path: root/src/backend/utils/adt/selfuncs.c

/*-------------------------------------------------------------------------
 *
 * selfuncs.c
 *	  Selectivity functions and index cost estimation functions for
 *	  standard operators and index access methods.
 *
 *	  Selectivity routines are registered in the pg_operator catalog
 *	  in the "oprrest" and "oprjoin" attributes.
 *
 *	  Index cost functions are registered in the pg_am catalog
 *	  in the "amcostestimate" attribute.
 *
 * Portions Copyright (c) 1996-2003, PostgreSQL Global Development Group
 * Portions Copyright (c) 1994, Regents of the University of California
 *
 *
 * IDENTIFICATION
 *	  $PostgreSQL: pgsql/src/backend/utils/adt/selfuncs.c,v 1.149 2003/11/29 19:51:59 pgsql Exp $
 *
 *-------------------------------------------------------------------------
 */

/*----------
 * Operator selectivity estimation functions are called to estimate the
 * selectivity of WHERE clauses whose top-level operator is their operator.
 * We divide the problem into two cases:
 *		Restriction clause estimation: the clause involves vars of just
 *			one relation.
 *		Join clause estimation: the clause involves vars of multiple rels.
 * Join selectivity estimation is far more difficult and usually less accurate
 * than restriction estimation.
 *
 * When dealing with the inner scan of a nestloop join, we consider the
 * join's joinclauses as restriction clauses for the inner relation, and
 * treat vars of the outer relation as parameters (a/k/a constants of unknown
 * values).  So, restriction estimators need to be able to accept an argument
 * telling which relation is to be treated as the variable.
 *
 * The call convention for a restriction estimator (oprrest function) is
 *
 *		Selectivity oprrest (Query *root,
 *							 Oid operator,
 *							 List *args,
 *							 int varRelid);
 *
 * root: general information about the query (rtable and RelOptInfo lists
 * are particularly important for the estimator).
 * operator: OID of the specific operator in question.
 * args: argument list from the operator clause.
 * varRelid: if not zero, the relid (rtable index) of the relation to
 * be treated as the variable relation.  May be zero if the args list
 * is known to contain vars of only one relation.
 *
 * This is represented at the SQL level (in pg_proc) as
 *
 *		float8 oprrest (internal, oid, internal, int4);
 *
 * The call convention for a join estimator (oprjoin function) is similar
 * except that varRelid is not needed, and instead the join type is
 * supplied:
 *
 *		Selectivity oprjoin (Query *root,
 *							 Oid operator,
 *							 List *args,
 *							 JoinType jointype);
 *
 *		float8 oprjoin (internal, oid, internal, int2);
 *
 * (We deliberately make the SQL signature different to facilitate
 * catching errors.)
 *----------
 */

#include "postgres.h"

#include <ctype.h>
#include <math.h>

#include "access/heapam.h"
#include "access/nbtree.h"
#include "access/tuptoaster.h"
#include "catalog/catname.h"
#include "catalog/pg_namespace.h"
#include "catalog/pg_opclass.h"
#include "catalog/pg_operator.h"
#include "catalog/pg_proc.h"
#include "catalog/pg_statistic.h"
#include "catalog/pg_type.h"
#include "mb/pg_wchar.h"
#include "nodes/makefuncs.h"
#include "optimizer/clauses.h"
#include "optimizer/cost.h"
#include "optimizer/pathnode.h"
#include "optimizer/paths.h"
#include "optimizer/plancat.h"
#include "optimizer/prep.h"
#include "optimizer/tlist.h"
#include "optimizer/var.h"
#include "parser/parse_expr.h"
#include "parser/parse_func.h"
#include "parser/parse_oper.h"
#include "parser/parsetree.h"
#include "utils/builtins.h"
#include "utils/date.h"
#include "utils/datum.h"
#include "utils/int8.h"
#include "utils/lsyscache.h"
#include "utils/pg_locale.h"
#include "utils/selfuncs.h"
#include "utils/syscache.h"


/*
 * Note: the default selectivity estimates are not chosen entirely at random.
 * We want them to be small enough to ensure that indexscans will be used if
 * available, for typical table densities of ~100 tuples/page.	Thus, for
 * example, 0.01 is not quite small enough, since that makes it appear that
 * nearly all pages will be hit anyway.  Also, since we sometimes estimate
 * eqsel as 1/num_distinct, we probably want DEFAULT_NUM_DISTINCT to equal
 * 1/DEFAULT_EQ_SEL.
 */

/* default selectivity estimate for equalities such as "A = b" */
#define DEFAULT_EQ_SEL	0.005

/* default selectivity estimate for inequalities such as "A < b" */
#define DEFAULT_INEQ_SEL  (1.0 / 3.0)

/* default selectivity estimate for pattern-match operators such as LIKE */
#define DEFAULT_MATCH_SEL	0.005

/* default number of distinct values in a table */
#define DEFAULT_NUM_DISTINCT  200

/* default selectivity estimate for boolean and null test nodes */
#define DEFAULT_UNK_SEL			0.005
#define DEFAULT_NOT_UNK_SEL		(1.0 - DEFAULT_UNK_SEL)
#define DEFAULT_BOOL_SEL		0.5

/*
 * Clamp a computed probability estimate (which may suffer from roundoff or
 * estimation errors) to valid range.  Argument must be a float variable.
 */
#define CLAMP_PROBABILITY(p) \
	do { \
		if (p < 0.0) \
			p = 0.0; \
		else if (p > 1.0) \
			p = 1.0; \
	} while (0)


static bool get_var_maximum(Query *root, Var *var, Oid sortop, Datum *max);
static bool convert_to_scalar(Datum value, Oid valuetypid, double *scaledvalue,
				  Datum lobound, Datum hibound, Oid boundstypid,
				  double *scaledlobound, double *scaledhibound);
static double convert_numeric_to_scalar(Datum value, Oid typid);
static void convert_string_to_scalar(unsigned char *value,
						 double *scaledvalue,
						 unsigned char *lobound,
						 double *scaledlobound,
						 unsigned char *hibound,
						 double *scaledhibound);
static void convert_bytea_to_scalar(Datum value,
						double *scaledvalue,
						Datum lobound,
						double *scaledlobound,
						Datum hibound,
						double *scaledhibound);
static double convert_one_string_to_scalar(unsigned char *value,
							 int rangelo, int rangehi);
static double convert_one_bytea_to_scalar(unsigned char *value, int valuelen,
							int rangelo, int rangehi);
static unsigned char *convert_string_datum(Datum value, Oid typid);
static double convert_timevalue_to_scalar(Datum value, Oid typid);
static double get_att_numdistinct(Query *root, Var *var,
					Form_pg_statistic stats);
static bool get_restriction_var(List *args, int varRelid,
					Var **var, Node **other,
					bool *varonleft);
static void get_join_vars(List *args, Var **var1, Var **var2);
static Selectivity prefix_selectivity(Query *root, Var *var,
				   Oid opclass, Const *prefix);
static Selectivity pattern_selectivity(Const *patt, Pattern_Type ptype);
static Datum string_to_datum(const char *str, Oid datatype);
static Const *string_to_const(const char *str, Oid datatype);


/*
 *		eqsel			- Selectivity of "=" for any data types.
 *
 * Note: this routine is also used to estimate selectivity for some
 * operators that are not "=" but have comparable selectivity behavior,
 * such as "~=" (geometric approximate-match).	Even for "=", we must
 * keep in mind that the left and right datatypes may differ.
 */
Datum
eqsel(PG_FUNCTION_ARGS)
{
	Query	   *root = (Query *) PG_GETARG_POINTER(0);
	Oid			operator = PG_GETARG_OID(1);
	List	   *args = (List *) PG_GETARG_POINTER(2);
	int			varRelid = PG_GETARG_INT32(3);
	Var		   *var;
	Node	   *other;
	bool		varonleft;
	Oid			relid;
	HeapTuple	statsTuple;
	Datum	   *values;
	int			nvalues;
	float4	   *numbers;
	int			nnumbers;
	double		selec;

	/*
	 * If expression is not var = something or something = var for a
	 * simple var of a real relation (no subqueries, for now), then punt
	 * and return a default estimate.
	 */
	if (!get_restriction_var(args, varRelid,
							 &var, &other, &varonleft))
		PG_RETURN_FLOAT8(DEFAULT_EQ_SEL);
	relid = getrelid(var->varno, root->rtable);
	if (relid == InvalidOid)
		PG_RETURN_FLOAT8(DEFAULT_EQ_SEL);

	/*
	 * If the something is a NULL constant, assume operator is strict and
	 * return zero, ie, operator will never return TRUE.
	 */
	if (IsA(other, Const) &&
		((Const *) other)->constisnull)
		PG_RETURN_FLOAT8(0.0);

	/* get stats for the attribute, if available */
	statsTuple = SearchSysCache(STATRELATT,
								ObjectIdGetDatum(relid),
								Int16GetDatum(var->varattno),
								0, 0);
	if (HeapTupleIsValid(statsTuple))
	{
		Form_pg_statistic stats;

		stats = (Form_pg_statistic) GETSTRUCT(statsTuple);

		if (IsA(other, Const))
		{
			/* Var is being compared to a known non-null constant */
			Datum		constval = ((Const *) other)->constvalue;
			bool		match = false;
			int			i;

			/*
			 * Is the constant "=" to any of the column's most common
			 * values?	(Although the given operator may not really be
			 * "=", we will assume that seeing whether it returns TRUE is
			 * an appropriate test.  If you don't like this, maybe you
			 * shouldn't be using eqsel for your operator...)
			 */
			if (get_attstatsslot(statsTuple, var->vartype, var->vartypmod,
								 STATISTIC_KIND_MCV, InvalidOid,
								 &values, &nvalues,
								 &numbers, &nnumbers))
			{
				FmgrInfo	eqproc;

				fmgr_info(get_opcode(operator), &eqproc);

				for (i = 0; i < nvalues; i++)
				{
					/* be careful to apply operator right way 'round */
					if (varonleft)
						match = DatumGetBool(FunctionCall2(&eqproc,
														   values[i],
														   constval));
					else
						match = DatumGetBool(FunctionCall2(&eqproc,
														   constval,
														   values[i]));
					if (match)
						break;
				}
			}
			else
			{
				/* no most-common-value info available */
				values = NULL;
				numbers = NULL;
				i = nvalues = nnumbers = 0;
			}

			if (match)
			{
				/*
				 * Constant is "=" to this common value.  We know
				 * selectivity exactly (or as exactly as VACUUM could
				 * calculate it, anyway).
				 */
				selec = numbers[i];
			}
			else
			{
				/*
				 * Comparison is against a constant that is neither NULL
				 * nor any of the common values.  Its selectivity cannot
				 * be more than this:
				 */
				double		sumcommon = 0.0;
				double		otherdistinct;

				for (i = 0; i < nnumbers; i++)
					sumcommon += numbers[i];
				selec = 1.0 - sumcommon - stats->stanullfrac;
				CLAMP_PROBABILITY(selec);

				/*
				 * and in fact it's probably a good deal less. We
				 * approximate that all the not-common values share this
				 * remaining fraction equally, so we divide by the number
				 * of other distinct values.
				 */
				otherdistinct = get_att_numdistinct(root, var, stats)
					- nnumbers;
				if (otherdistinct > 1)
					selec /= otherdistinct;

				/*
				 * Another cross-check: selectivity shouldn't be estimated
				 * as more than the least common "most common value".
				 */
				if (nnumbers > 0 && selec > numbers[nnumbers - 1])
					selec = numbers[nnumbers - 1];
			}

			free_attstatsslot(var->vartype, values, nvalues,
							  numbers, nnumbers);
		}
		else
		{
			double		ndistinct;

			/*
			 * Search is for a value that we do not know a priori, but we
			 * will assume it is not NULL.	Estimate the selectivity as
			 * non-null fraction divided by number of distinct values, so
			 * that we get a result averaged over all possible values
			 * whether common or uncommon.	(Essentially, we are assuming
			 * that the not-yet-known comparison value is equally likely
			 * to be any of the possible values, regardless of their
			 * frequency in the table.	Is that a good idea?)
			 */
			selec = 1.0 - stats->stanullfrac;
			ndistinct = get_att_numdistinct(root, var, stats);
			if (ndistinct > 1)
				selec /= ndistinct;

			/*
			 * Cross-check: selectivity should never be estimated as more
			 * than the most common value's.
			 */
			if (get_attstatsslot(statsTuple, var->vartype, var->vartypmod,
								 STATISTIC_KIND_MCV, InvalidOid,
								 NULL, NULL,
								 &numbers, &nnumbers))
			{
				if (nnumbers > 0 && selec > numbers[0])
					selec = numbers[0];
				free_attstatsslot(var->vartype, NULL, 0, numbers, nnumbers);
			}
		}

		ReleaseSysCache(statsTuple);
	}
	else
	{
		/*
		 * No VACUUM ANALYZE stats available, so make a guess using
		 * estimated number of distinct values and assuming they are
		 * equally common.	(The guess is unlikely to be very good, but we
		 * do know a few special cases.)
		 */
		selec = 1.0 / get_att_numdistinct(root, var, NULL);
	}

	/* result should be in range, but make sure... */
	CLAMP_PROBABILITY(selec);

	PG_RETURN_FLOAT8((float8) selec);
}

/*
 *		neqsel			- Selectivity of "!=" for any data types.
 *
 * This routine is also used for some operators that are not "!="
 * but have comparable selectivity behavior.  See above comments
 * for eqsel().
 */
Datum
neqsel(PG_FUNCTION_ARGS)
{
	Query	   *root = (Query *) PG_GETARG_POINTER(0);
	Oid			operator = PG_GETARG_OID(1);
	List	   *args = (List *) PG_GETARG_POINTER(2);
	int			varRelid = PG_GETARG_INT32(3);
	Oid			eqop;
	float8		result;

	/*
	 * We want 1 - eqsel() where the equality operator is the one
	 * associated with this != operator, that is, its negator.
	 */
	eqop = get_negator(operator);
	if (eqop)
	{
		result = DatumGetFloat8(DirectFunctionCall4(eqsel,
													PointerGetDatum(root),
												  ObjectIdGetDatum(eqop),
													PointerGetDatum(args),
											   Int32GetDatum(varRelid)));
	}
	else
	{
		/* Use default selectivity (should we raise an error instead?) */
		result = DEFAULT_EQ_SEL;
	}
	result = 1.0 - result;
	PG_RETURN_FLOAT8(result);
}

/*
 *	scalarineqsel		- Selectivity of "<", "<=", ">", ">=" for scalars.
 *
 * This is the guts of both scalarltsel and scalargtsel.  The caller has
 * commuted the clause, if necessary, so that we can treat the Var as
 * being on the left.  The caller must also make sure that the other side
 * of the clause is a non-null Const, and dissect same into a value and
 * datatype.
 *
 * This routine works for any datatype (or pair of datatypes) known to
 * convert_to_scalar().  If it is applied to some other datatype,
 * it will return a default estimate.
 */
static double
scalarineqsel(Query *root, Oid operator, bool isgt,
			  Var *var, Datum constval, Oid consttype)
{
	Oid			relid;
	HeapTuple	statsTuple;
	Form_pg_statistic stats;
	FmgrInfo	opproc;
	Datum	   *values;
	int			nvalues;
	float4	   *numbers;
	int			nnumbers;
	double		mcv_selec,
				hist_selec,
				sumcommon;
	double		selec;
	int			i;

	/*
	 * If expression is not var op something or something op var for a
	 * simple var of a real relation (no subqueries, for now), then punt
	 * and return a default estimate.
	 */
	relid = getrelid(var->varno, root->rtable);
	if (relid == InvalidOid)
		return DEFAULT_INEQ_SEL;

