postgres/src/backend/statistics/mcv.c

/*-------------------------------------------------------------------------
 *
 * mcv.c
 *	  POSTGRES multivariate MCV lists
 *
 *
 * Portions Copyright (c) 1996-2022, PostgreSQL Global Development Group
 * Portions Copyright (c) 1994, Regents of the University of California
 *
 * IDENTIFICATION
 *	  src/backend/statistics/mcv.c
 *
 *-------------------------------------------------------------------------
 */
#include "postgres.h"

#include <math.h>

#include "access/htup_details.h"
#include "catalog/pg_collation.h"
#include "catalog/pg_statistic_ext.h"
#include "catalog/pg_statistic_ext_data.h"
#include "fmgr.h"
#include "funcapi.h"
#include "nodes/nodeFuncs.h"
#include "optimizer/clauses.h"
#include "statistics/extended_stats_internal.h"
#include "statistics/statistics.h"
#include "utils/array.h"
#include "utils/builtins.h"
#include "utils/bytea.h"
#include "utils/fmgroids.h"
#include "utils/fmgrprotos.h"
#include "utils/lsyscache.h"
#include "utils/selfuncs.h"
#include "utils/syscache.h"
#include "utils/typcache.h"

/*
 * Computes size of a serialized MCV item, depending on the number of
 * dimensions (columns) the statistic is defined on. The datum values are
 * stored in a separate array (deduplicated, to minimize the size), and
 * so the serialized items only store uint16 indexes into that array.
 *
 * Each serialized item stores (in this order):
 *
 * - indexes to values	  (ndim * sizeof(uint16))
 * - null flags			  (ndim * sizeof(bool))
 * - frequency			  (sizeof(double))
 * - base_frequency		  (sizeof(double))
 *
 * There is no alignment padding within an MCV item.
 * So in total each MCV item requires this many bytes:
 *
 *	 ndim * (sizeof(uint16) + sizeof(bool)) + 2 * sizeof(double)
 */
#define ITEM_SIZE(ndims)	\
	((ndims) * (sizeof(uint16) + sizeof(bool)) + 2 * sizeof(double))

/*
 * Used to compute size of serialized MCV list representation.
 */
#define MinSizeOfMCVList		\
	(VARHDRSZ + sizeof(uint32) * 3 + sizeof(AttrNumber))

/*
 * Size of the serialized MCV list, excluding the space needed for
 * deduplicated per-dimension values. The macro is meant to be used
 * when it's not yet safe to access the serialized info about amount
 * of data for each column.
 */
#define SizeOfMCVList(ndims,nitems)	\
	((MinSizeOfMCVList + sizeof(Oid) * (ndims)) + \
	 ((ndims) * sizeof(DimensionInfo)) + \
	 ((nitems) * ITEM_SIZE(ndims)))

static MultiSortSupport build_mss(StatsBuildData *data);

static SortItem *build_distinct_groups(int numrows, SortItem *items,
									   MultiSortSupport mss, int *ndistinct);

static SortItem **build_column_frequencies(SortItem *groups, int ngroups,
										   MultiSortSupport mss, int *ncounts);

static int	count_distinct_groups(int numrows, SortItem *items,
								  MultiSortSupport mss);

/*
 * Compute new value for bitmap item, considering whether it's used for
 * clauses connected by AND/OR.
 */
#define RESULT_MERGE(value, is_or, match) \
	((is_or) ? ((value) || (match)) : ((value) && (match)))

/*
 * When processing a list of clauses, the bitmap item may get set to a value
 * such that additional clauses can't change it. For example, when processing
 * a list of clauses connected to AND, as soon as the item gets set to 'false'
 * then it'll remain like that. Similarly clauses connected by OR and 'true'.
 *
 * Returns true when the value in the bitmap can't change no matter how the
 * remaining clauses are evaluated.
 */
#define RESULT_IS_FINAL(value, is_or)	((is_or) ? (value) : (!(value)))

/*
 * get_mincount_for_mcv_list
 * 		Determine the minimum number of times a value needs to appear in
 * 		the sample for it to be included in the MCV list.
 *
 * We want to keep only values that appear sufficiently often in the
 * sample that it is reasonable to extrapolate their sample frequencies to
 * the entire table.  We do this by placing an upper bound on the relative
 * standard error of the sample frequency, so that any estimates the
 * planner generates from the MCV statistics can be expected to be
 * reasonably accurate.
 *
 * Since we are sampling without replacement, the sample frequency of a
 * particular value is described by a hypergeometric distribution.  A
 * common rule of thumb when estimating errors in this situation is to
 * require at least 10 instances of the value in the sample, in which case
 * the distribution can be approximated by a normal distribution, and
 * standard error analysis techniques can be applied.  Given a sample size
 * of n, a population size of N, and a sample frequency of p=cnt/n, the
 * standard error of the proportion p is given by
 *		SE = sqrt(p*(1-p)/n) * sqrt((N-n)/(N-1))
 * where the second term is the finite population correction.  To get
 * reasonably accurate planner estimates, we impose an upper bound on the
 * relative standard error of 20% -- i.e., SE/p < 0.2.  This 20% relative
 * error bound is fairly arbitrary, but has been found empirically to work
 * well.  Rearranging this formula gives a lower bound on the number of
 * instances of the value seen:
 *		cnt > n*(N-n) / (N-n+0.04*n*(N-1))
 * This bound is at most 25, and approaches 0 as n approaches 0 or N. The
 * case where n approaches 0 cannot happen in practice, since the sample
 * size is at least 300.  The case where n approaches N corresponds to
 * sampling the whole table, in which case it is reasonable to keep
 * the whole MCV list (have no lower bound), so it makes sense to apply
 * this formula for all inputs, even though the above derivation is
 * technically only valid when the right hand side is at least around 10.
 *
 * An alternative way to look at this formula is as follows -- assume that
 * the number of instances of the value seen scales up to the entire
 * table, so that the population count is K=N*cnt/n. Then the distribution
 * in the sample is a hypergeometric distribution parameterised by N, n
 * and K, and the bound above is mathematically equivalent to demanding
 * that the standard deviation of that distribution is less than 20% of
 * its mean.  Thus the relative errors in any planner estimates produced
 * from the MCV statistics are likely to be not too large.
 */
static double
get_mincount_for_mcv_list(int samplerows, double totalrows)
{
	double		n = samplerows;
	double		N = totalrows;
	double		numer,
				denom;

	numer = n * (N - n);
	denom = N - n + 0.04 * n * (N - 1);

	/* Guard against division by zero (possible if n = N = 1) */
	if (denom == 0.0)
		return 0.0;

	return numer / denom;
}

/*
 * Builds MCV list from the set of sampled rows.
 *
 * The algorithm is quite simple:
 *
 *	   (1) sort the data (default collation, '<' for the data type)
 *
 *	   (2) count distinct groups, decide how many to keep
 *
 *	   (3) build the MCV list using the threshold determined in (2)
 *
 *	   (4) remove rows represented by the MCV from the sample
 *
 */
MCVList *
statext_mcv_build(StatsBuildData *data, double totalrows, int stattarget)
{
	int			i,
				numattrs,
				numrows,
				ngroups,
				nitems;
	double		mincount;
	SortItem   *items;
	SortItem   *groups;
	MCVList    *mcvlist = NULL;
	MultiSortSupport mss;

	/* comparator for all the columns */
	mss = build_mss(data);

	/* sort the rows */
	items = build_sorted_items(data, &nitems, mss,
							   data->nattnums, data->attnums);

	if (!items)
		return NULL;

	/* for convenience */
	numattrs = data->nattnums;
	numrows = data->numrows;

	/* transform the sorted rows into groups (sorted by frequency) */
	groups = build_distinct_groups(nitems, items, mss, &ngroups);

	/*
	 * The maximum number of MCV items to store, based on the statistics
	 * target we computed for the statistics object (from the target set for
	 * the object itself, attributes and the system default). In any case, we
	 * can't keep more groups than we have available.
	 */
	nitems = stattarget;
	if (nitems > ngroups)
		nitems = ngroups;

	/*
	 * Decide how many items to keep in the MCV list. We can't use the same
	 * algorithm as per-column MCV lists, because that only considers the
	 * actual group frequency - but we're primarily interested in how the
	 * actual frequency differs from the base frequency (product of simple
	 * per-column frequencies, as if the columns were independent).
	 *
	 * Using the same algorithm might exclude items that are close to the
	 * "average" frequency of the sample. But that does not say whether the
	 * observed frequency is close to the base frequency or not. We also need
	 * to consider unexpectedly uncommon items (again, compared to the base
	 * frequency), and the single-column algorithm does not have to.
	 *
	 * We simply decide how many items to keep by computing the minimum count
	 * using get_mincount_for_mcv_list() and then keep all items that seem to
	 * be more common than that.
	 */
	mincount = get_mincount_for_mcv_list(numrows, totalrows);

	/*
	 * Walk the groups until we find the first group with a count below the
	 * mincount threshold (the index of that group is the number of groups we
	 * want to keep).
	 */
	for (i = 0; i < nitems; i++)
	{
		if (groups[i].count < mincount)
		{
			nitems = i;
			break;
		}
	}

