/*-------------------------------------------------------------------------
 *
 * nbtsearch.c
 *	  Search code for postgres btrees.
 *
 *
 * 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/nbtsearch.c,v 1.112 2007/04/06 22:33:42 tgl Exp $
 *
 *-------------------------------------------------------------------------
 */

#include "postgres.h"

#include "access/genam.h"
#include "access/nbtree.h"
#include "pgstat.h"
#include "utils/lsyscache.h"


static bool _bt_readpage(IndexScanDesc scan, ScanDirection dir,
			 OffsetNumber offnum);
static bool _bt_steppage(IndexScanDesc scan, ScanDirection dir);
static Buffer _bt_walk_left(Relation rel, Buffer buf);
static bool _bt_endpoint(IndexScanDesc scan, ScanDirection dir);


/*
 *	_bt_search() -- Search the tree for a particular scankey,
 *		or more precisely for the first leaf page it could be on.
 *
 * The passed scankey must be an insertion-type scankey (see nbtree/README),
 * but it can omit the rightmost column(s) of the index.
 *
 * When nextkey is false (the usual case), we are looking for the first
 * item >= scankey.  When nextkey is true, we are looking for the first
 * item strictly greater than scankey.
 *
 * Return value is a stack of parent-page pointers.  *bufP is set to the
 * address of the leaf-page buffer, which is read-locked and pinned.
 * No locks are held on the parent pages, however!
 *
 * NOTE that the returned buffer is read-locked regardless of the access
 * parameter.  However, access = BT_WRITE will allow an empty root page
 * to be created and returned.	When access = BT_READ, an empty index
 * will result in *bufP being set to InvalidBuffer.
 */
BTStack
_bt_search(Relation rel, int keysz, ScanKey scankey, bool nextkey,
		   Buffer *bufP, int access)
{
	BTStack		stack_in = NULL;

	/* Get the root page to start with */
	*bufP = _bt_getroot(rel, access);

	/* If index is empty and access = BT_READ, no root page is created. */
	if (!BufferIsValid(*bufP))
		return (BTStack) NULL;

	/* Loop iterates once per level descended in the tree */
	for (;;)
	{
		Page		page;
		BTPageOpaque opaque;
		OffsetNumber offnum;
		ItemId		itemid;
		IndexTuple	itup;
		BlockNumber blkno;
		BlockNumber par_blkno;
		BTStack		new_stack;

		/*
		 * Race -- the page we just grabbed may have split since we read its
		 * pointer in the parent (or metapage).  If it has, we may need to
		 * move right to its new sibling.  Do that.
		 */
		*bufP = _bt_moveright(rel, *bufP, keysz, scankey, nextkey, BT_READ);

		/* if this is a leaf page, we're done */
		page = BufferGetPage(*bufP);
		opaque = (BTPageOpaque) PageGetSpecialPointer(page);
		if (P_ISLEAF(opaque))
			break;

		/*
		 * Find the appropriate item on the internal page, and get the child
		 * page that it points to.
		 */
		offnum = _bt_binsrch(rel, *bufP, keysz, scankey, nextkey);
		itemid = PageGetItemId(page, offnum);
		itup = (IndexTuple) PageGetItem(page, itemid);
		blkno = ItemPointerGetBlockNumber(&(itup->t_tid));
		par_blkno = BufferGetBlockNumber(*bufP);

		/*
		 * We need to save the location of the index entry we chose in the
		 * parent page on a stack. In case we split the tree, we'll use the
		 * stack to work back up to the parent page.  We also save the actual
		 * downlink (TID) to uniquely identify the index entry, in case it
		 * moves right while we're working lower in the tree.  See the paper
		 * by Lehman and Yao for how this is detected and handled. (We use the
		 * child link to disambiguate duplicate keys in the index -- Lehman
		 * and Yao disallow duplicate keys.)
		 */
		new_stack = (BTStack) palloc(sizeof(BTStackData));
		new_stack->bts_blkno = par_blkno;
		new_stack->bts_offset = offnum;
		memcpy(&new_stack->bts_btentry, itup, sizeof(IndexTupleData));
		new_stack->bts_parent = stack_in;

		/* drop the read lock on the parent page, acquire one on the child */
		*bufP = _bt_relandgetbuf(rel, *bufP, blkno, BT_READ);

		/* okay, all set to move down a level */
		stack_in = new_stack;
	}

	return stack_in;
}

/*
 *	_bt_moveright() -- move right in the btree if necessary.
 *
 * When we follow a pointer to reach a page, it is possible that
 * the page has changed in the meanwhile.  If this happens, we're
 * guaranteed that the page has "split right" -- that is, that any
 * data that appeared on the page originally is either on the page
 * or strictly to the right of it.
 *
 * This routine decides whether or not we need to move right in the
 * tree by examining the high key entry on the page.  If that entry
 * is strictly less than the scankey, or <= the scankey in the nextkey=true
 * case, then we followed the wrong link and we need to move right.
 *
 * The passed scankey must be an insertion-type scankey (see nbtree/README),
 * but it can omit the rightmost column(s) of the index.
 *
 * When nextkey is false (the usual case), we are looking for the first
 * item >= scankey.  When nextkey is true, we are looking for the first
 * item strictly greater than scankey.
 *
 * On entry, we have the buffer pinned and a lock of the type specified by
 * 'access'.  If we move right, we release the buffer and lock and acquire
 * the same on the right sibling.  Return value is the buffer we stop at.
 */
Buffer
_bt_moveright(Relation rel,
			  Buffer buf,
			  int keysz,
			  ScanKey scankey,
			  bool nextkey,
			  int access)
{
	Page		page;
	BTPageOpaque opaque;
	int32		cmpval;

