                item = (IndexTuple) PageGetItem(origpage, itemid);
                if (PageAddItem(rightpage, (Item) item, itemsz, rightoff,
                                                false, false) == InvalidOffsetNumber)
                        elog(PANIC, "failed to add hikey to the right sibling"
                                 " while splitting block %u of index \"%s\"",
                                 BufferGetBlockNumber(buf), RelationGetRelationName(rel));
                rightoff = OffsetNumberNext(rightoff);
        }

        /*
         * The "high key" for the new left page will be the first key that's going
         * to go into the new right page.  This might be either the existing data
         * item at position firstright, or the incoming tuple.
         */
        leftoff = P_HIKEY;
        if (!newitemonleft && newitemoff == firstright)
        {
                /* incoming tuple will become first on right page */
                itemsz = newitemsz;
                item = newitem;
        }
        else
        {
                /* existing item at firstright will become first on right page */
                itemid = PageGetItemId(origpage, firstright);
                itemsz = ItemIdGetLength(itemid);
                item = (IndexTuple) PageGetItem(origpage, itemid);
        }
        if (PageAddItem(leftpage, (Item) item, itemsz, leftoff,
                                        false, false) == InvalidOffsetNumber)
                elog(PANIC, "failed to add hikey to the left sibling"
                         " while splitting block %u of index \"%s\"",
                         BufferGetBlockNumber(buf), RelationGetRelationName(rel));
        leftoff = OffsetNumberNext(leftoff);

        /*
         * Now transfer all the data items to the appropriate page.
         *
         * Note: we *must* insert at least the right page's items in item-number
         * order, for the benefit of _bt_restore_page().
         */
        maxoff = PageGetMaxOffsetNumber(origpage);

        for (i = P_FIRSTDATAKEY(oopaque); i <= maxoff; i = OffsetNumberNext(i))
        {
                itemid = PageGetItemId(origpage, i);
                itemsz = ItemIdGetLength(itemid);
                item = (IndexTuple) PageGetItem(origpage, itemid);

                /* does new item belong before this one? */
                if (i == newitemoff)
                {
                        if (newitemonleft)
                        {
                                _bt_pgaddtup(rel, leftpage, newitemsz, newitem, leftoff,
                                                         "left sibling");
                                leftoff = OffsetNumberNext(leftoff);
                        }
                        else
                        {
                                _bt_pgaddtup(rel, rightpage, newitemsz, newitem, rightoff,
                                                         "right sibling");
                                rightoff = OffsetNumberNext(rightoff);
                        }
                }

                /* decide which page to put it on */
                if (i < firstright)
                {
                        _bt_pgaddtup(rel, leftpage, itemsz, item, leftoff,
                                                 "left sibling");
                        leftoff = OffsetNumberNext(leftoff);
                }
                else
                {
                        _bt_pgaddtup(rel, rightpage, itemsz, item, rightoff,
                                                 "right sibling");
                        rightoff = OffsetNumberNext(rightoff);
                }
        }

        /* cope with possibility that newitem goes at the end */
        if (i <= newitemoff)
        {
                /*
                 * Can't have newitemonleft here; that would imply we were told to put
                 * *everything* on the left page, which cannot fit (if it could, we'd
                 * not be splitting the page).
                 */
                Assert(!newitemonleft);
                _bt_pgaddtup(rel, rightpage, newitemsz, newitem, rightoff,
                                         "right sibling");
                rightoff = OffsetNumberNext(rightoff);
        }

        /*
         * We have to grab the right sibling (if any) and fix the prev pointer
         * there. We are guaranteed that this is deadlock-free since no other
         * writer will be holding a lock on that page and trying to move left, and
         * all readers release locks on a page before trying to fetch its
         * neighbors.
         */

        if (!P_RIGHTMOST(ropaque))
        {
                sbuf = _bt_getbuf(rel, ropaque->btpo_next, BT_WRITE);
                spage = BufferGetPage(sbuf);
                sopaque = (BTPageOpaque) PageGetSpecialPointer(spage);
                if (sopaque->btpo_prev != ropaque->btpo_prev)
                        elog(PANIC, "right sibling's left-link doesn't match: "
                                 "block %u links to %u instead of expected %u in index \"%s\"",
                                 ropaque->btpo_next, sopaque->btpo_prev, ropaque->btpo_prev,
                                 RelationGetRelationName(rel));

                /*
                 * Check to see if we can set the SPLIT_END flag in the right-hand
                 * split page; this can save some I/O for vacuum since it need not
                 * proceed to the right sibling.  We can set the flag if the right
                 * sibling has a different cycleid: that means it could not be part of
                 * a group of pages that were all split off from the same ancestor
                 * page.  If you're confused, imagine that page A splits to A B and
                 * then again, yielding A C B, while vacuum is in progress.  Tuples
                 * originally in A could now be in either B or C, hence vacuum must
                 * examine both pages.  But if D, our right sibling, has a different
                 * cycleid then it could not contain any tuples that were in A when
                 * the vacuum started.
                 */
                if (sopaque->btpo_cycleid != ropaque->btpo_cycleid)
                        ropaque->btpo_flags |= BTP_SPLIT_END;
        }