	/*
	 * At this point, we know the number of items for the MCV list. There
	 * might be none (for uniform distribution with many groups), and in that
	 * case, there will be no MCV list. Otherwise, construct the MCV list.
	 */
	if (nitems > 0)
	{
		int			j;
		SortItem	key;
		MultiSortSupport tmp;

		/* frequencies for values in each attribute */
		SortItem  **freqs;
		int		   *nfreqs;

		/* used to search values */
		tmp = (MultiSortSupport) palloc(offsetof(MultiSortSupportData, ssup)
										+ sizeof(SortSupportData));

		/* compute frequencies for values in each column */
		nfreqs = (int *) palloc0(sizeof(int) * numattrs);
		freqs = build_column_frequencies(groups, ngroups, mss, nfreqs);

		/*
		 * Allocate the MCV list structure, set the global parameters.
		 */
		mcvlist = (MCVList *) palloc0(offsetof(MCVList, items) +
									  sizeof(MCVItem) * nitems);

		mcvlist->magic = STATS_MCV_MAGIC;
		mcvlist->type = STATS_MCV_TYPE_BASIC;
		mcvlist->ndimensions = numattrs;
		mcvlist->nitems = nitems;

		/* store info about data type OIDs */
		for (i = 0; i < numattrs; i++)
			mcvlist->types[i] = data->stats[i]->attrtypid;

		/* Copy the first chunk of groups into the result. */
		for (i = 0; i < nitems; i++)
		{
			/* just point to the proper place in the list */
			MCVItem    *item = &mcvlist->items[i];

			item->values = (Datum *) palloc(sizeof(Datum) * numattrs);
			item->isnull = (bool *) palloc(sizeof(bool) * numattrs);

			/* copy values for the group */
			memcpy(item->values, groups[i].values, sizeof(Datum) * numattrs);
			memcpy(item->isnull, groups[i].isnull, sizeof(bool) * numattrs);

			/* groups should be sorted by frequency in descending order */
			Assert((i == 0) || (groups[i - 1].count >= groups[i].count));

			/* group frequency */
			item->frequency = (double) groups[i].count / numrows;

			/* base frequency, if the attributes were independent */
			item->base_frequency = 1.0;
			for (j = 0; j < numattrs; j++)
			{
				SortItem   *freq;

				/* single dimension */
				tmp->ndims = 1;
				tmp->ssup[0] = mss->ssup[j];

				/* fill search key */
				key.values = &groups[i].values[j];
				key.isnull = &groups[i].isnull[j];

				freq = (SortItem *) bsearch_arg(&key, freqs[j], nfreqs[j],
												sizeof(SortItem),
												multi_sort_compare, tmp);

				item->base_frequency *= ((double) freq->count) / numrows;
			}
		}

		pfree(nfreqs);
		pfree(freqs);
	}

	pfree(items);
	pfree(groups);

	return mcvlist;
}

/*
 * build_mss
 *		Build a MultiSortSupport for the given StatsBuildData.
 */
static MultiSortSupport
build_mss(StatsBuildData *data)
{
	int			i;
	int			numattrs = data->nattnums;

	/* Sort by multiple columns (using array of SortSupport) */
	MultiSortSupport mss = multi_sort_init(numattrs);

	/* prepare the sort functions for all the attributes */
	for (i = 0; i < numattrs; i++)
	{
		VacAttrStats *colstat = data->stats[i];
		TypeCacheEntry *type;

		type = lookup_type_cache(colstat->attrtypid, TYPECACHE_LT_OPR);
		if (type->lt_opr == InvalidOid) /* shouldn't happen */
			elog(ERROR, "cache lookup failed for ordering operator for type %u",
				 colstat->attrtypid);

		multi_sort_add_dimension(mss, i, type->lt_opr, colstat->attrcollid);
	}

	return mss;
}

/*
 * count_distinct_groups
 *		Count distinct combinations of SortItems in the array.
 *
 * The array is assumed to be sorted according to the MultiSortSupport.
 */
static int
count_distinct_groups(int numrows, SortItem *items, MultiSortSupport mss)
{
	int			i;
	int			ndistinct;

	ndistinct = 1;
	for (i = 1; i < numrows; i++)
	{
		/* make sure the array really is sorted */
		Assert(multi_sort_compare(&items[i], &items[i - 1], mss) >= 0);

		if (multi_sort_compare(&items[i], &items[i - 1], mss) != 0)
			ndistinct += 1;
	}

	return ndistinct;
}

/*
 * compare_sort_item_count
 *		Comparator for sorting items by count (frequencies) in descending
 *		order.
 */
static int
compare_sort_item_count(const void *a, const void *b, void *arg)
{
	SortItem   *ia = (SortItem *) a;
	SortItem   *ib = (SortItem *) b;

	if (ia->count == ib->count)
		return 0;
	else if (ia->count > ib->count)
		return -1;

	return 1;
}

/*
 * build_distinct_groups
 *		Build an array of SortItems for distinct groups and counts matching
 *		items.
 *
 * The 'items' array is assumed to be sorted.
 */
static SortItem *
build_distinct_groups(int numrows, SortItem *items, MultiSortSupport mss,
					  int *ndistinct)
{
	int			i,
				j;
	int			ngroups = count_distinct_groups(numrows, items, mss);

	SortItem   *groups = (SortItem *) palloc(ngroups * sizeof(SortItem));

	j = 0;
	groups[0] = items[0];
	groups[0].count = 1;

	for (i = 1; i < numrows; i++)
	{
		/* Assume sorted in ascending order. */
		Assert(multi_sort_compare(&items[i], &items[i - 1], mss) >= 0);

		/* New distinct group detected. */
		if (multi_sort_compare(&items[i], &items[i - 1], mss) != 0)
		{
			groups[++j] = items[i];
			groups[j].count = 0;
		}

		groups[j].count++;
	}

	/* ensure we filled the expected number of distinct groups */
	Assert(j + 1 == ngroups);

	/* Sort the distinct groups by frequency (in descending order). */
	qsort_interruptible((void *) groups, ngroups, sizeof(SortItem),
						compare_sort_item_count, NULL);

	*ndistinct = ngroups;
	return groups;
}

/* compare sort items (single dimension) */
static int
sort_item_compare(const void *a, const void *b, void *arg)
{
	SortSupport ssup = (SortSupport) arg;
	SortItem   *ia = (SortItem *) a;
	SortItem   *ib = (SortItem *) b;

	return ApplySortComparator(ia->values[0], ia->isnull[0],
							   ib->values[0], ib->isnull[0],
							   ssup);
}

/*
 * build_column_frequencies
 *		Compute frequencies of values in each column.
 *
 * This returns an array of SortItems for each attribute the MCV is built
 * on, with a frequency (number of occurrences) for each value. This is
 * then used to compute "base" frequency of MCV items.
 *
 * All the memory is allocated in a single chunk, so that a single pfree
 * is enough to release it. We do not allocate space for values/isnull
 * arrays in the SortItems, because we can simply point into the input
 * groups directly.
 */
static SortItem **
build_column_frequencies(SortItem *groups, int ngroups,
						 MultiSortSupport mss, int *ncounts)
{
	int			i,
				dim;
	SortItem  **result;
	char	   *ptr;

	Assert(groups);
	Assert(ncounts);

	/* allocate arrays for all columns as a single chunk */
	ptr = palloc(MAXALIGN(sizeof(SortItem *) * mss->ndims) +
				 mss->ndims * MAXALIGN(sizeof(SortItem) * ngroups));

	/* initial array of pointers */
	result = (SortItem **) ptr;
	ptr += MAXALIGN(sizeof(SortItem *) * mss->ndims);

	for (dim = 0; dim < mss->ndims; dim++)
	{
		SortSupport ssup = &mss->ssup[dim];

		/* array of values for a single column */
		result[dim] = (SortItem *) ptr;
		ptr += MAXALIGN(sizeof(SortItem) * ngroups);

		/* extract data for the dimension */
		for (i = 0; i < ngroups; i++)
		{
			/* point into the input groups */
			result[dim][i].values = &groups[i].values[dim];
			result[dim][i].isnull = &groups[i].isnull[dim];
			result[dim][i].count = groups[i].count;
		}

		/* sort the values, deduplicate */
		qsort_interruptible((void *) result[dim], ngroups, sizeof(SortItem),
							sort_item_compare, ssup);

