array_selfuncs.c

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/*-------------------------------------------------------------------------
 *
 * array_selfuncs.c
 *    Functions for selectivity estimation of array operators
 *
 * Portions Copyright (c) 1996-2015, PostgreSQL Global Development Group
 * Portions Copyright (c) 1994, Regents of the University of California
 *
 *
 * IDENTIFICATION
 *    src/backend/utils/adt/array_selfuncs.c
 *
 *-------------------------------------------------------------------------
 */
#include "postgres.h"

#include <math.h>

#include "access/htup_details.h"
#include "catalog/pg_collation.h"
#include "catalog/pg_operator.h"
#include "catalog/pg_statistic.h"
#include "optimizer/clauses.h"
#include "utils/array.h"
#include "utils/lsyscache.h"
#include "utils/selfuncs.h"
#include "utils/typcache.h"


/* Default selectivity constant for "@>" and "<@" operators */
#define DEFAULT_CONTAIN_SEL 0.005

/* Default selectivity constant for "&&" operator */
#define DEFAULT_OVERLAP_SEL 0.01

/* Default selectivity for given operator */
#define DEFAULT_SEL(operator) \
    ((operator) == OID_ARRAY_OVERLAP_OP ? \
        DEFAULT_OVERLAP_SEL : DEFAULT_CONTAIN_SEL)

static Selectivity calc_arraycontsel(VariableStatData *vardata, Datum constval,
                  Oid elemtype, Oid operator);
static Selectivity mcelem_array_selec(ArrayType *array,
                   TypeCacheEntry *typentry,
                   Datum *mcelem, int nmcelem,
                   float4 *numbers, int nnumbers,
                   float4 *hist, int nhist,
                   Oid operator, FmgrInfo *cmpfunc);
static Selectivity mcelem_array_contain_overlap_selec(Datum *mcelem, int nmcelem,
                                   float4 *numbers, int nnumbers,
                                   Datum *array_data, int nitems,
                                   Oid operator, FmgrInfo *cmpfunc);
static Selectivity mcelem_array_contained_selec(Datum *mcelem, int nmcelem,
                             float4 *numbers, int nnumbers,
                             Datum *array_data, int nitems,
                             float4 *hist, int nhist,
                             Oid operator, FmgrInfo *cmpfunc);
static float *calc_hist(const float4 *hist, int nhist, int n);
static float *calc_distr(const float *p, int n, int m, float rest);
static int  floor_log2(uint32 n);
static bool find_next_mcelem(Datum *mcelem, int nmcelem, Datum value,
                 int *index, FmgrInfo *cmpfunc);
static int  element_compare(const void *key1, const void *key2, void *arg);
static int  float_compare_desc(const void *key1, const void *key2);


/*
 * scalararraysel_containment
 *      Estimate selectivity of ScalarArrayOpExpr via array containment.
 *
 * If we have const =/<> ANY/ALL (array_var) then we can estimate the
 * selectivity as though this were an array containment operator,
 * array_var op ARRAY[const].
 *
 * scalararraysel() has already verified that the ScalarArrayOpExpr's operator
 * is the array element type's default equality or inequality operator, and
 * has aggressively simplified both inputs to constants.
 *
 * Returns selectivity (0..1), or -1 if we fail to estimate selectivity.
 */
Selectivity
scalararraysel_containment(PlannerInfo *root,
                           Node *leftop, Node *rightop,
                           Oid elemtype, bool isEquality, bool useOr,
                           int varRelid)
{
    Selectivity selec;
    VariableStatData vardata;
    Datum       constval;
    TypeCacheEntry *typentry;
    FmgrInfo   *cmpfunc;

    /*
     * rightop must be a variable, else punt.
     */
    examine_variable(root, rightop, varRelid, &vardata);
    if (!vardata.rel)
    {
        ReleaseVariableStats(vardata);
        return -1.0;
    }

