/*-------------------------------------------------------------------------
 *
 * analyze.c
 *	  the Postgres statistics generator
 *
 * Portions Copyright (c) 1996-2008, PostgreSQL Global Development Group
 * Portions Copyright (c) 1994, Regents of the University of California
 *
 *
 * IDENTIFICATION
 *	  $PostgreSQL: pgsql/src/backend/commands/analyze.c,v 1.120 2008/05/12 00:00:47 alvherre Exp $
 *
 *-------------------------------------------------------------------------
 */
#include "postgres.h"

#include <math.h>

#include "access/heapam.h"
#include "access/transam.h"
#include "access/tuptoaster.h"
#include "access/xact.h"
#include "catalog/index.h"
#include "catalog/indexing.h"
#include "catalog/namespace.h"
#include "catalog/pg_namespace.h"
#include "commands/dbcommands.h"
#include "commands/vacuum.h"
#include "executor/executor.h"
#include "miscadmin.h"
#include "parser/parse_expr.h"
#include "parser/parse_oper.h"
#include "parser/parse_relation.h"
#include "pgstat.h"
#include "postmaster/autovacuum.h"
#include "storage/bufmgr.h"
#include "storage/proc.h"
#include "storage/procarray.h"
#include "utils/acl.h"
#include "utils/datum.h"
#include "utils/lsyscache.h"
#include "utils/memutils.h"
#include "utils/pg_rusage.h"
#include "utils/syscache.h"
#include "utils/tuplesort.h"
#include "utils/tqual.h"


/* Data structure for Algorithm S from Knuth 3.4.2 */
typedef struct
{
	BlockNumber N;				/* number of blocks, known in advance */
	int			n;				/* desired sample size */
	BlockNumber t;				/* current block number */
	int			m;				/* blocks selected so far */
} BlockSamplerData;
typedef BlockSamplerData *BlockSampler;

/* Per-index data for ANALYZE */
typedef struct AnlIndexData
{
	IndexInfo  *indexInfo;		/* BuildIndexInfo result */
	double		tupleFract;		/* fraction of rows for partial index */
	VacAttrStats **vacattrstats;	/* index attrs to analyze */
	int			attr_cnt;
} AnlIndexData;


/* Default statistics target (GUC parameter) */
int			default_statistics_target = 10;

/* A few variables that don't seem worth passing around as parameters */
static int	elevel = -1;

static MemoryContext anl_context = NULL;

static BufferAccessStrategy vac_strategy;


static void BlockSampler_Init(BlockSampler bs, BlockNumber nblocks,
				  int samplesize);
static bool BlockSampler_HasMore(BlockSampler bs);
static BlockNumber BlockSampler_Next(BlockSampler bs);
static void compute_index_stats(Relation onerel, double totalrows,
					AnlIndexData *indexdata, int nindexes,
					HeapTuple *rows, int numrows,
					MemoryContext col_context);
static VacAttrStats *examine_attribute(Relation onerel, int attnum);
static int acquire_sample_rows(Relation onerel, HeapTuple *rows,
					int targrows, double *totalrows, double *totaldeadrows);
static double random_fract(void);
static double init_selection_state(int n);
static double get_next_S(double t, int n, double *stateptr);
static int	compare_rows(const void *a, const void *b);
static void update_attstats(Oid relid, int natts, VacAttrStats **vacattrstats);
static Datum std_fetch_func(VacAttrStatsP stats, int rownum, bool *isNull);
static Datum ind_fetch_func(VacAttrStatsP stats, int rownum, bool *isNull);

static bool std_typanalyze(VacAttrStats *stats);


/*
 *	analyze_rel() -- analyze one relation
 */
void
analyze_rel(Oid relid, VacuumStmt *vacstmt,
			BufferAccessStrategy bstrategy)
{
	Relation	onerel;
	int			attr_cnt,
				tcnt,
				i,
				ind;
	Relation   *Irel;
	int			nindexes;
	bool		hasindex;
	bool		analyzableindex;
	VacAttrStats **vacattrstats;
	AnlIndexData *indexdata;
	int			targrows,
				numrows;
	double		totalrows,
				totaldeadrows;
	HeapTuple  *rows;
	PGRUsage	ru0;
	TimestampTz starttime = 0;
	Oid			save_userid;
	bool		save_secdefcxt;

	if (vacstmt->verbose)
		elevel = INFO;
	else
		elevel = DEBUG2;

	vac_strategy = bstrategy;

	/*
	 * Use the current context for storing analysis info.  vacuum.c ensures
	 * that this context will be cleared when I return, thus releasing the
	 * memory allocated here.
	 */
	anl_context = CurrentMemoryContext;

	/*
	 * Check for user-requested abort.	Note we want this to be inside a
	 * transaction, so xact.c doesn't issue useless WARNING.
	 */
	CHECK_FOR_INTERRUPTS();

	/*
	 * Open the relation, getting ShareUpdateExclusiveLock to ensure that two
	 * ANALYZEs don't run on it concurrently.  (This also locks out a
	 * concurrent VACUUM, which doesn't matter much at the moment but might
	 * matter if we ever try to accumulate stats on dead tuples.) If the rel
	 * has been dropped since we last saw it, we don't need to process it.
	 */
	onerel = try_relation_open(relid, ShareUpdateExclusiveLock);
	if (!onerel)
		return;

	/*
	 * Check permissions --- this should match vacuum's check!
	 */
	if (!(pg_class_ownercheck(RelationGetRelid(onerel), GetUserId()) ||
		  (pg_database_ownercheck(MyDatabaseId, GetUserId()) && !onerel->rd_rel->relisshared)))
	{
		/* No need for a WARNING if we already complained during VACUUM */
		if (!vacstmt->vacuum)
		{
			if (onerel->rd_rel->relisshared)
				ereport(WARNING,
						(errmsg("skipping \"%s\" --- only superuser can analyze it",
								RelationGetRelationName(onerel))));
			else if (onerel->rd_rel->relnamespace == PG_CATALOG_NAMESPACE)
				ereport(WARNING,
						(errmsg("skipping \"%s\" --- only superuser or database owner can analyze it",
								RelationGetRelationName(onerel))));
			else
				ereport(WARNING,
						(errmsg("skipping \"%s\" --- only table or database owner can analyze it",
								RelationGetRelationName(onerel))));
		}
		relation_close(onerel, ShareUpdateExclusiveLock);
		return;
	}

	/*
	 * Check that it's a plain table; we used to do this in get_rel_oids() but
	 * seems safer to check after we've locked the relation.
	 */
	if (onerel->rd_rel->relkind != RELKIND_RELATION)
	{
		/* No need for a WARNING if we already complained during VACUUM */
		if (!vacstmt->vacuum)
			ereport(WARNING,
					(errmsg("skipping \"%s\" --- cannot analyze indexes, views, or special system tables",
							RelationGetRelationName(onerel))));
		relation_close(onerel, ShareUpdateExclusiveLock);
		return;
	}

	/*
	 * Silently ignore tables that are temp tables of other backends ---
	 * trying to analyze these is rather pointless, since their contents are
	 * probably not up-to-date on disk.  (We don't throw a warning here; it
	 * would just lead to chatter during a database-wide ANALYZE.)
	 */
	if (isOtherTempNamespace(RelationGetNamespace(onerel)))
	{
		relation_close(onerel, ShareUpdateExclusiveLock);
		return;
	}

	/*
	 * We can ANALYZE any table except pg_statistic. See update_attstats
	 */
	if (RelationGetRelid(onerel) == StatisticRelationId)
	{
		relation_close(onerel, ShareUpdateExclusiveLock);
		return;
	}

	ereport(elevel,
			(errmsg("analyzing \"%s.%s\"",
					get_namespace_name(RelationGetNamespace(onerel)),
					RelationGetRelationName(onerel))));

	/*
	 * Switch to the table owner's userid, so that any index functions are
	 * run as that user.
	 */
	GetUserIdAndContext(&save_userid, &save_secdefcxt);
	SetUserIdAndContext(onerel->rd_rel->relowner, true);

	/* let others know what I'm doing */
	LWLockAcquire(ProcArrayLock, LW_EXCLUSIVE);
	MyProc->vacuumFlags |= PROC_IN_ANALYZE;
	LWLockRelease(ProcArrayLock);

	/* measure elapsed time iff autovacuum logging requires it */
	if (IsAutoVacuumWorkerProcess() && Log_autovacuum_min_duration >= 0)
	{
		pg_rusage_init(&ru0);
		if (Log_autovacuum_min_duration > 0)
			starttime = GetCurrentTimestamp();
	}

	/*
	 * Determine which columns to analyze
	 *
	 * Note that system attributes are never analyzed.
	 */
	if (vacstmt->va_cols != NIL)
	{
		ListCell   *le;

		vacattrstats = (VacAttrStats **) palloc(list_length(vacstmt->va_cols) *
												sizeof(VacAttrStats *));
		tcnt = 0;
		foreach(le, vacstmt->va_cols)
		{
			char	   *col = strVal(lfirst(le));

			i = attnameAttNum(onerel, col, false);
			if (i == InvalidAttrNumber)
				ereport(ERROR,
						(errcode(ERRCODE_UNDEFINED_COLUMN),
					errmsg("column \"%s\" of relation \"%s\" does not exist",
						   col, RelationGetRelationName(onerel))));
			vacattrstats[tcnt] = examine_attribute(onerel, i);
			if (vacattrstats[tcnt] != NULL)
				tcnt++;
		}
		attr_cnt = tcnt;
	}
	else
	{
		attr_cnt = onerel->rd_att->natts;
		vacattrstats = (VacAttrStats **)
			palloc(attr_cnt * sizeof(VacAttrStats *));
		tcnt = 0;
		for (i = 1; i <= attr_cnt; i++)
		{
			vacattrstats[tcnt] = examine_attribute(onerel, i);
			if (vacattrstats[tcnt] != NULL)
				tcnt++;
		}
		attr_cnt = tcnt;
	}