		/*
		 * Identify distinct values, compute frequency (there might be
		 * multiple MCV items containing this value, so we need to sum counts
		 * from all of them.
		 */
		ncounts[dim] = 1;
		for (i = 1; i < ngroups; i++)
		{
			if (sort_item_compare(&result[dim][i - 1], &result[dim][i], ssup) == 0)
			{
				result[dim][ncounts[dim] - 1].count += result[dim][i].count;
				continue;
			}

			result[dim][ncounts[dim]] = result[dim][i];

			ncounts[dim]++;
		}
	}

	return result;
}

/*
 * statext_mcv_load
 *		Load the MCV list for the indicated pg_statistic_ext_data tuple.
 */
MCVList *
statext_mcv_load(Oid mvoid, bool inh)
{
	MCVList    *result;
	bool		isnull;
	Datum		mcvlist;
	HeapTuple	htup = SearchSysCache2(STATEXTDATASTXOID,
									   ObjectIdGetDatum(mvoid), BoolGetDatum(inh));

	if (!HeapTupleIsValid(htup))
		elog(ERROR, "cache lookup failed for statistics object %u", mvoid);

	mcvlist = SysCacheGetAttr(STATEXTDATASTXOID, htup,
							  Anum_pg_statistic_ext_data_stxdmcv, &isnull);

	if (isnull)
		elog(ERROR,
			 "requested statistics kind \"%c\" is not yet built for statistics object %u",
			 STATS_EXT_DEPENDENCIES, mvoid);

	result = statext_mcv_deserialize(DatumGetByteaP(mcvlist));

	ReleaseSysCache(htup);

	return result;
}


/*
 * statext_mcv_serialize
 *		Serialize MCV list into a pg_mcv_list value.
 *
 * The MCV items may include values of various data types, and it's reasonable
 * to expect redundancy (values for a given attribute, repeated for multiple
 * MCV list items). So we deduplicate the values into arrays, and then replace
 * the values by indexes into those arrays.
 *
 * The overall structure of the serialized representation looks like this:
 *
 * +---------------+----------------+---------------------+-------+
 * | header fields | dimension info | deduplicated values | items |
 * +---------------+----------------+---------------------+-------+
 *
 * Where dimension info stores information about the type of the K-th
 * attribute (e.g. typlen, typbyval and length of deduplicated values).
 * Deduplicated values store deduplicated values for each attribute.  And
 * items store the actual MCV list items, with values replaced by indexes into
 * the arrays.
 *
 * When serializing the items, we use uint16 indexes. The number of MCV items
 * is limited by the statistics target (which is capped to 10k at the moment).
 * We might increase this to 65k and still fit into uint16, so there's a bit of
 * slack. Furthermore, this limit is on the number of distinct values per column,
 * and we usually have few of those (and various combinations of them for the
 * those MCV list). So uint16 seems fine for now.
 *
 * We don't really expect the serialization to save as much space as for
 * histograms, as we are not doing any bucket splits (which is the source
 * of high redundancy in histograms).
 *
 * TODO: Consider packing boolean flags (NULL) for each item into a single char
 * (or a longer type) instead of using an array of bool items.
 */
bytea *
statext_mcv_serialize(MCVList *mcvlist, VacAttrStats **stats)
{
	int			i;
	int			dim;
	int			ndims = mcvlist->ndimensions;

	SortSupport ssup;
	DimensionInfo *info;

	Size		total_length;

	/* serialized items (indexes into arrays, etc.) */
	bytea	   *raw;
	char	   *ptr;
	char	   *endptr PG_USED_FOR_ASSERTS_ONLY;

	/* values per dimension (and number of non-NULL values) */
	Datum	  **values = (Datum **) palloc0(sizeof(Datum *) * ndims);
	int		   *counts = (int *) palloc0(sizeof(int) * ndims);

	/*
	 * We'll include some rudimentary information about the attribute types
	 * (length, by-val flag), so that we don't have to look them up while
	 * deserializing the MCV list (we already have the type OID in the
	 * header).  This is safe because when changing the type of the attribute
	 * the statistics gets dropped automatically.  We need to store the info
	 * about the arrays of deduplicated values anyway.
	 */
	info = (DimensionInfo *) palloc0(sizeof(DimensionInfo) * ndims);

	/* sort support data for all attributes included in the MCV list */
	ssup = (SortSupport) palloc0(sizeof(SortSupportData) * ndims);

	/* collect and deduplicate values for each dimension (attribute) */
	for (dim = 0; dim < ndims; dim++)
	{
		int			ndistinct;
		TypeCacheEntry *typentry;

		/*
		 * Lookup the LT operator (can't get it from stats extra_data, as we
		 * don't know how to interpret that - scalar vs. array etc.).
		 */
		typentry = lookup_type_cache(stats[dim]->attrtypid, TYPECACHE_LT_OPR);

		/* copy important info about the data type (length, by-value) */
		info[dim].typlen = stats[dim]->attrtype->typlen;
		info[dim].typbyval = stats[dim]->attrtype->typbyval;

		/* allocate space for values in the attribute and collect them */
		values[dim] = (Datum *) palloc0(sizeof(Datum) * mcvlist->nitems);

		for (i = 0; i < mcvlist->nitems; i++)
		{
			/* skip NULL values - we don't need to deduplicate those */
			if (mcvlist->items[i].isnull[dim])
				continue;

			/* append the value at the end */
			values[dim][counts[dim]] = mcvlist->items[i].values[dim];
			counts[dim] += 1;
		}

		/* if there are just NULL values in this dimension, we're done */
		if (counts[dim] == 0)
			continue;

		/* sort and deduplicate the data */
		ssup[dim].ssup_cxt = CurrentMemoryContext;
		ssup[dim].ssup_collation = stats[dim]->attrcollid;
		ssup[dim].ssup_nulls_first = false;

		PrepareSortSupportFromOrderingOp(typentry->lt_opr, &ssup[dim]);

		qsort_interruptible(values[dim], counts[dim], sizeof(Datum),
							compare_scalars_simple, &ssup[dim]);

		/*
		 * Walk through the array and eliminate duplicate values, but keep the
		 * ordering (so that we can do a binary search later). We know there's
		 * at least one item as (counts[dim] != 0), so we can skip the first
		 * element.
		 */
		ndistinct = 1;			/* number of distinct values */
		for (i = 1; i < counts[dim]; i++)
		{
			/* expect sorted array */
			Assert(compare_datums_simple(values[dim][i - 1], values[dim][i], &ssup[dim]) <= 0);

			/* if the value is the same as the previous one, we can skip it */
			if (!compare_datums_simple(values[dim][i - 1], values[dim][i], &ssup[dim]))
				continue;

			values[dim][ndistinct] = values[dim][i];
			ndistinct += 1;
		}

		/* we must not exceed PG_UINT16_MAX, as we use uint16 indexes */
		Assert(ndistinct <= PG_UINT16_MAX);

		/*
		 * Store additional info about the attribute - number of deduplicated
		 * values, and also size of the serialized data. For fixed-length data
		 * types this is trivial to compute, for varwidth types we need to
		 * actually walk the array and sum the sizes.
		 */
		info[dim].nvalues = ndistinct;

		if (info[dim].typbyval) /* by-value data types */
		{
			info[dim].nbytes = info[dim].nvalues * info[dim].typlen;

			/*
			 * We copy the data into the MCV item during deserialization, so
			 * we don't need to allocate any extra space.
			 */
			info[dim].nbytes_aligned = 0;
		}
		else if (info[dim].typlen > 0)	/* fixed-length by-ref */
		{
			/*
			 * We don't care about alignment in the serialized data, so we
			 * pack the data as much as possible. But we also track how much
			 * data will be needed after deserialization, and in that case we
			 * need to account for alignment of each item.
			 *
			 * Note: As the items are fixed-length, we could easily compute
			 * this during deserialization, but we do it here anyway.
			 */
			info[dim].nbytes = info[dim].nvalues * info[dim].typlen;
			info[dim].nbytes_aligned = info[dim].nvalues * MAXALIGN(info[dim].typlen);
		}
		else if (info[dim].typlen == -1)	/* varlena */
		{
			info[dim].nbytes = 0;
			info[dim].nbytes_aligned = 0;
			for (i = 0; i < info[dim].nvalues; i++)
			{
				Size		len;

				/*
				 * For varlena values, we detoast the values and store the
				 * length and data separately. We don't bother with alignment
				 * here, which means that during deserialization we need to
				 * copy the fields and only access the copies.
				 */
				values[dim][i] = PointerGetDatum(PG_DETOAST_DATUM(values[dim][i]));

				/* serialized length (uint32 length + data) */
				len = VARSIZE_ANY_EXHDR(values[dim][i]);
				info[dim].nbytes += sizeof(uint32); /* length */
				info[dim].nbytes += len;	/* value (no header) */

				/*
				 * During deserialization we'll build regular varlena values
				 * with full headers, and we need to align them properly.
				 */
				info[dim].nbytes_aligned += MAXALIGN(VARHDRSZ + len);
			}
		}
		else if (info[dim].typlen == -2)	/* cstring */
		{
			info[dim].nbytes = 0;
			info[dim].nbytes_aligned = 0;
			for (i = 0; i < info[dim].nvalues; i++)
			{
				Size		len;

				/*
				 * cstring is handled similar to varlena - first we store the
				 * length as uint32 and then the data. We don't care about
				 * alignment, which means that during deserialization we need
				 * to copy the fields and only access the copies.
				 */

				/* c-strings include terminator, so +1 byte */
				len = strlen(DatumGetCString(values[dim][i])) + 1;
				info[dim].nbytes += sizeof(uint32); /* length */
				info[dim].nbytes += len;	/* value */