    /*
     * leftop must be a constant, else punt.
     */
    if (!IsA(leftop, Const))
    {
        ReleaseVariableStats(vardata);
        return -1.0;
    }
    if (((Const *) leftop)->constisnull)
    {
        /* qual can't succeed if null on left */
        ReleaseVariableStats(vardata);
        return (Selectivity) 0.0;
    }
    constval = ((Const *) leftop)->constvalue;

    /* Get element type's default comparison function */
    typentry = lookup_type_cache(elemtype, TYPECACHE_CMP_PROC_FINFO);
    if (!OidIsValid(typentry->cmp_proc_finfo.fn_oid))
    {
        ReleaseVariableStats(vardata);
        return -1.0;
    }
    cmpfunc = &typentry->cmp_proc_finfo;

    /*
     * If the operator is <>, swap ANY/ALL, then invert the result later.
     */
    if (!isEquality)
        useOr = !useOr;

    /* Get array element stats for var, if available */
    if (HeapTupleIsValid(vardata.statsTuple))
    {
        Form_pg_statistic stats;
        Datum      *values;
        int         nvalues;
        float4     *numbers;
        int         nnumbers;
        float4     *hist;
        int         nhist;

        stats = (Form_pg_statistic) GETSTRUCT(vardata.statsTuple);

        /* MCELEM will be an array of same type as element */
        if (get_attstatsslot(vardata.statsTuple,
                             elemtype, vardata.atttypmod,
                             STATISTIC_KIND_MCELEM, InvalidOid,
                             NULL,
                             &values, &nvalues,
                             &numbers, &nnumbers))
        {
            /* For ALL case, also get histogram of distinct-element counts */
            if (useOr ||
                !get_attstatsslot(vardata.statsTuple,
                                  elemtype, vardata.atttypmod,
                                  STATISTIC_KIND_DECHIST, InvalidOid,
                                  NULL,
                                  NULL, NULL,
                                  &hist, &nhist))
            {
                hist = NULL;
                nhist = 0;
            }

            /*
             * For = ANY, estimate as var @> ARRAY[const].
             *
             * For = ALL, estimate as var <@ ARRAY[const].
             */
            if (useOr)
                selec = mcelem_array_contain_overlap_selec(values, nvalues,
                                                           numbers, nnumbers,
                                                           &constval, 1,
                                                       OID_ARRAY_CONTAINS_OP,
                                                           cmpfunc);
            else
                selec = mcelem_array_contained_selec(values, nvalues,
                                                     numbers, nnumbers,
                                                     &constval, 1,
                                                     hist, nhist,
                                                     OID_ARRAY_CONTAINED_OP,
                                                     cmpfunc);

            if (hist)
                free_attstatsslot(elemtype, NULL, 0, hist, nhist);
            free_attstatsslot(elemtype, values, nvalues, numbers, nnumbers);
        }
        else
        {
            /* No most-common-elements info, so do without */
            if (useOr)
                selec = mcelem_array_contain_overlap_selec(NULL, 0,
                                                           NULL, 0,
                                                           &constval, 1,
                                                       OID_ARRAY_CONTAINS_OP,
                                                           cmpfunc);
            else
                selec = mcelem_array_contained_selec(NULL, 0,
                                                     NULL, 0,
                                                     &constval, 1,
                                                     NULL, 0,
                                                     OID_ARRAY_CONTAINED_OP,
                                                     cmpfunc);
        }

        /*
         * MCE stats count only non-null rows, so adjust for null rows.
         */
        selec *= (1.0 - stats->stanullfrac);
    }
    else
    {
        /* No stats at all, so do without */
        if (useOr)
            selec = mcelem_array_contain_overlap_selec(NULL, 0,
                                                       NULL, 0,
                                                       &constval, 1,
                                                       OID_ARRAY_CONTAINS_OP,
                                                       cmpfunc);
        else
            selec = mcelem_array_contained_selec(NULL, 0,
                                                 NULL, 0,
                                                 &constval, 1,
                                                 NULL, 0,
                                                 OID_ARRAY_CONTAINED_OP,
                                                 cmpfunc);
        /* we assume no nulls here, so no stanullfrac correction */
    }