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

	/*
	 * When nextkey = false (normal case): if the scan key that brought us to
	 * this page is > the high key stored on the page, then the page has split
	 * and we need to move right.  (If the scan key is equal to the high key,
	 * we might or might not need to move right; have to scan the page first
	 * anyway.)
	 *
	 * When nextkey = true: move right if the scan key is >= page's high key.
	 *
	 * The page could even have split more than once, so scan as far as
	 * needed.
	 *
	 * We also have to move right if we followed a link that brought us to a
	 * dead page.
	 */
	cmpval = nextkey ? 0 : 1;

	while (!P_RIGHTMOST(opaque) &&
		   (P_IGNORE(opaque) ||
			_bt_compare(rel, keysz, scankey, page, P_HIKEY) >= cmpval))
	{
		/* step right one page */
		BlockNumber rblkno = opaque->btpo_next;

		buf = _bt_relandgetbuf(rel, buf, rblkno, access);
		page = BufferGetPage(buf);
		opaque = (BTPageOpaque) PageGetSpecialPointer(page);
	}

	if (P_IGNORE(opaque))
		elog(ERROR, "fell off the end of \"%s\"",
			 RelationGetRelationName(rel));

	return buf;
}

/*
 *	_bt_binsrch() -- Do a binary search for a key on a particular page.
 *
 * The passed scankey must be an insertion-type scankey (see nbtree/README),
 * but it can omit the rightmost column(s) of the index.
 *
 * When nextkey is false (the usual case), we are looking for the first
 * item >= scankey.  When nextkey is true, we are looking for the first
 * item strictly greater than scankey.
 *
 * On a leaf page, _bt_binsrch() returns the OffsetNumber of the first
 * key >= given scankey, or > scankey if nextkey is true.  (NOTE: in
 * particular, this means it is possible to return a value 1 greater than the
 * number of keys on the page, if the scankey is > all keys on the page.)
 *
 * On an internal (non-leaf) page, _bt_binsrch() returns the OffsetNumber
 * of the last key < given scankey, or last key <= given scankey if nextkey
 * is true.  (Since _bt_compare treats the first data key of such a page as
 * minus infinity, there will be at least one key < scankey, so the result
 * always points at one of the keys on the page.)  This key indicates the
 * right place to descend to be sure we find all leaf keys >= given scankey
 * (or leaf keys > given scankey when nextkey is true).
 *
 * This procedure is not responsible for walking right, it just examines
 * the given page.	_bt_binsrch() has no lock or refcount side effects
 * on the buffer.
 */
OffsetNumber
_bt_binsrch(Relation rel,
			Buffer buf,
			int keysz,
			ScanKey scankey,
			bool nextkey)
{
	Page		page;
	BTPageOpaque opaque;
	OffsetNumber low,
				high;
	int32		result,
				cmpval;

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

	low = P_FIRSTDATAKEY(opaque);
	high = PageGetMaxOffsetNumber(page);

	/*
	 * If there are no keys on the page, return the first available slot. Note
	 * this covers two cases: the page is really empty (no keys), or it
	 * contains only a high key.  The latter case is possible after vacuuming.
	 * This can never happen on an internal page, however, since they are
	 * never empty (an internal page must have children).
	 */
	if (high < low)
		return low;

	/*
	 * Binary search to find the first key on the page >= scan key, or first
	 * key > scankey when nextkey is true.
	 *
	 * For nextkey=false (cmpval=1), the loop invariant is: all slots before
	 * 'low' are < scan key, all slots at or after 'high' are >= scan key.
	 *
	 * For nextkey=true (cmpval=0), the loop invariant is: all slots before
	 * 'low' are <= scan key, all slots at or after 'high' are > scan key.
	 *
	 * We can fall out when high == low.
	 */
	high++;						/* establish the loop invariant for high */

	cmpval = nextkey ? 0 : 1;	/* select comparison value */

	while (high > low)
	{
		OffsetNumber mid = low + ((high - low) / 2);

		/* We have low <= mid < high, so mid points at a real slot */

		result = _bt_compare(rel, keysz, scankey, page, mid);

		if (result >= cmpval)
			low = mid + 1;
		else
			high = mid;
	}

	/*
	 * At this point we have high == low, but be careful: they could point
	 * past the last slot on the page.
	 *
	 * On a leaf page, we always return the first key >= scan key (resp. >
	 * scan key), which could be the last slot + 1.
	 */
	if (P_ISLEAF(opaque))
		return low;

	/*
	 * On a non-leaf page, return the last key < scan key (resp. <= scan key).
	 * There must be one if _bt_compare() is playing by the rules.
	 */
	Assert(low > P_FIRSTDATAKEY(opaque));

	return OffsetNumberPrev(low);
}

/*----------
 *	_bt_compare() -- Compare scankey to a particular tuple on the page.
 *
 * The passed scankey must be an insertion-type scankey (see nbtree/README),
 * but it can omit the rightmost column(s) of the index.
 *
 *	keysz: number of key conditions to be checked (might be less than the
 *		number of index columns!)
 *	page/offnum: location of btree item to be compared to.
 *
 *		This routine returns:
 *			<0 if scankey < tuple at offnum;
 *			 0 if scankey == tuple at offnum;
 *			>0 if scankey > tuple at offnum.
 *		NULLs in the keys are treated as sortable values.  Therefore
 *		"equality" does not necessarily mean that the item should be
 *		returned to the caller as a matching key!
 *
 * CRUCIAL NOTE: on a non-leaf page, the first data key is assumed to be
 * "minus infinity": this routine will always claim it is less than the
 * scankey.  The actual key value stored (if any, which there probably isn't)
 * does not matter.  This convention allows us to implement the Lehman and
 * Yao convention that the first down-link pointer is before the first key.
 * See backend/access/nbtree/README for details.
 *----------
 */
int32
_bt_compare(Relation rel,
			int keysz,
			ScanKey scankey,
			Page page,
			OffsetNumber offnum)
{
	TupleDesc	itupdesc = RelationGetDescr(rel);
	BTPageOpaque opaque = (BTPageOpaque) PageGetSpecialPointer(page);
	IndexTuple	itup;
	int			i;

	/*
	 * Force result ">" if target item is first data item on an internal page
	 * --- see NOTE above.
	 */
	if (!P_ISLEAF(opaque) && offnum == P_FIRSTDATAKEY(opaque))
		return 1;

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

	/*
	 * The scan key is set up with the attribute number associated with each
	 * term in the key.  It is important that, if the index is multi-key, the
	 * scan contain the first k key attributes, and that they be in order.	If
	 * you think about how multi-key ordering works, you'll understand why
	 * this is.
	 *
	 * We don't test for violation of this condition here, however.  The
	 * initial setup for the index scan had better have gotten it right (see
	 * _bt_first).
	 */