	/* get stats for the attribute */
	statsTuple = SearchSysCache(STATRELATT,
								ObjectIdGetDatum(relid),
								Int16GetDatum(var->varattno),
								0, 0);
	if (!HeapTupleIsValid(statsTuple))
	{
		/* no stats available, so default result */
		return DEFAULT_INEQ_SEL;
	}
	stats = (Form_pg_statistic) GETSTRUCT(statsTuple);

	fmgr_info(get_opcode(operator), &opproc);

	/*
	 * If we have most-common-values info, add up the fractions of the MCV
	 * entries that satisfy MCV OP CONST.  These fractions contribute
	 * directly to the result selectivity.	Also add up the total fraction
	 * represented by MCV entries.
	 */
	mcv_selec = 0.0;
	sumcommon = 0.0;

	if (get_attstatsslot(statsTuple, var->vartype, var->vartypmod,
						 STATISTIC_KIND_MCV, InvalidOid,
						 &values, &nvalues,
						 &numbers, &nnumbers))
	{
		for (i = 0; i < nvalues; i++)
		{
			if (DatumGetBool(FunctionCall2(&opproc,
										   values[i],
										   constval)))
				mcv_selec += numbers[i];
			sumcommon += numbers[i];
		}
		free_attstatsslot(var->vartype, values, nvalues, numbers, nnumbers);
	}

	/*
	 * If there is a histogram, determine which bin the constant falls in,
	 * and compute the resulting contribution to selectivity.
	 *
	 * Someday, VACUUM might store more than one histogram per rel/att,
	 * corresponding to more than one possible sort ordering defined for
	 * the column type.  However, to make that work we will need to figure
	 * out which staop to search for --- it's not necessarily the one we
	 * have at hand!  (For example, we might have a '<=' operator rather
	 * than the '<' operator that will appear in staop.)  For now, assume
	 * that whatever appears in pg_statistic is sorted the same way our
	 * operator sorts, or the reverse way if isgt is TRUE.
	 */
	hist_selec = 0.0;

	if (get_attstatsslot(statsTuple, var->vartype, var->vartypmod,
						 STATISTIC_KIND_HISTOGRAM, InvalidOid,
						 &values, &nvalues,
						 NULL, NULL))
	{
		if (nvalues > 1)
		{
			double		histfrac;
			bool		ltcmp;

			ltcmp = DatumGetBool(FunctionCall2(&opproc,
											   values[0],
											   constval));
			if (isgt)
				ltcmp = !ltcmp;
			if (!ltcmp)
			{
				/* Constant is below lower histogram boundary. */
				histfrac = 0.0;
			}
			else
			{
				/*
				 * Scan to find proper location.  This could be made
				 * faster by using a binary-search method, but it's
				 * probably not worth the trouble for typical histogram
				 * sizes.
				 */
				for (i = 1; i < nvalues; i++)
				{
					ltcmp = DatumGetBool(FunctionCall2(&opproc,
													   values[i],
													   constval));
					if (isgt)
						ltcmp = !ltcmp;
					if (!ltcmp)
						break;
				}
				if (i >= nvalues)
				{
					/* Constant is above upper histogram boundary. */
					histfrac = 1.0;
				}
				else
				{
					double		val,
								high,
								low;
					double		binfrac;

					/*
					 * We have values[i-1] < constant < values[i].
					 *
					 * Convert the constant and the two nearest bin boundary
					 * values to a uniform comparison scale, and do a
					 * linear interpolation within this bin.
					 */
					if (convert_to_scalar(constval, consttype, &val,
										  values[i - 1], values[i],
										  var->vartype,
										  &low, &high))
					{
						if (high <= low)
						{
							/* cope if bin boundaries appear identical */
							binfrac = 0.5;
						}
						else if (val <= low)
							binfrac = 0.0;
						else if (val >= high)
							binfrac = 1.0;
						else
						{
							binfrac = (val - low) / (high - low);

							/*
							 * Watch out for the possibility that we got a
							 * NaN or Infinity from the division.  This
							 * can happen despite the previous checks, if
							 * for example "low" is -Infinity.
							 */
							if (isnan(binfrac) ||
								binfrac < 0.0 || binfrac > 1.0)
								binfrac = 0.5;
						}
					}
					else
					{
						/*
						 * Ideally we'd produce an error here, on the
						 * grounds that the given operator shouldn't have
						 * scalarXXsel registered as its selectivity func
						 * unless we can deal with its operand types.  But
						 * currently, all manner of stuff is invoking
						 * scalarXXsel, so give a default estimate until
						 * that can be fixed.
						 */
						binfrac = 0.5;
					}

					/*
					 * Now, compute the overall selectivity across the
					 * values represented by the histogram.  We have i-1
					 * full bins and binfrac partial bin below the
					 * constant.
					 */
					histfrac = (double) (i - 1) + binfrac;
					histfrac /= (double) (nvalues - 1);
				}
			}

			/*
			 * Now histfrac = fraction of histogram entries below the
			 * constant.
			 *
			 * Account for "<" vs ">"
			 */
			hist_selec = isgt ? (1.0 - histfrac) : histfrac;

			/*
			 * The histogram boundaries are only approximate to begin
			 * with, and may well be out of date anyway.  Therefore, don't
			 * believe extremely small or large selectivity estimates.
			 */
			if (hist_selec < 0.0001)
				hist_selec = 0.0001;
			else if (hist_selec > 0.9999)
				hist_selec = 0.9999;
		}

		free_attstatsslot(var->vartype, values, nvalues, NULL, 0);
	}

	/*
	 * Now merge the results from the MCV and histogram calculations,
	 * realizing that the histogram covers only the non-null values that
	 * are not listed in MCV.
	 */
	selec = 1.0 - stats->stanullfrac - sumcommon;

	if (hist_selec > 0.0)
		selec *= hist_selec;
	else
	{
		/*
		 * If no histogram but there are values not accounted for by MCV,
		 * arbitrarily assume half of them will match.
		 */
		selec *= 0.5;
	}

	selec += mcv_selec;

	ReleaseSysCache(statsTuple);

	/* result should be in range, but make sure... */
	CLAMP_PROBABILITY(selec);

	return selec;
}

/*
 *		scalarltsel		- Selectivity of "<" (also "<=") for scalars.
 */
Datum
scalarltsel(PG_FUNCTION_ARGS)
{
	Query	   *root = (Query *) PG_GETARG_POINTER(0);
	Oid			operator = PG_GETARG_OID(1);
	List	   *args = (List *) PG_GETARG_POINTER(2);
	int			varRelid = PG_GETARG_INT32(3);
	Var		   *var;
	Node	   *other;
	Datum		constval;
	Oid			consttype;
	bool		varonleft;
	bool		isgt;
	double		selec;

	/*
	 * If expression is not var op something or something op var for a
	 * simple var of a real relation (no subqueries, for now), then punt
	 * and return a default estimate.
	 */
	if (!get_restriction_var(args, varRelid,
							 &var, &other, &varonleft))
		PG_RETURN_FLOAT8(DEFAULT_INEQ_SEL);

	/*
	 * Can't do anything useful if the something is not a constant,
	 * either.
	 */
	if (!IsA(other, Const))
		PG_RETURN_FLOAT8(DEFAULT_INEQ_SEL);

	/*
	 * If the constant is NULL, assume operator is strict and return zero,
	 * ie, operator will never return TRUE.
	 */
	if (((Const *) other)->constisnull)
		PG_RETURN_FLOAT8(0.0);
	constval = ((Const *) other)->constvalue;
	consttype = ((Const *) other)->consttype;

	/*
	 * Force the var to be on the left to simplify logic in scalarineqsel.
	 */
	if (varonleft)
	{
		/* we have var < other */
		isgt = false;
	}
	else
	{
		/* we have other < var, commute to make var > other */
		operator = get_commutator(operator);
		if (!operator)
		{
			/* Use default selectivity (should we raise an error instead?) */
			PG_RETURN_FLOAT8(DEFAULT_INEQ_SEL);
		}
		isgt = true;
	}

	selec = scalarineqsel(root, operator, isgt, var, constval, consttype);

	PG_RETURN_FLOAT8((float8) selec);
}

/*
 *		scalargtsel		- Selectivity of ">" (also ">=") for integers.
 */
Datum
scalargtsel(PG_FUNCTION_ARGS)
{
	Query	   *root = (Query *) PG_GETARG_POINTER(0);
	Oid			operator = PG_GETARG_OID(1);
	List	   *args = (List *) PG_GETARG_POINTER(2);
	int			varRelid = PG_GETARG_INT32(3);
	Var		   *var;
	Node	   *other;
	Datum		constval;
	Oid			consttype;
	bool		varonleft;
	bool		isgt;
	double		selec;

	/*
	 * If expression is not var op something or something op var for a
	 * simple var of a real relation (no subqueries, for now), then punt
	 * and return a default estimate.
	 */
	if (!get_restriction_var(args, varRelid,
							 &var, &other, &varonleft))
		PG_RETURN_FLOAT8(DEFAULT_INEQ_SEL);

	/*
	 * Can't do anything useful if the something is not a constant,
	 * either.
	 */
	if (!IsA(other, Const))
		PG_RETURN_FLOAT8(DEFAULT_INEQ_SEL);

	/*
	 * If the constant is NULL, assume operator is strict and return zero,
	 * ie, operator will never return TRUE.
	 */
	if (((Const *) other)->constisnull)
		PG_RETURN_FLOAT8(0.0);
	constval = ((Const *) other)->constvalue;
	consttype = ((Const *) other)->consttype;

	/*
	 * Force the var to be on the left to simplify logic in scalarineqsel.
	 */
	if (varonleft)
	{
		/* we have var > other */
		isgt = true;
	}
	else
	{
		/* we have other > var, commute to make var < other */
		operator = get_commutator(operator);
		if (!operator)
		{
			/* Use default selectivity (should we raise an error instead?) */
			PG_RETURN_FLOAT8(DEFAULT_INEQ_SEL);
		}
		isgt = false;
	}

	selec = scalarineqsel(root, operator, isgt, var, constval, consttype);

	PG_RETURN_FLOAT8((float8) selec);
}

/*
 * patternsel			- Generic code for pattern-match selectivity.
 */
static double
patternsel(PG_FUNCTION_ARGS, Pattern_Type ptype)
{
	Query	   *root = (Query *) PG_GETARG_POINTER(0);

#ifdef NOT_USED
	Oid			operator = PG_GETARG_OID(1);
#endif
	List	   *args = (List *) PG_GETARG_POINTER(2);
	int			varRelid = PG_GETARG_INT32(3);
	Var		   *var;
	Node	   *other;
	bool		varonleft;
	Oid			relid;
	Datum		constval;
	Oid			consttype;
	Oid			vartype;
	Oid			opclass;
	Pattern_Prefix_Status pstatus;
	Const	   *patt = NULL;
	Const	   *prefix = NULL;
	Const	   *rest = NULL;
	double		result;

	/*
	 * If expression is not var op constant for a simple var of a real
	 * relation (no subqueries, for now), then punt and return a default
	 * estimate.
	 */
	if (!get_restriction_var(args, varRelid,
							 &var, &other, &varonleft))
		return DEFAULT_MATCH_SEL;
	if (!varonleft || !IsA(other, Const))
		return DEFAULT_MATCH_SEL;
	relid = getrelid(var->varno, root->rtable);
	if (relid == InvalidOid)
		return DEFAULT_MATCH_SEL;

	/*
	 * If the constant is NULL, assume operator is strict and return zero,
	 * ie, operator will never return TRUE.
	 */
	if (((Const *) other)->constisnull)
		return 0.0;
	constval = ((Const *) other)->constvalue;
	consttype = ((Const *) other)->consttype;

	/*
	 * The right-hand const is type text or bytea for all supported
	 * operators.  We do not expect to see binary-compatible types here,
	 * since const-folding should have relabeled the const to exactly
	 * match the operator's declared type.
	 */
	if (consttype != TEXTOID && consttype != BYTEAOID)
		return DEFAULT_MATCH_SEL;

	/*
	 * The var, on the other hand, might be a binary-compatible type;
	 * particularly a domain.  Try to fold it if it's not recognized
	 * immediately.
	 */
	vartype = var->vartype;
	if (vartype != consttype)
		vartype = getBaseType(vartype);

	/*
	 * We should now be able to recognize the var's datatype.  Choose the
	 * index opclass from which we must draw the comparison operators.
	 *
	 * NOTE: It would be more correct to use the PATTERN opclasses than the
	 * simple ones, but at the moment ANALYZE will not generate statistics
	 * for the PATTERN operators.  But our results are so approximate
	 * anyway that it probably hardly matters.
	 */
	switch (vartype)
	{
		case TEXTOID:
			opclass = TEXT_BTREE_OPS_OID;
			break;
		case VARCHAROID:
			opclass = VARCHAR_BTREE_OPS_OID;
			break;
		case BPCHAROID:
			opclass = BPCHAR_BTREE_OPS_OID;
			break;
		case NAMEOID:
			opclass = NAME_BTREE_OPS_OID;
			break;
		case BYTEAOID:
			opclass = BYTEA_BTREE_OPS_OID;
			break;
		default:
			return DEFAULT_MATCH_SEL;
	}

	/* divide pattern into fixed prefix and remainder */
	patt = (Const *) other;
	pstatus = pattern_fixed_prefix(patt, ptype, &prefix, &rest);

	/*
	 * If necessary, coerce the prefix constant to the right type. (The
	 * "rest" constant need not be changed.)
	 */
	if (prefix && prefix->consttype != vartype)
	{
		char	   *prefixstr;

		switch (prefix->consttype)
		{
			case TEXTOID:
				prefixstr = DatumGetCString(DirectFunctionCall1(textout,
													prefix->constvalue));
				break;
			case BYTEAOID:
				prefixstr = DatumGetCString(DirectFunctionCall1(byteaout,
													prefix->constvalue));
				break;
			default:
				elog(ERROR, "unrecognized consttype: %u",
					 prefix->consttype);
				return DEFAULT_MATCH_SEL;
		}
		prefix = string_to_const(prefixstr, vartype);
		pfree(prefixstr);
	}

	if (pstatus == Pattern_Prefix_Exact)
	{
		/*
		 * Pattern specifies an exact match, so pretend operator is '='
		 */
		Oid			eqopr = get_opclass_member(opclass, InvalidOid,
											   BTEqualStrategyNumber);
		List	   *eqargs;

		if (eqopr == InvalidOid)
			elog(ERROR, "no = operator for opclass %u", opclass);
		eqargs = makeList2(var, prefix);
		result = DatumGetFloat8(DirectFunctionCall4(eqsel,
													PointerGetDatum(root),
												 ObjectIdGetDatum(eqopr),
												 PointerGetDatum(eqargs),
											   Int32GetDatum(varRelid)));
	}
	else
	{
		/*
		 * Not exact-match pattern.  We estimate selectivity of the fixed
		 * prefix and remainder of pattern separately, then combine the
		 * two.
		 */
		Selectivity prefixsel;
		Selectivity restsel;
		Selectivity selec;

		if (pstatus == Pattern_Prefix_Partial)
			prefixsel = prefix_selectivity(root, var, opclass, prefix);
		else
			prefixsel = 1.0;
		restsel = pattern_selectivity(rest, ptype);
		selec = prefixsel * restsel;
		/* result should be in range, but make sure... */
		CLAMP_PROBABILITY(selec);
		result = selec;
	}

	if (prefix)
	{
		pfree(DatumGetPointer(prefix->constvalue));
		pfree(prefix);
	}

	return result;
}

/*
 *		regexeqsel		- Selectivity of regular-expression pattern match.
 */
Datum
regexeqsel(PG_FUNCTION_ARGS)
{
	PG_RETURN_FLOAT8(patternsel(fcinfo, Pattern_Type_Regex));
}