	/*
	 * Open all indexes of the relation, and see if there are any analyzable
	 * columns in the indexes.	We do not analyze index columns if there was
	 * an explicit column list in the ANALYZE command, however.
	 */
	vac_open_indexes(onerel, AccessShareLock, &nindexes, &Irel);
	hasindex = (nindexes > 0);
	indexdata = NULL;
	analyzableindex = false;
	if (hasindex)
	{
		indexdata = (AnlIndexData *) palloc0(nindexes * sizeof(AnlIndexData));
		for (ind = 0; ind < nindexes; ind++)
		{
			AnlIndexData *thisdata = &indexdata[ind];
			IndexInfo  *indexInfo;

			thisdata->indexInfo = indexInfo = BuildIndexInfo(Irel[ind]);
			thisdata->tupleFract = 1.0; /* fix later if partial */
			if (indexInfo->ii_Expressions != NIL && vacstmt->va_cols == NIL)
			{
				ListCell   *indexpr_item = list_head(indexInfo->ii_Expressions);

				thisdata->vacattrstats = (VacAttrStats **)
					palloc(indexInfo->ii_NumIndexAttrs * sizeof(VacAttrStats *));
				tcnt = 0;
				for (i = 0; i < indexInfo->ii_NumIndexAttrs; i++)
				{
					int			keycol = indexInfo->ii_KeyAttrNumbers[i];

					if (keycol == 0)
					{
						/* Found an index expression */
						Node	   *indexkey;

						if (indexpr_item == NULL)		/* shouldn't happen */
							elog(ERROR, "too few entries in indexprs list");
						indexkey = (Node *) lfirst(indexpr_item);
						indexpr_item = lnext(indexpr_item);

						/*
						 * Can't analyze if the opclass uses a storage type
						 * different from the expression result type. We'd get
						 * confused because the type shown in pg_attribute for
						 * the index column doesn't match what we are getting
						 * from the expression. Perhaps this can be fixed
						 * someday, but for now, punt.
						 */
						if (exprType(indexkey) !=
							Irel[ind]->rd_att->attrs[i]->atttypid)
							continue;

						thisdata->vacattrstats[tcnt] =
							examine_attribute(Irel[ind], i + 1);
						if (thisdata->vacattrstats[tcnt] != NULL)
						{
							tcnt++;
							analyzableindex = true;
						}
					}
				}
				thisdata->attr_cnt = tcnt;
			}
		}
	}

	/*
	 * Quit if no analyzable columns
	 */
	if (attr_cnt <= 0 && !analyzableindex)
	{
		/*
		 * We report that the table is empty; this is just so that the
		 * autovacuum code doesn't go nuts trying to get stats about a
		 * zero-column table.
		 */
		if (!vacstmt->vacuum)
			pgstat_report_analyze(onerel, 0, 0);
		goto cleanup;
	}

	/*
	 * Determine how many rows we need to sample, using the worst case from
	 * all analyzable columns.	We use a lower bound of 100 rows to avoid
	 * possible overflow in Vitter's algorithm.
	 */
	targrows = 100;
	for (i = 0; i < attr_cnt; i++)
	{
		if (targrows < vacattrstats[i]->minrows)
			targrows = vacattrstats[i]->minrows;
	}
	for (ind = 0; ind < nindexes; ind++)
	{
		AnlIndexData *thisdata = &indexdata[ind];

		for (i = 0; i < thisdata->attr_cnt; i++)
		{
			if (targrows < thisdata->vacattrstats[i]->minrows)
				targrows = thisdata->vacattrstats[i]->minrows;
		}
	}

	/*
	 * Acquire the sample rows
	 */
	rows = (HeapTuple *) palloc(targrows * sizeof(HeapTuple));
	numrows = acquire_sample_rows(onerel, rows, targrows,
								  &totalrows, &totaldeadrows);

	/*
	 * Compute the statistics.	Temporary results during the calculations for
	 * each column are stored in a child context.  The calc routines are
	 * responsible to make sure that whatever they store into the VacAttrStats
	 * structure is allocated in anl_context.
	 */
	if (numrows > 0)
	{
		MemoryContext col_context,
					old_context;

		col_context = AllocSetContextCreate(anl_context,
											"Analyze Column",
											ALLOCSET_DEFAULT_MINSIZE,
											ALLOCSET_DEFAULT_INITSIZE,
											ALLOCSET_DEFAULT_MAXSIZE);
		old_context = MemoryContextSwitchTo(col_context);

		for (i = 0; i < attr_cnt; i++)
		{
			VacAttrStats *stats = vacattrstats[i];

			stats->rows = rows;
			stats->tupDesc = onerel->rd_att;
			(*stats->compute_stats) (stats,
									 std_fetch_func,
									 numrows,
									 totalrows);
			MemoryContextResetAndDeleteChildren(col_context);
		}

		if (hasindex)
			compute_index_stats(onerel, totalrows,
								indexdata, nindexes,
								rows, numrows,
								col_context);

		MemoryContextSwitchTo(old_context);
		MemoryContextDelete(col_context);

		/*
		 * Emit the completed stats rows into pg_statistic, replacing any
		 * previous statistics for the target columns.	(If there are stats in
		 * pg_statistic for columns we didn't process, we leave them alone.)
		 */
		update_attstats(relid, attr_cnt, vacattrstats);

		for (ind = 0; ind < nindexes; ind++)
		{
			AnlIndexData *thisdata = &indexdata[ind];

			update_attstats(RelationGetRelid(Irel[ind]),
							thisdata->attr_cnt, thisdata->vacattrstats);
		}
	}

	/*
	 * If we are running a standalone ANALYZE, update pages/tuples stats in
	 * pg_class.  We know the accurate page count from the smgr, but only an
	 * approximate number of tuples; therefore, if we are part of VACUUM
	 * ANALYZE do *not* overwrite the accurate count already inserted by
	 * VACUUM.	The same consideration applies to indexes.
	 */
	if (!vacstmt->vacuum)
	{
		vac_update_relstats(RelationGetRelid(onerel),
							RelationGetNumberOfBlocks(onerel),
							totalrows, hasindex,
							InvalidTransactionId);

		for (ind = 0; ind < nindexes; ind++)
		{
			AnlIndexData *thisdata = &indexdata[ind];
			double		totalindexrows;

			totalindexrows = ceil(thisdata->tupleFract * totalrows);
			vac_update_relstats(RelationGetRelid(Irel[ind]),
								RelationGetNumberOfBlocks(Irel[ind]),
								totalindexrows, false,
								InvalidTransactionId);
		}

		/* report results to the stats collector, too */
		pgstat_report_analyze(onerel, totalrows, totaldeadrows);
	}

	/* We skip to here if there were no analyzable columns */
cleanup:

	/* Done with indexes */
	vac_close_indexes(nindexes, Irel, NoLock);

	/*
	 * Close source relation now, but keep lock so that no one deletes it
	 * before we commit.  (If someone did, they'd fail to clean up the entries
	 * we made in pg_statistic.  Also, releasing the lock before commit would
	 * expose us to concurrent-update failures in update_attstats.)
	 */
	relation_close(onerel, NoLock);

	/* Log the action if appropriate */
	if (IsAutoVacuumWorkerProcess() && Log_autovacuum_min_duration >= 0)
	{
		if (Log_autovacuum_min_duration == 0 ||
			TimestampDifferenceExceeds(starttime, GetCurrentTimestamp(),
									   Log_autovacuum_min_duration))
			ereport(LOG,
					(errmsg("automatic analyze of table \"%s.%s.%s\" system usage: %s",
							get_database_name(MyDatabaseId),
							get_namespace_name(RelationGetNamespace(onerel)),
							RelationGetRelationName(onerel),
							pg_rusage_show(&ru0))));
	}

	/*
	 * Reset my PGPROC flag.  Note: we need this here, and not in vacuum_rel,
	 * because the vacuum flag is cleared by the end-of-xact code.
	 */
	LWLockAcquire(ProcArrayLock, LW_EXCLUSIVE);
	MyProc->vacuumFlags &= ~PROC_IN_ANALYZE;
	LWLockRelease(ProcArrayLock);

	/* Restore userid */
	SetUserIdAndContext(save_userid, save_secdefcxt);
}

/*
 * Compute statistics about indexes of a relation
 */
static void
compute_index_stats(Relation onerel, double totalrows,
					AnlIndexData *indexdata, int nindexes,
					HeapTuple *rows, int numrows,
					MemoryContext col_context)
{
	MemoryContext ind_context,
				old_context;
	Datum		values[INDEX_MAX_KEYS];
	bool		isnull[INDEX_MAX_KEYS];
	int			ind,
				i;

	ind_context = AllocSetContextCreate(anl_context,
										"Analyze Index",
										ALLOCSET_DEFAULT_MINSIZE,
										ALLOCSET_DEFAULT_INITSIZE,
										ALLOCSET_DEFAULT_MAXSIZE);
	old_context = MemoryContextSwitchTo(ind_context);

	for (ind = 0; ind < nindexes; ind++)
	{
		AnlIndexData *thisdata = &indexdata[ind];
		IndexInfo  *indexInfo = thisdata->indexInfo;
		int			attr_cnt = thisdata->attr_cnt;
		TupleTableSlot *slot;
		EState	   *estate;
		ExprContext *econtext;
		List	   *predicate;
		Datum	   *exprvals;
		bool	   *exprnulls;
		int			numindexrows,
					tcnt,
					rowno;
		double		totalindexrows;

		/* Ignore index if no columns to analyze and not partial */
		if (attr_cnt == 0 && indexInfo->ii_Predicate == NIL)
			continue;