				/* space needed for properly aligned deserialized copies */
				info[dim].nbytes_aligned += MAXALIGN(len);
			}
		}

		/* we know (count>0) so there must be some data */
		Assert(info[dim].nbytes > 0);
	}

	/*
	 * Now we can finally compute how much space we'll actually need for the
	 * whole serialized MCV list (varlena header, MCV header, dimension info
	 * for each attribute, deduplicated values and items).
	 */
	total_length = (3 * sizeof(uint32)) /* magic + type + nitems */
		+ sizeof(AttrNumber)	/* ndimensions */
		+ (ndims * sizeof(Oid));	/* attribute types */

	/* dimension info */
	total_length += ndims * sizeof(DimensionInfo);

	/* add space for the arrays of deduplicated values */
	for (i = 0; i < ndims; i++)
		total_length += info[i].nbytes;

	/*
	 * And finally account for the items (those are fixed-length, thanks to
	 * replacing values with uint16 indexes into the deduplicated arrays).
	 */
	total_length += mcvlist->nitems * ITEM_SIZE(dim);

	/*
	 * Allocate space for the whole serialized MCV list (we'll skip bytes, so
	 * we set them to zero to make the result more compressible).
	 */
	raw = (bytea *) palloc0(VARHDRSZ + total_length);
	SET_VARSIZE(raw, VARHDRSZ + total_length);

	ptr = VARDATA(raw);
	endptr = ptr + total_length;

	/* copy the MCV list header fields, one by one */
	memcpy(ptr, &mcvlist->magic, sizeof(uint32));
	ptr += sizeof(uint32);

	memcpy(ptr, &mcvlist->type, sizeof(uint32));
	ptr += sizeof(uint32);

	memcpy(ptr, &mcvlist->nitems, sizeof(uint32));
	ptr += sizeof(uint32);

	memcpy(ptr, &mcvlist->ndimensions, sizeof(AttrNumber));
	ptr += sizeof(AttrNumber);

	memcpy(ptr, mcvlist->types, sizeof(Oid) * ndims);
	ptr += (sizeof(Oid) * ndims);

	/* store information about the attributes (data amounts, ...) */
	memcpy(ptr, info, sizeof(DimensionInfo) * ndims);
	ptr += sizeof(DimensionInfo) * ndims;

	/* Copy the deduplicated values for all attributes to the output. */
	for (dim = 0; dim < ndims; dim++)
	{
		/* remember the starting point for Asserts later */
		char	   *start PG_USED_FOR_ASSERTS_ONLY = ptr;

		for (i = 0; i < info[dim].nvalues; i++)
		{
			Datum		value = values[dim][i];

			if (info[dim].typbyval) /* passed by value */
			{
				Datum		tmp;

				/*
				 * For byval types, we need to copy just the significant bytes
				 * - we can't use memcpy directly, as that assumes
				 * little-endian behavior.  store_att_byval does almost what
				 * we need, but it requires a properly aligned buffer - the
				 * output buffer does not guarantee that. So we simply use a
				 * local Datum variable (which guarantees proper alignment),
				 * and then copy the value from it.
				 */
				store_att_byval(&tmp, value, info[dim].typlen);

				memcpy(ptr, &tmp, info[dim].typlen);
				ptr += info[dim].typlen;
			}
			else if (info[dim].typlen > 0)	/* passed by reference */
			{
				/* no special alignment needed, treated as char array */
				memcpy(ptr, DatumGetPointer(value), info[dim].typlen);
				ptr += info[dim].typlen;
			}
			else if (info[dim].typlen == -1)	/* varlena */
			{
				uint32		len = VARSIZE_ANY_EXHDR(DatumGetPointer(value));

				/* copy the length */
				memcpy(ptr, &len, sizeof(uint32));
				ptr += sizeof(uint32);

				/* data from the varlena value (without the header) */
				memcpy(ptr, VARDATA_ANY(DatumGetPointer(value)), len);
				ptr += len;
			}
			else if (info[dim].typlen == -2)	/* cstring */
			{
				uint32		len = (uint32) strlen(DatumGetCString(value)) + 1;

				/* copy the length */
				memcpy(ptr, &len, sizeof(uint32));
				ptr += sizeof(uint32);

				/* value */
				memcpy(ptr, DatumGetCString(value), len);
				ptr += len;
			}

			/* no underflows or overflows */
			Assert((ptr > start) && ((ptr - start) <= info[dim].nbytes));
		}

		/* we should get exactly nbytes of data for this dimension */
		Assert((ptr - start) == info[dim].nbytes);
	}

	/* Serialize the items, with uint16 indexes instead of the values. */
	for (i = 0; i < mcvlist->nitems; i++)
	{
		MCVItem    *mcvitem = &mcvlist->items[i];

		/* don't write beyond the allocated space */
		Assert(ptr <= (endptr - ITEM_SIZE(dim)));

		/* copy NULL and frequency flags into the serialized MCV */
		memcpy(ptr, mcvitem->isnull, sizeof(bool) * ndims);
		ptr += sizeof(bool) * ndims;

		memcpy(ptr, &mcvitem->frequency, sizeof(double));
		ptr += sizeof(double);

		memcpy(ptr, &mcvitem->base_frequency, sizeof(double));
		ptr += sizeof(double);

		/* store the indexes last */
		for (dim = 0; dim < ndims; dim++)
		{
			uint16		index = 0;
			Datum	   *value;

			/* do the lookup only for non-NULL values */
			if (!mcvitem->isnull[dim])
			{
				value = (Datum *) bsearch_arg(&mcvitem->values[dim], values[dim],
											  info[dim].nvalues, sizeof(Datum),
											  compare_scalars_simple, &ssup[dim]);

				Assert(value != NULL);	/* serialization or deduplication
										 * error */

				/* compute index within the deduplicated array */
				index = (uint16) (value - values[dim]);

				/* check the index is within expected bounds */
				Assert(index < info[dim].nvalues);
			}

			/* copy the index into the serialized MCV */
			memcpy(ptr, &index, sizeof(uint16));
			ptr += sizeof(uint16);
		}

		/* make sure we don't overflow the allocated value */
		Assert(ptr <= endptr);
	}

	/* at this point we expect to match the total_length exactly */
	Assert(ptr == endptr);

	pfree(values);
	pfree(counts);

	return raw;
}

/*
 * statext_mcv_deserialize
 *		Reads serialized MCV list into MCVList structure.
 *
 * All the memory needed by the MCV list is allocated as a single chunk, so
 * it's possible to simply pfree() it at once.
 */
MCVList *
statext_mcv_deserialize(bytea *data)
{
	int			dim,
				i;
	Size		expected_size;
	MCVList    *mcvlist;
	char	   *raw;
	char	   *ptr;
	char	   *endptr PG_USED_FOR_ASSERTS_ONLY;

	int			ndims,
				nitems;
	DimensionInfo *info = NULL;

	/* local allocation buffer (used only for deserialization) */
	Datum	  **map = NULL;

	/* MCV list */
	Size		mcvlen;

	/* buffer used for the result */
	Size		datalen;
	char	   *dataptr;
	char	   *valuesptr;
	char	   *isnullptr;

	if (data == NULL)
		return NULL;

	/*
	 * We can't possibly deserialize a MCV list if there's not even a complete
	 * header. We need an explicit formula here, because we serialize the
	 * header fields one by one, so we need to ignore struct alignment.
	 */
	if (VARSIZE_ANY(data) < MinSizeOfMCVList)
		elog(ERROR, "invalid MCV size %zd (expected at least %zu)",
			 VARSIZE_ANY(data), MinSizeOfMCVList);

	/* read the MCV list header */
	mcvlist = (MCVList *) palloc0(offsetof(MCVList, items));

	/* pointer to the data part (skip the varlena header) */
	raw = (char *) data;
	ptr = VARDATA_ANY(raw);
	endptr = (char *) raw + VARSIZE_ANY(data);

	/* get the header and perform further sanity checks */
	memcpy(&mcvlist->magic, ptr, sizeof(uint32));
	ptr += sizeof(uint32);

	memcpy(&mcvlist->type, ptr, sizeof(uint32));
	ptr += sizeof(uint32);

	memcpy(&mcvlist->nitems, ptr, sizeof(uint32));
	ptr += sizeof(uint32);

	memcpy(&mcvlist->ndimensions, ptr, sizeof(AttrNumber));
	ptr += sizeof(AttrNumber);

	if (mcvlist->magic != STATS_MCV_MAGIC)
		elog(ERROR, "invalid MCV magic %u (expected %u)",
			 mcvlist->magic, STATS_MCV_MAGIC);

	if (mcvlist->type != STATS_MCV_TYPE_BASIC)
		elog(ERROR, "invalid MCV type %u (expected %u)",
			 mcvlist->type, STATS_MCV_TYPE_BASIC);

	if (mcvlist->ndimensions == 0)
		elog(ERROR, "invalid zero-length dimension array in MCVList");
	else if ((mcvlist->ndimensions > STATS_MAX_DIMENSIONS) ||
			 (mcvlist->ndimensions < 0))
		elog(ERROR, "invalid length (%d) dimension array in MCVList",
			 mcvlist->ndimensions);

	if (mcvlist->nitems == 0)
		elog(ERROR, "invalid zero-length item array in MCVList");
	else if (mcvlist->nitems > STATS_MCVLIST_MAX_ITEMS)
		elog(ERROR, "invalid length (%u) item array in MCVList",
			 mcvlist->nitems);

	nitems = mcvlist->nitems;
	ndims = mcvlist->ndimensions;