        /*
         * Right sibling is locked, new siblings are prepared, but original page
         * is not updated yet.
         *
         * NO EREPORT(ERROR) till right sibling is updated.  We can get away with
         * not starting the critical section till here because we haven't been
         * scribbling on the original page yet, and we don't care about the new
         * sibling until it's linked into the btree.
         */
        START_CRIT_SECTION();

        /*
         * By here, the original data page has been split into two new halves, and
         * these are correct.  The algorithm requires that the left page never
         * move during a split, so we copy the new left page back on top of the
         * original.  Note that this is not a waste of time, since we also require
         * (in the page management code) that the center of a page always be
         * clean, and the most efficient way to guarantee this is just to compact
         * the data by reinserting it into a new left page.  (XXX the latter
         * comment is probably obsolete.)
         *
         * We need to do this before writing the WAL record, so that XLogInsert
         * can WAL log an image of the page if necessary.
         */
        PageRestoreTempPage(leftpage, origpage);

        MarkBufferDirty(buf);
        MarkBufferDirty(rbuf);

        if (!P_RIGHTMOST(ropaque))
        {
                sopaque->btpo_prev = BufferGetBlockNumber(rbuf);
                MarkBufferDirty(sbuf);
        }

        /* XLOG stuff */
        if (!rel->rd_istemp)
        {
                xl_btree_split xlrec;
                uint8           xlinfo;
                XLogRecPtr      recptr;
                XLogRecData rdata[7];
                XLogRecData *lastrdata;

                xlrec.node = rel->rd_node;
                xlrec.leftsib = BufferGetBlockNumber(buf);
                xlrec.rightsib = BufferGetBlockNumber(rbuf);
                xlrec.rnext = ropaque->btpo_next;
                xlrec.level = ropaque->btpo.level;
                xlrec.firstright = firstright;

                rdata[0].data = (char *) &xlrec;
                rdata[0].len = SizeOfBtreeSplit;
                rdata[0].buffer = InvalidBuffer;

                lastrdata = &rdata[0];

                if (ropaque->btpo.level > 0)
                {
                        /* Log downlink on non-leaf pages */
                        lastrdata->next = lastrdata + 1;
                        lastrdata++;

                        lastrdata->data = (char *) &newitem->t_tid.ip_blkid;
                        lastrdata->len = sizeof(BlockIdData);
                        lastrdata->buffer = InvalidBuffer;

                        /*
                         * We must also log the left page's high key, because the right
                         * page's leftmost key is suppressed on non-leaf levels.  Show it
                         * as belonging to the left page buffer, so that it is not stored
                         * if XLogInsert decides it needs a full-page image of the left
                         * page.
                         */
                        lastrdata->next = lastrdata + 1;
                        lastrdata++;

                        itemid = PageGetItemId(origpage, P_HIKEY);
                        item = (IndexTuple) PageGetItem(origpage, itemid);
                        lastrdata->data = (char *) item;
                        lastrdata->len = MAXALIGN(IndexTupleSize(item));
                        lastrdata->buffer = buf;        /* backup block 1 */
                        lastrdata->buffer_std = true;
                }

                /*
                 * Log the new item and its offset, if it was inserted on the left
                 * page. (If it was put on the right page, we don't need to explicitly
                 * WAL log it because it's included with all the other items on the
                 * right page.) Show the new item as belonging to the left page
                 * buffer, so that it is not stored if XLogInsert decides it needs a
                 * full-page image of the left page.  We store the offset anyway,
                 * though, to support archive compression of these records.
                 */
                if (newitemonleft)
                {
                        lastrdata->next = lastrdata + 1;
                        lastrdata++;

                        lastrdata->data = (char *) &newitemoff;
                        lastrdata->len = sizeof(OffsetNumber);
                        lastrdata->buffer = InvalidBuffer;

                        lastrdata->next = lastrdata + 1;
                        lastrdata++;

                        lastrdata->data = (char *) newitem;
                        lastrdata->len = MAXALIGN(newitemsz);
                        lastrdata->buffer = buf;        /* backup block 1 */
                        lastrdata->buffer_std = true;
                }
                else if (ropaque->btpo.level == 0)
                {
                        /*
                         * Although we don't need to WAL-log the new item, we still need
                         * XLogInsert to consider storing a full-page image of the left
                         * page, so make an empty entry referencing that buffer. This also
                         * ensures that the left page is always backup block 1.
                         */
                        lastrdata->next = lastrdata + 1;
                        lastrdata++;

                        lastrdata->data = NULL;
                        lastrdata->len = 0;
                        lastrdata->buffer = buf;        /* backup block 1 */
                        lastrdata->buffer_std = true;
                }