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

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

		/* see comments about NULLs handling in btbuild */
		if (scankey->sk_flags & SK_ISNULL)		/* key is NULL */
		{
			if (isNull)
				result = 0;		/* NULL "=" NULL */
			else if (scankey->sk_flags & SK_BT_NULLS_FIRST)
				result = -1;	/* NULL "<" NOT_NULL */
			else
				result = 1;		/* NULL ">" NOT_NULL */
		}
		else if (isNull)		/* key is NOT_NULL and item is NULL */
		{
			if (scankey->sk_flags & SK_BT_NULLS_FIRST)
				result = 1;		/* NOT_NULL ">" NULL */
			else
				result = -1;	/* NOT_NULL "<" NULL */
		}
		else
		{
			/*
			 * The sk_func needs to be passed the index value as left arg and
			 * the sk_argument as right arg (they might be of different
			 * types).	Since it is convenient for callers to think of
			 * _bt_compare as comparing the scankey to the index item, we have
			 * to flip the sign of the comparison result.  (Unless it's a DESC
			 * column, in which case we *don't* flip the sign.)
			 */
			result = DatumGetInt32(FunctionCall2(&scankey->sk_func,
												 datum,
												 scankey->sk_argument));

			if (!(scankey->sk_flags & SK_BT_DESC))
				result = -result;
		}

		/* if the keys are unequal, return the difference */
		if (result != 0)
			return result;

		scankey++;
	}

	/* if we get here, the keys are equal */
	return 0;
}

/*
 *	_bt_first() -- Find the first item in a scan.
 *
 *		We need to be clever about the direction of scan, the search
 *		conditions, and the tree ordering.	We find the first item (or,
 *		if backwards scan, the last item) in the tree that satisfies the
 *		qualifications in the scan key.  On success exit, the page containing
 *		the current index tuple is pinned but not locked, and data about
 *		the matching tuple(s) on the page has been loaded into so->currPos,
 *		and scan->xs_ctup.t_self is set to the heap TID of the current tuple.
 *
 * If there are no matching items in the index, we return FALSE, with no
 * pins or locks held.
 *
 * Note that scan->keyData[], and the so->keyData[] scankey built from it,
 * are both search-type scankeys (see nbtree/README for more about this).
 * Within this routine, we build a temporary insertion-type scankey to use
 * in locating the scan start position.
 */
bool
_bt_first(IndexScanDesc scan, ScanDirection dir)
{
	Relation	rel = scan->indexRelation;
	BTScanOpaque so = (BTScanOpaque) scan->opaque;
	Buffer		buf;
	BTStack		stack;
	OffsetNumber offnum;
	StrategyNumber strat;
	bool		nextkey;
	bool		goback;
	ScanKey		startKeys[INDEX_MAX_KEYS];
	ScanKeyData scankeys[INDEX_MAX_KEYS];
	int			keysCount = 0;
	int			i;
	StrategyNumber strat_total;

	pgstat_count_index_scan(&scan->xs_pgstat_info);

	/*
	 * Examine the scan keys and eliminate any redundant keys; also mark the
	 * keys that must be matched to continue the scan.
	 */
	_bt_preprocess_keys(scan);

	/*
	 * Quit now if _bt_preprocess_keys() discovered that the scan keys can
	 * never be satisfied (eg, x == 1 AND x > 2).
	 */
	if (!so->qual_ok)
		return false;

	/*----------
	 * Examine the scan keys to discover where we need to start the scan.
	 *
	 * We want to identify the keys that can be used as starting boundaries;
	 * these are =, >, or >= keys for a forward scan or =, <, <= keys for
	 * a backwards scan.  We can use keys for multiple attributes so long as
	 * the prior attributes had only =, >= (resp. =, <=) keys.	Once we accept
	 * a > or < boundary or find an attribute with no boundary (which can be
	 * thought of as the same as "> -infinity"), we can't use keys for any
	 * attributes to its right, because it would break our simplistic notion
	 * of what initial positioning strategy to use.
	 *
	 * When the scan keys include cross-type operators, _bt_preprocess_keys
	 * may not be able to eliminate redundant keys; in such cases we will
	 * arbitrarily pick a usable one for each attribute.  This is correct
	 * but possibly not optimal behavior.  (For example, with keys like
	 * "x >= 4 AND x >= 5" we would elect to scan starting at x=4 when
	 * x=5 would be more efficient.)  Since the situation only arises given
	 * a poorly-worded query plus an incomplete opfamily, live with it.
	 *
	 * When both equality and inequality keys appear for a single attribute
	 * (again, only possible when cross-type operators appear), we *must*
	 * select one of the equality keys for the starting point, because
	 * _bt_checkkeys() will stop the scan as soon as an equality qual fails.
	 * For example, if we have keys like "x >= 4 AND x = 10" and we elect to
	 * start at x=4, we will fail and stop before reaching x=10.  If multiple
	 * equality quals survive preprocessing, however, it doesn't matter which
	 * one we use --- by definition, they are either redundant or
	 * contradictory.
	 *
	 * In this loop, row-comparison keys are treated the same as keys on their
	 * first (leftmost) columns.  We'll add on lower-order columns of the row
	 * comparison below, if possible.
	 *
	 * The selected scan keys (at most one per index column) are remembered by
	 * storing their addresses into the local startKeys[] array.
	 *----------
	 */
	strat_total = BTEqualStrategyNumber;
	if (so->numberOfKeys > 0)
	{
		AttrNumber	curattr;
		ScanKey		chosen;
		ScanKey		cur;

		/*
		 * chosen is the so-far-chosen key for the current attribute, if any.
		 * We don't cast the decision in stone until we reach keys for the
		 * next attribute.
		 */
		curattr = 1;
		chosen = NULL;