    ReleaseVariableStats(vardata);

    /*
     * If the operator is <>, invert the results.
     */
    if (!isEquality)
        selec = 1.0 - selec;

    CLAMP_PROBABILITY(selec);

    return selec;
}

/*
 * arraycontsel -- restriction selectivity for array @>, &&, <@ operators
 */
Datum
arraycontsel(PG_FUNCTION_ARGS)
{
    PlannerInfo *root = (PlannerInfo *) PG_GETARG_POINTER(0);
    Oid         operator = PG_GETARG_OID(1);
    List       *args = (List *) PG_GETARG_POINTER(2);
    int         varRelid = PG_GETARG_INT32(3);
    VariableStatData vardata;
    Node       *other;
    bool        varonleft;
    Selectivity selec;
    Oid         element_typeid;

    /*
     * If expression is not (variable op something) or (something op
     * variable), then punt and return a default estimate.
     */
    if (!get_restriction_variable(root, args, varRelid,
                                  &vardata, &other, &varonleft))
        PG_RETURN_FLOAT8(DEFAULT_SEL(operator));

    /*
     * Can't do anything useful if the something is not a constant, either.
     */
    if (!IsA(other, Const))
    {
        ReleaseVariableStats(vardata);
        PG_RETURN_FLOAT8(DEFAULT_SEL(operator));
    }

    /*
     * The "&&", "@>" and "<@" operators are strict, so we can cope with a
     * NULL constant right away.
     */
    if (((Const *) other)->constisnull)
    {
        ReleaseVariableStats(vardata);
        PG_RETURN_FLOAT8(0.0);
    }

    /*
     * If var is on the right, commute the operator, so that we can assume the
     * var is on the left in what follows.
     */
    if (!varonleft)
    {
        if (operator == OID_ARRAY_CONTAINS_OP)
            operator = OID_ARRAY_CONTAINED_OP;
        else if (operator == OID_ARRAY_CONTAINED_OP)
            operator = OID_ARRAY_CONTAINS_OP;
    }

    /*
     * OK, there's a Var and a Const we're dealing with here.  We need the
     * Const to be an array with same element type as column, else we can't do
     * anything useful.  (Such cases will likely fail at runtime, but here
     * we'd rather just return a default estimate.)
     */
    element_typeid = get_base_element_type(((Const *) other)->consttype);
    if (element_typeid != InvalidOid &&
        element_typeid == get_base_element_type(vardata.vartype))
    {
        selec = calc_arraycontsel(&vardata, ((Const *) other)->constvalue,
                                  element_typeid, operator);
    }
    else
    {
        selec = DEFAULT_SEL(operator);
    }

    ReleaseVariableStats(vardata);

    CLAMP_PROBABILITY(selec);

    PG_RETURN_FLOAT8((float8) selec);
}

/*
 * arraycontjoinsel -- join selectivity for array @>, &&, <@ operators
 */
Datum
arraycontjoinsel(PG_FUNCTION_ARGS)
{
    /* For the moment this is just a stub */
    Oid         operator = PG_GETARG_OID(1);

    PG_RETURN_FLOAT8(DEFAULT_SEL(operator));
}

/*
 * Calculate selectivity for "arraycolumn @> const", "arraycolumn && const"
 * or "arraycolumn <@ const" based on the statistics
 *
 * This function is mainly responsible for extracting the pg_statistic data
 * to be used; we then pass the problem on to mcelem_array_selec().
 */
static Selectivity
calc_arraycontsel(VariableStatData *vardata, Datum constval,
                  Oid elemtype, Oid operator)
{
    Selectivity selec;
    TypeCacheEntry *typentry;
    FmgrInfo   *cmpfunc;
    ArrayType  *array;

    /* Get element type's default comparison function */
    typentry = lookup_type_cache(elemtype, TYPECACHE_CMP_PROC_FINFO);
    if (!OidIsValid(typentry->cmp_proc_finfo.fn_oid))
        return DEFAULT_SEL(operator);
    cmpfunc = &typentry->cmp_proc_finfo;