                /*
                 * Log the contents of the right page in the format understood by
                 * _bt_restore_page(). We set lastrdata->buffer to InvalidBuffer,
                 * because we're going to recreate the whole page anyway, so it should
                 * never be stored by XLogInsert.
                 *
                 * Direct access to page is not good but faster - we should implement
                 * some new func in page API.  Note we only store the tuples
                 * themselves, knowing that they were inserted in item-number order
                 * and so the item pointers can be reconstructed.  See comments for
                 * _bt_restore_page().
                 */
                lastrdata->next = lastrdata + 1;
                lastrdata++;

                lastrdata->data = (char *) rightpage +
                        ((PageHeader) rightpage)->pd_upper;
                lastrdata->len = ((PageHeader) rightpage)->pd_special -
                        ((PageHeader) rightpage)->pd_upper;
                lastrdata->buffer = InvalidBuffer;

                /* Log the right sibling, because we've changed its' prev-pointer. */
                if (!P_RIGHTMOST(ropaque))
                {
                        lastrdata->next = lastrdata + 1;
                        lastrdata++;

                        lastrdata->data = NULL;
                        lastrdata->len = 0;
                        lastrdata->buffer = sbuf;       /* backup block 2 */
                        lastrdata->buffer_std = true;
                }

                lastrdata->next = NULL;

                if (isroot)
                        xlinfo = newitemonleft ? XLOG_BTREE_SPLIT_L_ROOT : XLOG_BTREE_SPLIT_R_ROOT;
                else
                        xlinfo = newitemonleft ? XLOG_BTREE_SPLIT_L : XLOG_BTREE_SPLIT_R;

                recptr = XLogInsert(RM_BTREE_ID, xlinfo, rdata);

                PageSetLSN(origpage, recptr);
                PageSetTLI(origpage, ThisTimeLineID);
                PageSetLSN(rightpage, recptr);
                PageSetTLI(rightpage, ThisTimeLineID);
                if (!P_RIGHTMOST(ropaque))
                {
                        PageSetLSN(spage, recptr);
                        PageSetTLI(spage, ThisTimeLineID);
                }
        }

        END_CRIT_SECTION();

        /* release the old right sibling */
        if (!P_RIGHTMOST(ropaque))
                _bt_relbuf(rel, sbuf);

        /* split's done */
        return rbuf;
}

/*
 *      _bt_findsplitloc() -- find an appropriate place to split a page.
 *
 * The idea here is to equalize the free space that will be on each split
 * page, *after accounting for the inserted tuple*.  (If we fail to account
 * for it, we might find ourselves with too little room on the page that
 * it needs to go into!)
 *
 * If the page is the rightmost page on its level, we instead try to arrange
 * to leave the left split page fillfactor% full.  In this way, when we are
 * inserting successively increasing keys (consider sequences, timestamps,
 * etc) we will end up with a tree whose pages are about fillfactor% full,
 * instead of the 50% full result that we'd get without this special case.
 * This is the same as nbtsort.c produces for a newly-created tree.  Note
 * that leaf and nonleaf pages use different fillfactors.
 *
 * We are passed the intended insert position of the new tuple, expressed as
 * the offsetnumber of the tuple it must go in front of.  (This could be
 * maxoff+1 if the tuple is to go at the end.)
 *
 * We return the index of the first existing tuple that should go on the
 * righthand page, plus a boolean indicating whether the new tuple goes on
 * the left or right page.      The bool is necessary to disambiguate the case
 * where firstright == newitemoff.
 */
static OffsetNumber
_bt_findsplitloc(Relation rel,
                                 Page page,
                                 OffsetNumber newitemoff,
                                 Size newitemsz,
                                 bool *newitemonleft)
{
        BTPageOpaque opaque;
        OffsetNumber offnum;
        OffsetNumber maxoff;
        ItemId          itemid;
        FindSplitData state;
        int                     leftspace,
                                rightspace,
                                goodenough,
                                olddataitemstotal,
                                olddataitemstoleft;
        bool            goodenoughfound;

        opaque = (BTPageOpaque) PageGetSpecialPointer(page);

        /* Passed-in newitemsz is MAXALIGNED but does not include line pointer */
        newitemsz += sizeof(ItemIdData);

        /* Total free space available on a btree page, after fixed overhead */
        leftspace = rightspace =
                PageGetPageSize(page) - SizeOfPageHeaderData -
                MAXALIGN(sizeof(BTPageOpaqueData));

        /* The right page will have the same high key as the old page */
        if (!P_RIGHTMOST(opaque))
        {
                itemid = PageGetItemId(page, P_HIKEY);
                rightspace -= (int) (MAXALIGN(ItemIdGetLength(itemid)) +
                                                         sizeof(ItemIdData));
        }