		/*
		 * Need an EState for evaluation of index expressions and
		 * partial-index predicates.  Create it in the per-index context to be
		 * sure it gets cleaned up at the bottom of the loop.
		 */
		estate = CreateExecutorState();
		econtext = GetPerTupleExprContext(estate);
		/* Need a slot to hold the current heap tuple, too */
		slot = MakeSingleTupleTableSlot(RelationGetDescr(onerel));

		/* Arrange for econtext's scan tuple to be the tuple under test */
		econtext->ecxt_scantuple = slot;

		/* Set up execution state for predicate. */
		predicate = (List *)
			ExecPrepareExpr((Expr *) indexInfo->ii_Predicate,
							estate);

		/* Compute and save index expression values */
		exprvals = (Datum *) palloc(numrows * attr_cnt * sizeof(Datum));
		exprnulls = (bool *) palloc(numrows * attr_cnt * sizeof(bool));
		numindexrows = 0;
		tcnt = 0;
		for (rowno = 0; rowno < numrows; rowno++)
		{
			HeapTuple	heapTuple = rows[rowno];

			/* Set up for predicate or expression evaluation */
			ExecStoreTuple(heapTuple, slot, InvalidBuffer, false);

			/* If index is partial, check predicate */
			if (predicate != NIL)
			{
				if (!ExecQual(predicate, econtext, false))
					continue;
			}
			numindexrows++;

			if (attr_cnt > 0)
			{
				/*
				 * Evaluate the index row to compute expression values. We
				 * could do this by hand, but FormIndexDatum is convenient.
				 */
				FormIndexDatum(indexInfo,
							   slot,
							   estate,
							   values,
							   isnull);

				/*
				 * Save just the columns we care about.
				 */
				for (i = 0; i < attr_cnt; i++)
				{
					VacAttrStats *stats = thisdata->vacattrstats[i];
					int			attnum = stats->attr->attnum;

					exprvals[tcnt] = values[attnum - 1];
					exprnulls[tcnt] = isnull[attnum - 1];
					tcnt++;
				}
			}
		}

		/*
		 * Having counted the number of rows that pass the predicate in the
		 * sample, we can estimate the total number of rows in the index.
		 */
		thisdata->tupleFract = (double) numindexrows / (double) numrows;
		totalindexrows = ceil(thisdata->tupleFract * totalrows);

		/*
		 * Now we can compute the statistics for the expression columns.
		 */
		if (numindexrows > 0)
		{
			MemoryContextSwitchTo(col_context);
			for (i = 0; i < attr_cnt; i++)
			{
				VacAttrStats *stats = thisdata->vacattrstats[i];

				stats->exprvals = exprvals + i;
				stats->exprnulls = exprnulls + i;
				stats->rowstride = attr_cnt;
				(*stats->compute_stats) (stats,
										 ind_fetch_func,
										 numindexrows,
										 totalindexrows);
				MemoryContextResetAndDeleteChildren(col_context);
			}
		}

		/* And clean up */
		MemoryContextSwitchTo(ind_context);

		ExecDropSingleTupleTableSlot(slot);
		FreeExecutorState(estate);
		MemoryContextResetAndDeleteChildren(ind_context);
	}

	MemoryContextSwitchTo(old_context);
	MemoryContextDelete(ind_context);
}

/*
 * examine_attribute -- pre-analysis of a single column
 *
 * Determine whether the column is analyzable; if so, create and initialize
 * a VacAttrStats struct for it.  If not, return NULL.
 */
static VacAttrStats *
examine_attribute(Relation onerel, int attnum)
{
	Form_pg_attribute attr = onerel->rd_att->attrs[attnum - 1];
	HeapTuple	typtuple;
	VacAttrStats *stats;
	bool		ok;

	/* Never analyze dropped columns */
	if (attr->attisdropped)
		return NULL;

	/* Don't analyze column if user has specified not to */
	if (attr->attstattarget == 0)
		return NULL;

	/*
	 * Create the VacAttrStats struct.
	 */
	stats = (VacAttrStats *) palloc0(sizeof(VacAttrStats));
	stats->attr = (Form_pg_attribute) palloc(ATTRIBUTE_TUPLE_SIZE);
	memcpy(stats->attr, attr, ATTRIBUTE_TUPLE_SIZE);
	typtuple = SearchSysCache(TYPEOID,
							  ObjectIdGetDatum(attr->atttypid),
							  0, 0, 0);
	if (!HeapTupleIsValid(typtuple))
		elog(ERROR, "cache lookup failed for type %u", attr->atttypid);
	stats->attrtype = (Form_pg_type) palloc(sizeof(FormData_pg_type));
	memcpy(stats->attrtype, GETSTRUCT(typtuple), sizeof(FormData_pg_type));
	ReleaseSysCache(typtuple);
	stats->anl_context = anl_context;
	stats->tupattnum = attnum;

	/*
	 * Call the type-specific typanalyze function.	If none is specified, use
	 * std_typanalyze().
	 */
	if (OidIsValid(stats->attrtype->typanalyze))
		ok = DatumGetBool(OidFunctionCall1(stats->attrtype->typanalyze,
										   PointerGetDatum(stats)));
	else
		ok = std_typanalyze(stats);

	if (!ok || stats->compute_stats == NULL || stats->minrows <= 0)
	{
		pfree(stats->attrtype);
		pfree(stats->attr);
		pfree(stats);
		return NULL;
	}

	return stats;
}

/*
 * BlockSampler_Init -- prepare for random sampling of blocknumbers
 *
 * BlockSampler is used for stage one of our new two-stage tuple
 * sampling mechanism as discussed on pgsql-hackers 2004-04-02 (subject
 * "Large DB").  It selects a random sample of samplesize blocks out of
 * the nblocks blocks in the table.  If the table has less than
 * samplesize blocks, all blocks are selected.
 *
 * Since we know the total number of blocks in advance, we can use the
 * straightforward Algorithm S from Knuth 3.4.2, rather than Vitter's
 * algorithm.
 */
static void
BlockSampler_Init(BlockSampler bs, BlockNumber nblocks, int samplesize)
{
	bs->N = nblocks;			/* measured table size */

	/*
	 * If we decide to reduce samplesize for tables that have less or not much
	 * more than samplesize blocks, here is the place to do it.
	 */
	bs->n = samplesize;
	bs->t = 0;					/* blocks scanned so far */
	bs->m = 0;					/* blocks selected so far */
}

static bool
BlockSampler_HasMore(BlockSampler bs)
{
	return (bs->t < bs->N) && (bs->m < bs->n);
}

static BlockNumber
BlockSampler_Next(BlockSampler bs)
{
	BlockNumber K = bs->N - bs->t;		/* remaining blocks */
	int			k = bs->n - bs->m;		/* blocks still to sample */
	double		p;				/* probability to skip block */
	double		V;				/* random */

	Assert(BlockSampler_HasMore(bs));	/* hence K > 0 and k > 0 */

	if ((BlockNumber) k >= K)
	{
		/* need all the rest */
		bs->m++;
		return bs->t++;
	}

	/*----------
	 * It is not obvious that this code matches Knuth's Algorithm S.
	 * Knuth says to skip the current block with probability 1 - k/K.
	 * If we are to skip, we should advance t (hence decrease K), and
	 * repeat the same probabilistic test for the next block.  The naive
	 * implementation thus requires a random_fract() call for each block
	 * number.	But we can reduce this to one random_fract() call per
	 * selected block, by noting that each time the while-test succeeds,
	 * we can reinterpret V as a uniform random number in the range 0 to p.
	 * Therefore, instead of choosing a new V, we just adjust p to be
	 * the appropriate fraction of its former value, and our next loop
	 * makes the appropriate probabilistic test.
	 *
	 * We have initially K > k > 0.  If the loop reduces K to equal k,
	 * the next while-test must fail since p will become exactly zero
	 * (we assume there will not be roundoff error in the division).
	 * (Note: Knuth suggests a "<=" loop condition, but we use "<" just
	 * to be doubly sure about roundoff error.)  Therefore K cannot become
	 * less than k, which means that we cannot fail to select enough blocks.
	 *----------
	 */
	V = random_fract();
	p = 1.0 - (double) k / (double) K;
	while (V < p)
	{
		/* skip */
		bs->t++;
		K--;					/* keep K == N - t */

		/* adjust p to be new cutoff point in reduced range */
		p *= 1.0 - (double) k / (double) K;
	}

	/* select */
	bs->m++;
	return bs->t++;
}

/*
 * acquire_sample_rows -- acquire a random sample of rows from the table
 *
 * As of May 2004 we use a new two-stage method:  Stage one selects up
 * to targrows random blocks (or all blocks, if there aren't so many).
 * Stage two scans these blocks and uses the Vitter algorithm to create
 * a random sample of targrows rows (or less, if there are less in the
 * sample of blocks).  The two stages are executed simultaneously: each
 * block is processed as soon as stage one returns its number and while
 * the rows are read stage two controls which ones are to be inserted
 * into the sample.
 *
 * Although every row has an equal chance of ending up in the final
 * sample, this sampling method is not perfect: not every possible
 * sample has an equal chance of being selected.  For large relations
 * the number of different blocks represented by the sample tends to be
 * too small.  We can live with that for now.  Improvements are welcome.
 *
 * We also estimate the total numbers of live and dead rows in the table,
 * and return them into *totalrows and *totaldeadrows, respectively.
 *
 * An important property of this sampling method is that because we do
 * look at a statistically unbiased set of blocks, we should get
 * unbiased estimates of the average numbers of live and dead rows per
 * block.  The previous sampling method put too much credence in the row
 * density near the start of the table.
 *
 * The returned list of tuples is in order by physical position in the table.
 * (We will rely on this later to derive correlation estimates.)
 */
static int
acquire_sample_rows(Relation onerel, HeapTuple *rows, int targrows,
					double *totalrows, double *totaldeadrows)
{
	int			numrows = 0;	/* # rows now in reservoir */
	double		samplerows = 0;	/* total # rows collected */
	double		liverows = 0;	/* # live rows seen */
	double		deadrows = 0;	/* # dead rows seen */
	double		rowstoskip = -1;	/* -1 means not set yet */
	BlockNumber totalblocks;
	TransactionId OldestXmin;
	BlockSamplerData bs;
	double		rstate;