    /*
     * The caller made sure the const is an array with same element type, so
     * get it now
     */
    array = DatumGetArrayTypeP(constval);

    if (HeapTupleIsValid(vardata->statsTuple))
    {
        Form_pg_statistic stats;
        Datum      *values;
        int         nvalues;
        float4     *numbers;
        int         nnumbers;
        float4     *hist;
        int         nhist;

        stats = (Form_pg_statistic) GETSTRUCT(vardata->statsTuple);

        /* MCELEM will be an array of same type as column */
        if (get_attstatsslot(vardata->statsTuple,
                             elemtype, vardata->atttypmod,
                             STATISTIC_KIND_MCELEM, InvalidOid,
                             NULL,
                             &values, &nvalues,
                             &numbers, &nnumbers))
        {
            /*
             * For "array <@ const" case we also need histogram of distinct
             * element counts.
             */
            if (operator != OID_ARRAY_CONTAINED_OP ||
                !get_attstatsslot(vardata->statsTuple,
                                  elemtype, vardata->atttypmod,
                                  STATISTIC_KIND_DECHIST, InvalidOid,
                                  NULL,
                                  NULL, NULL,
                                  &hist, &nhist))
            {
                hist = NULL;
                nhist = 0;
            }

            /* Use the most-common-elements slot for the array Var. */
            selec = mcelem_array_selec(array, typentry,
                                       values, nvalues,
                                       numbers, nnumbers,
                                       hist, nhist,
                                       operator, cmpfunc);

            if (hist)
                free_attstatsslot(elemtype, NULL, 0, hist, nhist);
            free_attstatsslot(elemtype, values, nvalues, numbers, nnumbers);
        }
        else
        {
            /* No most-common-elements info, so do without */
            selec = mcelem_array_selec(array, typentry,
                                       NULL, 0, NULL, 0, NULL, 0,
                                       operator, cmpfunc);
        }

        /*
         * MCE stats count only non-null rows, so adjust for null rows.
         */
        selec *= (1.0 - stats->stanullfrac);
    }
    else
    {
        /* No stats at all, so do without */
        selec = mcelem_array_selec(array, typentry,
                                   NULL, 0, NULL, 0, NULL, 0,
                                   operator, cmpfunc);
        /* we assume no nulls here, so no stanullfrac correction */
    }

    /* If constant was toasted, release the copy we made */
    if (PointerGetDatum(array) != constval)
        pfree(array);

    return selec;
}

/*
 * Array selectivity estimation based on most common elements statistics
 *
 * This function just deconstructs and sorts the array constant's contents,
 * and then passes the problem on to mcelem_array_contain_overlap_selec or
 * mcelem_array_contained_selec depending on the operator.
 */
static Selectivity
mcelem_array_selec(ArrayType *array, TypeCacheEntry *typentry,
                   Datum *mcelem, int nmcelem,
                   float4 *numbers, int nnumbers,
                   float4 *hist, int nhist,
                   Oid operator, FmgrInfo *cmpfunc)
{
    Selectivity selec;
    int         num_elems;
    Datum      *elem_values;
    bool       *elem_nulls;
    bool        null_present;
    int         nonnull_nitems;
    int         i;

    /*
     * Prepare constant array data for sorting.  Sorting lets us find unique
     * elements and efficiently merge with the MCELEM array.
     */
    deconstruct_array(array,
                      typentry->type_id,
                      typentry->typlen,
                      typentry->typbyval,
                      typentry->typalign,
                      &elem_values, &elem_nulls, &num_elems);

    /* Collapse out any null elements */
    nonnull_nitems = 0;
    null_present = false;
    for (i = 0; i < num_elems; i++)
    {
        if (elem_nulls[i])
            null_present = true;
        else
            elem_values[nonnull_nitems++] = elem_values[i];
    }