		/*
		 * Loop iterates from 0 to numberOfKeys inclusive; we use the last
		 * pass to handle after-last-key processing.  Actual exit from the
		 * loop is at one of the "break" statements below.
		 */
		for (cur = so->keyData, i = 0;; cur++, i++)
		{
			if (i >= so->numberOfKeys || cur->sk_attno != curattr)
			{
				/*
				 * Done looking at keys for curattr.  If we didn't find a
				 * usable boundary key, quit; else save the boundary key
				 * pointer in startKeys.
				 */
				if (chosen == NULL)
					break;
				startKeys[keysCount++] = chosen;

				/*
				 * Adjust strat_total, and quit if we have stored a > or <
				 * key.
				 */
				strat = chosen->sk_strategy;
				if (strat != BTEqualStrategyNumber)
				{
					strat_total = strat;
					if (strat == BTGreaterStrategyNumber ||
						strat == BTLessStrategyNumber)
						break;
				}

				/*
				 * Done if that was the last attribute, or if next key is not
				 * in sequence (implying no boundary key is available for the
				 * next attribute).
				 */
				if (i >= so->numberOfKeys ||
					cur->sk_attno != curattr + 1)
					break;

				/*
				 * Reset for next attr.
				 */
				curattr = cur->sk_attno;
				chosen = NULL;
			}

			/* Can we use this key as a starting boundary for this attr? */
			switch (cur->sk_strategy)
			{
				case BTLessStrategyNumber:
				case BTLessEqualStrategyNumber:
					if (chosen == NULL && ScanDirectionIsBackward(dir))
						chosen = cur;
					break;
				case BTEqualStrategyNumber:
					/* override any non-equality choice */
					chosen = cur;
					break;
				case BTGreaterEqualStrategyNumber:
				case BTGreaterStrategyNumber:
					if (chosen == NULL && ScanDirectionIsForward(dir))
						chosen = cur;
					break;
			}
		}
	}

	/*
	 * If we found no usable boundary keys, we have to start from one end of
	 * the tree.  Walk down that edge to the first or last key, and scan from
	 * there.
	 */
	if (keysCount == 0)
		return _bt_endpoint(scan, dir);

	/*
	 * We want to start the scan somewhere within the index.  Set up an
	 * insertion scankey we can use to search for the boundary point we
	 * identified above.  The insertion scankey is built in the local
	 * scankeys[] array, using the keys identified by startKeys[].
	 */
	Assert(keysCount <= INDEX_MAX_KEYS);
	for (i = 0; i < keysCount; i++)
	{
		ScanKey		cur = startKeys[i];

		Assert(cur->sk_attno == i + 1);

		if (cur->sk_flags & SK_ROW_HEADER)
		{
			/*
			 * Row comparison header: look to the first row member instead.
			 *
			 * The member scankeys are already in insertion format (ie, they
			 * have sk_func = 3-way-comparison function), but we have to watch
			 * out for nulls, which _bt_preprocess_keys didn't check. A null
			 * in the first row member makes the condition unmatchable, just
			 * like qual_ok = false.
			 */
			ScanKey		subkey = (ScanKey) DatumGetPointer(cur->sk_argument);

			Assert(subkey->sk_flags & SK_ROW_MEMBER);
			if (subkey->sk_flags & SK_ISNULL)
				return false;
			memcpy(scankeys + i, subkey, sizeof(ScanKeyData));

			/*
			 * If the row comparison is the last positioning key we accepted,
			 * try to add additional keys from the lower-order row members.
			 * (If we accepted independent conditions on additional index
			 * columns, we use those instead --- doesn't seem worth trying to
			 * determine which is more restrictive.)  Note that this is OK
			 * even if the row comparison is of ">" or "<" type, because the
			 * condition applied to all but the last row member is effectively
			 * ">=" or "<=", and so the extra keys don't break the positioning
			 * scheme.  But, by the same token, if we aren't able to use all
			 * the row members, then the part of the row comparison that we
			 * did use has to be treated as just a ">=" or "<=" condition,
			 * and so we'd better adjust strat_total accordingly.
			 */
			if (i == keysCount - 1)
			{
				bool		used_all_subkeys = false;

				Assert(!(subkey->sk_flags & SK_ROW_END));
				for(;;)
				{
					subkey++;
					Assert(subkey->sk_flags & SK_ROW_MEMBER);
					if (subkey->sk_attno != keysCount + 1)
						break;	/* out-of-sequence, can't use it */
					if (subkey->sk_strategy != cur->sk_strategy)
						break;	/* wrong direction, can't use it */
					if (subkey->sk_flags & SK_ISNULL)
						break;	/* can't use null keys */
					Assert(keysCount < INDEX_MAX_KEYS);
					memcpy(scankeys + keysCount, subkey, sizeof(ScanKeyData));
					keysCount++;
					if (subkey->sk_flags & SK_ROW_END)
					{
						used_all_subkeys = true;
						break;
					}
				}
				if (!used_all_subkeys)
				{
					switch (strat_total)
					{
						case BTLessStrategyNumber:
							strat_total = BTLessEqualStrategyNumber;
							break;
						case BTGreaterStrategyNumber:
							strat_total = BTGreaterEqualStrategyNumber;
							break;
					}
				}
				break;			/* done with outer loop */
			}
		}
		else
		{
			/*
			 * Ordinary comparison key.  Transform the search-style scan key
			 * to an insertion scan key by replacing the sk_func with the
			 * appropriate btree comparison function.
			 *
			 * If scankey operator is not a cross-type comparison, we can use
			 * the cached comparison function; otherwise gotta look it up in
			 * the catalogs.  (That can't lead to infinite recursion, since no
			 * indexscan initiated by syscache lookup will use cross-data-type
			 * operators.)
			 *
			 * We support the convention that sk_subtype == InvalidOid means
			 * the opclass input type; this is a hack to simplify life for
			 * ScanKeyInit().
			 */
			if (cur->sk_subtype == rel->rd_opcintype[i] ||
				cur->sk_subtype == InvalidOid)
			{
				FmgrInfo   *procinfo;

				procinfo = index_getprocinfo(rel, cur->sk_attno, BTORDER_PROC);
				ScanKeyEntryInitializeWithInfo(scankeys + i,
											   cur->sk_flags,
											   cur->sk_attno,
											   InvalidStrategy,
											   cur->sk_subtype,
											   procinfo,
											   cur->sk_argument);
			}
			else
			{
				RegProcedure cmp_proc;

				cmp_proc = get_opfamily_proc(rel->rd_opfamily[i],
											 rel->rd_opcintype[i],
											 cur->sk_subtype,
											 BTORDER_PROC);
				if (!RegProcedureIsValid(cmp_proc))
					elog(ERROR, "missing support function %d(%u,%u) for attribute %d of index \"%s\"",
						 BTORDER_PROC, rel->rd_opcintype[i], cur->sk_subtype,
						 cur->sk_attno, RelationGetRelationName(rel));
				ScanKeyEntryInitialize(scankeys + i,
									   cur->sk_flags,
									   cur->sk_attno,
									   InvalidStrategy,
									   cur->sk_subtype,
									   cmp_proc,
									   cur->sk_argument);
			}
		}
	}