	Assert(targrows > 1);

	totalblocks = RelationGetNumberOfBlocks(onerel);

	/* Need a cutoff xmin for HeapTupleSatisfiesVacuum */
	OldestXmin = GetOldestXmin(onerel->rd_rel->relisshared, true);

	/* Prepare for sampling block numbers */
	BlockSampler_Init(&bs, totalblocks, targrows);
	/* Prepare for sampling rows */
	rstate = init_selection_state(targrows);

	/* Outer loop over blocks to sample */
	while (BlockSampler_HasMore(&bs))
	{
		BlockNumber targblock = BlockSampler_Next(&bs);
		Buffer		targbuffer;
		Page		targpage;
		OffsetNumber targoffset,
					maxoffset;

		vacuum_delay_point();

		/*
		 * We must maintain a pin on the target page's buffer to ensure that
		 * the maxoffset value stays good (else concurrent VACUUM might delete
		 * tuples out from under us).  Hence, pin the page until we are done
		 * looking at it.  We also choose to hold sharelock on the buffer
		 * throughout --- we could release and re-acquire sharelock for
		 * each tuple, but since we aren't doing much work per tuple, the
		 * extra lock traffic is probably better avoided.
		 */
		targbuffer = ReadBufferWithStrategy(onerel, targblock, vac_strategy);
		LockBuffer(targbuffer, BUFFER_LOCK_SHARE);
		targpage = BufferGetPage(targbuffer);
		maxoffset = PageGetMaxOffsetNumber(targpage);

		/* Inner loop over all tuples on the selected page */
		for (targoffset = FirstOffsetNumber; targoffset <= maxoffset; targoffset++)
		{
			ItemId		itemid;
			HeapTupleData targtuple;
			bool		sample_it = false;

			itemid = PageGetItemId(targpage, targoffset);

			/*
			 * We ignore unused and redirect line pointers.  DEAD line
			 * pointers should be counted as dead, because we need vacuum
			 * to run to get rid of them.  Note that this rule agrees with
			 * the way that heap_page_prune() counts things.
			 */
			if (!ItemIdIsNormal(itemid))
			{
				if (ItemIdIsDead(itemid))
					deadrows += 1;
				continue;
			}

			ItemPointerSet(&targtuple.t_self, targblock, targoffset);

			targtuple.t_data = (HeapTupleHeader) PageGetItem(targpage, itemid);
			targtuple.t_len = ItemIdGetLength(itemid);

			switch (HeapTupleSatisfiesVacuum(targtuple.t_data,
											 OldestXmin,
											 targbuffer))
			{
				case HEAPTUPLE_LIVE:
					sample_it = true;
					liverows += 1;
					break;

				case HEAPTUPLE_DEAD:
				case HEAPTUPLE_RECENTLY_DEAD:
					/* Count dead and recently-dead rows */
					deadrows += 1;
					break;

				case HEAPTUPLE_INSERT_IN_PROGRESS:
					/*
					 * Insert-in-progress rows are not counted.  We assume
					 * that when the inserting transaction commits or aborts,
					 * it will send a stats message to increment the proper
					 * count.  This works right only if that transaction ends
					 * after we finish analyzing the table; if things happen
					 * in the other order, its stats update will be
					 * overwritten by ours.  However, the error will be
					 * large only if the other transaction runs long enough
					 * to insert many tuples, so assuming it will finish
					 * after us is the safer option.
					 *
					 * A special case is that the inserting transaction might
					 * be our own.  In this case we should count and sample
					 * the row, to accommodate users who load a table and
					 * analyze it in one transaction.  (pgstat_report_analyze
					 * has to adjust the numbers we send to the stats collector
					 * to make this come out right.)
					 */
					if (TransactionIdIsCurrentTransactionId(HeapTupleHeaderGetXmin(targtuple.t_data)))
					{
						sample_it = true;
						liverows += 1;
					}
					break;

				case HEAPTUPLE_DELETE_IN_PROGRESS:
					/*
					 * We count delete-in-progress rows as still live, using
					 * the same reasoning given above; but we don't bother to
					 * include them in the sample.
					 *
					 * If the delete was done by our own transaction, however,
					 * we must count the row as dead to make
					 * pgstat_report_analyze's stats adjustments come out
					 * right.  (Note: this works out properly when the row
					 * was both inserted and deleted in our xact.)
					 */
					if (TransactionIdIsCurrentTransactionId(HeapTupleHeaderGetXmax(targtuple.t_data)))
						deadrows += 1;
					else
						liverows += 1;
					break;

				default:
					elog(ERROR, "unexpected HeapTupleSatisfiesVacuum result");
					break;
			}

			if (sample_it)
			{
				/*
				 * The first targrows sample rows are simply copied into the
				 * reservoir. Then we start replacing tuples in the sample
				 * until we reach the end of the relation.	This algorithm is
				 * from Jeff Vitter's paper (see full citation below). It
				 * works by repeatedly computing the number of tuples to skip
				 * before selecting a tuple, which replaces a randomly chosen
				 * element of the reservoir (current set of tuples).  At all
				 * times the reservoir is a true random sample of the tuples
				 * we've passed over so far, so when we fall off the end of
				 * the relation we're done.
				 */
				if (numrows < targrows)
					rows[numrows++] = heap_copytuple(&targtuple);
				else
				{
					/*
					 * t in Vitter's paper is the number of records already
					 * processed.  If we need to compute a new S value, we
					 * must use the not-yet-incremented value of samplerows
					 * as t.
					 */
					if (rowstoskip < 0)
						rowstoskip = get_next_S(samplerows, targrows, &rstate);

					if (rowstoskip <= 0)
					{
						/*
						 * Found a suitable tuple, so save it, replacing one
						 * old tuple at random
						 */
						int			k = (int) (targrows * random_fract());

						Assert(k >= 0 && k < targrows);
						heap_freetuple(rows[k]);
						rows[k] = heap_copytuple(&targtuple);
					}

					rowstoskip -= 1;
				}

				samplerows += 1;
			}
		}

		/* Now release the lock and pin on the page */
		UnlockReleaseBuffer(targbuffer);
	}

	/*
	 * If we didn't find as many tuples as we wanted then we're done. No sort
	 * is needed, since they're already in order.
	 *
	 * Otherwise we need to sort the collected tuples by position
	 * (itempointer). It's not worth worrying about corner cases where the
	 * tuples are already sorted.
	 */
	if (numrows == targrows)
		qsort((void *) rows, numrows, sizeof(HeapTuple), compare_rows);

	/*
	 * Estimate total numbers of rows in relation.
	 */
	if (bs.m > 0)
	{
		*totalrows = floor((liverows * totalblocks) / bs.m + 0.5);
		*totaldeadrows = floor((deadrows * totalblocks) / bs.m + 0.5);
	}
	else
	{
		*totalrows = 0.0;
		*totaldeadrows = 0.0;
	}

	/*
	 * Emit some interesting relation info
	 */
	ereport(elevel,
			(errmsg("\"%s\": scanned %d of %u pages, "
					"containing %.0f live rows and %.0f dead rows; "
					"%d rows in sample, %.0f estimated total rows",
					RelationGetRelationName(onerel),
					bs.m, totalblocks,
					liverows, deadrows,
					numrows, *totalrows)));

	return numrows;
}

/* Select a random value R uniformly distributed in (0 - 1) */
static double
random_fract(void)
{
	return ((double) random() + 1) / ((double) MAX_RANDOM_VALUE + 2);
}

/*
 * These two routines embody Algorithm Z from "Random sampling with a
 * reservoir" by Jeffrey S. Vitter, in ACM Trans. Math. Softw. 11, 1
 * (Mar. 1985), Pages 37-57.  Vitter describes his algorithm in terms
 * of the count S of records to skip before processing another record.
 * It is computed primarily based on t, the number of records already read.
 * The only extra state needed between calls is W, a random state variable.
 *
 * init_selection_state computes the initial W value.
 *
 * Given that we've already read t records (t >= n), get_next_S
 * determines the number of records to skip before the next record is
 * processed.
 */
static double
init_selection_state(int n)
{
	/* Initial value of W (for use when Algorithm Z is first applied) */
	return exp(-log(random_fract()) / n);
}

static double
get_next_S(double t, int n, double *stateptr)
{
	double		S;

	/* The magic constant here is T from Vitter's paper */
	if (t <= (22.0 * n))
	{
		/* Process records using Algorithm X until t is large enough */
		double		V,
					quot;

		V = random_fract();		/* Generate V */
		S = 0;
		t += 1;
		/* Note: "num" in Vitter's code is always equal to t - n */
		quot = (t - (double) n) / t;
		/* Find min S satisfying (4.1) */
		while (quot > V)
		{
			S += 1;
			t += 1;
			quot *= (t - (double) n) / t;
		}
	}
	else
	{
		/* Now apply Algorithm Z */
		double		W = *stateptr;
		double		term = t - (double) n + 1;

		for (;;)
		{
			double		numer,
						numer_lim,
						denom;
			double		U,
						X,
						lhs,
						rhs,
						y,
						tmp;

			/* Generate U and X */
			U = random_fract();
			X = t * (W - 1.0);
			S = floor(X);		/* S is tentatively set to floor(X) */
			/* Test if U <= h(S)/cg(X) in the manner of (6.3) */
			tmp = (t + 1) / term;
			lhs = exp(log(((U * tmp * tmp) * (term + S)) / (t + X)) / n);
			rhs = (((t + X) / (term + S)) * term) / t;
			if (lhs <= rhs)
			{
				W = rhs / lhs;
				break;
			}
			/* Test if U <= f(S)/cg(X) */
			y = (((U * (t + 1)) / term) * (t + S + 1)) / (t + X);
			if ((double) n < S)
			{
				denom = t;
				numer_lim = term + S;
			}
			else
			{
				denom = t - (double) n + S;
				numer_lim = t + 1;
			}
			for (numer = t + S; numer >= numer_lim; numer -= 1)
			{
				y *= numer / denom;
				denom -= 1;
			}
			W = exp(-log(random_fract()) / n);	/* Generate W in advance */
			if (exp(log(y) / n) <= (t + X) / t)
				break;
		}
		*stateptr = W;
	}
	return S;
}