        /* Count up total space in data items without actually scanning 'em */
        olddataitemstotal = rightspace - (int) PageGetExactFreeSpace(page);

        state.newitemsz = newitemsz;
        state.is_leaf = P_ISLEAF(opaque);
        state.is_rightmost = P_RIGHTMOST(opaque);
        state.have_split = false;
        if (state.is_leaf)
                state.fillfactor = RelationGetFillFactor(rel,
                                                                                                 BTREE_DEFAULT_FILLFACTOR);
        else
                state.fillfactor = BTREE_NONLEAF_FILLFACTOR;
        state.newitemonleft = false;    /* these just to keep compiler quiet */
        state.firstright = 0;
        state.best_delta = 0;
        state.leftspace = leftspace;
        state.rightspace = rightspace;
        state.olddataitemstotal = olddataitemstotal;
        state.newitemoff = newitemoff;

        /*
         * Finding the best possible split would require checking all the possible
         * split points, because of the high-key and left-key special cases.
         * That's probably more work than it's worth; instead, stop as soon as we
         * find a "good-enough" split, where good-enough is defined as an
         * imbalance in free space of no more than pagesize/16 (arbitrary...) This
         * should let us stop near the middle on most pages, instead of plowing to
         * the end.
         */
        goodenough = leftspace / 16;

        /*
         * Scan through the data items and calculate space usage for a split at
         * each possible position.
         */
        olddataitemstoleft = 0;
        goodenoughfound = false;
        maxoff = PageGetMaxOffsetNumber(page);

        for (offnum = P_FIRSTDATAKEY(opaque);
                 offnum <= maxoff;
                 offnum = OffsetNumberNext(offnum))
        {
                Size            itemsz;

                itemid = PageGetItemId(page, offnum);
                itemsz = MAXALIGN(ItemIdGetLength(itemid)) + sizeof(ItemIdData);

                /*
                 * Will the new item go to left or right of split?
                 */
                if (offnum > newitemoff)
                        _bt_checksplitloc(&state, offnum, true,
                                                          olddataitemstoleft, itemsz);

                else if (offnum < newitemoff)
                        _bt_checksplitloc(&state, offnum, false,
                                                          olddataitemstoleft, itemsz);
                else
                {
                        /* need to try it both ways! */
                        _bt_checksplitloc(&state, offnum, true,
                                                          olddataitemstoleft, itemsz);

                        _bt_checksplitloc(&state, offnum, false,
                                                          olddataitemstoleft, itemsz);
                }

                /* Abort scan once we find a good-enough choice */
                if (state.have_split && state.best_delta <= goodenough)
                {
                        goodenoughfound = true;
                        break;
                }

                olddataitemstoleft += itemsz;
        }

        /*
         * If the new item goes as the last item, check for splitting so that all
         * the old items go to the left page and the new item goes to the right
         * page.
         */
        if (newitemoff > maxoff && !goodenoughfound)
                _bt_checksplitloc(&state, newitemoff, false, olddataitemstotal, 0);

        /*
         * I believe it is not possible to fail to find a feasible split, but just
         * in case ...
         */
        if (!state.have_split)
                elog(ERROR, "could not find a feasible split point for index \"%s\"",
                         RelationGetRelationName(rel));

        *newitemonleft = state.newitemonleft;
        return state.firstright;
}

/*
 * Subroutine to analyze a particular possible split choice (ie, firstright
 * and newitemonleft settings), and record the best split so far in *state.
 *
 * firstoldonright is the offset of the first item on the original page
 * that goes to the right page, and firstoldonrightsz is the size of that
 * tuple. firstoldonright can be > max offset, which means that all the old
 * items go to the left page and only the new item goes to the right page.
 * In that case, firstoldonrightsz is not used.
 *
 * olddataitemstoleft is the total size of all old items to the left of
 * firstoldonright.
 */
static void
_bt_checksplitloc(FindSplitData *state,
                                  OffsetNumber firstoldonright,
                                  bool newitemonleft,
                                  int olddataitemstoleft,
                                  Size firstoldonrightsz)
{
        int                     leftfree,
                                rightfree;
        Size            firstrightitemsz;
        bool            newitemisfirstonright;

        /* Is the new item going to be the first item on the right page? */
        newitemisfirstonright = (firstoldonright == state->newitemoff
                                                         && !newitemonleft);

        if (newitemisfirstonright)
                firstrightitemsz = state->newitemsz;
        else
                firstrightitemsz = firstoldonrightsz;

        /* Account for all the old tuples */
        leftfree = state->leftspace - olddataitemstoleft;
        rightfree = state->rightspace -
                (state->olddataitemstotal - olddataitemstoleft);

        /*
         * The first item on the right page becomes the high key of the left page;
         * therefore it counts against left space as well as right space.
         */
        leftfree -= firstrightitemsz;

        /* account for the new item */
        if (newitemonleft)
                leftfree -= (int) state->newitemsz;
        else
                rightfree -= (int) state->newitemsz;