	/*----------
	 * Examine the selected initial-positioning strategy to determine exactly
	 * where we need to start the scan, and set flag variables to control the
	 * code below.
	 *
	 * If nextkey = false, _bt_search and _bt_binsrch will locate the first
	 * item >= scan key.  If nextkey = true, they will locate the first
	 * item > scan key.
	 *
	 * If goback = true, we will then step back one item, while if
	 * goback = false, we will start the scan on the located item.
	 *----------
	 */
	switch (strat_total)
	{
		case BTLessStrategyNumber:

			/*
			 * Find first item >= scankey, then back up one to arrive at last
			 * item < scankey.	(Note: this positioning strategy is only used
			 * for a backward scan, so that is always the correct starting
			 * position.)
			 */
			nextkey = false;
			goback = true;
			break;

		case BTLessEqualStrategyNumber:

			/*
			 * Find first item > scankey, then back up one to arrive at last
			 * item <= scankey.  (Note: this positioning strategy is only used
			 * for a backward scan, so that is always the correct starting
			 * position.)
			 */
			nextkey = true;
			goback = true;
			break;

		case BTEqualStrategyNumber:

			/*
			 * If a backward scan was specified, need to start with last equal
			 * item not first one.
			 */
			if (ScanDirectionIsBackward(dir))
			{
				/*
				 * This is the same as the <= strategy.  We will check at the
				 * end whether the found item is actually =.
				 */
				nextkey = true;
				goback = true;
			}
			else
			{
				/*
				 * This is the same as the >= strategy.  We will check at the
				 * end whether the found item is actually =.
				 */
				nextkey = false;
				goback = false;
			}
			break;

		case BTGreaterEqualStrategyNumber:

			/*
			 * Find first item >= scankey.	(This is only used for forward
			 * scans.)
			 */
			nextkey = false;
			goback = false;
			break;

		case BTGreaterStrategyNumber:

			/*
			 * Find first item > scankey.  (This is only used for forward
			 * scans.)
			 */
			nextkey = true;
			goback = false;
			break;

		default:
			/* can't get here, but keep compiler quiet */
			elog(ERROR, "unrecognized strat_total: %d", (int) strat_total);
			return false;
	}

	/*
	 * Use the manufactured insertion scan key to descend the tree and
	 * position ourselves on the target leaf page.
	 */
	stack = _bt_search(rel, keysCount, scankeys, nextkey, &buf, BT_READ);

	/* don't need to keep the stack around... */
	_bt_freestack(stack);

	/* remember which buffer we have pinned, if any */
	so->currPos.buf = buf;

	if (!BufferIsValid(buf))
	{
		/* Only get here if index is completely empty */
		return false;
	}

	/* initialize moreLeft/moreRight appropriately for scan direction */
	if (ScanDirectionIsForward(dir))
	{
		so->currPos.moreLeft = false;
		so->currPos.moreRight = true;
	}
	else
	{
		so->currPos.moreLeft = true;
		so->currPos.moreRight = false;
	}
	so->numKilled = 0;			/* just paranoia */
	so->markItemIndex = -1;		/* ditto */

	/* position to the precise item on the page */
	offnum = _bt_binsrch(rel, buf, keysCount, scankeys, nextkey);

	/*
	 * If nextkey = false, we are positioned at the first item >= scan key, or
	 * possibly at the end of a page on which all the existing items are less
	 * than the scan key and we know that everything on later pages is greater
	 * than or equal to scan key.
	 *
	 * If nextkey = true, we are positioned at the first item > scan key, or
	 * possibly at the end of a page on which all the existing items are less
	 * than or equal to the scan key and we know that everything on later
	 * pages is greater than scan key.
	 *
	 * The actually desired starting point is either this item or the prior
	 * one, or in the end-of-page case it's the first item on the next page or
	 * the last item on this page.	Adjust the starting offset if needed. (If
	 * this results in an offset before the first item or after the last one,
	 * _bt_readpage will report no items found, and then we'll step to the
	 * next page as needed.)
	 */
	if (goback)
		offnum = OffsetNumberPrev(offnum);

	/*
	 * Now load data from the first page of the scan.
	 */
	if (!_bt_readpage(scan, dir, offnum))
	{
		/*
		 * There's no actually-matching data on this page.  Try to advance to
		 * the next page.  Return false if there's no matching data at all.
		 */
		if (!_bt_steppage(scan, dir))
			return false;
	}

	/* Drop the lock, but not pin, on the current page */
	LockBuffer(so->currPos.buf, BUFFER_LOCK_UNLOCK);

	/* OK, itemIndex says what to return */
	scan->xs_ctup.t_self = so->currPos.items[so->currPos.itemIndex].heapTid;

	return true;
}

/*
 *	_bt_next() -- Get the next item in a scan.
 *
 *		On entry, so->currPos describes the current page, which is pinned
 *		but not locked, and so->currPos.itemIndex identifies which item was
 *		previously returned.
 *
 *		On successful exit, scan->xs_ctup.t_self is set to the TID of the
 *		next heap tuple, and so->currPos is updated as needed.
 *
 *		On failure exit (no more tuples), we release pin and set
 *		so->currPos.buf to InvalidBuffer.
 */
bool
_bt_next(IndexScanDesc scan, ScanDirection dir)
{
	BTScanOpaque so = (BTScanOpaque) scan->opaque;