/*
 *		icregexeqsel	- Selectivity of case-insensitive regex match.
 */
Datum
icregexeqsel(PG_FUNCTION_ARGS)
{
	PG_RETURN_FLOAT8(patternsel(fcinfo, Pattern_Type_Regex_IC));
}

/*
 *		likesel			- Selectivity of LIKE pattern match.
 */
Datum
likesel(PG_FUNCTION_ARGS)
{
	PG_RETURN_FLOAT8(patternsel(fcinfo, Pattern_Type_Like));
}

/*
 *		iclikesel			- Selectivity of ILIKE pattern match.
 */
Datum
iclikesel(PG_FUNCTION_ARGS)
{
	PG_RETURN_FLOAT8(patternsel(fcinfo, Pattern_Type_Like_IC));
}

/*
 *		regexnesel		- Selectivity of regular-expression pattern non-match.
 */
Datum
regexnesel(PG_FUNCTION_ARGS)
{
	double		result;

	result = patternsel(fcinfo, Pattern_Type_Regex);
	result = 1.0 - result;
	PG_RETURN_FLOAT8(result);
}

/*
 *		icregexnesel	- Selectivity of case-insensitive regex non-match.
 */
Datum
icregexnesel(PG_FUNCTION_ARGS)
{
	double		result;

	result = patternsel(fcinfo, Pattern_Type_Regex_IC);
	result = 1.0 - result;
	PG_RETURN_FLOAT8(result);
}

/*
 *		nlikesel		- Selectivity of LIKE pattern non-match.
 */
Datum
nlikesel(PG_FUNCTION_ARGS)
{
	double		result;

	result = patternsel(fcinfo, Pattern_Type_Like);
	result = 1.0 - result;
	PG_RETURN_FLOAT8(result);
}

/*
 *		icnlikesel		- Selectivity of ILIKE pattern non-match.
 */
Datum
icnlikesel(PG_FUNCTION_ARGS)
{
	double		result;

	result = patternsel(fcinfo, Pattern_Type_Like_IC);
	result = 1.0 - result;
	PG_RETURN_FLOAT8(result);
}

/*
 *		booltestsel		- Selectivity of BooleanTest Node.
 */
Selectivity
booltestsel(Query *root, BoolTestType booltesttype, Node *arg,
			int varRelid, JoinType jointype)
{
	Var		   *var;
	Oid			relid;
	HeapTuple	statsTuple;
	Datum	   *values;
	int			nvalues;
	float4	   *numbers;
	int			nnumbers;
	double		selec;

	/*
	 * Ignore any binary-compatible relabeling (probably unnecessary, but
	 * can't hurt)
	 */
	if (IsA(arg, RelabelType))
		arg = (Node *) ((RelabelType *) arg)->arg;

	if (IsA(arg, Var) &&
		(varRelid == 0 || varRelid == ((Var *) arg)->varno))
		var = (Var *) arg;
	else
	{
		/*
		 * If argument is not a Var, we can't get statistics for it, but
		 * perhaps clause_selectivity can do something with it.  We ignore
		 * the possibility of a NULL value when using clause_selectivity,
		 * and just assume the value is either TRUE or FALSE.
		 */
		switch (booltesttype)
		{
			case IS_UNKNOWN:
				selec = DEFAULT_UNK_SEL;
				break;
			case IS_NOT_UNKNOWN:
				selec = DEFAULT_NOT_UNK_SEL;
				break;
			case IS_TRUE:
			case IS_NOT_FALSE:
				selec = (double) clause_selectivity(root, arg,
													varRelid, jointype);
				break;
			case IS_FALSE:
			case IS_NOT_TRUE:
				selec = 1.0 - (double) clause_selectivity(root, arg,
													 varRelid, jointype);
				break;
			default:
				elog(ERROR, "unrecognized booltesttype: %d",
					 (int) booltesttype);
				selec = 0.0;	/* Keep compiler quiet */
				break;
		}
		return (Selectivity) selec;
	}

	/* get stats for the attribute, if available */
	relid = getrelid(var->varno, root->rtable);
	if (relid == InvalidOid)
		statsTuple = NULL;
	else
		statsTuple = SearchSysCache(STATRELATT,
									ObjectIdGetDatum(relid),
									Int16GetDatum(var->varattno),
									0, 0);

	if (HeapTupleIsValid(statsTuple))
	{
		Form_pg_statistic stats;
		double		freq_null;

		stats = (Form_pg_statistic) GETSTRUCT(statsTuple);

		freq_null = stats->stanullfrac;

		if (get_attstatsslot(statsTuple, var->vartype, var->vartypmod,
							 STATISTIC_KIND_MCV, InvalidOid,
							 &values, &nvalues,
							 &numbers, &nnumbers)
			&& nnumbers > 0)
		{
			double		freq_true;
			double		freq_false;

			/*
			 * Get first MCV frequency and derive frequency for true.
			 */
			if (DatumGetBool(values[0]))
				freq_true = numbers[0];
			else
				freq_true = 1.0 - numbers[0] - freq_null;

			/*
			 * Next derive freqency for false. Then use these as
			 * appropriate to derive frequency for each case.
			 */
			freq_false = 1.0 - freq_true - freq_null;

			switch (booltesttype)
			{
				case IS_UNKNOWN:
					/* select only NULL values */
					selec = freq_null;
					break;
				case IS_NOT_UNKNOWN:
					/* select non-NULL values */
					selec = 1.0 - freq_null;
					break;
				case IS_TRUE:
					/* select only TRUE values */
					selec = freq_true;
					break;
				case IS_NOT_TRUE:
					/* select non-TRUE values */
					selec = 1.0 - freq_true;
					break;
				case IS_FALSE:
					/* select only FALSE values */
					selec = freq_false;
					break;
				case IS_NOT_FALSE:
					/* select non-FALSE values */
					selec = 1.0 - freq_false;
					break;
				default:
					elog(ERROR, "unrecognized booltesttype: %d",
						 (int) booltesttype);
					selec = 0.0;	/* Keep compiler quiet */
					break;
			}

			free_attstatsslot(var->vartype, values, nvalues,
							  numbers, nnumbers);
		}
		else
		{
			/*
			 * No most-common-value info available. Still have null
			 * fraction information, so use it for IS [NOT] UNKNOWN.
			 * Otherwise adjust for null fraction and assume an even split
			 * for boolean tests.
			 */
			switch (booltesttype)
			{
				case IS_UNKNOWN:

					/*
					 * Use freq_null directly.
					 */
					selec = freq_null;
					break;
				case IS_NOT_UNKNOWN:

					/*
					 * Select not unknown (not null) values. Calculate
					 * from freq_null.
					 */
					selec = 1.0 - freq_null;
					break;
				case IS_TRUE:
				case IS_NOT_TRUE:
				case IS_FALSE:
				case IS_NOT_FALSE:
					selec = (1.0 - freq_null) / 2.0;
					break;
				default:
					elog(ERROR, "unrecognized booltesttype: %d",
						 (int) booltesttype);
					selec = 0.0;	/* Keep compiler quiet */
					break;
			}
		}

		ReleaseSysCache(statsTuple);
	}
	else
	{
		/*
		 * No VACUUM ANALYZE stats available, so use a default value.
		 * (Note: not much point in recursing to clause_selectivity here.)
		 */
		switch (booltesttype)
		{
			case IS_UNKNOWN:
				selec = DEFAULT_UNK_SEL;
				break;
			case IS_NOT_UNKNOWN:
				selec = DEFAULT_NOT_UNK_SEL;
				break;
			case IS_TRUE:
			case IS_NOT_TRUE:
			case IS_FALSE:
			case IS_NOT_FALSE:
				selec = DEFAULT_BOOL_SEL;
				break;
			default:
				elog(ERROR, "unrecognized booltesttype: %d",
					 (int) booltesttype);
				selec = 0.0;	/* Keep compiler quiet */
				break;
		}
	}

	/* result should be in range, but make sure... */
	CLAMP_PROBABILITY(selec);

	return (Selectivity) selec;
}

/*
 *		nulltestsel		- Selectivity of NullTest Node.
 */
Selectivity
nulltestsel(Query *root, NullTestType nulltesttype, Node *arg, int varRelid)
{
	Var		   *var;
	Oid			relid;
	HeapTuple	statsTuple;
	double		selec;
	double		defselec;
	double		freq_null;

	switch (nulltesttype)
	{
		case IS_NULL:
			defselec = DEFAULT_UNK_SEL;
			break;
		case IS_NOT_NULL:
			defselec = DEFAULT_NOT_UNK_SEL;
			break;
		default:
			elog(ERROR, "unrecognized nulltesttype: %d",
				 (int) nulltesttype);
			return (Selectivity) 0;		/* keep compiler quiet */
	}

	/*
	 * Ignore any binary-compatible relabeling
	 */
	if (IsA(arg, RelabelType))
		arg = (Node *) ((RelabelType *) arg)->arg;

	if (IsA(arg, Var) &&
		(varRelid == 0 || varRelid == ((Var *) arg)->varno))
		var = (Var *) arg;
	else
	{
		/* punt if non-Var argument */
		return (Selectivity) defselec;
	}

	relid = getrelid(var->varno, root->rtable);
	if (relid == InvalidOid)
		return (Selectivity) defselec;

	/* get stats for the attribute, if available */
	statsTuple = SearchSysCache(STATRELATT,
								ObjectIdGetDatum(relid),
								Int16GetDatum(var->varattno),
								0, 0);
	if (HeapTupleIsValid(statsTuple))
	{
		Form_pg_statistic stats;

		stats = (Form_pg_statistic) GETSTRUCT(statsTuple);
		freq_null = stats->stanullfrac;

		switch (nulltesttype)
		{
			case IS_NULL:

				/*
				 * Use freq_null directly.
				 */
				selec = freq_null;
				break;
			case IS_NOT_NULL:

				/*
				 * Select not unknown (not null) values. Calculate from
				 * freq_null.
				 */
				selec = 1.0 - freq_null;
				break;
			default:
				elog(ERROR, "unrecognized nulltesttype: %d",
					 (int) nulltesttype);
				return (Selectivity) 0; /* keep compiler quiet */
		}

		ReleaseSysCache(statsTuple);
	}
	else
	{
		/*
		 * No VACUUM ANALYZE stats available, so make a guess
		 */
		selec = defselec;
	}

	/* result should be in range, but make sure... */
	CLAMP_PROBABILITY(selec);

	return (Selectivity) selec;
}

/*
 *		eqjoinsel		- Join selectivity of "="
 */
Datum
eqjoinsel(PG_FUNCTION_ARGS)
{
	Query	   *root = (Query *) PG_GETARG_POINTER(0);
	Oid			operator = PG_GETARG_OID(1);
	List	   *args = (List *) PG_GETARG_POINTER(2);
	JoinType	jointype = (JoinType) PG_GETARG_INT16(3);
	Var		   *var1;
	Var		   *var2;
	double		selec;

	get_join_vars(args, &var1, &var2);

	if (var1 == NULL && var2 == NULL)
		selec = DEFAULT_EQ_SEL;
	else
	{
		HeapTuple	statsTuple1 = NULL;
		HeapTuple	statsTuple2 = NULL;
		Form_pg_statistic stats1 = NULL;
		Form_pg_statistic stats2 = NULL;
		double		nd1 = DEFAULT_NUM_DISTINCT;
		double		nd2 = DEFAULT_NUM_DISTINCT;
		bool		have_mcvs1 = false;
		Datum	   *values1 = NULL;
		int			nvalues1 = 0;
		float4	   *numbers1 = NULL;
		int			nnumbers1 = 0;
		bool		have_mcvs2 = false;
		Datum	   *values2 = NULL;
		int			nvalues2 = 0;
		float4	   *numbers2 = NULL;
		int			nnumbers2 = 0;

		if (var1 != NULL)
		{
			/* get stats for the attribute, if available */
			Oid			relid1 = getrelid(var1->varno, root->rtable);

			if (relid1 != InvalidOid)
			{
				statsTuple1 = SearchSysCache(STATRELATT,
											 ObjectIdGetDatum(relid1),
										   Int16GetDatum(var1->varattno),
											 0, 0);
				if (HeapTupleIsValid(statsTuple1))
				{
					stats1 = (Form_pg_statistic) GETSTRUCT(statsTuple1);
					have_mcvs1 = get_attstatsslot(statsTuple1,
												  var1->vartype,
												  var1->vartypmod,
												  STATISTIC_KIND_MCV,
												  InvalidOid,
												  &values1, &nvalues1,
												  &numbers1, &nnumbers1);
				}

				nd1 = get_att_numdistinct(root, var1, stats1);
			}
		}

		if (var2 != NULL)
		{
			/* get stats for the attribute, if available */
			Oid			relid2 = getrelid(var2->varno, root->rtable);

			if (relid2 != InvalidOid)
			{
				statsTuple2 = SearchSysCache(STATRELATT,
											 ObjectIdGetDatum(relid2),
										   Int16GetDatum(var2->varattno),
											 0, 0);
				if (HeapTupleIsValid(statsTuple2))
				{
					stats2 = (Form_pg_statistic) GETSTRUCT(statsTuple2);
					have_mcvs2 = get_attstatsslot(statsTuple2,
												  var2->vartype,
												  var2->vartypmod,
												  STATISTIC_KIND_MCV,
												  InvalidOid,
												  &values2, &nvalues2,
												  &numbers2, &nnumbers2);
				}

				nd2 = get_att_numdistinct(root, var2, stats2);
			}
		}

		if (have_mcvs1 && have_mcvs2)
		{
			/*
			 * We have most-common-value lists for both relations.	Run
			 * through the lists to see which MCVs actually join to each
			 * other with the given operator.  This allows us to determine
			 * the exact join selectivity for the portion of the relations
			 * represented by the MCV lists.  We still have to estimate
			 * for the remaining population, but in a skewed distribution
			 * this gives us a big leg up in accuracy.	For motivation see
			 * the analysis in Y. Ioannidis and S. Christodoulakis, "On
			 * the propagation of errors in the size of join results",
			 * Technical Report 1018, Computer Science Dept., University
			 * of Wisconsin, Madison, March 1991 (available from
			 * ftp.cs.wisc.edu).
			 */
			FmgrInfo	eqproc;
			bool	   *hasmatch1;
			bool	   *hasmatch2;
			double		nullfrac1 = stats1->stanullfrac;
			double		nullfrac2 = stats2->stanullfrac;
			double		matchprodfreq,
						matchfreq1,
						matchfreq2,
						unmatchfreq1,
						unmatchfreq2,
						otherfreq1,
						otherfreq2,
						totalsel1,
						totalsel2;
			int			i,
						nmatches;

			fmgr_info(get_opcode(operator), &eqproc);
			hasmatch1 = (bool *) palloc0(nvalues1 * sizeof(bool));
			hasmatch2 = (bool *) palloc0(nvalues2 * sizeof(bool));

			/*
			 * If we are doing any variant of JOIN_IN, pretend all the
			 * values of the righthand relation are unique (ie, act as if
			 * it's been DISTINCT'd).
			 *
			 * NOTE: it might seem that we should unique-ify the lefthand
			 * input when considering JOIN_REVERSE_IN.	But this is not
			 * so, because the join clause we've been handed has not been
			 * commuted from the way the parser originally wrote it.  We
			 * know that the unique side of the IN clause is *always* on
			 * the right.
			 *
			 * NOTE: it would be dangerous to try to be smart about JOIN_LEFT
			 * or JOIN_RIGHT here, because we do not have enough
			 * information to determine which var is really on which side
			 * of the join. Perhaps someday we should pass in more
			 * information.
			 */
			if (jointype == JOIN_IN ||
				jointype == JOIN_REVERSE_IN ||
				jointype == JOIN_UNIQUE_INNER ||
				jointype == JOIN_UNIQUE_OUTER)
			{
				float4		oneovern = 1.0 / nd2;

				for (i = 0; i < nvalues2; i++)
					numbers2[i] = oneovern;
				nullfrac2 = oneovern;
			}