        /*
         * If we are not on the leaf level, we will be able to discard the key
         * data from the first item that winds up on the right page.
         */
        if (!state->is_leaf)
                rightfree += (int) firstrightitemsz -
                        (int) (MAXALIGN(sizeof(IndexTupleData)) + sizeof(ItemIdData));

        /*
         * If feasible split point, remember best delta.
         */
        if (leftfree >= 0 && rightfree >= 0)
        {
                int                     delta;

                if (state->is_rightmost)
                {
                        /*
                         * If splitting a rightmost page, try to put (100-fillfactor)% of
                         * free space on left page. See comments for _bt_findsplitloc.
                         */
                        delta = (state->fillfactor * leftfree)
                                - ((100 - state->fillfactor) * rightfree);
                }
                else
                {
                        /* Otherwise, aim for equal free space on both sides */
                        delta = leftfree - rightfree;
                }

                if (delta < 0)
                        delta = -delta;
                if (!state->have_split || delta < state->best_delta)
                {
                        state->have_split = true;
                        state->newitemonleft = newitemonleft;
                        state->firstright = firstoldonright;
                        state->best_delta = delta;
                }
        }
}

/*
 * _bt_insert_parent() -- Insert downlink into parent after a page split.
 *
 * On entry, buf and rbuf are the left and right split pages, which we
 * still hold write locks on per the L&Y algorithm.  We release the
 * write locks once we have write lock on the parent page.      (Any sooner,
 * and it'd be possible for some other process to try to split or delete
 * one of these pages, and get confused because it cannot find the downlink.)
 *
 * stack - stack showing how we got here.  May be NULL in cases that don't
 *                      have to be efficient (concurrent ROOT split, WAL recovery)
 * is_root - we split the true root
 * is_only - we split a page alone on its level (might have been fast root)
 *
 * This is exported so it can be called by nbtxlog.c.
 */
void
_bt_insert_parent(Relation rel,
                                  Buffer buf,
                                  Buffer rbuf,
                                  BTStack stack,
                                  bool is_root,
                                  bool is_only)
{
        /*
         * Here we have to do something Lehman and Yao don't talk about: deal with
         * a root split and construction of a new root.  If our stack is empty
         * then we have just split a node on what had been the root level when we
         * descended the tree.  If it was still the root then we perform a
         * new-root construction.  If it *wasn't* the root anymore, search to find
         * the next higher level that someone constructed meanwhile, and find the
         * right place to insert as for the normal case.
         *
         * If we have to search for the parent level, we do so by re-descending
         * from the root.  This is not super-efficient, but it's rare enough not
         * to matter.  (This path is also taken when called from WAL recovery ---
         * we have no stack in that case.)
         */
        if (is_root)
        {
                Buffer          rootbuf;

                Assert(stack == NULL);
                Assert(is_only);
                /* create a new root node and update the metapage */
                rootbuf = _bt_newroot(rel, buf, rbuf);
                /* release the split buffers */
                _bt_relbuf(rel, rootbuf);
                _bt_relbuf(rel, rbuf);
                _bt_relbuf(rel, buf);
        }
        else
        {
                BlockNumber bknum = BufferGetBlockNumber(buf);
                BlockNumber rbknum = BufferGetBlockNumber(rbuf);
                Page            page = BufferGetPage(buf);
                IndexTuple      new_item;
                BTStackData fakestack;
                IndexTuple      ritem;
                Buffer          pbuf;

                if (stack == NULL)
                {
                        BTPageOpaque lpageop;

                        if (!InRecovery)
                                elog(DEBUG2, "concurrent ROOT page split");
                        lpageop = (BTPageOpaque) PageGetSpecialPointer(page);
                        /* Find the leftmost page at the next level up */
                        pbuf = _bt_get_endpoint(rel, lpageop->btpo.level + 1, false);
                        /* Set up a phony stack entry pointing there */
                        stack = &fakestack;
                        stack->bts_blkno = BufferGetBlockNumber(pbuf);
                        stack->bts_offset = InvalidOffsetNumber;
                        /* bts_btentry will be initialized below */
                        stack->bts_parent = NULL;
                        _bt_relbuf(rel, pbuf);
                }

                /* get high key from left page == lowest key on new right page */
                ritem = (IndexTuple) PageGetItem(page,
                                                                                 PageGetItemId(page, P_HIKEY));

                /* form an index tuple that points at the new right page */
                new_item = CopyIndexTuple(ritem);
                ItemPointerSet(&(new_item->t_tid), rbknum, P_HIKEY);

                /*
                 * Find the parent buffer and get the parent page.
                 *
                 * Oops - if we were moved right then we need to change stack item! We
                 * want to find parent pointing to where we are, right ?        - vadim
                 * 05/27/97
                 */
                ItemPointerSet(&(stack->bts_btentry.t_tid), bknum, P_HIKEY);

                pbuf = _bt_getstackbuf(rel, stack, BT_WRITE);