	/*
	 * Advance to next tuple on current page; or if there's no more, try to
	 * step to the next page with data.
	 */
	if (ScanDirectionIsForward(dir))
	{
		if (++so->currPos.itemIndex > so->currPos.lastItem)
		{
			/* We must acquire lock before applying _bt_steppage */
			Assert(BufferIsValid(so->currPos.buf));
			LockBuffer(so->currPos.buf, BT_READ);
			if (!_bt_steppage(scan, dir))
				return false;
			/* Drop the lock, but not pin, on the new page */
			LockBuffer(so->currPos.buf, BUFFER_LOCK_UNLOCK);
		}
	}
	else
	{
		if (--so->currPos.itemIndex < so->currPos.firstItem)
		{
			/* We must acquire lock before applying _bt_steppage */
			Assert(BufferIsValid(so->currPos.buf));
			LockBuffer(so->currPos.buf, BT_READ);
			if (!_bt_steppage(scan, dir))
				return false;
			/* Drop the lock, but not pin, on the new page */
			LockBuffer(so->currPos.buf, BUFFER_LOCK_UNLOCK);
		}
	}

	/* OK, itemIndex says what to return */
	scan->xs_ctup.t_self = so->currPos.items[so->currPos.itemIndex].heapTid;

	return true;
}

/*
 *	_bt_readpage() -- Load data from current index page into so->currPos
 *
 * Caller must have pinned and read-locked so->currPos.buf; the buffer's state
 * is not changed here.  Also, currPos.moreLeft and moreRight must be valid;
 * they are updated as appropriate.  All other fields of so->currPos are
 * initialized from scratch here.
 *
 * We scan the current page starting at offnum and moving in the indicated
 * direction.  All items matching the scan keys are loaded into currPos.items.
 * moreLeft or moreRight (as appropriate) is cleared if _bt_checkkeys reports
 * that there can be no more matching tuples in the current scan direction.
 *
 * Returns true if any matching items found on the page, false if none.
 */
static bool
_bt_readpage(IndexScanDesc scan, ScanDirection dir, OffsetNumber offnum)
{
	BTScanOpaque so = (BTScanOpaque) scan->opaque;
	Page		page;
	BTPageOpaque opaque;
	OffsetNumber minoff;
	OffsetNumber maxoff;
	int			itemIndex;
	bool		continuescan;

	/* we must have the buffer pinned and locked */
	Assert(BufferIsValid(so->currPos.buf));

	page = BufferGetPage(so->currPos.buf);
	opaque = (BTPageOpaque) PageGetSpecialPointer(page);
	minoff = P_FIRSTDATAKEY(opaque);
	maxoff = PageGetMaxOffsetNumber(page);

	/*
	 * we must save the page's right-link while scanning it; this tells us
	 * where to step right to after we're done with these items.  There is no
	 * corresponding need for the left-link, since splits always go right.
	 */
	so->currPos.nextPage = opaque->btpo_next;

	if (ScanDirectionIsForward(dir))
	{
		/* load items[] in ascending order */
		itemIndex = 0;

		offnum = Max(offnum, minoff);

		while (offnum <= maxoff)
		{
			if (_bt_checkkeys(scan, page, offnum, dir, &continuescan))
			{
				/* tuple passes all scan key conditions, so remember it */
				/* _bt_checkkeys put the heap ptr into scan->xs_ctup.t_self */
				so->currPos.items[itemIndex].heapTid = scan->xs_ctup.t_self;
				so->currPos.items[itemIndex].indexOffset = offnum;
				itemIndex++;
			}
			if (!continuescan)
			{
				/* there can't be any more matches, so stop */
				so->currPos.moreRight = false;
				break;
			}

			offnum = OffsetNumberNext(offnum);
		}

		Assert(itemIndex <= MaxIndexTuplesPerPage);
		so->currPos.firstItem = 0;
		so->currPos.lastItem = itemIndex - 1;
		so->currPos.itemIndex = 0;
	}
	else
	{
		/* load items[] in descending order */
		itemIndex = MaxIndexTuplesPerPage;

		offnum = Min(offnum, maxoff);

		while (offnum >= minoff)
		{
			if (_bt_checkkeys(scan, page, offnum, dir, &continuescan))
			{
				/* tuple passes all scan key conditions, so remember it */
				/* _bt_checkkeys put the heap ptr into scan->xs_ctup.t_self */
				itemIndex--;
				so->currPos.items[itemIndex].heapTid = scan->xs_ctup.t_self;
				so->currPos.items[itemIndex].indexOffset = offnum;
			}
			if (!continuescan)
			{
				/* there can't be any more matches, so stop */
				so->currPos.moreLeft = false;
				break;
			}

			offnum = OffsetNumberPrev(offnum);
		}

		Assert(itemIndex >= 0);
		so->currPos.firstItem = itemIndex;
		so->currPos.lastItem = MaxIndexTuplesPerPage - 1;
		so->currPos.itemIndex = MaxIndexTuplesPerPage - 1;
	}

	return (so->currPos.firstItem <= so->currPos.lastItem);
}

/*
 *	_bt_steppage() -- Step to next page containing valid data for scan
 *
 * On entry, so->currPos.buf must be pinned and read-locked.  We'll drop
 * the lock and pin before moving to next page.
 *
 * On success exit, we hold pin and read-lock on the next interesting page,
 * and so->currPos is updated to contain data from that page.
 *
 * If there are no more matching records in the given direction, we drop all
 * locks and pins, set so->currPos.buf to InvalidBuffer, and return FALSE.
 */
static bool
_bt_steppage(IndexScanDesc scan, ScanDirection dir)
{
	BTScanOpaque so = (BTScanOpaque) scan->opaque;
	Relation	rel;
	Page		page;
	BTPageOpaque opaque;

	/* we must have the buffer pinned and locked */
	Assert(BufferIsValid(so->currPos.buf));

	/* Before leaving current page, deal with any killed items */
	if (so->numKilled > 0)
		_bt_killitems(scan, true);