			/*
			 * Note we assume that each MCV will match at most one member
			 * of the other MCV list.  If the operator isn't really
			 * equality, there could be multiple matches --- but we don't
			 * look for them, both for speed and because the math wouldn't
			 * add up...
			 */
			matchprodfreq = 0.0;
			nmatches = 0;
			for (i = 0; i < nvalues1; i++)
			{
				int			j;

				for (j = 0; j < nvalues2; j++)
				{
					if (hasmatch2[j])
						continue;
					if (DatumGetBool(FunctionCall2(&eqproc,
												   values1[i],
												   values2[j])))
					{
						hasmatch1[i] = hasmatch2[j] = true;
						matchprodfreq += numbers1[i] * numbers2[j];
						nmatches++;
						break;
					}
				}
			}
			CLAMP_PROBABILITY(matchprodfreq);
			/* Sum up frequencies of matched and unmatched MCVs */
			matchfreq1 = unmatchfreq1 = 0.0;
			for (i = 0; i < nvalues1; i++)
			{
				if (hasmatch1[i])
					matchfreq1 += numbers1[i];
				else
					unmatchfreq1 += numbers1[i];
			}
			CLAMP_PROBABILITY(matchfreq1);
			CLAMP_PROBABILITY(unmatchfreq1);
			matchfreq2 = unmatchfreq2 = 0.0;
			for (i = 0; i < nvalues2; i++)
			{
				if (hasmatch2[i])
					matchfreq2 += numbers2[i];
				else
					unmatchfreq2 += numbers2[i];
			}
			CLAMP_PROBABILITY(matchfreq2);
			CLAMP_PROBABILITY(unmatchfreq2);
			pfree(hasmatch1);
			pfree(hasmatch2);

			/*
			 * Compute total frequency of non-null values that are not in
			 * the MCV lists.
			 */
			otherfreq1 = 1.0 - nullfrac1 - matchfreq1 - unmatchfreq1;
			otherfreq2 = 1.0 - nullfrac2 - matchfreq2 - unmatchfreq2;
			CLAMP_PROBABILITY(otherfreq1);
			CLAMP_PROBABILITY(otherfreq2);

			/*
			 * We can estimate the total selectivity from the point of
			 * view of relation 1 as: the known selectivity for matched
			 * MCVs, plus unmatched MCVs that are assumed to match against
			 * random members of relation 2's non-MCV population, plus
			 * non-MCV values that are assumed to match against random
			 * members of relation 2's unmatched MCVs plus non-MCV values.
			 */
			totalsel1 = matchprodfreq;
			if (nd2 > nvalues2)
				totalsel1 += unmatchfreq1 * otherfreq2 / (nd2 - nvalues2);
			if (nd2 > nmatches)
				totalsel1 += otherfreq1 * (otherfreq2 + unmatchfreq2) /
					(nd2 - nmatches);
			/* Same estimate from the point of view of relation 2. */
			totalsel2 = matchprodfreq;
			if (nd1 > nvalues1)
				totalsel2 += unmatchfreq2 * otherfreq1 / (nd1 - nvalues1);
			if (nd1 > nmatches)
				totalsel2 += otherfreq2 * (otherfreq1 + unmatchfreq1) /
					(nd1 - nmatches);

			/*
			 * Use the smaller of the two estimates.  This can be
			 * justified in essentially the same terms as given below for
			 * the no-stats case: to a first approximation, we are
			 * estimating from the point of view of the relation with
			 * smaller nd.
			 */
			selec = (totalsel1 < totalsel2) ? totalsel1 : totalsel2;
		}
		else
		{
			/*
			 * We do not have MCV lists for both sides.  Estimate the join
			 * selectivity as
			 * MIN(1/nd1,1/nd2)*(1-nullfrac1)*(1-nullfrac2). This is
			 * plausible if we assume that the join operator is strict and
			 * the non-null values are about equally distributed: a given
			 * non-null tuple of rel1 will join to either zero or
			 * N2*(1-nullfrac2)/nd2 rows of rel2, so total join rows are
			 * at most N1*(1-nullfrac1)*N2*(1-nullfrac2)/nd2 giving a join
			 * selectivity of not more than
			 * (1-nullfrac1)*(1-nullfrac2)/nd2. By the same logic it is
			 * not more than (1-nullfrac1)*(1-nullfrac2)/nd1, so the
			 * expression with MIN() is an upper bound.  Using the MIN()
			 * means we estimate from the point of view of the relation
			 * with smaller nd (since the larger nd is determining the
			 * MIN).  It is reasonable to assume that most tuples in this
			 * rel will have join partners, so the bound is probably
			 * reasonably tight and should be taken as-is.
			 *
			 * XXX Can we be smarter if we have an MCV list for just one
			 * side? It seems that if we assume equal distribution for the
			 * other side, we end up with the same answer anyway.
			 */
			double		nullfrac1 = stats1 ? stats1->stanullfrac : 0.0;
			double		nullfrac2 = stats2 ? stats2->stanullfrac : 0.0;

			selec = (1.0 - nullfrac1) * (1.0 - nullfrac2);
			if (nd1 > nd2)
				selec /= nd1;
			else
				selec /= nd2;
		}

		if (have_mcvs1)
			free_attstatsslot(var1->vartype, values1, nvalues1,
							  numbers1, nnumbers1);
		if (have_mcvs2)
			free_attstatsslot(var2->vartype, values2, nvalues2,
							  numbers2, nnumbers2);
		if (HeapTupleIsValid(statsTuple1))
			ReleaseSysCache(statsTuple1);
		if (HeapTupleIsValid(statsTuple2))
			ReleaseSysCache(statsTuple2);
	}

	CLAMP_PROBABILITY(selec);

	PG_RETURN_FLOAT8((float8) selec);
}

/*
 *		neqjoinsel		- Join selectivity of "!="
 */
Datum
neqjoinsel(PG_FUNCTION_ARGS)
{
	Query	   *root = (Query *) PG_GETARG_POINTER(0);
	Oid			operator = PG_GETARG_OID(1);
	List	   *args = (List *) PG_GETARG_POINTER(2);
	JoinType	jointype = (JoinType) PG_GETARG_INT16(3);
	Oid			eqop;
	float8		result;

	/*
	 * We want 1 - eqjoinsel() where the equality operator is the one
	 * associated with this != operator, that is, its negator.
	 */
	eqop = get_negator(operator);
	if (eqop)
	{
		result = DatumGetFloat8(DirectFunctionCall4(eqjoinsel,
													PointerGetDatum(root),
												  ObjectIdGetDatum(eqop),
													PointerGetDatum(args),
											   Int16GetDatum(jointype)));
	}
	else
	{
		/* Use default selectivity (should we raise an error instead?) */
		result = DEFAULT_EQ_SEL;
	}
	result = 1.0 - result;
	PG_RETURN_FLOAT8(result);
}

/*
 *		scalarltjoinsel - Join selectivity of "<" and "<=" for scalars
 */
Datum
scalarltjoinsel(PG_FUNCTION_ARGS)
{
	PG_RETURN_FLOAT8(DEFAULT_INEQ_SEL);
}

/*
 *		scalargtjoinsel - Join selectivity of ">" and ">=" for scalars
 */
Datum
scalargtjoinsel(PG_FUNCTION_ARGS)
{
	PG_RETURN_FLOAT8(DEFAULT_INEQ_SEL);
}

/*
 *		regexeqjoinsel	- Join selectivity of regular-expression pattern match.
 */
Datum
regexeqjoinsel(PG_FUNCTION_ARGS)
{
	PG_RETURN_FLOAT8(DEFAULT_MATCH_SEL);
}

/*
 *		icregexeqjoinsel	- Join selectivity of case-insensitive regex match.
 */
Datum
icregexeqjoinsel(PG_FUNCTION_ARGS)
{
	PG_RETURN_FLOAT8(DEFAULT_MATCH_SEL);
}

/*
 *		likejoinsel			- Join selectivity of LIKE pattern match.
 */
Datum
likejoinsel(PG_FUNCTION_ARGS)
{
	PG_RETURN_FLOAT8(DEFAULT_MATCH_SEL);
}

/*
 *		iclikejoinsel			- Join selectivity of ILIKE pattern match.
 */
Datum
iclikejoinsel(PG_FUNCTION_ARGS)
{
	PG_RETURN_FLOAT8(DEFAULT_MATCH_SEL);
}

/*
 *		regexnejoinsel	- Join selectivity of regex non-match.
 */
Datum
regexnejoinsel(PG_FUNCTION_ARGS)
{
	float8		result;

	result = DatumGetFloat8(regexeqjoinsel(fcinfo));
	result = 1.0 - result;
	PG_RETURN_FLOAT8(result);
}

/*
 *		icregexnejoinsel	- Join selectivity of case-insensitive regex non-match.
 */
Datum
icregexnejoinsel(PG_FUNCTION_ARGS)
{
	float8		result;

	result = DatumGetFloat8(icregexeqjoinsel(fcinfo));
	result = 1.0 - result;
	PG_RETURN_FLOAT8(result);
}

/*
 *		nlikejoinsel		- Join selectivity of LIKE pattern non-match.
 */
Datum
nlikejoinsel(PG_FUNCTION_ARGS)
{
	float8		result;

	result = DatumGetFloat8(likejoinsel(fcinfo));
	result = 1.0 - result;
	PG_RETURN_FLOAT8(result);
}

/*
 *		icnlikejoinsel		- Join selectivity of ILIKE pattern non-match.
 */
Datum
icnlikejoinsel(PG_FUNCTION_ARGS)
{
	float8		result;

	result = DatumGetFloat8(iclikejoinsel(fcinfo));
	result = 1.0 - result;
	PG_RETURN_FLOAT8(result);
}

/*
 * mergejoinscansel			- Scan selectivity of merge join.
 *
 * A merge join will stop as soon as it exhausts either input stream.
 * Therefore, if we can estimate the ranges of both input variables,
 * we can estimate how much of the input will actually be read.  This
 * can have a considerable impact on the cost when using indexscans.
 *
 * clause should be a clause already known to be mergejoinable.
 *
 * *leftscan is set to the fraction of the left-hand variable expected
 * to be scanned (0 to 1), and similarly *rightscan for the right-hand
 * variable.
 */
void
mergejoinscansel(Query *root, Node *clause,
				 Selectivity *leftscan,
				 Selectivity *rightscan)
{
	Var		   *left,
			   *right;
	Oid			lefttype,
				righttype;
	Oid			opno,
				lsortop,
				rsortop,
				ltop,
				gtop,
				leop,
				revgtop,
				revleop;
	Datum		leftmax,
				rightmax;
	double		selec;

	/* Set default results if we can't figure anything out. */
	*leftscan = *rightscan = 1.0;

	/* Deconstruct the merge clause */
	if (!is_opclause(clause))
		return;					/* shouldn't happen */
	opno = ((OpExpr *) clause)->opno;
	left = (Var *) get_leftop((Expr *) clause);
	right = (Var *) get_rightop((Expr *) clause);
	if (!right)
		return;					/* shouldn't happen */

	/* Save the direct input types of the operator */
	lefttype = exprType((Node *) left);
	righttype = exprType((Node *) right);

	/*
	 * Now skip any binary-compatible relabeling; there can only be one
	 * level since constant-expression folder eliminates adjacent
	 * RelabelTypes.
	 */
	if (IsA(left, RelabelType))
		left = (Var *) ((RelabelType *) left)->arg;
	if (IsA(right, RelabelType))
		right = (Var *) ((RelabelType *) right)->arg;

	/* Can't do anything if inputs are not Vars */
	if (!IsA(left, Var) ||
		!IsA(right, Var))
		return;

	/* Verify mergejoinability and get left and right "<" operators */
	if (!op_mergejoinable(opno,
						  &lsortop,
						  &rsortop))
		return;					/* shouldn't happen */

	/* Try to get maximum values of both vars */
	if (!get_var_maximum(root, left, lsortop, &leftmax))
		return;					/* no max available from stats */

	if (!get_var_maximum(root, right, rsortop, &rightmax))
		return;					/* no max available from stats */

	/* Look up the "left < right" and "left > right" operators */
	op_mergejoin_crossops(opno, &ltop, &gtop, NULL, NULL);

	/* Look up the "left <= right" operator */
	leop = get_negator(gtop);
	if (!OidIsValid(leop))
		return;					/* insufficient info in catalogs */

	/* Look up the "right > left" operator */
	revgtop = get_commutator(ltop);
	if (!OidIsValid(revgtop))
		return;					/* insufficient info in catalogs */

	/* Look up the "right <= left" operator */
	revleop = get_negator(revgtop);
	if (!OidIsValid(revleop))
		return;					/* insufficient info in catalogs */

	/*
	 * Now, the fraction of the left variable that will be scanned is the
	 * fraction that's <= the right-side maximum value.  But only believe
	 * non-default estimates, else stick with our 1.0.
	 */
	selec = scalarineqsel(root, leop, false, left,
						  rightmax, righttype);
	if (selec != DEFAULT_INEQ_SEL)
		*leftscan = selec;

	/* And similarly for the right variable. */
	selec = scalarineqsel(root, revleop, false, right,
						  leftmax, lefttype);
	if (selec != DEFAULT_INEQ_SEL)
		*rightscan = selec;

	/*
	 * Only one of the two fractions can really be less than 1.0; believe
	 * the smaller estimate and reset the other one to exactly 1.0.  If we
	 * get exactly equal estimates (as can easily happen with self-joins),
	 * believe neither.
	 */
	if (*leftscan > *rightscan)
		*leftscan = 1.0;
	else if (*leftscan < *rightscan)
		*rightscan = 1.0;
	else
		*leftscan = *rightscan = 1.0;
}