	/*
	 * Check amount of data including DimensionInfo for all dimensions and
	 * also the serialized items (including uint16 indexes). Also, walk
	 * through the dimension information and add it to the sum.
	 */
	expected_size = SizeOfMCVList(ndims, nitems);

	/*
	 * Check that we have at least the dimension and info records, along with
	 * the items. We don't know the size of the serialized values yet. We need
	 * to do this check first, before accessing the dimension info.
	 */
	if (VARSIZE_ANY(data) < expected_size)
		elog(ERROR, "invalid MCV size %zd (expected %zu)",
			 VARSIZE_ANY(data), expected_size);

	/* Now copy the array of type Oids. */
	memcpy(mcvlist->types, ptr, sizeof(Oid) * ndims);
	ptr += (sizeof(Oid) * ndims);

	/* Now it's safe to access the dimension info. */
	info = palloc(ndims * sizeof(DimensionInfo));

	memcpy(info, ptr, ndims * sizeof(DimensionInfo));
	ptr += (ndims * sizeof(DimensionInfo));

	/* account for the value arrays */
	for (dim = 0; dim < ndims; dim++)
	{
		/*
		 * XXX I wonder if we can/should rely on asserts here. Maybe those
		 * checks should be done every time?
		 */
		Assert(info[dim].nvalues >= 0);
		Assert(info[dim].nbytes >= 0);

		expected_size += info[dim].nbytes;
	}

	/*
	 * Now we know the total expected MCV size, including all the pieces
	 * (header, dimension info. items and deduplicated data). So do the final
	 * check on size.
	 */
	if (VARSIZE_ANY(data) != expected_size)
		elog(ERROR, "invalid MCV size %zd (expected %zu)",
			 VARSIZE_ANY(data), expected_size);

	/*
	 * We need an array of Datum values for each dimension, so that we can
	 * easily translate the uint16 indexes later. We also need a top-level
	 * array of pointers to those per-dimension arrays.
	 *
	 * While allocating the arrays for dimensions, compute how much space we
	 * need for a copy of the by-ref data, as we can't simply point to the
	 * original values (it might go away).
	 */
	datalen = 0;				/* space for by-ref data */
	map = (Datum **) palloc(ndims * sizeof(Datum *));

	for (dim = 0; dim < ndims; dim++)
	{
		map[dim] = (Datum *) palloc(sizeof(Datum) * info[dim].nvalues);

		/* space needed for a copy of data for by-ref types */
		datalen += info[dim].nbytes_aligned;
	}

	/*
	 * Now resize the MCV list so that the allocation includes all the data.
	 *
	 * Allocate space for a copy of the data, as we can't simply reference the
	 * serialized data - it's not aligned properly, and it may disappear while
	 * we're still using the MCV list, e.g. due to catcache release.
	 *
	 * We do care about alignment here, because we will allocate all the
	 * pieces at once, but then use pointers to different parts.
	 */
	mcvlen = MAXALIGN(offsetof(MCVList, items) + (sizeof(MCVItem) * nitems));

	/* arrays of values and isnull flags for all MCV items */
	mcvlen += nitems * MAXALIGN(sizeof(Datum) * ndims);
	mcvlen += nitems * MAXALIGN(sizeof(bool) * ndims);

	/* we don't quite need to align this, but it makes some asserts easier */
	mcvlen += MAXALIGN(datalen);

	/* now resize the deserialized MCV list, and compute pointers to parts */
	mcvlist = repalloc(mcvlist, mcvlen);

	/* pointer to the beginning of values/isnull arrays */
	valuesptr = (char *) mcvlist
		+ MAXALIGN(offsetof(MCVList, items) + (sizeof(MCVItem) * nitems));

	isnullptr = valuesptr + (nitems * MAXALIGN(sizeof(Datum) * ndims));

	dataptr = isnullptr + (nitems * MAXALIGN(sizeof(bool) * ndims));

	/*
	 * Build mapping (index => value) for translating the serialized data into
	 * the in-memory representation.
	 */
	for (dim = 0; dim < ndims; dim++)
	{
		/* remember start position in the input array */
		char	   *start PG_USED_FOR_ASSERTS_ONLY = ptr;

		if (info[dim].typbyval)
		{
			/* for by-val types we simply copy data into the mapping */
			for (i = 0; i < info[dim].nvalues; i++)
			{
				Datum		v = 0;

				memcpy(&v, ptr, info[dim].typlen);
				ptr += info[dim].typlen;

				map[dim][i] = fetch_att(&v, true, info[dim].typlen);

				/* no under/overflow of input array */
				Assert(ptr <= (start + info[dim].nbytes));
			}
		}
		else
		{
			/* for by-ref types we need to also make a copy of the data */

			/* passed by reference, but fixed length (name, tid, ...) */
			if (info[dim].typlen > 0)
			{
				for (i = 0; i < info[dim].nvalues; i++)
				{
					memcpy(dataptr, ptr, info[dim].typlen);
					ptr += info[dim].typlen;

					/* just point into the array */
					map[dim][i] = PointerGetDatum(dataptr);
					dataptr += MAXALIGN(info[dim].typlen);
				}
			}
			else if (info[dim].typlen == -1)
			{
				/* varlena */
				for (i = 0; i < info[dim].nvalues; i++)
				{
					uint32		len;

					/* read the uint32 length */
					memcpy(&len, ptr, sizeof(uint32));
					ptr += sizeof(uint32);

					/* the length is data-only */
					SET_VARSIZE(dataptr, len + VARHDRSZ);
					memcpy(VARDATA(dataptr), ptr, len);
					ptr += len;

					/* just point into the array */
					map[dim][i] = PointerGetDatum(dataptr);

					/* skip to place of the next deserialized value */
					dataptr += MAXALIGN(len + VARHDRSZ);
				}
			}
			else if (info[dim].typlen == -2)
			{
				/* cstring */
				for (i = 0; i < info[dim].nvalues; i++)
				{
					uint32		len;

					memcpy(&len, ptr, sizeof(uint32));
					ptr += sizeof(uint32);

					memcpy(dataptr, ptr, len);
					ptr += len;

					/* just point into the array */
					map[dim][i] = PointerGetDatum(dataptr);
					dataptr += MAXALIGN(len);
				}
			}

			/* no under/overflow of input array */
			Assert(ptr <= (start + info[dim].nbytes));

			/* no overflow of the output mcv value */
			Assert(dataptr <= ((char *) mcvlist + mcvlen));
		}

		/* check we consumed input data for this dimension exactly */
		Assert(ptr == (start + info[dim].nbytes));
	}

	/* we should have also filled the MCV list exactly */
	Assert(dataptr == ((char *) mcvlist + mcvlen));

	/* deserialize the MCV items and translate the indexes to Datums */
	for (i = 0; i < nitems; i++)
	{
		MCVItem    *item = &mcvlist->items[i];

		item->values = (Datum *) valuesptr;
		valuesptr += MAXALIGN(sizeof(Datum) * ndims);

		item->isnull = (bool *) isnullptr;
		isnullptr += MAXALIGN(sizeof(bool) * ndims);

		memcpy(item->isnull, ptr, sizeof(bool) * ndims);
		ptr += sizeof(bool) * ndims;

		memcpy(&item->frequency, ptr, sizeof(double));
		ptr += sizeof(double);

		memcpy(&item->base_frequency, ptr, sizeof(double));
		ptr += sizeof(double);

		/* finally translate the indexes (for non-NULL only) */
		for (dim = 0; dim < ndims; dim++)
		{
			uint16		index;

			memcpy(&index, ptr, sizeof(uint16));
			ptr += sizeof(uint16);

			if (item->isnull[dim])
				continue;

			item->values[dim] = map[dim][index];
		}

		/* check we're not overflowing the input */
		Assert(ptr <= endptr);
	}

	/* check that we processed all the data */
	Assert(ptr == endptr);

	/* release the buffers used for mapping */
	for (dim = 0; dim < ndims; dim++)
		pfree(map[dim]);

	pfree(map);

	return mcvlist;
}

/*
 * SRF with details about buckets of a histogram:
 *
 * - item ID (0...nitems)
 * - values (string array)
 * - nulls only (boolean array)
 * - frequency (double precision)
 * - base_frequency (double precision)
 *
 * The input is the OID of the statistics, and there are no rows returned if
 * the statistics contains no histogram.
 */
Datum
pg_stats_ext_mcvlist_items(PG_FUNCTION_ARGS)
{
	FuncCallContext *funcctx;

	/* stuff done only on the first call of the function */
	if (SRF_IS_FIRSTCALL())
	{
		MemoryContext oldcontext;
		MCVList    *mcvlist;
		TupleDesc	tupdesc;