    /*
     * Query "column @> '{anything, null}'" matches nothing.  For the other
     * two operators, presence of a null in the constant can be ignored.
     */
    if (null_present && operator == OID_ARRAY_CONTAINS_OP)
    {
        pfree(elem_values);
        pfree(elem_nulls);
        return (Selectivity) 0.0;
    }

    /* Sort extracted elements using their default comparison function. */
    qsort_arg(elem_values, nonnull_nitems, sizeof(Datum),
              element_compare, cmpfunc);

    /* Separate cases according to operator */
    if (operator == OID_ARRAY_CONTAINS_OP || operator == OID_ARRAY_OVERLAP_OP)
        selec = mcelem_array_contain_overlap_selec(mcelem, nmcelem,
                                                   numbers, nnumbers,
                                                 elem_values, nonnull_nitems,
                                                   operator, cmpfunc);
    else if (operator == OID_ARRAY_CONTAINED_OP)
        selec = mcelem_array_contained_selec(mcelem, nmcelem,
                                             numbers, nnumbers,
                                             elem_values, nonnull_nitems,
                                             hist, nhist,
                                             operator, cmpfunc);
    else
    {
        elog(ERROR, "arraycontsel called for unrecognized operator %u",
             operator);
        selec = 0.0;            /* keep compiler quiet */
    }

    pfree(elem_values);
    pfree(elem_nulls);
    return selec;
}

/*
 * Estimate selectivity of "column @> const" and "column && const" based on
 * most common element statistics.  This estimation assumes element
 * occurrences are independent.
 *
 * mcelem (of length nmcelem) and numbers (of length nnumbers) are from
 * the array column's MCELEM statistics slot, or are NULL/0 if stats are
 * not available.  array_data (of length nitems) is the constant's elements.
 *
 * Both the mcelem and array_data arrays are assumed presorted according
 * to the element type's cmpfunc.  Null elements are not present.
 *
 * TODO: this estimate probably could be improved by using the distinct
 * elements count histogram.  For example, excepting the special case of
 * "column @> '{}'", we can multiply the calculated selectivity by the
 * fraction of nonempty arrays in the column.
 */
static Selectivity
mcelem_array_contain_overlap_selec(Datum *mcelem, int nmcelem,
                                   float4 *numbers, int nnumbers,
                                   Datum *array_data, int nitems,
                                   Oid operator, FmgrInfo *cmpfunc)
{
    Selectivity selec,
                elem_selec;
    int         mcelem_index,
                i;
    bool        use_bsearch;
    float4      minfreq;

    /*
     * There should be three more Numbers than Values, because the last three
     * cells should hold minimal and maximal frequency among the non-null
     * elements, and then the frequency of null elements.  Ignore the Numbers
     * if not right.
     */
    if (nnumbers != nmcelem + 3)
    {
        numbers = NULL;
        nnumbers = 0;
    }

    if (numbers)
    {
        /* Grab the lowest observed frequency */
        minfreq = numbers[nmcelem];
    }
    else
    {
        /* Without statistics make some default assumptions */
        minfreq = 2 * (float4) DEFAULT_CONTAIN_SEL;
    }

    /* Decide whether it is faster to use binary search or not. */
    if (nitems * floor_log2((uint32) nmcelem) < nmcelem + nitems)
        use_bsearch = true;
    else
        use_bsearch = false;

    if (operator == OID_ARRAY_CONTAINS_OP)
    {
        /*
         * Initial selectivity for "column @> const" query is 1.0, and it will
         * be decreased with each element of constant array.
         */
        selec = 1.0;
    }
    else
    {
        /*
         * Initial selectivity for "column && const" query is 0.0, and it will
         * be increased with each element of constant array.
         */
        selec = 0.0;
    }

    /* Scan mcelem and array in parallel. */
    mcelem_index = 0;
    for (i = 0; i < nitems; i++)
    {
        bool        match = false;

        /* Ignore any duplicates in the array data. */
        if (i > 0 &&
            element_compare(&array_data[i - 1], &array_data[i], cmpfunc) == 0)
            continue;