/*
 * estimate_num_groups		- Estimate number of groups in a grouped query
 *
 * Given a query having a GROUP BY clause, estimate how many groups there
 * will be --- ie, the number of distinct combinations of the GROUP BY
 * expressions.
 *
 * This routine is also used to estimate the number of rows emitted by
 * a DISTINCT filtering step; that is an isomorphic problem.  (Note:
 * actually, we only use it for DISTINCT when there's no grouping or
 * aggregation ahead of the DISTINCT.)
 *
 * Inputs:
 *	root - the query
 *	groupExprs - list of expressions being grouped by
 *	input_rows - number of rows estimated to arrive at the group/unique
 *		filter step
 *
 * Given the lack of any cross-correlation statistics in the system, it's
 * impossible to do anything really trustworthy with GROUP BY conditions
 * involving multiple Vars.  We should however avoid assuming the worst
 * case (all possible cross-product terms actually appear as groups) since
 * very often the grouped-by Vars are highly correlated.  Our current approach
 * is as follows:
 *	1.	Reduce the given expressions to a list of unique Vars used.  For
 *		example, GROUP BY a, a + b is treated the same as GROUP BY a, b.
 *		It is clearly correct not to count the same Var more than once.
 *		It is also reasonable to treat f(x) the same as x: f() cannot
 *		increase the number of distinct values (unless it is volatile,
 *		which we consider unlikely for grouping), but it probably won't
 *		reduce the number of distinct values much either.
 *	2.	If the list contains Vars of different relations that are known equal
 *		due to equijoin clauses, then drop all but one of the Vars from each
 *		known-equal set, keeping the one with smallest estimated # of values
 *		(since the extra values of the others can't appear in joined rows).
 *		Note the reason we only consider Vars of different relations is that
 *		if we considered ones of the same rel, we'd be double-counting the
 *		restriction selectivity of the equality in the next step.
 *	3.	For Vars within a single source rel, we multiply together the numbers
 *		of values, clamp to the number of rows in the rel, and then multiply
 *		by the selectivity of the restriction clauses for that rel.  The
 *		initial product is probably too high (it's the worst case) but since
 *		we can clamp to the rel's rows it won't be hugely bad.	Multiplying
 *		by the restriction selectivity is effectively assuming that the
 *		restriction clauses are independent of the grouping, which is a crummy
 *		assumption, but it's hard to do better.
 *	4.	If there are Vars from multiple rels, we repeat step 3 for each such
 *		rel, and multiply the results together.
 * Note that rels not containing grouped Vars are ignored completely, as are
 * join clauses other than the equijoin clauses used in step 2.  Such rels
 * cannot increase the number of groups, and we assume such clauses do not
 * reduce the number either (somewhat bogus, but we don't have the info to
 * do better).
 */
double
estimate_num_groups(Query *root, List *groupExprs, double input_rows)
{
	List	   *allvars = NIL;
	List	   *varinfos = NIL;
	double		numdistinct;
	List	   *l;
	typedef struct
	{							/* varinfos is a List of these */
		Var		   *var;
		double		ndistinct;
	} MyVarInfo;

	/* We should not be called unless query has GROUP BY (or DISTINCT) */
	Assert(groupExprs != NIL);

	/* Step 1: get the unique Vars used */
	foreach(l, groupExprs)
	{
		Node	   *groupexpr = (Node *) lfirst(l);
		List	   *varshere;

		varshere = pull_var_clause(groupexpr, false);

		/*
		 * If we find any variable-free GROUP BY item, then either it is a
		 * constant (and we can ignore it) or it contains a volatile
		 * function; in the latter case we punt and assume that each input
		 * row will yield a distinct group.
		 */
		if (varshere == NIL)
		{
			if (contain_volatile_functions(groupexpr))
				return input_rows;
			continue;
		}
		allvars = nconc(allvars, varshere);
	}

	/* If now no Vars, we must have an all-constant GROUP BY list. */
	if (allvars == NIL)
		return 1.0;

	/* Use set_union() to discard duplicates */
	allvars = set_union(NIL, allvars);

	/*
	 * Step 2: acquire statistical estimate of number of distinct values
	 * of each Var (total in its table, without regard for filtering).
	 * Also, detect known-equal Vars and discard the ones we don't want.
	 */
	foreach(l, allvars)
	{
		Var		   *var = (Var *) lfirst(l);
		Oid			relid = getrelid(var->varno, root->rtable);
		HeapTuple	statsTuple = NULL;
		Form_pg_statistic stats = NULL;
		double		ndistinct;
		bool		keep = true;
		List	   *l2;

		if (OidIsValid(relid))
		{
			statsTuple = SearchSysCache(STATRELATT,
										ObjectIdGetDatum(relid),
										Int16GetDatum(var->varattno),
										0, 0);
			if (HeapTupleIsValid(statsTuple))
				stats = (Form_pg_statistic) GETSTRUCT(statsTuple);
		}
		ndistinct = get_att_numdistinct(root, var, stats);
		if (HeapTupleIsValid(statsTuple))
			ReleaseSysCache(statsTuple);

		/* cannot use foreach here because of possible lremove */
		l2 = varinfos;
		while (l2)
		{
			MyVarInfo  *varinfo = (MyVarInfo *) lfirst(l2);

			/* must advance l2 before lremove possibly pfree's it */
			l2 = lnext(l2);

			if (var->varno != varinfo->var->varno &&
			exprs_known_equal(root, (Node *) var, (Node *) varinfo->var))
			{
				/* Found a match */
				if (varinfo->ndistinct <= ndistinct)
				{
					/* Keep older item, forget new one */
					keep = false;
					break;
				}
				else
				{
					/* Delete the older item */
					varinfos = lremove(varinfo, varinfos);
				}
			}
		}

		if (keep)
		{
			MyVarInfo  *varinfo = (MyVarInfo *) palloc(sizeof(MyVarInfo));

			varinfo->var = var;
			varinfo->ndistinct = ndistinct;
			varinfos = lcons(varinfo, varinfos);
		}
	}

	/*
	 * Steps 3/4: group Vars by relation and estimate total numdistinct.
	 *
	 * For each iteration of the outer loop, we process the frontmost Var in
	 * varinfos, plus all other Vars in the same relation.	We remove
	 * these Vars from the newvarinfos list for the next iteration. This
	 * is the easiest way to group Vars of same rel together.
	 */
	Assert(varinfos != NIL);
	numdistinct = 1.0;

	do
	{
		MyVarInfo  *varinfo1 = (MyVarInfo *) lfirst(varinfos);
		RelOptInfo *rel = find_base_rel(root, varinfo1->var->varno);
		double		reldistinct = varinfo1->ndistinct;
		List	   *newvarinfos = NIL;

		/*
		 * Get the largest numdistinct estimate of the Vars for this rel.
		 * Also, construct new varinfos list of remaining Vars.
		 */
		foreach(l, lnext(varinfos))
		{
			MyVarInfo  *varinfo2 = (MyVarInfo *) lfirst(l);

			if (varinfo2->var->varno == varinfo1->var->varno)
				reldistinct *= varinfo2->ndistinct;
			else
			{
				/* not time to process varinfo2 yet */
				newvarinfos = lcons(varinfo2, newvarinfos);
			}
		}

		/*
		 * Sanity check --- don't divide by zero if empty relation.
		 */
		Assert(rel->reloptkind == RELOPT_BASEREL);
		if (rel->tuples > 0)
		{
			/*
			 * Clamp to size of rel, multiply by restriction selectivity.
			 */
			if (reldistinct > rel->tuples)
				reldistinct = rel->tuples;
			reldistinct *= rel->rows / rel->tuples;

			/*
			 * Update estimate of total distinct groups.
			 */
			numdistinct *= reldistinct;
		}

		varinfos = newvarinfos;
	} while (varinfos != NIL);

	numdistinct = ceil(numdistinct);

	/* Guard against out-of-range answers */
	if (numdistinct > input_rows)
		numdistinct = input_rows;
	if (numdistinct < 1.0)
		numdistinct = 1.0;

	return numdistinct;
}


/*-------------------------------------------------------------------------
 *
 * Support routines
 *
 *-------------------------------------------------------------------------
 */

/*
 * get_var_maximum
 *		Estimate the maximum value of the specified variable.
 *		If successful, store value in *max and return TRUE.
 *		If no data available, return FALSE.
 *
 * sortop is the "<" comparison operator to use.  (To extract the
 * minimum instead of the maximum, just pass the ">" operator instead.)
 */
static bool
get_var_maximum(Query *root, Var *var, Oid sortop, Datum *max)
{
	Datum		tmax = 0;
	bool		have_max = false;
	Oid			relid;
	HeapTuple	statsTuple;
	Form_pg_statistic stats;
	int16		typLen;
	bool		typByVal;
	Datum	   *values;
	int			nvalues;
	int			i;

	relid = getrelid(var->varno, root->rtable);
	if (relid == InvalidOid)
		return false;

	/* get stats for the attribute */
	statsTuple = SearchSysCache(STATRELATT,
								ObjectIdGetDatum(relid),
								Int16GetDatum(var->varattno),
								0, 0);
	if (!HeapTupleIsValid(statsTuple))
	{
		/* no stats available, so default result */
		return false;
	}
	stats = (Form_pg_statistic) GETSTRUCT(statsTuple);

	get_typlenbyval(var->vartype, &typLen, &typByVal);

	/*
	 * If there is a histogram, grab the last or first value as
	 * appropriate.
	 *
	 * If there is a histogram that is sorted with some other operator than
	 * the one we want, fail --- this suggests that there is data we can't
	 * use.
	 */
	if (get_attstatsslot(statsTuple, var->vartype, var->vartypmod,
						 STATISTIC_KIND_HISTOGRAM, sortop,
						 &values, &nvalues,
						 NULL, NULL))
	{
		if (nvalues > 0)
		{
			tmax = datumCopy(values[nvalues - 1], typByVal, typLen);
			have_max = true;
		}
		free_attstatsslot(var->vartype, values, nvalues, NULL, 0);
	}
	else
	{
		Oid			rsortop = get_commutator(sortop);

		if (OidIsValid(rsortop) &&
			get_attstatsslot(statsTuple, var->vartype, var->vartypmod,
							 STATISTIC_KIND_HISTOGRAM, rsortop,
							 &values, &nvalues,
							 NULL, NULL))
		{
			if (nvalues > 0)
			{
				tmax = datumCopy(values[0], typByVal, typLen);
				have_max = true;
			}
			free_attstatsslot(var->vartype, values, nvalues, NULL, 0);
		}
		else if (get_attstatsslot(statsTuple, var->vartype, var->vartypmod,
								  STATISTIC_KIND_HISTOGRAM, InvalidOid,
								  &values, &nvalues,
								  NULL, NULL))
		{
			free_attstatsslot(var->vartype, values, nvalues, NULL, 0);
			ReleaseSysCache(statsTuple);
			return false;
		}
	}

	/*
	 * If we have most-common-values info, look for a large MCV.  This is
	 * needed even if we also have a histogram, since the histogram
	 * excludes the MCVs.  However, usually the MCVs will not be the
	 * extreme values, so avoid unnecessary data copying.
	 */
	if (get_attstatsslot(statsTuple, var->vartype, var->vartypmod,
						 STATISTIC_KIND_MCV, InvalidOid,
						 &values, &nvalues,
						 NULL, NULL))
	{
		bool		large_mcv = false;
		FmgrInfo	opproc;

		fmgr_info(get_opcode(sortop), &opproc);

		for (i = 0; i < nvalues; i++)
		{
			if (!have_max)
			{
				tmax = values[i];
				large_mcv = have_max = true;
			}
			else if (DatumGetBool(FunctionCall2(&opproc, tmax, values[i])))
			{
				tmax = values[i];
				large_mcv = true;
			}
		}
		if (large_mcv)
			tmax = datumCopy(tmax, typByVal, typLen);
		free_attstatsslot(var->vartype, values, nvalues, NULL, 0);
	}

	ReleaseSysCache(statsTuple);

	*max = tmax;
	return have_max;
}

/*
 * convert_to_scalar
 *	  Convert non-NULL values of the indicated types to the comparison
 *	  scale needed by scalarltsel()/scalargtsel().
 *	  Returns "true" if successful.
 *
 * XXX this routine is a hack: ideally we should look up the conversion
 * subroutines in pg_type.
 *
 * All numeric datatypes are simply converted to their equivalent
 * "double" values.  (NUMERIC values that are outside the range of "double"
 * are clamped to +/- HUGE_VAL.)
 *
 * String datatypes are converted by convert_string_to_scalar(),
 * which is explained below.  The reason why this routine deals with
 * three values at a time, not just one, is that we need it for strings.
 *
 * The bytea datatype is just enough different from strings that it has
 * to be treated separately.
 *
 * The several datatypes representing absolute times are all converted
 * to Timestamp, which is actually a double, and then we just use that
 * double value.  Note this will give correct results even for the "special"
 * values of Timestamp, since those are chosen to compare correctly;
 * see timestamp_cmp.
 *
 * The several datatypes representing relative times (intervals) are all
 * converted to measurements expressed in seconds.
 */
static bool
convert_to_scalar(Datum value, Oid valuetypid, double *scaledvalue,
				  Datum lobound, Datum hibound, Oid boundstypid,
				  double *scaledlobound, double *scaledhibound)
{
	/*
	 * In present usage, we can assume that the valuetypid exactly matches
	 * the declared input type of the operator we are invoked for (because
	 * constant-folding will ensure that any Const passed to the operator
	 * has been reduced to the correct type).  However, the boundstypid is
	 * the type of some variable that might be only binary-compatible with
	 * the declared type; in particular it might be a domain type.	Must
	 * fold the variable type down to base type so we can recognize it.
	 * (But we can skip that lookup if the variable type matches the
	 * const.)
	 */
	if (boundstypid != valuetypid)
		boundstypid = getBaseType(boundstypid);

	switch (valuetypid)
	{
			/*
			 * Built-in numeric types
			 */
		case BOOLOID:
		case INT2OID:
		case INT4OID:
		case INT8OID:
		case FLOAT4OID:
		case FLOAT8OID:
		case NUMERICOID:
		case OIDOID:
		case REGPROCOID:
		case REGPROCEDUREOID:
		case REGOPEROID:
		case REGOPERATOROID:
		case REGCLASSOID:
		case REGTYPEOID:
			*scaledvalue = convert_numeric_to_scalar(value, valuetypid);
			*scaledlobound = convert_numeric_to_scalar(lobound, boundstypid);
			*scaledhibound = convert_numeric_to_scalar(hibound, boundstypid);
			return true;

			/*
			 * Built-in string types
			 */
		case CHAROID:
		case BPCHAROID:
		case VARCHAROID:
		case TEXTOID:
		case NAMEOID:
			{
				unsigned char *valstr = convert_string_datum(value, valuetypid);
				unsigned char *lostr = convert_string_datum(lobound, boundstypid);
				unsigned char *histr = convert_string_datum(hibound, boundstypid);

				convert_string_to_scalar(valstr, scaledvalue,
										 lostr, scaledlobound,
										 histr, scaledhibound);
				pfree(valstr);
				pfree(lostr);
				pfree(histr);
				return true;
			}

			/*
			 * Built-in bytea type
			 */
		case BYTEAOID:
			{
				convert_bytea_to_scalar(value, scaledvalue,
										lobound, scaledlobound,
										hibound, scaledhibound);
				return true;
			}

			/*
			 * Built-in time types
			 */
		case TIMESTAMPOID:
		case TIMESTAMPTZOID:
		case ABSTIMEOID:
		case DATEOID:
		case INTERVALOID:
		case RELTIMEOID:
		case TINTERVALOID:
		case TIMEOID:
		case TIMETZOID:
			*scaledvalue = convert_timevalue_to_scalar(value, valuetypid);
			*scaledlobound = convert_timevalue_to_scalar(lobound, boundstypid);
			*scaledhibound = convert_timevalue_to_scalar(hibound, boundstypid);
			return true;

			/*
			 * Built-in network types
			 */
		case INETOID:
		case CIDROID:
		case MACADDROID:
			*scaledvalue = convert_network_to_scalar(value, valuetypid);
			*scaledlobound = convert_network_to_scalar(lobound, boundstypid);
			*scaledhibound = convert_network_to_scalar(hibound, boundstypid);
			return true;
	}
	/* Don't know how to convert */
	return false;
}

/*
 * Do convert_to_scalar()'s work for any numeric data type.
 */
static double
convert_numeric_to_scalar(Datum value, Oid typid)
{
	switch (typid)
	{
		case BOOLOID:
			return (double) DatumGetBool(value);
		case INT2OID:
			return (double) DatumGetInt16(value);
		case INT4OID:
			return (double) DatumGetInt32(value);
		case INT8OID:
			return (double) DatumGetInt64(value);
		case FLOAT4OID:
			return (double) DatumGetFloat4(value);
		case FLOAT8OID:
			return (double) DatumGetFloat8(value);
		case NUMERICOID:
			/* Note: out-of-range values will be clamped to +-HUGE_VAL */
			return (double)
				DatumGetFloat8(DirectFunctionCall1(numeric_float8_no_overflow,
												   value));
		case OIDOID:
		case REGPROCOID:
		case REGPROCEDUREOID:
		case REGOPEROID:
		case REGOPERATOROID:
		case REGCLASSOID:
		case REGTYPEOID:
			/* we can treat OIDs as integers... */
			return (double) DatumGetObjectId(value);
	}