/*
 * qsort comparator for sorting rows[] array
 */
static int
compare_rows(const void *a, const void *b)
{
	HeapTuple	ha = *(HeapTuple *) a;
	HeapTuple	hb = *(HeapTuple *) b;
	BlockNumber ba = ItemPointerGetBlockNumber(&ha->t_self);
	OffsetNumber oa = ItemPointerGetOffsetNumber(&ha->t_self);
	BlockNumber bb = ItemPointerGetBlockNumber(&hb->t_self);
	OffsetNumber ob = ItemPointerGetOffsetNumber(&hb->t_self);

	if (ba < bb)
		return -1;
	if (ba > bb)
		return 1;
	if (oa < ob)
		return -1;
	if (oa > ob)
		return 1;
	return 0;
}


/*
 *	update_attstats() -- update attribute statistics for one relation
 *
 *		Statistics are stored in several places: the pg_class row for the
 *		relation has stats about the whole relation, and there is a
 *		pg_statistic row for each (non-system) attribute that has ever
 *		been analyzed.	The pg_class values are updated by VACUUM, not here.
 *
 *		pg_statistic rows are just added or updated normally.  This means
 *		that pg_statistic will probably contain some deleted rows at the
 *		completion of a vacuum cycle, unless it happens to get vacuumed last.
 *
 *		To keep things simple, we punt for pg_statistic, and don't try
 *		to compute or store rows for pg_statistic itself in pg_statistic.
 *		This could possibly be made to work, but it's not worth the trouble.
 *		Note analyze_rel() has seen to it that we won't come here when
 *		vacuuming pg_statistic itself.
 *
 *		Note: there would be a race condition here if two backends could
 *		ANALYZE the same table concurrently.  Presently, we lock that out
 *		by taking a self-exclusive lock on the relation in analyze_rel().
 */
static void
update_attstats(Oid relid, int natts, VacAttrStats **vacattrstats)
{
	Relation	sd;
	int			attno;

	if (natts <= 0)
		return;					/* nothing to do */

	sd = heap_open(StatisticRelationId, RowExclusiveLock);

	for (attno = 0; attno < natts; attno++)
	{
		VacAttrStats *stats = vacattrstats[attno];
		HeapTuple	stup,
					oldtup;
		int			i,
					k,
					n;
		Datum		values[Natts_pg_statistic];
		char		nulls[Natts_pg_statistic];
		char		replaces[Natts_pg_statistic];

		/* Ignore attr if we weren't able to collect stats */
		if (!stats->stats_valid)
			continue;

		/*
		 * Construct a new pg_statistic tuple
		 */
		for (i = 0; i < Natts_pg_statistic; ++i)
		{
			nulls[i] = ' ';
			replaces[i] = 'r';
		}

		i = 0;
		values[i++] = ObjectIdGetDatum(relid);	/* starelid */
		values[i++] = Int16GetDatum(stats->attr->attnum);		/* staattnum */
		values[i++] = Float4GetDatum(stats->stanullfrac);		/* stanullfrac */
		values[i++] = Int32GetDatum(stats->stawidth);	/* stawidth */
		values[i++] = Float4GetDatum(stats->stadistinct);		/* stadistinct */
		for (k = 0; k < STATISTIC_NUM_SLOTS; k++)
		{
			values[i++] = Int16GetDatum(stats->stakind[k]);		/* stakindN */
		}
		for (k = 0; k < STATISTIC_NUM_SLOTS; k++)
		{
			values[i++] = ObjectIdGetDatum(stats->staop[k]);	/* staopN */
		}
		for (k = 0; k < STATISTIC_NUM_SLOTS; k++)
		{
			int			nnum = stats->numnumbers[k];

			if (nnum > 0)
			{
				Datum	   *numdatums = (Datum *) palloc(nnum * sizeof(Datum));
				ArrayType  *arry;

				for (n = 0; n < nnum; n++)
					numdatums[n] = Float4GetDatum(stats->stanumbers[k][n]);
				/* XXX knows more than it should about type float4: */
				arry = construct_array(numdatums, nnum,
									   FLOAT4OID,
									   sizeof(float4), FLOAT4PASSBYVAL, 'i');
				values[i++] = PointerGetDatum(arry);	/* stanumbersN */
			}
			else
			{
				nulls[i] = 'n';
				values[i++] = (Datum) 0;
			}
		}
		for (k = 0; k < STATISTIC_NUM_SLOTS; k++)
		{
			if (stats->numvalues[k] > 0)
			{
				ArrayType  *arry;

				arry = construct_array(stats->stavalues[k],
									   stats->numvalues[k],
									   stats->attr->atttypid,
									   stats->attrtype->typlen,
									   stats->attrtype->typbyval,
									   stats->attrtype->typalign);
				values[i++] = PointerGetDatum(arry);	/* stavaluesN */
			}
			else
			{
				nulls[i] = 'n';
				values[i++] = (Datum) 0;
			}
		}

		/* Is there already a pg_statistic tuple for this attribute? */
		oldtup = SearchSysCache(STATRELATT,
								ObjectIdGetDatum(relid),
								Int16GetDatum(stats->attr->attnum),
								0, 0);

		if (HeapTupleIsValid(oldtup))
		{
			/* Yes, replace it */
			stup = heap_modifytuple(oldtup,
									RelationGetDescr(sd),
									values,
									nulls,
									replaces);
			ReleaseSysCache(oldtup);
			simple_heap_update(sd, &stup->t_self, stup);
		}
		else
		{
			/* No, insert new tuple */
			stup = heap_formtuple(RelationGetDescr(sd), values, nulls);
			simple_heap_insert(sd, stup);
		}

		/* update indexes too */
		CatalogUpdateIndexes(sd, stup);

		heap_freetuple(stup);
	}

	heap_close(sd, RowExclusiveLock);
}

/*
 * Standard fetch function for use by compute_stats subroutines.
 *
 * This exists to provide some insulation between compute_stats routines
 * and the actual storage of the sample data.
 */
static Datum
std_fetch_func(VacAttrStatsP stats, int rownum, bool *isNull)
{
	int			attnum = stats->tupattnum;
	HeapTuple	tuple = stats->rows[rownum];
	TupleDesc	tupDesc = stats->tupDesc;

	return heap_getattr(tuple, attnum, tupDesc, isNull);
}

/*
 * Fetch function for analyzing index expressions.
 *
 * We have not bothered to construct index tuples, instead the data is
 * just in Datum arrays.
 */
static Datum
ind_fetch_func(VacAttrStatsP stats, int rownum, bool *isNull)
{
	int			i;

	/* exprvals and exprnulls are already offset for proper column */
	i = rownum * stats->rowstride;
	*isNull = stats->exprnulls[i];
	return stats->exprvals[i];
}


/*==========================================================================
 *
 * Code below this point represents the "standard" type-specific statistics
 * analysis algorithms.  This code can be replaced on a per-data-type basis
 * by setting a nonzero value in pg_type.typanalyze.
 *
 *==========================================================================
 */


/*
 * To avoid consuming too much memory during analysis and/or too much space
 * in the resulting pg_statistic rows, we ignore varlena datums that are wider
 * than WIDTH_THRESHOLD (after detoasting!).  This is legitimate for MCV
 * and distinct-value calculations since a wide value is unlikely to be
 * duplicated at all, much less be a most-common value.  For the same reason,
 * ignoring wide values will not affect our estimates of histogram bin
 * boundaries very much.
 */
#define WIDTH_THRESHOLD  1024

#define swapInt(a,b)	do {int _tmp; _tmp=a; a=b; b=_tmp;} while(0)
#define swapDatum(a,b)	do {Datum _tmp; _tmp=a; a=b; b=_tmp;} while(0)

/*
 * Extra information used by the default analysis routines
 */
typedef struct
{
	Oid			eqopr;			/* '=' operator for datatype, if any */
	Oid			eqfunc;			/* and associated function */
	Oid			ltopr;			/* '<' operator for datatype, if any */
} StdAnalyzeData;

typedef struct
{
	Datum		value;			/* a data value */
	int			tupno;			/* position index for tuple it came from */
} ScalarItem;

typedef struct
{
	int			count;			/* # of duplicates */
	int			first;			/* values[] index of first occurrence */
} ScalarMCVItem;

typedef struct
{
	FmgrInfo   *cmpFn;
	int			cmpFlags;
	int		   *tupnoLink;
} CompareScalarsContext;


static void compute_minimal_stats(VacAttrStatsP stats,
					  AnalyzeAttrFetchFunc fetchfunc,
					  int samplerows,
					  double totalrows);
static void compute_scalar_stats(VacAttrStatsP stats,
					 AnalyzeAttrFetchFunc fetchfunc,
					 int samplerows,
					 double totalrows);
static int	compare_scalars(const void *a, const void *b, void *arg);
static int	compare_mcvs(const void *a, const void *b);


/*
 * std_typanalyze -- the default type-specific typanalyze function
 */
static bool
std_typanalyze(VacAttrStats *stats)
{
	Form_pg_attribute attr = stats->attr;
	Operator	func_operator;
	Oid			eqopr = InvalidOid;
	Oid			eqfunc = InvalidOid;
	Oid			ltopr = InvalidOid;
	StdAnalyzeData *mystats;