                /* Now we can unlock the children */
                _bt_relbuf(rel, rbuf);
                _bt_relbuf(rel, buf);

                /* Check for error only after writing children */
                if (pbuf == InvalidBuffer)
                        elog(ERROR, "failed to re-find parent key in index \"%s\" for split pages %u/%u",
                                 RelationGetRelationName(rel), bknum, rbknum);

                /* Recursively update the parent */
                _bt_insertonpg(rel, pbuf, stack->bts_parent,
                                           new_item, stack->bts_offset + 1,
                                           is_only);

                /* be tidy */
                pfree(new_item);
        }
}

/*
 *      _bt_getstackbuf() -- Walk back up the tree one step, and find the item
 *                                               we last looked at in the parent.
 *
 *              This is possible because we save the downlink from the parent item,
 *              which is enough to uniquely identify it.  Insertions into the parent
 *              level could cause the item to move right; deletions could cause it
 *              to move left, but not left of the page we previously found it in.
 *
 *              Adjusts bts_blkno & bts_offset if changed.
 *
 *              Returns InvalidBuffer if item not found (should not happen).
 */
Buffer
_bt_getstackbuf(Relation rel, BTStack stack, int access)
{
        BlockNumber blkno;
        OffsetNumber start;

        blkno = stack->bts_blkno;
        start = stack->bts_offset;

        for (;;)
        {
                Buffer          buf;
                Page            page;
                BTPageOpaque opaque;

                buf = _bt_getbuf(rel, blkno, access);
                page = BufferGetPage(buf);
                opaque = (BTPageOpaque) PageGetSpecialPointer(page);

                if (!P_IGNORE(opaque))
                {
                        OffsetNumber offnum,
                                                minoff,
                                                maxoff;
                        ItemId          itemid;
                        IndexTuple      item;

                        minoff = P_FIRSTDATAKEY(opaque);
                        maxoff = PageGetMaxOffsetNumber(page);

                        /*
                         * start = InvalidOffsetNumber means "search the whole page". We
                         * need this test anyway due to possibility that page has a high
                         * key now when it didn't before.
                         */
                        if (start < minoff)
                                start = minoff;

                        /*
                         * Need this check too, to guard against possibility that page
                         * split since we visited it originally.
                         */
                        if (start > maxoff)
                                start = OffsetNumberNext(maxoff);

                        /*
                         * These loops will check every item on the page --- but in an
                         * order that's attuned to the probability of where it actually
                         * is.  Scan to the right first, then to the left.
                         */
                        for (offnum = start;
                                 offnum <= maxoff;
                                 offnum = OffsetNumberNext(offnum))
                        {
                                itemid = PageGetItemId(page, offnum);
                                item = (IndexTuple) PageGetItem(page, itemid);
                                if (BTEntrySame(item, &stack->bts_btentry))
                                {
                                        /* Return accurate pointer to where link is now */
                                        stack->bts_blkno = blkno;
                                        stack->bts_offset = offnum;
                                        return buf;
                                }
                        }

                        for (offnum = OffsetNumberPrev(start);
                                 offnum >= minoff;
                                 offnum = OffsetNumberPrev(offnum))
                        {
                                itemid = PageGetItemId(page, offnum);
                                item = (IndexTuple) PageGetItem(page, itemid);
                                if (BTEntrySame(item, &stack->bts_btentry))
                                {
                                        /* Return accurate pointer to where link is now */
                                        stack->bts_blkno = blkno;
                                        stack->bts_offset = offnum;
                                        return buf;
                                }
                        }
                }

                /*
                 * The item we're looking for moved right at least one page.
                 */
                if (P_RIGHTMOST(opaque))
                {
                        _bt_relbuf(rel, buf);
                        return InvalidBuffer;
                }
                blkno = opaque->btpo_next;
                start = InvalidOffsetNumber;
                _bt_relbuf(rel, buf);
        }
}

/*
 *      _bt_newroot() -- Create a new root page for the index.
 *
 *              We've just split the old root page and need to create a new one.
 *              In order to do this, we add a new root page to the file, then lock
 *              the metadata page and update it.  This is guaranteed to be deadlock-
 *              free, because all readers release their locks on the metadata page
 *              before trying to lock the root, and all writers lock the root before
 *              trying to lock the metadata page.  We have a write lock on the old
 *              root page, so we have not introduced any cycles into the waits-for
 *              graph.
 *
 *              On entry, lbuf (the old root) and rbuf (its new peer) are write-
 *              locked. On exit, a new root page exists with entries for the
 *              two new children, metapage is updated and unlocked/unpinned.
 *              The new root buffer is returned to caller which has to unlock/unpin
 *              lbuf, rbuf & rootbuf.
 */
static Buffer
_bt_newroot(Relation rel, Buffer lbuf, Buffer rbuf)
{
        Buffer          rootbuf;
        Page            lpage,
                                rootpage;
        BlockNumber lbkno,
                                rbkno;
        BlockNumber rootblknum;
        BTPageOpaque rootopaque;
        ItemId          itemid;
        IndexTuple      item;
        Size            itemsz;
        IndexTuple      new_item;
        Buffer          metabuf;
        Page            metapg;
        BTMetaPageData *metad;

        lbkno = BufferGetBlockNumber(lbuf);
        rbkno = BufferGetBlockNumber(rbuf);
        lpage = BufferGetPage(lbuf);