	/*
	 * Before we modify currPos, make a copy of the page data if there was a
	 * mark position that needs it.
	 */
	if (so->markItemIndex >= 0)
	{
		/* bump pin on current buffer for assignment to mark buffer */
		IncrBufferRefCount(so->currPos.buf);
		memcpy(&so->markPos, &so->currPos,
			   offsetof(BTScanPosData, items[1]) +
			   so->currPos.lastItem * sizeof(BTScanPosItem));
		so->markPos.itemIndex = so->markItemIndex;
		so->markItemIndex = -1;
	}

	rel = scan->indexRelation;

	if (ScanDirectionIsForward(dir))
	{
		/* Walk right to the next page with data */
		/* We must rely on the previously saved nextPage link! */
		BlockNumber blkno = so->currPos.nextPage;

		/* Remember we left a page with data */
		so->currPos.moreLeft = true;

		for (;;)
		{
			/* if we're at end of scan, release the buffer and return */
			if (blkno == P_NONE || !so->currPos.moreRight)
			{
				_bt_relbuf(rel, so->currPos.buf);
				so->currPos.buf = InvalidBuffer;
				return false;
			}
			/* step right one page */
			so->currPos.buf = _bt_relandgetbuf(rel, so->currPos.buf,
											   blkno, BT_READ);
			/* check for deleted page */
			page = BufferGetPage(so->currPos.buf);
			opaque = (BTPageOpaque) PageGetSpecialPointer(page);
			if (!P_IGNORE(opaque))
			{
				/* see if there are any matches on this page */
				/* note that this will clear moreRight if we can stop */
				if (_bt_readpage(scan, dir, P_FIRSTDATAKEY(opaque)))
					break;
			}
			/* nope, keep going */
			blkno = opaque->btpo_next;
		}
	}
	else
	{
		/* Remember we left a page with data */
		so->currPos.moreRight = true;

		/*
		 * Walk left to the next page with data.  This is much more complex
		 * than the walk-right case because of the possibility that the page
		 * to our left splits while we are in flight to it, plus the
		 * possibility that the page we were on gets deleted after we leave
		 * it.	See nbtree/README for details.
		 */
		for (;;)
		{
			/* Done if we know there are no matching keys to the left */
			if (!so->currPos.moreLeft)
			{
				_bt_relbuf(rel, so->currPos.buf);
				so->currPos.buf = InvalidBuffer;
				return false;
			}

			/* Step to next physical page */
			so->currPos.buf = _bt_walk_left(rel, so->currPos.buf);

			/* if we're physically at end of index, return failure */
			if (so->currPos.buf == InvalidBuffer)
				return false;

			/*
			 * Okay, we managed to move left to a non-deleted page. Done if
			 * it's not half-dead and contains matching tuples. Else loop back
			 * and do it all again.
			 */
			page = BufferGetPage(so->currPos.buf);
			opaque = (BTPageOpaque) PageGetSpecialPointer(page);
			if (!P_IGNORE(opaque))
			{
				/* see if there are any matches on this page */
				/* note that this will clear moreLeft if we can stop */
				if (_bt_readpage(scan, dir, PageGetMaxOffsetNumber(page)))
					break;
			}
		}
	}

	return true;
}

/*
 * _bt_walk_left() -- step left one page, if possible
 *
 * The given buffer must be pinned and read-locked.  This will be dropped
 * before stepping left.  On return, we have pin and read lock on the
 * returned page, instead.
 *
 * Returns InvalidBuffer if there is no page to the left (no lock is held
 * in that case).
 *
 * When working on a non-leaf level, it is possible for the returned page
 * to be half-dead; the caller should check that condition and step left
 * again if it's important.
 */
static Buffer
_bt_walk_left(Relation rel, Buffer buf)
{
	Page		page;
	BTPageOpaque opaque;

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

	for (;;)
	{
		BlockNumber obknum;
		BlockNumber lblkno;
		BlockNumber blkno;
		int			tries;

		/* if we're at end of tree, release buf and return failure */
		if (P_LEFTMOST(opaque))
		{
			_bt_relbuf(rel, buf);
			break;
		}
		/* remember original page we are stepping left from */
		obknum = BufferGetBlockNumber(buf);
		/* step left */
		blkno = lblkno = opaque->btpo_prev;
		buf = _bt_relandgetbuf(rel, buf, blkno, BT_READ);
		page = BufferGetPage(buf);
		opaque = (BTPageOpaque) PageGetSpecialPointer(page);

		/*
		 * If this isn't the page we want, walk right till we find what we
		 * want --- but go no more than four hops (an arbitrary limit). If we
		 * don't find the correct page by then, the most likely bet is that
		 * the original page got deleted and isn't in the sibling chain at all
		 * anymore, not that its left sibling got split more than four times.
		 *
		 * Note that it is correct to test P_ISDELETED not P_IGNORE here,
		 * because half-dead pages are still in the sibling chain.	Caller
		 * must reject half-dead pages if wanted.
		 */
		tries = 0;
		for (;;)
		{
			if (!P_ISDELETED(opaque) && opaque->btpo_next == obknum)
			{
				/* Found desired page, return it */
				return buf;
			}
			if (P_RIGHTMOST(opaque) || ++tries > 4)
				break;
			blkno = opaque->btpo_next;
			buf = _bt_relandgetbuf(rel, buf, blkno, BT_READ);
			page = BufferGetPage(buf);
			opaque = (BTPageOpaque) PageGetSpecialPointer(page);
		}

		/* Return to the original page to see what's up */
		buf = _bt_relandgetbuf(rel, buf, obknum, BT_READ);
		page = BufferGetPage(buf);
		opaque = (BTPageOpaque) PageGetSpecialPointer(page);
		if (P_ISDELETED(opaque))
		{
			/*
			 * It was deleted.	Move right to first nondeleted page (there
			 * must be one); that is the page that has acquired the deleted
			 * one's keyspace, so stepping left from it will take us where we
			 * want to be.
			 */
			for (;;)
			{
				if (P_RIGHTMOST(opaque))
					elog(ERROR, "fell off the end of \"%s\"",
						 RelationGetRelationName(rel));
				blkno = opaque->btpo_next;
				buf = _bt_relandgetbuf(rel, buf, blkno, BT_READ);
				page = BufferGetPage(buf);
				opaque = (BTPageOpaque) PageGetSpecialPointer(page);
				if (!P_ISDELETED(opaque))
					break;
			}