		/* create a function context for cross-call persistence */
		funcctx = SRF_FIRSTCALL_INIT();

		/* switch to memory context appropriate for multiple function calls */
		oldcontext = MemoryContextSwitchTo(funcctx->multi_call_memory_ctx);

		mcvlist = statext_mcv_deserialize(PG_GETARG_BYTEA_P(0));

		funcctx->user_fctx = mcvlist;

		/* total number of tuples to be returned */
		funcctx->max_calls = 0;
		if (funcctx->user_fctx != NULL)
			funcctx->max_calls = mcvlist->nitems;

		/* Build a tuple descriptor for our result type */
		if (get_call_result_type(fcinfo, NULL, &tupdesc) != TYPEFUNC_COMPOSITE)
			ereport(ERROR,
					(errcode(ERRCODE_FEATURE_NOT_SUPPORTED),
					 errmsg("function returning record called in context "
							"that cannot accept type record")));
		tupdesc = BlessTupleDesc(tupdesc);

		/*
		 * generate attribute metadata needed later to produce tuples from raw
		 * C strings
		 */
		funcctx->attinmeta = TupleDescGetAttInMetadata(tupdesc);

		MemoryContextSwitchTo(oldcontext);
	}

	/* stuff done on every call of the function */
	funcctx = SRF_PERCALL_SETUP();

	if (funcctx->call_cntr < funcctx->max_calls)	/* do when there is more
													 * left to send */
	{
		Datum		values[5];
		bool		nulls[5];
		HeapTuple	tuple;
		Datum		result;
		ArrayBuildState *astate_values = NULL;
		ArrayBuildState *astate_nulls = NULL;

		int			i;
		MCVList    *mcvlist;
		MCVItem    *item;

		mcvlist = (MCVList *) funcctx->user_fctx;

		Assert(funcctx->call_cntr < mcvlist->nitems);

		item = &mcvlist->items[funcctx->call_cntr];

		for (i = 0; i < mcvlist->ndimensions; i++)
		{

			astate_nulls = accumArrayResult(astate_nulls,
											BoolGetDatum(item->isnull[i]),
											false,
											BOOLOID,
											CurrentMemoryContext);

			if (!item->isnull[i])
			{
				bool		isvarlena;
				Oid			outfunc;
				FmgrInfo	fmgrinfo;
				Datum		val;
				text	   *txt;

				/* lookup output func for the type */
				getTypeOutputInfo(mcvlist->types[i], &outfunc, &isvarlena);
				fmgr_info(outfunc, &fmgrinfo);

				val = FunctionCall1(&fmgrinfo, item->values[i]);
				txt = cstring_to_text(DatumGetPointer(val));

				astate_values = accumArrayResult(astate_values,
												 PointerGetDatum(txt),
												 false,
												 TEXTOID,
												 CurrentMemoryContext);
			}
			else
				astate_values = accumArrayResult(astate_values,
												 (Datum) 0,
												 true,
												 TEXTOID,
												 CurrentMemoryContext);
		}

		values[0] = Int32GetDatum(funcctx->call_cntr);
		values[1] = PointerGetDatum(makeArrayResult(astate_values, CurrentMemoryContext));
		values[2] = PointerGetDatum(makeArrayResult(astate_nulls, CurrentMemoryContext));
		values[3] = Float8GetDatum(item->frequency);
		values[4] = Float8GetDatum(item->base_frequency);

		/* no NULLs in the tuple */
		memset(nulls, 0, sizeof(nulls));

		/* build a tuple */
		tuple = heap_form_tuple(funcctx->attinmeta->tupdesc, values, nulls);

		/* make the tuple into a datum */
		result = HeapTupleGetDatum(tuple);

		SRF_RETURN_NEXT(funcctx, result);
	}
	else						/* do when there is no more left */
	{
		SRF_RETURN_DONE(funcctx);
	}
}

/*
 * pg_mcv_list_in		- input routine for type pg_mcv_list.
 *
 * pg_mcv_list is real enough to be a table column, but it has no operations
 * of its own, and disallows input too
 */
Datum
pg_mcv_list_in(PG_FUNCTION_ARGS)
{
	/*
	 * pg_mcv_list stores the data in binary form and parsing text input is
	 * not needed, so disallow this.
	 */
	ereport(ERROR,
			(errcode(ERRCODE_FEATURE_NOT_SUPPORTED),
			 errmsg("cannot accept a value of type %s", "pg_mcv_list")));

	PG_RETURN_VOID();			/* keep compiler quiet */
}


/*
 * pg_mcv_list_out		- output routine for type pg_mcv_list.
 *
 * MCV lists are serialized into a bytea value, so we simply call byteaout()
 * to serialize the value into text. But it'd be nice to serialize that into
 * a meaningful representation (e.g. for inspection by people).
 *
 * XXX This should probably return something meaningful, similar to what
 * pg_dependencies_out does. Not sure how to deal with the deduplicated
 * values, though - do we want to expand that or not?
 */
Datum
pg_mcv_list_out(PG_FUNCTION_ARGS)
{
	return byteaout(fcinfo);
}

/*
 * pg_mcv_list_recv		- binary input routine for type pg_mcv_list.
 */
Datum
pg_mcv_list_recv(PG_FUNCTION_ARGS)
{
	ereport(ERROR,
			(errcode(ERRCODE_FEATURE_NOT_SUPPORTED),
			 errmsg("cannot accept a value of type %s", "pg_mcv_list")));

	PG_RETURN_VOID();			/* keep compiler quiet */
}

/*
 * pg_mcv_list_send		- binary output routine for type pg_mcv_list.
 *
 * MCV lists are serialized in a bytea value (although the type is named
 * differently), so let's just send that.
 */
Datum
pg_mcv_list_send(PG_FUNCTION_ARGS)
{
	return byteasend(fcinfo);
}

/*
 * match the attribute/expression to a dimension of the statistic
 *
 * Returns the zero-based index of the matching statistics dimension.
 * Optionally determines the collation.
 */
static int
mcv_match_expression(Node *expr, Bitmapset *keys, List *exprs, Oid *collid)
{
	int			idx;

	if (IsA(expr, Var))
	{
		/* simple Var, so just lookup using varattno */
		Var		   *var = (Var *) expr;

		if (collid)
			*collid = var->varcollid;

		idx = bms_member_index(keys, var->varattno);

		if (idx < 0)
			elog(ERROR, "variable not found in statistics object");
	}
	else
	{
		/* expression - lookup in stats expressions */
		ListCell   *lc;

		if (collid)
			*collid = exprCollation(expr);

		/* expressions are stored after the simple columns */
		idx = bms_num_members(keys);
		foreach(lc, exprs)
		{
			Node	   *stat_expr = (Node *) lfirst(lc);

			if (equal(expr, stat_expr))
				break;

			idx++;
		}

		if (lc == NULL)
			elog(ERROR, "expression not found in statistics object");
	}

	return idx;
}

/*
 * mcv_get_match_bitmap
 *	Evaluate clauses using the MCV list, and update the match bitmap.
 *
 * A match bitmap keeps match/mismatch status for each MCV item, and we
 * update it based on additional clauses. We also use it to skip items
 * that can't possibly match (e.g. item marked as "mismatch" can't change
 * to "match" when evaluating AND clause list).
 *
 * The function also returns a flag indicating whether there was an
 * equality condition for all attributes, the minimum frequency in the MCV
 * list, and a total MCV frequency (sum of frequencies for all items).
 *
 * XXX Currently the match bitmap uses a bool for each MCV item, which is
 * somewhat wasteful as we could do with just a single bit, thus reducing
 * the size to ~1/8. It would also allow us to combine bitmaps simply using
 * & and |, which should be faster than min/max. The bitmaps are fairly
 * small, though (thanks to the cap on the MCV list size).
 */
static bool *
mcv_get_match_bitmap(PlannerInfo *root, List *clauses,
					 Bitmapset *keys, List *exprs,
					 MCVList *mcvlist, bool is_or)
{
	int			i;
	ListCell   *l;
	bool	   *matches;

	/* The bitmap may be partially built. */
	Assert(clauses != NIL);
	Assert(mcvlist != NULL);
	Assert(mcvlist->nitems > 0);
	Assert(mcvlist->nitems <= STATS_MCVLIST_MAX_ITEMS);

	matches = palloc(sizeof(bool) * mcvlist->nitems);
	memset(matches, !is_or, sizeof(bool) * mcvlist->nitems);

	/*
	 * Loop through the list of clauses, and for each of them evaluate all the
	 * MCV items not yet eliminated by the preceding clauses.
	 */
	foreach(l, clauses)
	{
		Node	   *clause = (Node *) lfirst(l);

		/* if it's a RestrictInfo, then extract the clause */
		if (IsA(clause, RestrictInfo))
			clause = (Node *) ((RestrictInfo *) clause)->clause;

		/*
		 * Handle the various types of clauses - OpClause, NullTest and
		 * AND/OR/NOT
		 */
		if (is_opclause(clause))
		{
			OpExpr	   *expr = (OpExpr *) clause;
			FmgrInfo	opproc;