	/* If the attstattarget column is negative, use the default value */
	/* NB: it is okay to scribble on stats->attr since it's a copy */
	if (attr->attstattarget < 0)
		attr->attstattarget = default_statistics_target;

	/* If column has no "=" operator, we can't do much of anything */
	func_operator = equality_oper(attr->atttypid, true);
	if (func_operator != NULL)
	{
		eqopr = oprid(func_operator);
		eqfunc = oprfuncid(func_operator);
		ReleaseSysCache(func_operator);
	}
	if (!OidIsValid(eqfunc))
		return false;

	/* Is there a "<" operator with suitable semantics? */
	func_operator = ordering_oper(attr->atttypid, true);
	if (func_operator != NULL)
	{
		ltopr = oprid(func_operator);
		ReleaseSysCache(func_operator);
	}

	/* Save the operator info for compute_stats routines */
	mystats = (StdAnalyzeData *) palloc(sizeof(StdAnalyzeData));
	mystats->eqopr = eqopr;
	mystats->eqfunc = eqfunc;
	mystats->ltopr = ltopr;
	stats->extra_data = mystats;

	/*
	 * Determine which standard statistics algorithm to use
	 */
	if (OidIsValid(ltopr))
	{
		/* Seems to be a scalar datatype */
		stats->compute_stats = compute_scalar_stats;
		/*--------------------
		 * The following choice of minrows is based on the paper
		 * "Random sampling for histogram construction: how much is enough?"
		 * by Surajit Chaudhuri, Rajeev Motwani and Vivek Narasayya, in
		 * Proceedings of ACM SIGMOD International Conference on Management
		 * of Data, 1998, Pages 436-447.  Their Corollary 1 to Theorem 5
		 * says that for table size n, histogram size k, maximum relative
		 * error in bin size f, and error probability gamma, the minimum
		 * random sample size is
		 *		r = 4 * k * ln(2*n/gamma) / f^2
		 * Taking f = 0.5, gamma = 0.01, n = 1 million rows, we obtain
		 *		r = 305.82 * k
		 * Note that because of the log function, the dependence on n is
		 * quite weak; even at n = 1 billion, a 300*k sample gives <= 0.59
		 * bin size error with probability 0.99.  So there's no real need to
		 * scale for n, which is a good thing because we don't necessarily
		 * know it at this point.
		 *--------------------
		 */
		stats->minrows = 300 * attr->attstattarget;
	}
	else
	{
		/* Can't do much but the minimal stuff */
		stats->compute_stats = compute_minimal_stats;
		/* Might as well use the same minrows as above */
		stats->minrows = 300 * attr->attstattarget;
	}

	return true;
}

/*
 *	compute_minimal_stats() -- compute minimal column statistics
 *
 *	We use this when we can find only an "=" operator for the datatype.
 *
 *	We determine the fraction of non-null rows, the average width, the
 *	most common values, and the (estimated) number of distinct values.
 *
 *	The most common values are determined by brute force: we keep a list
 *	of previously seen values, ordered by number of times seen, as we scan
 *	the samples.  A newly seen value is inserted just after the last
 *	multiply-seen value, causing the bottommost (oldest) singly-seen value
 *	to drop off the list.  The accuracy of this method, and also its cost,
 *	depend mainly on the length of the list we are willing to keep.
 */
static void
compute_minimal_stats(VacAttrStatsP stats,
					  AnalyzeAttrFetchFunc fetchfunc,
					  int samplerows,
					  double totalrows)
{
	int			i;
	int			null_cnt = 0;
	int			nonnull_cnt = 0;
	int			toowide_cnt = 0;
	double		total_width = 0;
	bool		is_varlena = (!stats->attr->attbyval &&
							  stats->attr->attlen == -1);
	bool		is_varwidth = (!stats->attr->attbyval &&
							   stats->attr->attlen < 0);
	FmgrInfo	f_cmpeq;
	typedef struct
	{
		Datum		value;
		int			count;
	} TrackItem;
	TrackItem  *track;
	int			track_cnt,
				track_max;
	int			num_mcv = stats->attr->attstattarget;
	StdAnalyzeData *mystats = (StdAnalyzeData *) stats->extra_data;

	/*
	 * We track up to 2*n values for an n-element MCV list; but at least 10
	 */
	track_max = 2 * num_mcv;
	if (track_max < 10)
		track_max = 10;
	track = (TrackItem *) palloc(track_max * sizeof(TrackItem));
	track_cnt = 0;

	fmgr_info(mystats->eqfunc, &f_cmpeq);

	for (i = 0; i < samplerows; i++)
	{
		Datum		value;
		bool		isnull;
		bool		match;
		int			firstcount1,
					j;

		vacuum_delay_point();

		value = fetchfunc(stats, i, &isnull);

		/* Check for null/nonnull */
		if (isnull)
		{
			null_cnt++;
			continue;
		}
		nonnull_cnt++;

		/*
		 * If it's a variable-width field, add up widths for average width
		 * calculation.  Note that if the value is toasted, we use the toasted
		 * width.  We don't bother with this calculation if it's a fixed-width
		 * type.
		 */
		if (is_varlena)
		{
			total_width += VARSIZE_ANY(DatumGetPointer(value));

			/*
			 * If the value is toasted, we want to detoast it just once to
			 * avoid repeated detoastings and resultant excess memory usage
			 * during the comparisons.	Also, check to see if the value is
			 * excessively wide, and if so don't detoast at all --- just
			 * ignore the value.
			 */
			if (toast_raw_datum_size(value) > WIDTH_THRESHOLD)
			{
				toowide_cnt++;
				continue;
			}
			value = PointerGetDatum(PG_DETOAST_DATUM(value));
		}
		else if (is_varwidth)
		{
			/* must be cstring */
			total_width += strlen(DatumGetCString(value)) + 1;
		}

		/*
		 * See if the value matches anything we're already tracking.
		 */
		match = false;
		firstcount1 = track_cnt;
		for (j = 0; j < track_cnt; j++)
		{
			if (DatumGetBool(FunctionCall2(&f_cmpeq, value, track[j].value)))
			{
				match = true;
				break;
			}
			if (j < firstcount1 && track[j].count == 1)
				firstcount1 = j;
		}

		if (match)
		{
			/* Found a match */
			track[j].count++;
			/* This value may now need to "bubble up" in the track list */
			while (j > 0 && track[j].count > track[j - 1].count)
			{
				swapDatum(track[j].value, track[j - 1].value);
				swapInt(track[j].count, track[j - 1].count);
				j--;
			}
		}
		else
		{
			/* No match.  Insert at head of count-1 list */
			if (track_cnt < track_max)
				track_cnt++;
			for (j = track_cnt - 1; j > firstcount1; j--)
			{
				track[j].value = track[j - 1].value;
				track[j].count = track[j - 1].count;
			}
			if (firstcount1 < track_cnt)
			{
				track[firstcount1].value = value;
				track[firstcount1].count = 1;
			}
		}
	}

	/* We can only compute real stats if we found some non-null values. */
	if (nonnull_cnt > 0)
	{
		int			nmultiple,
					summultiple;

		stats->stats_valid = true;
		/* Do the simple null-frac and width stats */
		stats->stanullfrac = (double) null_cnt / (double) samplerows;
		if (is_varwidth)
			stats->stawidth = total_width / (double) nonnull_cnt;
		else
			stats->stawidth = stats->attrtype->typlen;

		/* Count the number of values we found multiple times */
		summultiple = 0;
		for (nmultiple = 0; nmultiple < track_cnt; nmultiple++)
		{
			if (track[nmultiple].count == 1)
				break;
			summultiple += track[nmultiple].count;
		}

		if (nmultiple == 0)
		{
			/* If we found no repeated values, assume it's a unique column */
			stats->stadistinct = -1.0;
		}
		else if (track_cnt < track_max && toowide_cnt == 0 &&
				 nmultiple == track_cnt)
		{
			/*
			 * Our track list includes every value in the sample, and every
			 * value appeared more than once.  Assume the column has just
			 * these values.
			 */
			stats->stadistinct = track_cnt;
		}
		else
		{
			/*----------
			 * Estimate the number of distinct values using the estimator
			 * proposed by Haas and Stokes in IBM Research Report RJ 10025:
			 *		n*d / (n - f1 + f1*n/N)
			 * where f1 is the number of distinct values that occurred
			 * exactly once in our sample of n rows (from a total of N),
			 * and d is the total number of distinct values in the sample.
			 * This is their Duj1 estimator; the other estimators they
			 * recommend are considerably more complex, and are numerically
			 * very unstable when n is much smaller than N.
			 *
			 * We assume (not very reliably!) that all the multiply-occurring
			 * values are reflected in the final track[] list, and the other
			 * nonnull values all appeared but once.  (XXX this usually
			 * results in a drastic overestimate of ndistinct.	Can we do
			 * any better?)
			 *----------
			 */
			int			f1 = nonnull_cnt - summultiple;
			int			d = f1 + nmultiple;
			double		numer,
						denom,
						stadistinct;

			numer = (double) samplerows *(double) d;

			denom = (double) (samplerows - f1) +
				(double) f1 *(double) samplerows / totalrows;

			stadistinct = numer / denom;
			/* Clamp to sane range in case of roundoff error */
			if (stadistinct < (double) d)
				stadistinct = (double) d;
			if (stadistinct > totalrows)
				stadistinct = totalrows;
			stats->stadistinct = floor(stadistinct + 0.5);
		}

		/*
		 * If we estimated the number of distinct values at more than 10% of
		 * the total row count (a very arbitrary limit), then assume that
		 * stadistinct should scale with the row count rather than be a fixed
		 * value.
		 */
		if (stats->stadistinct > 0.1 * totalrows)
			stats->stadistinct = -(stats->stadistinct / totalrows);