	/*
	 * Can't get here unless someone tries to use scalarltsel/scalargtsel
	 * on an operator with one numeric and one non-numeric operand.
	 */
	elog(ERROR, "unsupported type: %u", typid);
	return 0;
}

/*
 * Do convert_to_scalar()'s work for any character-string data type.
 *
 * String datatypes are converted to a scale that ranges from 0 to 1,
 * where we visualize the bytes of the string as fractional digits.
 *
 * We do not want the base to be 256, however, since that tends to
 * generate inflated selectivity estimates; few databases will have
 * occurrences of all 256 possible byte values at each position.
 * Instead, use the smallest and largest byte values seen in the bounds
 * as the estimated range for each byte, after some fudging to deal with
 * the fact that we probably aren't going to see the full range that way.
 *
 * An additional refinement is that we discard any common prefix of the
 * three strings before computing the scaled values.  This allows us to
 * "zoom in" when we encounter a narrow data range.  An example is a phone
 * number database where all the values begin with the same area code.
 * (Actually, the bounds will be adjacent histogram-bin-boundary values,
 * so this is more likely to happen than you might think.)
 */
static void
convert_string_to_scalar(unsigned char *value,
						 double *scaledvalue,
						 unsigned char *lobound,
						 double *scaledlobound,
						 unsigned char *hibound,
						 double *scaledhibound)
{
	int			rangelo,
				rangehi;
	unsigned char *sptr;

	rangelo = rangehi = hibound[0];
	for (sptr = lobound; *sptr; sptr++)
	{
		if (rangelo > *sptr)
			rangelo = *sptr;
		if (rangehi < *sptr)
			rangehi = *sptr;
	}
	for (sptr = hibound; *sptr; sptr++)
	{
		if (rangelo > *sptr)
			rangelo = *sptr;
		if (rangehi < *sptr)
			rangehi = *sptr;
	}
	/* If range includes any upper-case ASCII chars, make it include all */
	if (rangelo <= 'Z' && rangehi >= 'A')
	{
		if (rangelo > 'A')
			rangelo = 'A';
		if (rangehi < 'Z')
			rangehi = 'Z';
	}
	/* Ditto lower-case */
	if (rangelo <= 'z' && rangehi >= 'a')
	{
		if (rangelo > 'a')
			rangelo = 'a';
		if (rangehi < 'z')
			rangehi = 'z';
	}
	/* Ditto digits */
	if (rangelo <= '9' && rangehi >= '0')
	{
		if (rangelo > '0')
			rangelo = '0';
		if (rangehi < '9')
			rangehi = '9';
	}

	/*
	 * If range includes less than 10 chars, assume we have not got enough
	 * data, and make it include regular ASCII set.
	 */
	if (rangehi - rangelo < 9)
	{
		rangelo = ' ';
		rangehi = 127;
	}

	/*
	 * Now strip any common prefix of the three strings.
	 */
	while (*lobound)
	{
		if (*lobound != *hibound || *lobound != *value)
			break;
		lobound++, hibound++, value++;
	}

	/*
	 * Now we can do the conversions.
	 */
	*scaledvalue = convert_one_string_to_scalar(value, rangelo, rangehi);
	*scaledlobound = convert_one_string_to_scalar(lobound, rangelo, rangehi);
	*scaledhibound = convert_one_string_to_scalar(hibound, rangelo, rangehi);
}

static double
convert_one_string_to_scalar(unsigned char *value, int rangelo, int rangehi)
{
	int			slen = strlen((char *) value);
	double		num,
				denom,
				base;

	if (slen <= 0)
		return 0.0;				/* empty string has scalar value 0 */

	/*
	 * Since base is at least 10, need not consider more than about 20
	 * chars
	 */
	if (slen > 20)
		slen = 20;

	/* Convert initial characters to fraction */
	base = rangehi - rangelo + 1;
	num = 0.0;
	denom = base;
	while (slen-- > 0)
	{
		int			ch = *value++;

		if (ch < rangelo)
			ch = rangelo - 1;
		else if (ch > rangehi)
			ch = rangehi + 1;
		num += ((double) (ch - rangelo)) / denom;
		denom *= base;
	}

	return num;
}

/*
 * Convert a string-type Datum into a palloc'd, null-terminated string.
 *
 * When using a non-C locale, we must pass the string through strxfrm()
 * before continuing, so as to generate correct locale-specific results.
 */
static unsigned char *
convert_string_datum(Datum value, Oid typid)
{
	char	   *val;

	switch (typid)
	{
		case CHAROID:
			val = (char *) palloc(2);
			val[0] = DatumGetChar(value);
			val[1] = '\0';
			break;
		case BPCHAROID:
		case VARCHAROID:
		case TEXTOID:
			{
				char	   *str = (char *) VARDATA(DatumGetPointer(value));
				int			strlength = VARSIZE(DatumGetPointer(value)) - VARHDRSZ;

				val = (char *) palloc(strlength + 1);
				memcpy(val, str, strlength);
				val[strlength] = '\0';
				break;
			}
		case NAMEOID:
			{
				NameData   *nm = (NameData *) DatumGetPointer(value);

				val = pstrdup(NameStr(*nm));
				break;
			}
		default:

			/*
			 * Can't get here unless someone tries to use scalarltsel on
			 * an operator with one string and one non-string operand.
			 */
			elog(ERROR, "unsupported type: %u", typid);
			return NULL;
	}

	if (!lc_collate_is_c())
	{
		char	   *xfrmstr;
		size_t		xfrmlen;
		size_t		xfrmlen2;

		/*
		 * Note: originally we guessed at a suitable output buffer size,
		 * and only needed to call strxfrm twice if our guess was too
		 * small. However, it seems that some versions of Solaris have
		 * buggy strxfrm that can write past the specified buffer length
		 * in that scenario.  So, do it the dumb way for portability.
		 *
		 * Yet other systems (e.g., glibc) sometimes return a smaller value
		 * from the second call than the first; thus the Assert must be <=
		 * not == as you'd expect.  Can't any of these people program
		 * their way out of a paper bag?
		 */
		xfrmlen = strxfrm(NULL, val, 0);
		xfrmstr = (char *) palloc(xfrmlen + 1);
		xfrmlen2 = strxfrm(xfrmstr, val, xfrmlen + 1);
		Assert(xfrmlen2 <= xfrmlen);
		pfree(val);
		val = xfrmstr;
	}

	return (unsigned char *) val;
}

/*
 * Do convert_to_scalar()'s work for any bytea data type.
 *
 * Very similar to convert_string_to_scalar except we can't assume
 * null-termination and therefore pass explicit lengths around.
 *
 * Also, assumptions about likely "normal" ranges of characters have been
 * removed - a data range of 0..255 is always used, for now.  (Perhaps
 * someday we will add information about actual byte data range to
 * pg_statistic.)
 */
static void
convert_bytea_to_scalar(Datum value,
						double *scaledvalue,
						Datum lobound,
						double *scaledlobound,
						Datum hibound,
						double *scaledhibound)
{
	int			rangelo,
				rangehi,
				valuelen = VARSIZE(DatumGetPointer(value)) - VARHDRSZ,
				loboundlen = VARSIZE(DatumGetPointer(lobound)) - VARHDRSZ,
				hiboundlen = VARSIZE(DatumGetPointer(hibound)) - VARHDRSZ,
				i,
				minlen;
	unsigned char *valstr = (unsigned char *) VARDATA(DatumGetPointer(value)),
			   *lostr = (unsigned char *) VARDATA(DatumGetPointer(lobound)),
			   *histr = (unsigned char *) VARDATA(DatumGetPointer(hibound));

	/*
	 * Assume bytea data is uniformly distributed across all byte values.
	 */
	rangelo = 0;
	rangehi = 255;

	/*
	 * Now strip any common prefix of the three strings.
	 */
	minlen = Min(Min(valuelen, loboundlen), hiboundlen);
	for (i = 0; i < minlen; i++)
	{
		if (*lostr != *histr || *lostr != *valstr)
			break;
		lostr++, histr++, valstr++;
		loboundlen--, hiboundlen--, valuelen--;
	}

	/*
	 * Now we can do the conversions.
	 */
	*scaledvalue = convert_one_bytea_to_scalar(valstr, valuelen, rangelo, rangehi);
	*scaledlobound = convert_one_bytea_to_scalar(lostr, loboundlen, rangelo, rangehi);
	*scaledhibound = convert_one_bytea_to_scalar(histr, hiboundlen, rangelo, rangehi);
}

static double
convert_one_bytea_to_scalar(unsigned char *value, int valuelen,
							int rangelo, int rangehi)
{
	double		num,
				denom,
				base;

	if (valuelen <= 0)
		return 0.0;				/* empty string has scalar value 0 */

	/*
	 * Since base is 256, need not consider more than about 10 chars (even
	 * this many seems like overkill)
	 */
	if (valuelen > 10)
		valuelen = 10;

	/* Convert initial characters to fraction */
	base = rangehi - rangelo + 1;
	num = 0.0;
	denom = base;
	while (valuelen-- > 0)
	{
		int			ch = *value++;

		if (ch < rangelo)
			ch = rangelo - 1;
		else if (ch > rangehi)
			ch = rangehi + 1;
		num += ((double) (ch - rangelo)) / denom;
		denom *= base;
	}

	return num;
}

/*
 * Do convert_to_scalar()'s work for any timevalue data type.
 */
static double
convert_timevalue_to_scalar(Datum value, Oid typid)
{
	switch (typid)
	{
		case TIMESTAMPOID:
			return DatumGetTimestamp(value);
		case TIMESTAMPTZOID:
			return DatumGetTimestampTz(value);
		case ABSTIMEOID:
			return DatumGetTimestamp(DirectFunctionCall1(abstime_timestamp,
														 value));
		case DATEOID:
			return DatumGetTimestamp(DirectFunctionCall1(date_timestamp,
														 value));
		case INTERVALOID:
			{
				Interval   *interval = DatumGetIntervalP(value);

				/*
				 * Convert the month part of Interval to days using
				 * assumed average month length of 365.25/12.0 days.  Not
				 * too accurate, but plenty good enough for our purposes.
				 */
#ifdef HAVE_INT64_TIMESTAMP
				return (interval->time + (interval->month * ((365.25 / 12.0) * 86400000000.0)));
#else
				return interval->time +
					interval->month * (365.25 / 12.0 * 24.0 * 60.0 * 60.0);
#endif
			}
		case RELTIMEOID:
#ifdef HAVE_INT64_TIMESTAMP
			return (DatumGetRelativeTime(value) * 1000000.0);
#else
			return DatumGetRelativeTime(value);
#endif
		case TINTERVALOID:
			{
				TimeInterval interval = DatumGetTimeInterval(value);

#ifdef HAVE_INT64_TIMESTAMP
				if (interval->status != 0)
					return ((interval->data[1] - interval->data[0]) * 1000000.0);
#else
				if (interval->status != 0)
					return interval->data[1] - interval->data[0];
#endif
				return 0;		/* for lack of a better idea */
			}
		case TIMEOID:
			return DatumGetTimeADT(value);
		case TIMETZOID:
			{
				TimeTzADT  *timetz = DatumGetTimeTzADTP(value);

				/* use GMT-equivalent time */
#ifdef HAVE_INT64_TIMESTAMP
				return (double) (timetz->time + (timetz->zone * 1000000.0));
#else
				return (double) (timetz->time + timetz->zone);
#endif
			}
	}

	/*
	 * Can't get here unless someone tries to use scalarltsel/scalargtsel
	 * on an operator with one timevalue and one non-timevalue operand.
	 */
	elog(ERROR, "unsupported type: %u", typid);
	return 0;
}


/*
 * get_att_numdistinct
 *	  Estimate the number of distinct values of an attribute.
 *
 * var: identifies the attribute to examine.
 * stats: pg_statistic tuple for attribute, or NULL if not available.
 *
 * NB: be careful to produce an integral result, since callers may compare
 * the result to exact integer counts.
 */
static double
get_att_numdistinct(Query *root, Var *var, Form_pg_statistic stats)
{
	RelOptInfo *rel;
	double		ntuples;

	/*
	 * Special-case boolean columns: presumably, two distinct values.
	 *
	 * Are there any other cases we should wire in special estimates for?
	 */
	if (var->vartype == BOOLOID)
		return 2.0;

	/*
	 * Otherwise we need to get the relation size.
	 */
	rel = find_base_rel(root, var->varno);
	ntuples = rel->tuples;

	if (ntuples <= 0.0)
		return DEFAULT_NUM_DISTINCT;	/* no data available; return a
										 * default */

	/*
	 * Look to see if there is a unique index on the attribute. If so, we
	 * assume it's distinct, ignoring pg_statistic info which could be out
	 * of date.
	 */
	if (has_unique_index(rel, var->varattno))
		return ntuples;

	/*
	 * If ANALYZE determined a fixed or scaled estimate, use it.
	 */
	if (stats)
	{
		if (stats->stadistinct > 0.0)
			return stats->stadistinct;
		if (stats->stadistinct < 0.0)
			return floor((-stats->stadistinct * ntuples) + 0.5);
	}

	/*
	 * ANALYZE does not compute stats for system attributes, but some of
	 * them can reasonably be assumed unique anyway.
	 */
	switch (var->varattno)
	{
		case ObjectIdAttributeNumber:
		case SelfItemPointerAttributeNumber:
			return ntuples;
		case TableOidAttributeNumber:
			return 1.0;
	}

	/*
	 * Estimate ndistinct = ntuples if the table is small, else use
	 * default.
	 */
	if (ntuples < DEFAULT_NUM_DISTINCT)
		return ntuples;

	return DEFAULT_NUM_DISTINCT;
}

/*
 * get_restriction_var
 *		Examine the args of a restriction clause to see if it's of the
 *		form (var op something) or (something op var).	If so, extract
 *		and return the var and the other argument.
 *
 * Inputs:
 *	args: clause argument list
 *	varRelid: see specs for restriction selectivity functions
 *
 * Outputs: (these are set only if TRUE is returned)
 *	*var: gets Var node
 *	*other: gets other clause argument
 *	*varonleft: set TRUE if var is on the left, FALSE if on the right
 *
 * Returns TRUE if a Var is identified, otherwise FALSE.
 */
static bool
get_restriction_var(List *args,
					int varRelid,
					Var **var,
					Node **other,
					bool *varonleft)
{
	Node	   *left,
			   *right;

	if (length(args) != 2)
		return false;

	left = (Node *) lfirst(args);
	right = (Node *) lsecond(args);

	/* Ignore any binary-compatible relabeling */

	if (IsA(left, RelabelType))
		left = (Node *) ((RelabelType *) left)->arg;
	if (IsA(right, RelabelType))
		right = (Node *) ((RelabelType *) right)->arg;

	/* Look for the var */

	if (IsA(left, Var) &&
		(varRelid == 0 || varRelid == ((Var *) left)->varno))
	{
		*var = (Var *) left;
		*other = right;
		*varonleft = true;
	}
	else if (IsA(right, Var) &&
			 (varRelid == 0 || varRelid == ((Var *) right)->varno))
	{
		*var = (Var *) right;
		*other = left;
		*varonleft = false;
	}
	else
	{
		/* Duh, it's too complicated for me... */
		return false;
	}

	return true;
}

/*
 * get_join_vars
 *
 * Extract the two Vars from a join clause's argument list.  Returns
 * NULL for arguments that are not simple vars.
 */
static void
get_join_vars(List *args, Var **var1, Var **var2)
{
	Node	   *left,
			   *right;

	if (length(args) != 2)
	{
		*var1 = NULL;
		*var2 = NULL;
		return;
	}

	left = (Node *) lfirst(args);
	right = (Node *) lsecond(args);