        /* get a new root page */
        rootbuf = _bt_getbuf(rel, P_NEW, BT_WRITE);
        rootpage = BufferGetPage(rootbuf);
        rootblknum = BufferGetBlockNumber(rootbuf);

        /* acquire lock on the metapage */
        metabuf = _bt_getbuf(rel, BTREE_METAPAGE, BT_WRITE);
        metapg = BufferGetPage(metabuf);
        metad = BTPageGetMeta(metapg);

        /* NO EREPORT(ERROR) from here till newroot op is logged */
        START_CRIT_SECTION();

        /* set btree special data */
        rootopaque = (BTPageOpaque) PageGetSpecialPointer(rootpage);
        rootopaque->btpo_prev = rootopaque->btpo_next = P_NONE;
        rootopaque->btpo_flags = BTP_ROOT;
        rootopaque->btpo.level =
                ((BTPageOpaque) PageGetSpecialPointer(lpage))->btpo.level + 1;
        rootopaque->btpo_cycleid = 0;

        /* update metapage data */
        metad->btm_root = rootblknum;
        metad->btm_level = rootopaque->btpo.level;
        metad->btm_fastroot = rootblknum;
        metad->btm_fastlevel = rootopaque->btpo.level;

        /*
         * Create downlink item for left page (old root).  Since this will be the
         * first item in a non-leaf page, it implicitly has minus-infinity key
         * value, so we need not store any actual key in it.
         */
        itemsz = sizeof(IndexTupleData);
        new_item = (IndexTuple) palloc(itemsz);
        new_item->t_info = itemsz;
        ItemPointerSet(&(new_item->t_tid), lbkno, P_HIKEY);

        /*
         * Insert the left page pointer into the new root page.  The root page is
         * the rightmost page on its level so there is no "high key" in it; the
         * two items will go into positions P_HIKEY and P_FIRSTKEY.
         *
         * Note: we *must* insert the two items in item-number order, for the
         * benefit of _bt_restore_page().
         */
        if (PageAddItem(rootpage, (Item) new_item, itemsz, P_HIKEY,
                                        false, false) == InvalidOffsetNumber)
                elog(PANIC, "failed to add leftkey to new root page"
                         " while splitting block %u of index \"%s\"",
                         BufferGetBlockNumber(lbuf), RelationGetRelationName(rel));
        pfree(new_item);

        /*
         * Create downlink item for right page.  The key for it is obtained from
         * the "high key" position in the left page.
         */
        itemid = PageGetItemId(lpage, P_HIKEY);
        itemsz = ItemIdGetLength(itemid);
        item = (IndexTuple) PageGetItem(lpage, itemid);
        new_item = CopyIndexTuple(item);
        ItemPointerSet(&(new_item->t_tid), rbkno, P_HIKEY);

        /*
         * insert the right page pointer into the new root page.
         */
        if (PageAddItem(rootpage, (Item) new_item, itemsz, P_FIRSTKEY,
                                        false, false) == InvalidOffsetNumber)
                elog(PANIC, "failed to add rightkey to new root page"
                         " while splitting block %u of index \"%s\"",
                         BufferGetBlockNumber(lbuf), RelationGetRelationName(rel));
        pfree(new_item);

        MarkBufferDirty(rootbuf);
        MarkBufferDirty(metabuf);

        /* XLOG stuff */
        if (!rel->rd_istemp)
        {
                xl_btree_newroot xlrec;
                XLogRecPtr      recptr;
                XLogRecData rdata[2];

                xlrec.node = rel->rd_node;
                xlrec.rootblk = rootblknum;
                xlrec.level = metad->btm_level;

                rdata[0].data = (char *) &xlrec;
                rdata[0].len = SizeOfBtreeNewroot;
                rdata[0].buffer = InvalidBuffer;
                rdata[0].next = &(rdata[1]);

                /*
                 * Direct access to page is not good but faster - we should implement
                 * some new func in page API.
                 */
                rdata[1].data = (char *) rootpage + ((PageHeader) rootpage)->pd_upper;
                rdata[1].len = ((PageHeader) rootpage)->pd_special -
                        ((PageHeader) rootpage)->pd_upper;
                rdata[1].buffer = InvalidBuffer;
                rdata[1].next = NULL;

                recptr = XLogInsert(RM_BTREE_ID, XLOG_BTREE_NEWROOT, rdata);