			/* valid only after examine_opclause_args returns true */
			Node	   *clause_expr;
			Const	   *cst;
			bool		expronleft;
			int			idx;
			Oid			collid;

			fmgr_info(get_opcode(expr->opno), &opproc);

			/* extract the var/expr and const from the expression */
			if (!examine_opclause_args(expr->args, &clause_expr, &cst, &expronleft))
				elog(ERROR, "incompatible clause");

			/* match the attribute/expression to a dimension of the statistic */
			idx = mcv_match_expression(clause_expr, keys, exprs, &collid);

			/*
			 * Walk through the MCV items and evaluate the current clause. We
			 * can skip items that were already ruled out, and terminate if
			 * there are no remaining MCV items that might possibly match.
			 */
			for (i = 0; i < mcvlist->nitems; i++)
			{
				bool		match = true;
				MCVItem    *item = &mcvlist->items[i];

				Assert(idx >= 0);

				/*
				 * When the MCV item or the Const value is NULL we can treat
				 * this as a mismatch. We must not call the operator because
				 * of strictness.
				 */
				if (item->isnull[idx] || cst->constisnull)
				{
					matches[i] = RESULT_MERGE(matches[i], is_or, false);
					continue;
				}

				/*
				 * Skip MCV items that can't change result in the bitmap. Once
				 * the value gets false for AND-lists, or true for OR-lists,
				 * we don't need to look at more clauses.
				 */
				if (RESULT_IS_FINAL(matches[i], is_or))
					continue;

				/*
				 * First check whether the constant is below the lower
				 * boundary (in that case we can skip the bucket, because
				 * there's no overlap).
				 *
				 * We don't store collations used to build the statistics, but
				 * we can use the collation for the attribute itself, as
				 * stored in varcollid. We do reset the statistics after a
				 * type change (including collation change), so this is OK.
				 * For expressions, we use the collation extracted from the
				 * expression itself.
				 */
				if (expronleft)
					match = DatumGetBool(FunctionCall2Coll(&opproc,
														   collid,
														   item->values[idx],
														   cst->constvalue));
				else
					match = DatumGetBool(FunctionCall2Coll(&opproc,
														   collid,
														   cst->constvalue,
														   item->values[idx]));

				/* update the match bitmap with the result */
				matches[i] = RESULT_MERGE(matches[i], is_or, match);
			}
		}
		else if (IsA(clause, ScalarArrayOpExpr))
		{
			ScalarArrayOpExpr *expr = (ScalarArrayOpExpr *) clause;
			FmgrInfo	opproc;

			/* valid only after examine_opclause_args returns true */
			Node	   *clause_expr;
			Const	   *cst;
			bool		expronleft;
			Oid			collid;
			int			idx;

			/* array evaluation */
			ArrayType  *arrayval;
			int16		elmlen;
			bool		elmbyval;
			char		elmalign;
			int			num_elems;
			Datum	   *elem_values;
			bool	   *elem_nulls;

			fmgr_info(get_opcode(expr->opno), &opproc);

			/* extract the var/expr and const from the expression */
			if (!examine_opclause_args(expr->args, &clause_expr, &cst, &expronleft))
				elog(ERROR, "incompatible clause");

			/* We expect Var on left */
			if (!expronleft)
				elog(ERROR, "incompatible clause");

			/*
			 * Deconstruct the array constant, unless it's NULL (we'll cover
			 * that case below)
			 */
			if (!cst->constisnull)
			{
				arrayval = DatumGetArrayTypeP(cst->constvalue);
				get_typlenbyvalalign(ARR_ELEMTYPE(arrayval),
									 &elmlen, &elmbyval, &elmalign);
				deconstruct_array(arrayval,
								  ARR_ELEMTYPE(arrayval),
								  elmlen, elmbyval, elmalign,
								  &elem_values, &elem_nulls, &num_elems);
			}

			/* match the attribute/expression to a dimension of the statistic */
			idx = mcv_match_expression(clause_expr, keys, exprs, &collid);

			/*
			 * Walk through the MCV items and evaluate the current clause. We
			 * can skip items that were already ruled out, and terminate if
			 * there are no remaining MCV items that might possibly match.
			 */
			for (i = 0; i < mcvlist->nitems; i++)
			{
				int			j;
				bool		match = !expr->useOr;
				MCVItem    *item = &mcvlist->items[i];

				/*
				 * When the MCV item or the Const value is NULL we can treat
				 * this as a mismatch. We must not call the operator because
				 * of strictness.
				 */
				if (item->isnull[idx] || cst->constisnull)
				{
					matches[i] = RESULT_MERGE(matches[i], is_or, false);
					continue;
				}

				/*
				 * Skip MCV items that can't change result in the bitmap. Once
				 * the value gets false for AND-lists, or true for OR-lists,
				 * we don't need to look at more clauses.
				 */
				if (RESULT_IS_FINAL(matches[i], is_or))
					continue;

				for (j = 0; j < num_elems; j++)
				{
					Datum		elem_value = elem_values[j];
					bool		elem_isnull = elem_nulls[j];
					bool		elem_match;

					/* NULL values always evaluate as not matching. */
					if (elem_isnull)
					{
						match = RESULT_MERGE(match, expr->useOr, false);
						continue;
					}

					/*
					 * Stop evaluating the array elements once we reach a
					 * matching value that can't change - ALL() is the same as
					 * AND-list, ANY() is the same as OR-list.
					 */
					if (RESULT_IS_FINAL(match, expr->useOr))
						break;

					elem_match = DatumGetBool(FunctionCall2Coll(&opproc,
																collid,
																item->values[idx],
																elem_value));

					match = RESULT_MERGE(match, expr->useOr, elem_match);
				}

				/* update the match bitmap with the result */
				matches[i] = RESULT_MERGE(matches[i], is_or, match);
			}
		}
		else if (IsA(clause, NullTest))
		{
			NullTest   *expr = (NullTest *) clause;
			Node	   *clause_expr = (Node *) (expr->arg);

			/* match the attribute/expression to a dimension of the statistic */
			int			idx = mcv_match_expression(clause_expr, keys, exprs, NULL);

			/*
			 * Walk through the MCV items and evaluate the current clause. We
			 * can skip items that were already ruled out, and terminate if
			 * there are no remaining MCV items that might possibly match.
			 */
			for (i = 0; i < mcvlist->nitems; i++)
			{
				bool		match = false;	/* assume mismatch */
				MCVItem    *item = &mcvlist->items[i];

				/* if the clause mismatches the MCV item, update the bitmap */
				switch (expr->nulltesttype)
				{
					case IS_NULL:
						match = (item->isnull[idx]) ? true : match;
						break;

					case IS_NOT_NULL:
						match = (!item->isnull[idx]) ? true : match;
						break;
				}

				/* now, update the match bitmap, depending on OR/AND type */
				matches[i] = RESULT_MERGE(matches[i], is_or, match);
			}
		}
		else if (is_orclause(clause) || is_andclause(clause))
		{
			/* AND/OR clause, with all subclauses being compatible */

			int			i;
			BoolExpr   *bool_clause = ((BoolExpr *) clause);
			List	   *bool_clauses = bool_clause->args;

			/* match/mismatch bitmap for each MCV item */
			bool	   *bool_matches = NULL;

			Assert(bool_clauses != NIL);
			Assert(list_length(bool_clauses) >= 2);

			/* build the match bitmap for the OR-clauses */
			bool_matches = mcv_get_match_bitmap(root, bool_clauses, keys, exprs,
												mcvlist, is_orclause(clause));

			/*
			 * Merge the bitmap produced by mcv_get_match_bitmap into the
			 * current one. We need to consider if we're evaluating AND or OR
			 * condition when merging the results.
			 */
			for (i = 0; i < mcvlist->nitems; i++)
				matches[i] = RESULT_MERGE(matches[i], is_or, bool_matches[i]);

			pfree(bool_matches);
		}
		else if (is_notclause(clause))
		{
			/* NOT clause, with all subclauses compatible */

			int			i;
			BoolExpr   *not_clause = ((BoolExpr *) clause);
			List	   *not_args = not_clause->args;

			/* match/mismatch bitmap for each MCV item */
			bool	   *not_matches = NULL;

			Assert(not_args != NIL);
			Assert(list_length(not_args) == 1);

			/* build the match bitmap for the NOT-clause */
			not_matches = mcv_get_match_bitmap(root, not_args, keys, exprs,
											   mcvlist, false);

			/*
			 * Merge the bitmap produced by mcv_get_match_bitmap into the
			 * current one. We're handling a NOT clause, so invert the result
			 * before merging it into the global bitmap.
			 */
			for (i = 0; i < mcvlist->nitems; i++)
				matches[i] = RESULT_MERGE(matches[i], is_or, !not_matches[i]);

			pfree(not_matches);
		}
		else if (IsA(clause, Var))
		{
			/* Var (has to be a boolean Var, possibly from below NOT) */

			Var		   *var = (Var *) (clause);

			/* match the attribute to a dimension of the statistic */
			int			idx = bms_member_index(keys, var->varattno);

			Assert(var->vartype == BOOLOID);