			/*
			 * Now return to top of loop, resetting obknum to point to this
			 * nondeleted page, and try again.
			 */
		}
		else
		{
			/*
			 * It wasn't deleted; the explanation had better be that the page
			 * to the left got split or deleted. Without this check, we'd go
			 * into an infinite loop if there's anything wrong.
			 */
			if (opaque->btpo_prev == lblkno)
				elog(ERROR, "could not find left sibling in \"%s\"",
					 RelationGetRelationName(rel));
			/* Okay to try again with new lblkno value */
		}
	}

	return InvalidBuffer;
}

/*
 * _bt_get_endpoint() -- Find the first or last page on a given tree level
 *
 * If the index is empty, we will return InvalidBuffer; any other failure
 * condition causes ereport().	We will not return a dead page.
 *
 * The returned buffer is pinned and read-locked.
 */
Buffer
_bt_get_endpoint(Relation rel, uint32 level, bool rightmost)
{
	Buffer		buf;
	Page		page;
	BTPageOpaque opaque;
	OffsetNumber offnum;
	BlockNumber blkno;
	IndexTuple	itup;

	/*
	 * If we are looking for a leaf page, okay to descend from fast root;
	 * otherwise better descend from true root.  (There is no point in being
	 * smarter about intermediate levels.)
	 */
	if (level == 0)
		buf = _bt_getroot(rel, BT_READ);
	else
		buf = _bt_gettrueroot(rel);

	if (!BufferIsValid(buf))
	{
		/* empty index... */
		return InvalidBuffer;
	}

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

	for (;;)
	{
		/*
		 * If we landed on a deleted page, step right to find a live page
		 * (there must be one).  Also, if we want the rightmost page, step
		 * right if needed to get to it (this could happen if the page split
		 * since we obtained a pointer to it).
		 */
		while (P_IGNORE(opaque) ||
			   (rightmost && !P_RIGHTMOST(opaque)))
		{
			blkno = opaque->btpo_next;
			if (blkno == P_NONE)
				elog(ERROR, "fell off the end of \"%s\"",
					 RelationGetRelationName(rel));
			buf = _bt_relandgetbuf(rel, buf, blkno, BT_READ);
			page = BufferGetPage(buf);
			opaque = (BTPageOpaque) PageGetSpecialPointer(page);
		}

		/* Done? */
		if (opaque->btpo.level == level)
			break;
		if (opaque->btpo.level < level)
			elog(ERROR, "btree level %u not found", level);

		/* Descend to leftmost or rightmost child page */
		if (rightmost)
			offnum = PageGetMaxOffsetNumber(page);
		else
			offnum = P_FIRSTDATAKEY(opaque);

		itup = (IndexTuple) PageGetItem(page, PageGetItemId(page, offnum));
		blkno = ItemPointerGetBlockNumber(&(itup->t_tid));

		buf = _bt_relandgetbuf(rel, buf, blkno, BT_READ);
		page = BufferGetPage(buf);
		opaque = (BTPageOpaque) PageGetSpecialPointer(page);
	}

	return buf;
}

/*
 *	_bt_endpoint() -- Find the first or last page in the index, and scan
 * from there to the first key satisfying all the quals.
 *
 * This is used by _bt_first() to set up a scan when we've determined
 * that the scan must start at the beginning or end of the index (for
 * a forward or backward scan respectively).  Exit conditions are the
 * same as for _bt_first().
 */
static bool
_bt_endpoint(IndexScanDesc scan, ScanDirection dir)
{
	Relation	rel = scan->indexRelation;
	BTScanOpaque so = (BTScanOpaque) scan->opaque;
	Buffer		buf;
	Page		page;
	BTPageOpaque opaque;
	OffsetNumber start;

	/*
	 * Scan down to the leftmost or rightmost leaf page.  This is a simplified
	 * version of _bt_search().  We don't maintain a stack since we know we
	 * won't need it.
	 */
	buf = _bt_get_endpoint(rel, 0, ScanDirectionIsBackward(dir));

	if (!BufferIsValid(buf))
	{
		/* empty index... */
		so->currPos.buf = InvalidBuffer;
		return false;
	}

	page = BufferGetPage(buf);
	opaque = (BTPageOpaque) PageGetSpecialPointer(page);
	Assert(P_ISLEAF(opaque));

	if (ScanDirectionIsForward(dir))
	{
		/* There could be dead pages to the left, so not this: */
		/* Assert(P_LEFTMOST(opaque)); */

		start = P_FIRSTDATAKEY(opaque);
	}
	else if (ScanDirectionIsBackward(dir))
	{
		Assert(P_RIGHTMOST(opaque));

		start = PageGetMaxOffsetNumber(page);
	}
	else
	{
		elog(ERROR, "invalid scan direction: %d", (int) dir);
		start = 0;				/* keep compiler quiet */
	}

	/* remember which buffer we have pinned */
	so->currPos.buf = buf;

	/* initialize moreLeft/moreRight appropriately for scan direction */
	if (ScanDirectionIsForward(dir))
	{
		so->currPos.moreLeft = false;
		so->currPos.moreRight = true;
	}
	else
	{
		so->currPos.moreLeft = true;
		so->currPos.moreRight = false;
	}
	so->numKilled = 0;			/* just paranoia */
	so->markItemIndex = -1;		/* ditto */

	/*
	 * Now load data from the first page of the scan.
	 */
	if (!_bt_readpage(scan, dir, start))
	{
		/*
		 * There's no actually-matching data on this page.  Try to advance to
		 * the next page.  Return false if there's no matching data at all.
		 */
		if (!_bt_steppage(scan, dir))
			return false;
	}

	/* Drop the lock, but not pin, on the current page */
	LockBuffer(so->currPos.buf, BUFFER_LOCK_UNLOCK);

	/* OK, itemIndex says what to return */
	scan->xs_ctup.t_self = so->currPos.items[so->currPos.itemIndex].heapTid;

	return true;
}