		/*
		 * Decide how many values are worth storing as most-common values. If
		 * we are able to generate a complete MCV list (all the values in the
		 * sample will fit, and we think these are all the ones in the table),
		 * then do so.	Otherwise, store only those values that are
		 * significantly more common than the (estimated) average. We set the
		 * threshold rather arbitrarily at 25% more than average, with at
		 * least 2 instances in the sample.
		 */
		if (track_cnt < track_max && toowide_cnt == 0 &&
			stats->stadistinct > 0 &&
			track_cnt <= num_mcv)
		{
			/* Track list includes all values seen, and all will fit */
			num_mcv = track_cnt;
		}
		else
		{
			double		ndistinct = stats->stadistinct;
			double		avgcount,
						mincount;

			if (ndistinct < 0)
				ndistinct = -ndistinct * totalrows;
			/* estimate # of occurrences in sample of a typical value */
			avgcount = (double) samplerows / ndistinct;
			/* set minimum threshold count to store a value */
			mincount = avgcount * 1.25;
			if (mincount < 2)
				mincount = 2;
			if (num_mcv > track_cnt)
				num_mcv = track_cnt;
			for (i = 0; i < num_mcv; i++)
			{
				if (track[i].count < mincount)
				{
					num_mcv = i;
					break;
				}
			}
		}

		/* Generate MCV slot entry */
		if (num_mcv > 0)
		{
			MemoryContext old_context;
			Datum	   *mcv_values;
			float4	   *mcv_freqs;

			/* Must copy the target values into anl_context */
			old_context = MemoryContextSwitchTo(stats->anl_context);
			mcv_values = (Datum *) palloc(num_mcv * sizeof(Datum));
			mcv_freqs = (float4 *) palloc(num_mcv * sizeof(float4));
			for (i = 0; i < num_mcv; i++)
			{
				mcv_values[i] = datumCopy(track[i].value,
										  stats->attr->attbyval,
										  stats->attr->attlen);
				mcv_freqs[i] = (double) track[i].count / (double) samplerows;
			}
			MemoryContextSwitchTo(old_context);

			stats->stakind[0] = STATISTIC_KIND_MCV;
			stats->staop[0] = mystats->eqopr;
			stats->stanumbers[0] = mcv_freqs;
			stats->numnumbers[0] = num_mcv;
			stats->stavalues[0] = mcv_values;
			stats->numvalues[0] = num_mcv;
		}
	}
	else if (null_cnt > 0)
	{
		/* We found only nulls; assume the column is entirely null */
		stats->stats_valid = true;
		stats->stanullfrac = 1.0;
		if (is_varwidth)
			stats->stawidth = 0;	/* "unknown" */
		else
			stats->stawidth = stats->attrtype->typlen;
		stats->stadistinct = 0.0;		/* "unknown" */
	}

	/* We don't need to bother cleaning up any of our temporary palloc's */
}


/*
 *	compute_scalar_stats() -- compute column statistics
 *
 *	We use this when we can find "=" and "<" operators for the datatype.
 *
 *	We determine the fraction of non-null rows, the average width, the
 *	most common values, the (estimated) number of distinct values, the
 *	distribution histogram, and the correlation of physical to logical order.
 *
 *	The desired stats can be determined fairly easily after sorting the
 *	data values into order.
 */
static void
compute_scalar_stats(VacAttrStatsP stats,
					 AnalyzeAttrFetchFunc fetchfunc,
					 int samplerows,
					 double totalrows)
{
	int			i;
	int			null_cnt = 0;
	int			nonnull_cnt = 0;
	int			toowide_cnt = 0;
	double		total_width = 0;
	bool		is_varlena = (!stats->attr->attbyval &&
							  stats->attr->attlen == -1);
	bool		is_varwidth = (!stats->attr->attbyval &&
							   stats->attr->attlen < 0);
	double		corr_xysum;
	Oid			cmpFn;
	int			cmpFlags;
	FmgrInfo	f_cmpfn;
	ScalarItem *values;
	int			values_cnt = 0;
	int		   *tupnoLink;
	ScalarMCVItem *track;
	int			track_cnt = 0;
	int			num_mcv = stats->attr->attstattarget;
	int			num_bins = stats->attr->attstattarget;
	StdAnalyzeData *mystats = (StdAnalyzeData *) stats->extra_data;

	values = (ScalarItem *) palloc(samplerows * sizeof(ScalarItem));
	tupnoLink = (int *) palloc(samplerows * sizeof(int));
	track = (ScalarMCVItem *) palloc(num_mcv * sizeof(ScalarMCVItem));

	SelectSortFunction(mystats->ltopr, false, &cmpFn, &cmpFlags);
	fmgr_info(cmpFn, &f_cmpfn);

	/* Initial scan to find sortable values */
	for (i = 0; i < samplerows; i++)
	{
		Datum		value;
		bool		isnull;

		vacuum_delay_point();

		value = fetchfunc(stats, i, &isnull);

		/* Check for null/nonnull */
		if (isnull)
		{
			null_cnt++;
			continue;
		}
		nonnull_cnt++;

		/*
		 * If it's a variable-width field, add up widths for average width
		 * calculation.  Note that if the value is toasted, we use the toasted
		 * width.  We don't bother with this calculation if it's a fixed-width
		 * type.
		 */
		if (is_varlena)
		{
			total_width += VARSIZE_ANY(DatumGetPointer(value));

			/*
			 * If the value is toasted, we want to detoast it just once to
			 * avoid repeated detoastings and resultant excess memory usage
			 * during the comparisons.	Also, check to see if the value is
			 * excessively wide, and if so don't detoast at all --- just
			 * ignore the value.
			 */
			if (toast_raw_datum_size(value) > WIDTH_THRESHOLD)
			{
				toowide_cnt++;
				continue;
			}
			value = PointerGetDatum(PG_DETOAST_DATUM(value));
		}
		else if (is_varwidth)
		{
			/* must be cstring */
			total_width += strlen(DatumGetCString(value)) + 1;
		}

		/* Add it to the list to be sorted */
		values[values_cnt].value = value;
		values[values_cnt].tupno = values_cnt;
		tupnoLink[values_cnt] = values_cnt;
		values_cnt++;
	}

	/* We can only compute real stats if we found some sortable values. */
	if (values_cnt > 0)
	{
		int			ndistinct,	/* # distinct values in sample */
					nmultiple,	/* # that appear multiple times */
					num_hist,
					dups_cnt;
		int			slot_idx = 0;
		CompareScalarsContext cxt;

		/* Sort the collected values */
		cxt.cmpFn = &f_cmpfn;
		cxt.cmpFlags = cmpFlags;
		cxt.tupnoLink = tupnoLink;
		qsort_arg((void *) values, values_cnt, sizeof(ScalarItem),
				  compare_scalars, (void *) &cxt);

		/*
		 * Now scan the values in order, find the most common ones, and also
		 * accumulate ordering-correlation statistics.
		 *
		 * To determine which are most common, we first have to count the
		 * number of duplicates of each value.	The duplicates are adjacent in
		 * the sorted list, so a brute-force approach is to compare successive
		 * datum values until we find two that are not equal. However, that
		 * requires N-1 invocations of the datum comparison routine, which are
		 * completely redundant with work that was done during the sort.  (The
		 * sort algorithm must at some point have compared each pair of items
		 * that are adjacent in the sorted order; otherwise it could not know
		 * that it's ordered the pair correctly.) We exploit this by having
		 * compare_scalars remember the highest tupno index that each
		 * ScalarItem has been found equal to.	At the end of the sort, a
		 * ScalarItem's tupnoLink will still point to itself if and only if it
		 * is the last item of its group of duplicates (since the group will
		 * be ordered by tupno).
		 */
		corr_xysum = 0;
		ndistinct = 0;
		nmultiple = 0;
		dups_cnt = 0;
		for (i = 0; i < values_cnt; i++)
		{
			int			tupno = values[i].tupno;

			corr_xysum += ((double) i) * ((double) tupno);
			dups_cnt++;
			if (tupnoLink[tupno] == tupno)
			{
				/* Reached end of duplicates of this value */
				ndistinct++;
				if (dups_cnt > 1)
				{
					nmultiple++;
					if (track_cnt < num_mcv ||
						dups_cnt > track[track_cnt - 1].count)
					{
						/*
						 * Found a new item for the mcv list; find its
						 * position, bubbling down old items if needed. Loop
						 * invariant is that j points at an empty/ replaceable
						 * slot.
						 */
						int			j;

						if (track_cnt < num_mcv)
							track_cnt++;
						for (j = track_cnt - 1; j > 0; j--)
						{
							if (dups_cnt <= track[j - 1].count)
								break;
							track[j].count = track[j - 1].count;
							track[j].first = track[j - 1].first;
						}
						track[j].count = dups_cnt;
						track[j].first = i + 1 - dups_cnt;
					}
				}
				dups_cnt = 0;
			}
		}

		stats->stats_valid = true;
		/* Do the simple null-frac and width stats */
		stats->stanullfrac = (double) null_cnt / (double) samplerows;
		if (is_varwidth)
			stats->stawidth = total_width / (double) nonnull_cnt;
		else
			stats->stawidth = stats->attrtype->typlen;

		if (nmultiple == 0)
		{
			/* If we found no repeated values, assume it's a unique column */
			stats->stadistinct = -1.0;
		}
		else if (toowide_cnt == 0 && nmultiple == ndistinct)
		{
			/*
			 * Every value in the sample appeared more than once.  Assume the
			 * column has just these values.
			 */
			stats->stadistinct = ndistinct;
		}
		else
		{
			/*----------
			 * Estimate the number of distinct values using the estimator
			 * proposed by Haas and Stokes in IBM Research Report RJ 10025:
			 *		n*d / (n - f1 + f1*n/N)
			 * where f1 is the number of distinct values that occurred
			 * exactly once in our sample of n rows (from a total of N),
			 * and d is the total number of distinct values in the sample.
			 * This is their Duj1 estimator; the other estimators they
			 * recommend are considerably more complex, and are numerically
			 * very unstable when n is much smaller than N.
			 *
			 * Overwidth values are assumed to have been distinct.
			 *----------
			 */
			int			f1 = ndistinct - nmultiple + toowide_cnt;
			int			d = f1 + nmultiple;
			double		numer,
						denom,
						stadistinct;

			numer = (double) samplerows *(double) d;

			denom = (double) (samplerows - f1) +
				(double) f1 *(double) samplerows / totalrows;

			stadistinct = numer / denom;
			/* Clamp to sane range in case of roundoff error */
			if (stadistinct < (double) d)
				stadistinct = (double) d;
			if (stadistinct > totalrows)
				stadistinct = totalrows;
			stats->stadistinct = floor(stadistinct + 0.5);
		}