	/* Ignore any binary-compatible relabeling */
	if (IsA(left, RelabelType))
		left = (Node *) ((RelabelType *) left)->arg;
	if (IsA(right, RelabelType))
		right = (Node *) ((RelabelType *) right)->arg;

	if (IsA(left, Var))
		*var1 = (Var *) left;
	else
		*var1 = NULL;

	if (IsA(right, Var))
		*var2 = (Var *) right;
	else
		*var2 = NULL;
}

/*-------------------------------------------------------------------------
 *
 * Pattern analysis functions
 *
 * These routines support analysis of LIKE and regular-expression patterns
 * by the planner/optimizer.  It's important that they agree with the
 * regular-expression code in backend/regex/ and the LIKE code in
 * backend/utils/adt/like.c.
 *
 * Note that the prefix-analysis functions are called from
 * backend/optimizer/path/indxpath.c as well as from routines in this file.
 *
 *-------------------------------------------------------------------------
 */

/*
 * Extract the fixed prefix, if any, for a pattern.
 *
 * *prefix is set to a palloc'd prefix string (in the form of a Const node),
 *	or to NULL if no fixed prefix exists for the pattern.
 * *rest is set to a palloc'd Const representing the remainder of the pattern
 *	after the portion describing the fixed prefix.
 * Each of these has the same type (TEXT or BYTEA) as the given pattern Const.
 *
 * The return value distinguishes no fixed prefix, a partial prefix,
 * or an exact-match-only pattern.
 */

static Pattern_Prefix_Status
like_fixed_prefix(Const *patt_const, bool case_insensitive,
				  Const **prefix_const, Const **rest_const)
{
	char	   *match;
	char	   *patt;
	int			pattlen;
	char	   *rest;
	Oid			typeid = patt_const->consttype;
	int			pos,
				match_pos;

	/* the right-hand const is type text or bytea */
	Assert(typeid == BYTEAOID || typeid == TEXTOID);

	if (typeid == BYTEAOID && case_insensitive)
		ereport(ERROR,
				(errcode(ERRCODE_FEATURE_NOT_SUPPORTED),
		errmsg("case insensitive matching not supported on type bytea")));

	if (typeid != BYTEAOID)
	{
		patt = DatumGetCString(DirectFunctionCall1(textout, patt_const->constvalue));
		pattlen = strlen(patt);
	}
	else
	{
		patt = DatumGetCString(DirectFunctionCall1(byteaout, patt_const->constvalue));
		pattlen = toast_raw_datum_size(patt_const->constvalue) - VARHDRSZ;
	}

	match = palloc(pattlen + 1);
	match_pos = 0;

	for (pos = 0; pos < pattlen; pos++)
	{
		/* % and _ are wildcard characters in LIKE */
		if (patt[pos] == '%' ||
			patt[pos] == '_')
			break;
		/* Backslash quotes the next character */
		if (patt[pos] == '\\')
		{
			pos++;
			if (patt[pos] == '\0' && typeid != BYTEAOID)
				break;
		}

		/*
		 * XXX I suspect isalpha() is not an adequately locale-sensitive
		 * test for characters that can vary under case folding?
		 */
		if (case_insensitive && isalpha((unsigned char) patt[pos]))
			break;

		/*
		 * NOTE: this code used to think that %% meant a literal %, but
		 * textlike() itself does not think that, and the SQL92 spec
		 * doesn't say any such thing either.
		 */
		match[match_pos++] = patt[pos];
	}

	match[match_pos] = '\0';
	rest = &patt[pos];

	*prefix_const = string_to_const(match, typeid);
	*rest_const = string_to_const(rest, typeid);

	pfree(patt);
	pfree(match);

	/* in LIKE, an empty pattern is an exact match! */
	if (pos == pattlen)
		return Pattern_Prefix_Exact;	/* reached end of pattern, so
										 * exact */

	if (match_pos > 0)
		return Pattern_Prefix_Partial;

	return Pattern_Prefix_None;
}

static Pattern_Prefix_Status
regex_fixed_prefix(Const *patt_const, bool case_insensitive,
				   Const **prefix_const, Const **rest_const)
{
	char	   *match;
	int			pos,
				match_pos,
				paren_depth;
	char	   *patt;
	char	   *rest;
	Oid			typeid = patt_const->consttype;

	/*
	 * Should be unnecessary, there are no bytea regex operators defined.
	 * As such, it should be noted that the rest of this function has *not*
	 * been made safe for binary (possibly NULL containing) strings.
	 */
	if (typeid == BYTEAOID)
		ereport(ERROR,
				(errcode(ERRCODE_FEATURE_NOT_SUPPORTED),
				 errmsg("regular-expression matching not supported on type bytea")));

	/* the right-hand const is type text for all of these */
	patt = DatumGetCString(DirectFunctionCall1(textout, patt_const->constvalue));

	/* Pattern must be anchored left */
	if (patt[0] != '^')
	{
		rest = patt;

		*prefix_const = NULL;
		*rest_const = string_to_const(rest, typeid);

		return Pattern_Prefix_None;
	}

	/*
	 * If unquoted | is present at paren level 0 in pattern, then there
	 * are multiple alternatives for the start of the string.
	 */
	paren_depth = 0;
	for (pos = 1; patt[pos]; pos++)
	{
		if (patt[pos] == '|' && paren_depth == 0)
		{
			rest = patt;

			*prefix_const = NULL;
			*rest_const = string_to_const(rest, typeid);

			return Pattern_Prefix_None;
		}
		else if (patt[pos] == '(')
			paren_depth++;
		else if (patt[pos] == ')' && paren_depth > 0)
			paren_depth--;
		else if (patt[pos] == '\\')
		{
			/* backslash quotes the next character */
			pos++;
			if (patt[pos] == '\0')
				break;
		}
	}

	/* OK, allocate space for pattern */
	match = palloc(strlen(patt) + 1);
	match_pos = 0;

	/* note start at pos 1 to skip leading ^ */
	for (pos = 1; patt[pos]; pos++)
	{
		/*
		 * Check for characters that indicate multiple possible matches
		 * here. XXX I suspect isalpha() is not an adequately
		 * locale-sensitive test for characters that can vary under case
		 * folding?
		 */
		if (patt[pos] == '.' ||
			patt[pos] == '(' ||
			patt[pos] == '[' ||
			patt[pos] == '$' ||
			(case_insensitive && isalpha((unsigned char) patt[pos])))
			break;

		/*
		 * Check for quantifiers.  Except for +, this means the preceding
		 * character is optional, so we must remove it from the prefix
		 * too!
		 */
		if (patt[pos] == '*' ||
			patt[pos] == '?' ||
			patt[pos] == '{')
		{
			if (match_pos > 0)
				match_pos--;
			pos--;
			break;
		}
		if (patt[pos] == '+')
		{
			pos--;
			break;
		}
		if (patt[pos] == '\\')
		{
			/* backslash quotes the next character */
			pos++;
			if (patt[pos] == '\0')
				break;
		}
		match[match_pos++] = patt[pos];
	}

	match[match_pos] = '\0';
	rest = &patt[pos];

	if (patt[pos] == '$' && patt[pos + 1] == '\0')
	{
		rest = &patt[pos + 1];

		*prefix_const = string_to_const(match, typeid);
		*rest_const = string_to_const(rest, typeid);

		pfree(patt);
		pfree(match);

		return Pattern_Prefix_Exact;	/* pattern specifies exact match */
	}

	*prefix_const = string_to_const(match, typeid);
	*rest_const = string_to_const(rest, typeid);

	pfree(patt);
	pfree(match);

	if (match_pos > 0)
		return Pattern_Prefix_Partial;

	return Pattern_Prefix_None;
}

Pattern_Prefix_Status
pattern_fixed_prefix(Const *patt, Pattern_Type ptype,
					 Const **prefix, Const **rest)
{
	Pattern_Prefix_Status result;

	switch (ptype)
	{
		case Pattern_Type_Like:
			result = like_fixed_prefix(patt, false, prefix, rest);
			break;
		case Pattern_Type_Like_IC:
			result = like_fixed_prefix(patt, true, prefix, rest);
			break;
		case Pattern_Type_Regex:
			result = regex_fixed_prefix(patt, false, prefix, rest);
			break;
		case Pattern_Type_Regex_IC:
			result = regex_fixed_prefix(patt, true, prefix, rest);
			break;
		default:
			elog(ERROR, "unrecognized ptype: %d", (int) ptype);
			result = Pattern_Prefix_None;		/* keep compiler quiet */
			break;
	}
	return result;
}

/*
 * Estimate the selectivity of a fixed prefix for a pattern match.
 *
 * A fixed prefix "foo" is estimated as the selectivity of the expression
 * "var >= 'foo' AND var < 'fop'" (see also indxqual.c).
 *
 * We use the >= and < operators from the specified btree opclass to do the
 * estimation.	The given Var and Const must be of the associated datatype.
 *
 * XXX Note: we make use of the upper bound to estimate operator selectivity
 * even if the locale is such that we cannot rely on the upper-bound string.
 * The selectivity only needs to be approximately right anyway, so it seems
 * more useful to use the upper-bound code than not.
 */
static Selectivity
prefix_selectivity(Query *root, Var *var, Oid opclass, Const *prefixcon)
{
	Selectivity prefixsel;
	Oid			cmpopr;
	List	   *cmpargs;
	Const	   *greaterstrcon;

	cmpopr = get_opclass_member(opclass, InvalidOid,
								BTGreaterEqualStrategyNumber);
	if (cmpopr == InvalidOid)
		elog(ERROR, "no >= operator for opclass %u", opclass);
	cmpargs = makeList2(var, prefixcon);
	/* Assume scalargtsel is appropriate for all supported types */
	prefixsel = DatumGetFloat8(DirectFunctionCall4(scalargtsel,
												   PointerGetDatum(root),
												ObjectIdGetDatum(cmpopr),
												PointerGetDatum(cmpargs),
												   Int32GetDatum(0)));

	/*-------
	 * If we can create a string larger than the prefix, say
	 *	"x < greaterstr".
	 *-------
	 */
	greaterstrcon = make_greater_string(prefixcon);
	if (greaterstrcon)
	{
		Selectivity topsel;

		cmpopr = get_opclass_member(opclass, InvalidOid,
									BTLessStrategyNumber);
		if (cmpopr == InvalidOid)
			elog(ERROR, "no < operator for opclass %u", opclass);
		cmpargs = makeList2(var, greaterstrcon);
		/* Assume scalarltsel is appropriate for all supported types */
		topsel = DatumGetFloat8(DirectFunctionCall4(scalarltsel,
													PointerGetDatum(root),
												ObjectIdGetDatum(cmpopr),
												PointerGetDatum(cmpargs),
													Int32GetDatum(0)));

		/*
		 * Merge the two selectivities in the same way as for a range
		 * query (see clauselist_selectivity()).
		 */
		prefixsel = topsel + prefixsel - 1.0;

		/* Adjust for double-exclusion of NULLs */
		prefixsel += nulltestsel(root, IS_NULL, (Node *) var, var->varno);

		/*
		 * A zero or slightly negative prefixsel should be converted into
		 * a small positive value; we probably are dealing with a very
		 * tight range and got a bogus result due to roundoff errors.
		 * However, if prefixsel is very negative, then we probably have
		 * default selectivity estimates on one or both sides of the
		 * range.  In that case, insert a not-so-wildly-optimistic default
		 * estimate.
		 */
		if (prefixsel <= 0.0)
		{
			if (prefixsel < -0.01)
			{
				/*
				 * No data available --- use a default estimate that is
				 * small, but not real small.
				 */
				prefixsel = 0.005;
			}
			else
			{
				/*
				 * It's just roundoff error; use a small positive value
				 */
				prefixsel = 1.0e-10;
			}
		}
	}

	return prefixsel;
}


/*
 * Estimate the selectivity of a pattern of the specified type.
 * Note that any fixed prefix of the pattern will have been removed already.
 *
 * For now, we use a very simplistic approach: fixed characters reduce the
 * selectivity a good deal, character ranges reduce it a little,
 * wildcards (such as % for LIKE or .* for regex) increase it.
 */

#define FIXED_CHAR_SEL	0.20	/* about 1/5 */
#define CHAR_RANGE_SEL	0.25
#define ANY_CHAR_SEL	0.9		/* not 1, since it won't match
								 * end-of-string */
#define FULL_WILDCARD_SEL 5.0
#define PARTIAL_WILDCARD_SEL 2.0

static Selectivity
like_selectivity(Const *patt_const, bool case_insensitive)
{
	Selectivity sel = 1.0;
	int			pos;
	int			start;
	Oid			typeid = patt_const->consttype;
	char	   *patt;
	int			pattlen;

	/* the right-hand const is type text or bytea */
	Assert(typeid == BYTEAOID || typeid == TEXTOID);

	if (typeid == BYTEAOID && case_insensitive)
		ereport(ERROR,
				(errcode(ERRCODE_FEATURE_NOT_SUPPORTED),
		errmsg("case insensitive matching not supported on type bytea")));

	if (typeid != BYTEAOID)
	{
		patt = DatumGetCString(DirectFunctionCall1(textout, patt_const->constvalue));
		pattlen = strlen(patt);
	}
	else
	{
		patt = DatumGetCString(DirectFunctionCall1(byteaout, patt_const->constvalue));
		pattlen = toast_raw_datum_size(patt_const->constvalue) - VARHDRSZ;
	}

	/* Skip any leading %; it's already factored into initial sel */
	start = (*patt == '%') ? 1 : 0;
	for (pos = start; pos < pattlen; pos++)
	{
		/* % and _ are wildcard characters in LIKE */
		if (patt[pos] == '%')
			sel *= FULL_WILDCARD_SEL;
		else if (patt[pos] == '_')
			sel *= ANY_CHAR_SEL;
		else if (patt[pos] == '\\')
		{
			/* Backslash quotes the next character */
			pos++;
			if (patt[pos] == '\0' && typeid != BYTEAOID)
				break;
			sel *= FIXED_CHAR_SEL;
		}
		else
			sel *= FIXED_CHAR_SEL;
	}
	/* Could get sel > 1 if multiple wildcards */
	if (sel > 1.0)
		sel = 1.0;
	return sel;
}

static Selectivity
regex_selectivity_sub(char *patt, int pattlen, bool case_insensitive)
{
	Selectivity sel = 1.0;
	int			paren_depth = 0;
	int			paren_pos = 0;	/* dummy init to keep compiler quiet */
	int			pos;

	for (pos = 0; pos < pattlen; pos++)
	{
		if (patt[pos] == '(')
		{
			if (paren_depth == 0)
				paren_pos = pos;	/* remember start of parenthesized item */
			paren_depth++;
		}
		else if (patt[pos] == ')' && paren_depth > 0)
		{
			paren_depth--;
			if (paren_depth == 0)
				sel *= regex_selectivity_sub(patt + (paren_pos + 1),
											 pos - (paren_pos + 1),
											 case_insensitive);
		}
		else if (patt[pos] == '|' && paren_depth == 0)
		{
			/*
			 * If unquoted | is present at paren level 0 in pattern, we
			 * have multiple alternatives; sum their probabilities.
			 */
			sel += regex_selectivity_sub(patt + (pos + 1),
										 pattlen - (pos + 1),
										 case_insensitive);
			break;				/* rest of pattern is now processed */
		}
		else if (patt[pos] == '[')
		{
			bool		negclass = false;

			if (patt[++pos] == '^')
			{
				negclass = true;
				pos++;
			}
			if (patt[pos] == ']')		/* ']' at start of class is not
										 * special */
				pos++;
			while (pos < pattlen && patt[pos] != ']')
				pos++;
			if (paren_depth == 0)
				sel *= (negclass ? (1.0 - CHAR_RANGE_SEL) : CHAR_RANGE_SEL);
		}
		else if (patt[pos] == '.')
		{
			if (paren_depth == 0)
				sel *= ANY_CHAR_SEL;
		}
		else if (patt[pos] == '*' ||
				 patt[pos] == '?' ||
				 patt[pos] == '+')
		{
			/* Ought to be smarter about quantifiers... */
			if (paren_depth == 0)
				sel *= PARTIAL_WILDCARD_SEL;
		}
		else if (patt[pos] == '{')
		{
			while (pos < pattlen && patt[pos] != '}')
				pos++;
			if (paren_depth == 0)
				sel *= PARTIAL_WILDCARD_SEL;
		}
		else if (patt[pos] == '\\')
		{
			/* backslash quotes the next character */
			pos++;
			if (pos >= pattlen)
				break;
			if (paren_depth == 0)
				sel *= FIXED_CHAR_SEL;
		}
		else
		{
			if (paren_depth == 0)
				sel *= FIXED_CHAR_SEL;
		}
	}
	/* Could get sel > 1 if multiple wildcards */
	if (sel > 1.0)
		sel = 1.0;
	return sel;
}

static Selectivity
regex_selectivity(Const *patt_const, bool case_insensitive)
{
	Selectivity sel;
	char	   *patt;
	int			pattlen;
	Oid			typeid = patt_const->consttype;