			/*
			 * Walk through the MCV items and evaluate the current clause. We
			 * can skip items that were already ruled out, and terminate if
			 * there are no remaining MCV items that might possibly match.
			 */
			for (i = 0; i < mcvlist->nitems; i++)
			{
				MCVItem    *item = &mcvlist->items[i];
				bool		match = false;

				/* if the item is NULL, it's a mismatch */
				if (!item->isnull[idx] && DatumGetBool(item->values[idx]))
					match = true;

				/* update the result bitmap */
				matches[i] = RESULT_MERGE(matches[i], is_or, match);
			}
		}
		else
		{
			/* Otherwise, it must be a bare boolean-returning expression */
			int			idx;

			/* match the expression to a dimension of the statistic */
			idx = mcv_match_expression(clause, keys, exprs, NULL);

			/*
			 * Walk through the MCV items and evaluate the current clause. We
			 * can skip items that were already ruled out, and terminate if
			 * there are no remaining MCV items that might possibly match.
			 */
			for (i = 0; i < mcvlist->nitems; i++)
			{
				bool		match;
				MCVItem    *item = &mcvlist->items[i];

				/* "match" just means it's bool TRUE */
				match = !item->isnull[idx] && DatumGetBool(item->values[idx]);

				/* now, update the match bitmap, depending on OR/AND type */
				matches[i] = RESULT_MERGE(matches[i], is_or, match);
			}
		}
	}

	return matches;
}


/*
 * mcv_combine_selectivities
 * 		Combine per-column and multi-column MCV selectivity estimates.
 *
 * simple_sel is a "simple" selectivity estimate (produced without using any
 * extended statistics, essentially assuming independence of columns/clauses).
 *
 * mcv_sel and mcv_basesel are sums of the frequencies and base frequencies of
 * all matching MCV items.  The difference (mcv_sel - mcv_basesel) is then
 * essentially interpreted as a correction to be added to simple_sel, as
 * described below.
 *
 * mcv_totalsel is the sum of the frequencies of all MCV items (not just the
 * matching ones).  This is used as an upper bound on the portion of the
 * selectivity estimates not covered by the MCV statistics.
 *
 * Note: While simple and base selectivities are defined in a quite similar
 * way, the values are computed differently and are not therefore equal. The
 * simple selectivity is computed as a product of per-clause estimates, while
 * the base selectivity is computed by adding up base frequencies of matching
 * items of the multi-column MCV list. So the values may differ for two main
 * reasons - (a) the MCV list may not cover 100% of the data and (b) some of
 * the MCV items did not match the estimated clauses.
 *
 * As both (a) and (b) reduce the base selectivity value, it generally holds
 * that (simple_sel >= mcv_basesel). If the MCV list covers all the data, the
 * values may be equal.
 *
 * So, other_sel = (simple_sel - mcv_basesel) is an estimate for the part not
 * covered by the MCV list, and (mcv_sel - mcv_basesel) may be seen as a
 * correction for the part covered by the MCV list. Those two statements are
 * actually equivalent.
 */
Selectivity
mcv_combine_selectivities(Selectivity simple_sel,
						  Selectivity mcv_sel,
						  Selectivity mcv_basesel,
						  Selectivity mcv_totalsel)
{
	Selectivity other_sel;
	Selectivity sel;

	/* estimated selectivity of values not covered by MCV matches */
	other_sel = simple_sel - mcv_basesel;
	CLAMP_PROBABILITY(other_sel);

	/* this non-MCV selectivity cannot exceed 1 - mcv_totalsel */
	if (other_sel > 1.0 - mcv_totalsel)
		other_sel = 1.0 - mcv_totalsel;

	/* overall selectivity is the sum of the MCV and non-MCV parts */
	sel = mcv_sel + other_sel;
	CLAMP_PROBABILITY(sel);

	return sel;
}


/*
 * mcv_clauselist_selectivity
 *		Use MCV statistics to estimate the selectivity of an implicitly-ANDed
 *		list of clauses.
 *
 * This determines which MCV items match every clause in the list and returns
 * the sum of the frequencies of those items.
 *
 * In addition, it returns the sum of the base frequencies of each of those
 * items (that is the sum of the selectivities that each item would have if
 * the columns were independent of one another), and the total selectivity of
 * all the MCV items (not just the matching ones).  These are expected to be
 * used together with a "simple" selectivity estimate (one based only on
 * per-column statistics) to produce an overall selectivity estimate that
 * makes use of both per-column and multi-column statistics --- see
 * mcv_combine_selectivities().
 */
Selectivity
mcv_clauselist_selectivity(PlannerInfo *root, StatisticExtInfo *stat,
						   List *clauses, int varRelid,
						   JoinType jointype, SpecialJoinInfo *sjinfo,
						   RelOptInfo *rel,
						   Selectivity *basesel, Selectivity *totalsel)
{
	int			i;
	MCVList    *mcv;
	Selectivity s = 0.0;
	RangeTblEntry *rte = root->simple_rte_array[rel->relid];

	/* match/mismatch bitmap for each MCV item */
	bool	   *matches = NULL;

	/* load the MCV list stored in the statistics object */
	mcv = statext_mcv_load(stat->statOid, rte->inh);

	/* build a match bitmap for the clauses */
	matches = mcv_get_match_bitmap(root, clauses, stat->keys, stat->exprs,
								   mcv, false);

	/* sum frequencies for all the matching MCV items */
	*basesel = 0.0;
	*totalsel = 0.0;
	for (i = 0; i < mcv->nitems; i++)
	{
		*totalsel += mcv->items[i].frequency;

		if (matches[i] != false)
		{
			*basesel += mcv->items[i].base_frequency;
			s += mcv->items[i].frequency;
		}
	}

	return s;
}


/*
 * mcv_clause_selectivity_or
 *		Use MCV statistics to estimate the selectivity of a clause that
 *		appears in an ORed list of clauses.
 *
 * As with mcv_clauselist_selectivity() this determines which MCV items match
 * the clause and returns both the sum of the frequencies and the sum of the
 * base frequencies of those items, as well as the sum of the frequencies of
 * all MCV items (not just the matching ones) so that this information can be
 * used by mcv_combine_selectivities() to produce a selectivity estimate that
 * makes use of both per-column and multi-column statistics.
 *
 * Additionally, we return information to help compute the overall selectivity
 * of the ORed list of clauses assumed to contain this clause.  This function
 * is intended to be called for each clause in the ORed list of clauses,
 * allowing the overall selectivity to be computed using the following
 * algorithm:
 *
 * Suppose P[n] = P(C[1] OR C[2] OR ... OR C[n]) is the combined selectivity
 * of the first n clauses in the list.  Then the combined selectivity taking
 * into account the next clause C[n+1] can be written as
 *
 *		P[n+1] = P[n] + P(C[n+1]) - P((C[1] OR ... OR C[n]) AND C[n+1])
 *
 * The final term above represents the overlap between the clauses examined so
 * far and the (n+1)'th clause.  To estimate its selectivity, we track the
 * match bitmap for the ORed list of clauses examined so far and examine its
 * intersection with the match bitmap for the (n+1)'th clause.
 *
 * We then also return the sums of the MCV item frequencies and base
 * frequencies for the match bitmap intersection corresponding to the overlap
 * term above, so that they can be combined with a simple selectivity estimate
 * for that term.
 *
 * The parameter "or_matches" is an in/out parameter tracking the match bitmap
 * for the clauses examined so far.  The caller is expected to set it to NULL
 * the first time it calls this function.
 */
Selectivity
mcv_clause_selectivity_or(PlannerInfo *root, StatisticExtInfo *stat,
						  MCVList *mcv, Node *clause, bool **or_matches,
						  Selectivity *basesel, Selectivity *overlap_mcvsel,
						  Selectivity *overlap_basesel, Selectivity *totalsel)
{
	Selectivity s = 0.0;
	bool	   *new_matches;
	int			i;

	/* build the OR-matches bitmap, if not built already */
	if (*or_matches == NULL)
		*or_matches = palloc0(sizeof(bool) * mcv->nitems);

	/* build the match bitmap for the new clause */
	new_matches = mcv_get_match_bitmap(root, list_make1(clause), stat->keys,
									   stat->exprs, mcv, false);

	/*
	 * Sum the frequencies for all the MCV items matching this clause and also
	 * those matching the overlap between this clause and any of the preceding
	 * clauses as described above.
	 */
	*basesel = 0.0;
	*overlap_mcvsel = 0.0;
	*overlap_basesel = 0.0;
	*totalsel = 0.0;
	for (i = 0; i < mcv->nitems; i++)
	{
		*totalsel += mcv->items[i].frequency;

		if (new_matches[i])
		{
			s += mcv->items[i].frequency;
			*basesel += mcv->items[i].base_frequency;

			if ((*or_matches)[i])
			{
				*overlap_mcvsel += mcv->items[i].frequency;
				*overlap_basesel += mcv->items[i].base_frequency;
			}
		}

		/* update the OR-matches bitmap for the next clause */
		(*or_matches)[i] = (*or_matches)[i] || new_matches[i];
	}

	pfree(new_matches);

	return s;
}