                PageSetLSN(rootpage, recptr);
                PageSetTLI(rootpage, ThisTimeLineID);
                PageSetLSN(metapg, recptr);
                PageSetTLI(metapg, ThisTimeLineID);
        }

        END_CRIT_SECTION();

        /* send out relcache inval for metapage change */
        CacheInvalidateRelcache(rel);

        /* done with metapage */
        _bt_relbuf(rel, metabuf);

        return rootbuf;
}

/*
 *      _bt_pgaddtup() -- add a tuple to a particular page in the index.
 *
 *              This routine adds the tuple to the page as requested.  It does
 *              not affect pin/lock status, but you'd better have a write lock
 *              and pin on the target buffer!  Don't forget to write and release
 *              the buffer afterwards, either.
 *
 *              The main difference between this routine and a bare PageAddItem call
 *              is that this code knows that the leftmost index tuple on a non-leaf
 *              btree page doesn't need to have a key.  Therefore, it strips such
 *              tuples down to just the tuple header.  CAUTION: this works ONLY if
 *              we insert the tuples in order, so that the given itup_off does
 *              represent the final position of the tuple!
 */
static void
_bt_pgaddtup(Relation rel,
                         Page page,
                         Size itemsize,
                         IndexTuple itup,
                         OffsetNumber itup_off,
                         const char *where)
{
        BTPageOpaque opaque = (BTPageOpaque) PageGetSpecialPointer(page);
        IndexTupleData trunctuple;

        if (!P_ISLEAF(opaque) && itup_off == P_FIRSTDATAKEY(opaque))
        {
                trunctuple = *itup;
                trunctuple.t_info = sizeof(IndexTupleData);
                itup = &trunctuple;
                itemsize = sizeof(IndexTupleData);
        }

        if (PageAddItem(page, (Item) itup, itemsize, itup_off,
                                        false, false) == InvalidOffsetNumber)
                elog(PANIC, "failed to add item to the %s in index \"%s\"",
                         where, RelationGetRelationName(rel));
}

/*
 * _bt_isequal - used in _bt_doinsert in check for duplicates.
 *
 * This is very similar to _bt_compare, except for NULL handling.
 * Rule is simple: NOT_NULL not equal NULL, NULL not equal NULL too.
 */
static bool
_bt_isequal(TupleDesc itupdesc, Page page, OffsetNumber offnum,
                        int keysz, ScanKey scankey)
{
        IndexTuple      itup;
        int                     i;

        /* Better be comparing to a leaf item */
        Assert(P_ISLEAF((BTPageOpaque) PageGetSpecialPointer(page)));

        itup = (IndexTuple) PageGetItem(page, PageGetItemId(page, offnum));

        for (i = 1; i <= keysz; i++)
        {
                AttrNumber      attno;
                Datum           datum;
                bool            isNull;
                int32           result;

                attno = scankey->sk_attno;
                Assert(attno == i);
                datum = index_getattr(itup, attno, itupdesc, &isNull);

                /* NULLs are never equal to anything */
                if (isNull || (scankey->sk_flags & SK_ISNULL))
                        return false;

                result = DatumGetInt32(FunctionCall2(&scankey->sk_func,
                                                                                         datum,
                                                                                         scankey->sk_argument));

                if (result != 0)
                        return false;

                scankey++;
        }

        /* if we get here, the keys are equal */
        return true;
}

/*
 * _bt_vacuum_one_page - vacuum just one index page.
 *
 * Try to remove LP_DEAD items from the given page.  The passed buffer
 * must be exclusive-locked, but unlike a real VACUUM, we don't need a
 * super-exclusive "cleanup" lock (see nbtree/README).
 */
static void
_bt_vacuum_one_page(Relation rel, Buffer buffer)
{
        OffsetNumber deletable[MaxOffsetNumber];
        int                     ndeletable = 0;
        OffsetNumber offnum,
                                minoff,
                                maxoff;
        Page            page = BufferGetPage(buffer);
        BTPageOpaque opaque = (BTPageOpaque) PageGetSpecialPointer(page);

        /*
         * Scan over all items to see which ones need to be deleted according to
         * LP_DEAD flags.
         */
        minoff = P_FIRSTDATAKEY(opaque);
        maxoff = PageGetMaxOffsetNumber(page);
        for (offnum = minoff;
                 offnum <= maxoff;
                 offnum = OffsetNumberNext(offnum))
        {
                ItemId          itemId = PageGetItemId(page, offnum);

                if (ItemIdIsDead(itemId))
                        deletable[ndeletable++] = offnum;
        }

        if (ndeletable > 0)
                _bt_delitems(rel, buffer, deletable, ndeletable);

        /*
         * Note: if we didn't find any LP_DEAD items, then the page's
         * BTP_HAS_GARBAGE hint bit is falsely set.  We do not bother expending a
         * separate write to clear it, however.  We will clear it when we split
         * the page.
         */
}