	/*
	 * Should be unnecessary, there are no bytea regex operators defined.
	 * As such, it should be noted that the rest of this function has *not*
	 * been made safe for binary (possibly NULL containing) strings.
	 */
	if (typeid == BYTEAOID)
		ereport(ERROR,
				(errcode(ERRCODE_FEATURE_NOT_SUPPORTED),
				 errmsg("regular-expression matching not supported on type bytea")));

	/* the right-hand const is type text for all of these */
	patt = DatumGetCString(DirectFunctionCall1(textout, patt_const->constvalue));
	pattlen = strlen(patt);

	/* If patt doesn't end with $, consider it to have a trailing wildcard */
	if (pattlen > 0 && patt[pattlen - 1] == '$' &&
		(pattlen == 1 || patt[pattlen - 2] != '\\'))
	{
		/* has trailing $ */
		sel = regex_selectivity_sub(patt, pattlen - 1, case_insensitive);
	}
	else
	{
		/* no trailing $ */
		sel = regex_selectivity_sub(patt, pattlen, case_insensitive);
		sel *= FULL_WILDCARD_SEL;
		if (sel > 1.0)
			sel = 1.0;
	}
	return sel;
}

static Selectivity
pattern_selectivity(Const *patt, Pattern_Type ptype)
{
	Selectivity result;

	switch (ptype)
	{
		case Pattern_Type_Like:
			result = like_selectivity(patt, false);
			break;
		case Pattern_Type_Like_IC:
			result = like_selectivity(patt, true);
			break;
		case Pattern_Type_Regex:
			result = regex_selectivity(patt, false);
			break;
		case Pattern_Type_Regex_IC:
			result = regex_selectivity(patt, true);
			break;
		default:
			elog(ERROR, "unrecognized ptype: %d", (int) ptype);
			result = 1.0;		/* keep compiler quiet */
			break;
	}
	return result;
}


/*
 * Try to generate a string greater than the given string or any
 * string it is a prefix of.  If successful, return a palloc'd string;
 * else return NULL.
 *
 * The key requirement here is that given a prefix string, say "foo",
 * we must be able to generate another string "fop" that is greater
 * than all strings "foobar" starting with "foo".
 *
 * If we max out the righthand byte, truncate off the last character
 * and start incrementing the next.  For example, if "z" were the last
 * character in the sort order, then we could produce "foo" as a
 * string greater than "fonz".
 *
 * This could be rather slow in the worst case, but in most cases we
 * won't have to try more than one or two strings before succeeding.
 */
Const *
make_greater_string(const Const *str_const)
{
	Oid			datatype = str_const->consttype;
	char	   *str;
	char	   *workstr;
	int			len;

	/* Get the string and a modifiable copy */
	if (datatype == NAMEOID)
	{
		str = DatumGetCString(DirectFunctionCall1(nameout, str_const->constvalue));
		len = strlen(str);
	}
	else if (datatype == BYTEAOID)
	{
		str = DatumGetCString(DirectFunctionCall1(byteaout, str_const->constvalue));
		len = toast_raw_datum_size(str_const->constvalue) - VARHDRSZ;
	}
	else
	{
		str = DatumGetCString(DirectFunctionCall1(textout, str_const->constvalue));
		len = strlen(str);
	}
	workstr = pstrdup(str);

	while (len > 0)
	{
		unsigned char *lastchar = (unsigned char *) (workstr + len - 1);
		unsigned char savelastchar = *lastchar;

		/*
		 * Try to generate a larger string by incrementing the last byte.
		 */
		if (*lastchar < (unsigned char) 255)
		{
			Const	   *workstr_const;

			(*lastchar)++;
			workstr_const = string_to_const(workstr, datatype);

			pfree(str);
			pfree(workstr);
			return workstr_const;
		}

		/* restore last byte so we don't confuse pg_mbcliplen */
		*lastchar = savelastchar;

		/*
		 * Truncate off the last character, which might be more than 1
		 * byte, depending on the character encoding.
		 */
		if (datatype != BYTEAOID && pg_database_encoding_max_length() > 1)
			len = pg_mbcliplen((const unsigned char *) workstr, len, len - 1);
		else
			len -= 1;

		if (datatype != BYTEAOID)
			workstr[len] = '\0';
	}

	/* Failed... */
	pfree(str);
	pfree(workstr);

	return (Const *) NULL;
}

/*
 * Generate a Datum of the appropriate type from a C string.
 * Note that all of the supported types are pass-by-ref, so the
 * returned value should be pfree'd if no longer needed.
 */
static Datum
string_to_datum(const char *str, Oid datatype)
{
	Assert(str != NULL);

	/*
	 * We cheat a little by assuming that textin() will do for bpchar and
	 * varchar constants too...
	 */
	if (datatype == NAMEOID)
		return DirectFunctionCall1(namein, CStringGetDatum(str));
	else if (datatype == BYTEAOID)
		return DirectFunctionCall1(byteain, CStringGetDatum(str));
	else
		return DirectFunctionCall1(textin, CStringGetDatum(str));
}

/*
 * Generate a Const node of the appropriate type from a C string.
 */
static Const *
string_to_const(const char *str, Oid datatype)
{
	Datum		conval = string_to_datum(str, datatype);

	return makeConst(datatype, ((datatype == NAMEOID) ? NAMEDATALEN : -1),
					 conval, false, false);
}

/*-------------------------------------------------------------------------
 *
 * Index cost estimation functions
 *
 * genericcostestimate is a general-purpose estimator for use when we
 * don't have any better idea about how to estimate.  Index-type-specific
 * knowledge can be incorporated in the type-specific routines.
 *
 *-------------------------------------------------------------------------
 */

static void
genericcostestimate(Query *root, RelOptInfo *rel,
					IndexOptInfo *index, List *indexQuals,
					Cost *indexStartupCost,
					Cost *indexTotalCost,
					Selectivity *indexSelectivity,
					double *indexCorrelation)
{
	double		numIndexTuples;
	double		numIndexPages;
	QualCost	index_qual_cost;
	List	   *selectivityQuals = indexQuals;

	/*
	 * If the index is partial, AND the index predicate with the
	 * explicitly given indexquals to produce a more accurate idea of the
	 * index restriction.  This may produce redundant clauses, which we
	 * hope that cnfify and clauselist_selectivity will deal with
	 * intelligently.
	 *
	 * Note that index->indpred and indexQuals are both in implicit-AND form
	 * to start with, which we have to make explicit to hand to
	 * canonicalize_qual, and then we get back implicit-AND form again.
	 */
	if (index->indpred != NIL)
	{
		Expr	   *andedQuals;

		andedQuals = make_ands_explicit(nconc(listCopy(index->indpred),
											  indexQuals));
		selectivityQuals = canonicalize_qual(andedQuals, true);
	}

	/* Estimate the fraction of main-table tuples that will be visited */
	*indexSelectivity = clauselist_selectivity(root, selectivityQuals,
											   rel->relid,
											   JOIN_INNER);

	/*
	 * Estimate the number of tuples that will be visited.	We do it in
	 * this rather peculiar-looking way in order to get the right answer
	 * for partial indexes.  We can bound the number of tuples by the
	 * index size, in any case.
	 */
	numIndexTuples = *indexSelectivity * rel->tuples;

	if (numIndexTuples > index->tuples)
		numIndexTuples = index->tuples;

	/*
	 * Always estimate at least one tuple is touched, even when
	 * indexSelectivity estimate is tiny.
	 */
	if (numIndexTuples < 1.0)
		numIndexTuples = 1.0;

	/*
	 * Estimate the number of index pages that will be retrieved.
	 *
	 * For all currently-supported index types, the first page of the index
	 * is a metadata page, and we should figure on fetching that plus a
	 * pro-rated fraction of the remaining pages.
	 */
	if (index->pages > 1 && index->tuples > 0)
	{
		numIndexPages = (numIndexTuples / index->tuples) * (index->pages - 1);
		numIndexPages += 1;		/* count the metapage too */
		numIndexPages = ceil(numIndexPages);
	}
	else
		numIndexPages = 1.0;

	/*
	 * Compute the index access cost.
	 *
	 * Our generic assumption is that the index pages will be read
	 * sequentially, so they have cost 1.0 each, not random_page_cost.
	 * Also, we charge for evaluation of the indexquals at each index
	 * tuple. All the costs are assumed to be paid incrementally during
	 * the scan.
	 */
	cost_qual_eval(&index_qual_cost, indexQuals);
	*indexStartupCost = index_qual_cost.startup;
	*indexTotalCost = numIndexPages +
		(cpu_index_tuple_cost + index_qual_cost.per_tuple) * numIndexTuples;

	/*
	 * Generic assumption about index correlation: there isn't any.
	 */
	*indexCorrelation = 0.0;
}


Datum
btcostestimate(PG_FUNCTION_ARGS)
{
	Query	   *root = (Query *) PG_GETARG_POINTER(0);
	RelOptInfo *rel = (RelOptInfo *) PG_GETARG_POINTER(1);
	IndexOptInfo *index = (IndexOptInfo *) PG_GETARG_POINTER(2);
	List	   *indexQuals = (List *) PG_GETARG_POINTER(3);
	Cost	   *indexStartupCost = (Cost *) PG_GETARG_POINTER(4);
	Cost	   *indexTotalCost = (Cost *) PG_GETARG_POINTER(5);
	Selectivity *indexSelectivity = (Selectivity *) PG_GETARG_POINTER(6);
	double	   *indexCorrelation = (double *) PG_GETARG_POINTER(7);

	genericcostestimate(root, rel, index, indexQuals,
						indexStartupCost, indexTotalCost,
						indexSelectivity, indexCorrelation);

	/*
	 * If the first column is a simple variable, and we can get an
	 * estimate for its ordering correlation C from pg_statistic, estimate
	 * the index correlation as C / number-of-columns. (The idea here is
	 * that multiple columns dilute the importance of the first column's
	 * ordering, but don't negate it entirely.)
	 */
	if (index->indexkeys[0] != 0)
	{
		Oid			relid;
		HeapTuple	tuple;

		relid = getrelid(rel->relid, root->rtable);
		Assert(relid != InvalidOid);
		tuple = SearchSysCache(STATRELATT,
							   ObjectIdGetDatum(relid),
							   Int16GetDatum(index->indexkeys[0]),
							   0, 0);
		if (HeapTupleIsValid(tuple))
		{
			Oid			typid;
			int32		typmod;
			float4	   *numbers;
			int			nnumbers;

			get_atttypetypmod(relid, index->indexkeys[0],
							  &typid, &typmod);
			if (get_attstatsslot(tuple, typid, typmod,
								 STATISTIC_KIND_CORRELATION,
								 index->ordering[0],
								 NULL, NULL, &numbers, &nnumbers))
			{
				double		varCorrelation;
				int			nKeys;

				Assert(nnumbers == 1);
				varCorrelation = numbers[0];
				nKeys = index->ncolumns;

				*indexCorrelation = varCorrelation / nKeys;

				free_attstatsslot(typid, NULL, 0, numbers, nnumbers);
			}
			ReleaseSysCache(tuple);
		}
	}

	PG_RETURN_VOID();
}

Datum
rtcostestimate(PG_FUNCTION_ARGS)
{
	Query	   *root = (Query *) PG_GETARG_POINTER(0);
	RelOptInfo *rel = (RelOptInfo *) PG_GETARG_POINTER(1);
	IndexOptInfo *index = (IndexOptInfo *) PG_GETARG_POINTER(2);
	List	   *indexQuals = (List *) PG_GETARG_POINTER(3);
	Cost	   *indexStartupCost = (Cost *) PG_GETARG_POINTER(4);
	Cost	   *indexTotalCost = (Cost *) PG_GETARG_POINTER(5);
	Selectivity *indexSelectivity = (Selectivity *) PG_GETARG_POINTER(6);
	double	   *indexCorrelation = (double *) PG_GETARG_POINTER(7);

	genericcostestimate(root, rel, index, indexQuals,
						indexStartupCost, indexTotalCost,
						indexSelectivity, indexCorrelation);

	PG_RETURN_VOID();
}

Datum
hashcostestimate(PG_FUNCTION_ARGS)
{
	Query	   *root = (Query *) PG_GETARG_POINTER(0);
	RelOptInfo *rel = (RelOptInfo *) PG_GETARG_POINTER(1);
	IndexOptInfo *index = (IndexOptInfo *) PG_GETARG_POINTER(2);
	List	   *indexQuals = (List *) PG_GETARG_POINTER(3);
	Cost	   *indexStartupCost = (Cost *) PG_GETARG_POINTER(4);
	Cost	   *indexTotalCost = (Cost *) PG_GETARG_POINTER(5);
	Selectivity *indexSelectivity = (Selectivity *) PG_GETARG_POINTER(6);
	double	   *indexCorrelation = (double *) PG_GETARG_POINTER(7);

	genericcostestimate(root, rel, index, indexQuals,
						indexStartupCost, indexTotalCost,
						indexSelectivity, indexCorrelation);

	PG_RETURN_VOID();
}

Datum
gistcostestimate(PG_FUNCTION_ARGS)
{
	Query	   *root = (Query *) PG_GETARG_POINTER(0);
	RelOptInfo *rel = (RelOptInfo *) PG_GETARG_POINTER(1);
	IndexOptInfo *index = (IndexOptInfo *) PG_GETARG_POINTER(2);
	List	   *indexQuals = (List *) PG_GETARG_POINTER(3);
	Cost	   *indexStartupCost = (Cost *) PG_GETARG_POINTER(4);
	Cost	   *indexTotalCost = (Cost *) PG_GETARG_POINTER(5);
	Selectivity *indexSelectivity = (Selectivity *) PG_GETARG_POINTER(6);
	double	   *indexCorrelation = (double *) PG_GETARG_POINTER(7);

	genericcostestimate(root, rel, index, indexQuals,
						indexStartupCost, indexTotalCost,
						indexSelectivity, indexCorrelation);

	PG_RETURN_VOID();
}