		/*
		 * If we estimated the number of distinct values at more than 10% of
		 * the total row count (a very arbitrary limit), then assume that
		 * stadistinct should scale with the row count rather than be a fixed
		 * value.
		 */
		if (stats->stadistinct > 0.1 * totalrows)
			stats->stadistinct = -(stats->stadistinct / totalrows);

		/*
		 * Decide how many values are worth storing as most-common values. If
		 * we are able to generate a complete MCV list (all the values in the
		 * sample will fit, and we think these are all the ones in the table),
		 * then do so.	Otherwise, store only those values that are
		 * significantly more common than the (estimated) average. We set the
		 * threshold rather arbitrarily at 25% more than average, with at
		 * least 2 instances in the sample.  Also, we won't suppress values
		 * that have a frequency of at least 1/K where K is the intended
		 * number of histogram bins; such values might otherwise cause us to
		 * emit duplicate histogram bin boundaries.
		 */
		if (track_cnt == ndistinct && toowide_cnt == 0 &&
			stats->stadistinct > 0 &&
			track_cnt <= num_mcv)
		{
			/* Track list includes all values seen, and all will fit */
			num_mcv = track_cnt;
		}
		else
		{
			double		ndistinct = stats->stadistinct;
			double		avgcount,
						mincount,
						maxmincount;

			if (ndistinct < 0)
				ndistinct = -ndistinct * totalrows;
			/* estimate # of occurrences in sample of a typical value */
			avgcount = (double) samplerows / ndistinct;
			/* set minimum threshold count to store a value */
			mincount = avgcount * 1.25;
			if (mincount < 2)
				mincount = 2;
			/* don't let threshold exceed 1/K, however */
			maxmincount = (double) samplerows / (double) num_bins;
			if (mincount > maxmincount)
				mincount = maxmincount;
			if (num_mcv > track_cnt)
				num_mcv = track_cnt;
			for (i = 0; i < num_mcv; i++)
			{
				if (track[i].count < mincount)
				{
					num_mcv = i;
					break;
				}
			}
		}

		/* Generate MCV slot entry */
		if (num_mcv > 0)
		{
			MemoryContext old_context;
			Datum	   *mcv_values;
			float4	   *mcv_freqs;

			/* Must copy the target values into anl_context */
			old_context = MemoryContextSwitchTo(stats->anl_context);
			mcv_values = (Datum *) palloc(num_mcv * sizeof(Datum));
			mcv_freqs = (float4 *) palloc(num_mcv * sizeof(float4));
			for (i = 0; i < num_mcv; i++)
			{
				mcv_values[i] = datumCopy(values[track[i].first].value,
										  stats->attr->attbyval,
										  stats->attr->attlen);
				mcv_freqs[i] = (double) track[i].count / (double) samplerows;
			}
			MemoryContextSwitchTo(old_context);

			stats->stakind[slot_idx] = STATISTIC_KIND_MCV;
			stats->staop[slot_idx] = mystats->eqopr;
			stats->stanumbers[slot_idx] = mcv_freqs;
			stats->numnumbers[slot_idx] = num_mcv;
			stats->stavalues[slot_idx] = mcv_values;
			stats->numvalues[slot_idx] = num_mcv;
			slot_idx++;
		}

		/*
		 * Generate a histogram slot entry if there are at least two distinct
		 * values not accounted for in the MCV list.  (This ensures the
		 * histogram won't collapse to empty or a singleton.)
		 */
		num_hist = ndistinct - num_mcv;
		if (num_hist > num_bins)
			num_hist = num_bins + 1;
		if (num_hist >= 2)
		{
			MemoryContext old_context;
			Datum	   *hist_values;
			int			nvals;

			/* Sort the MCV items into position order to speed next loop */
			qsort((void *) track, num_mcv,
				  sizeof(ScalarMCVItem), compare_mcvs);

			/*
			 * Collapse out the MCV items from the values[] array.
			 *
			 * Note we destroy the values[] array here... but we don't need it
			 * for anything more.  We do, however, still need values_cnt.
			 * nvals will be the number of remaining entries in values[].
			 */
			if (num_mcv > 0)
			{
				int			src,
							dest;
				int			j;

				src = dest = 0;
				j = 0;			/* index of next interesting MCV item */
				while (src < values_cnt)
				{
					int			ncopy;

					if (j < num_mcv)
					{
						int			first = track[j].first;

						if (src >= first)
						{
							/* advance past this MCV item */
							src = first + track[j].count;
							j++;
							continue;
						}
						ncopy = first - src;
					}
					else
						ncopy = values_cnt - src;
					memmove(&values[dest], &values[src],
							ncopy * sizeof(ScalarItem));
					src += ncopy;
					dest += ncopy;
				}
				nvals = dest;
			}
			else
				nvals = values_cnt;
			Assert(nvals >= num_hist);

			/* Must copy the target values into anl_context */
			old_context = MemoryContextSwitchTo(stats->anl_context);
			hist_values = (Datum *) palloc(num_hist * sizeof(Datum));
			for (i = 0; i < num_hist; i++)
			{
				int			pos;

				pos = (i * (nvals - 1)) / (num_hist - 1);
				hist_values[i] = datumCopy(values[pos].value,
										   stats->attr->attbyval,
										   stats->attr->attlen);
			}
			MemoryContextSwitchTo(old_context);

			stats->stakind[slot_idx] = STATISTIC_KIND_HISTOGRAM;
			stats->staop[slot_idx] = mystats->ltopr;
			stats->stavalues[slot_idx] = hist_values;
			stats->numvalues[slot_idx] = num_hist;
			slot_idx++;
		}

		/* Generate a correlation entry if there are multiple values */
		if (values_cnt > 1)
		{
			MemoryContext old_context;
			float4	   *corrs;
			double		corr_xsum,
						corr_x2sum;

			/* Must copy the target values into anl_context */
			old_context = MemoryContextSwitchTo(stats->anl_context);
			corrs = (float4 *) palloc(sizeof(float4));
			MemoryContextSwitchTo(old_context);

			/*----------
			 * Since we know the x and y value sets are both
			 *		0, 1, ..., values_cnt-1
			 * we have sum(x) = sum(y) =
			 *		(values_cnt-1)*values_cnt / 2
			 * and sum(x^2) = sum(y^2) =
			 *		(values_cnt-1)*values_cnt*(2*values_cnt-1) / 6.
			 *----------
			 */
			corr_xsum = ((double) (values_cnt - 1)) *
				((double) values_cnt) / 2.0;
			corr_x2sum = ((double) (values_cnt - 1)) *
				((double) values_cnt) * (double) (2 * values_cnt - 1) / 6.0;

			/* And the correlation coefficient reduces to */
			corrs[0] = (values_cnt * corr_xysum - corr_xsum * corr_xsum) /
				(values_cnt * corr_x2sum - corr_xsum * corr_xsum);

			stats->stakind[slot_idx] = STATISTIC_KIND_CORRELATION;
			stats->staop[slot_idx] = mystats->ltopr;
			stats->stanumbers[slot_idx] = corrs;
			stats->numnumbers[slot_idx] = 1;
			slot_idx++;
		}
	}
	else if (nonnull_cnt == 0 && null_cnt > 0)
	{
		/* We found only nulls; assume the column is entirely null */
		stats->stats_valid = true;
		stats->stanullfrac = 1.0;
		if (is_varwidth)
			stats->stawidth = 0;	/* "unknown" */
		else
			stats->stawidth = stats->attrtype->typlen;
		stats->stadistinct = 0.0;		/* "unknown" */
	}

	/* We don't need to bother cleaning up any of our temporary palloc's */
}

/*
 * qsort_arg comparator for sorting ScalarItems
 *
 * Aside from sorting the items, we update the tupnoLink[] array
 * whenever two ScalarItems are found to contain equal datums.	The array
 * is indexed by tupno; for each ScalarItem, it contains the highest
 * tupno that that item's datum has been found to be equal to.  This allows
 * us to avoid additional comparisons in compute_scalar_stats().
 */
static int
compare_scalars(const void *a, const void *b, void *arg)
{
	Datum		da = ((ScalarItem *) a)->value;
	int			ta = ((ScalarItem *) a)->tupno;
	Datum		db = ((ScalarItem *) b)->value;
	int			tb = ((ScalarItem *) b)->tupno;
	CompareScalarsContext *cxt = (CompareScalarsContext *) arg;
	int32		compare;

	compare = ApplySortFunction(cxt->cmpFn, cxt->cmpFlags,
								da, false, db, false);
	if (compare != 0)
		return compare;

	/*
	 * The two datums are equal, so update cxt->tupnoLink[].
	 */
	if (cxt->tupnoLink[ta] < tb)
		cxt->tupnoLink[ta] = tb;
	if (cxt->tupnoLink[tb] < ta)
		cxt->tupnoLink[tb] = ta;

	/*
	 * For equal datums, sort by tupno
	 */
	return ta - tb;
}

/*
 * qsort comparator for sorting ScalarMCVItems by position
 */
static int
compare_mcvs(const void *a, const void *b)
{
	int			da = ((ScalarMCVItem *) a)->first;
	int			db = ((ScalarMCVItem *) b)->first;

	return da - db;
}