        /* Find the smallest MCELEM >= this array item. */
        if (use_bsearch)
        {
            match = find_next_mcelem(mcelem, nmcelem, array_data[i],
                                     &mcelem_index, cmpfunc);
        }
        else
        {
            while (mcelem_index < nmcelem)
            {
                int         cmp = element_compare(&mcelem[mcelem_index],
                                                  &array_data[i],
                                                  cmpfunc);

                if (cmp < 0)
                    mcelem_index++;
                else
                {
                    if (cmp == 0)
                        match = true;   /* mcelem is found */
                    break;
                }
            }
        }

        if (match && numbers)
        {
            /* MCELEM matches the array item; use its frequency. */
            elem_selec = numbers[mcelem_index];
            mcelem_index++;
        }
        else
        {
            /*
             * The element is not in MCELEM.  Punt, but assume that the
             * selectivity cannot be more than minfreq / 2.
             */
            elem_selec = Min(DEFAULT_CONTAIN_SEL, minfreq / 2);
        }

        /*
         * Update overall selectivity using the current element's selectivity
         * and an assumption of element occurrence independence.
         */
        if (operator == OID_ARRAY_CONTAINS_OP)
            selec *= elem_selec;
        else
            selec = selec + elem_selec - selec * elem_selec;

        /* Clamp intermediate results to stay sane despite roundoff error */
        CLAMP_PROBABILITY(selec);
    }

    return selec;
}

/*
 * Estimate selectivity of "column <@ const" based on most common element
 * statistics.
 *
 * mcelem (of length nmcelem) and numbers (of length nnumbers) are from
 * the array column's MCELEM statistics slot, or are NULL/0 if stats are
 * not available.  array_data (of length nitems) is the constant's elements.
 * hist (of length nhist) is from the array column's DECHIST statistics slot,
 * or is NULL/0 if those stats are not available.
 *
 * Both the mcelem and array_data arrays are assumed presorted according
 * to the element type's cmpfunc.  Null elements are not present.
 *
 * Independent element occurrence would imply a particular distribution of
 * distinct element counts among matching rows.  Real data usually falsifies
 * that assumption.  For example, in a set of 11-element integer arrays having
 * elements in the range [0..10], element occurrences are typically not
 * independent.  If they were, a sufficiently-large set would include all
 * distinct element counts 0 through 11.  We correct for this using the
 * histogram of distinct element counts.
 *
 * In the "column @> const" and "column && const" cases, we usually have a
 * "const" with low number of elements (otherwise we have selectivity close
 * to 0 or 1 respectively).  That's why the effect of dependence related
 * to distinct element count distribution is negligible there.  In the
 * "column <@ const" case, number of elements is usually high (otherwise we
 * have selectivity close to 0).  That's why we should do a correction with
 * the array distinct element count distribution here.
 *
 * Using the histogram of distinct element counts produces a different
 * distribution law than independent occurrences of elements.  This
 * distribution law can be described as follows:
 *
 * P(o1, o2, ..., on) = f1^o1 * (1 - f1)^(1 - o1) * f2^o2 *
 *    (1 - f2)^(1 - o2) * ... * fn^on * (1 - fn)^(1 - on) * hist[m] / ind[m]
 *
 * where:
 * o1, o2, ..., on - occurrences of elements 1, 2, ..., n
 *      (1 - occurrence, 0 - no occurrence) in row
 * f1, f2, ..., fn - frequencies of elements 1, 2, ..., n
 *      (scalar values in [0..1]) according to collected statistics
 * m = o1 + o2 + ... + on = total number of distinct elements in row
 * hist[m] - histogram data for occurrence of m elements.
 * ind[m] - probability of m occurrences from n events assuming their
 *    probabilities to be equal to frequencies of array elements.
 *
 * ind[m] = sum(f1^o1 * (1 - f1)^(1 - o1) * f2^o2 * (1 - f2)^(1 - o2) *
 * ... * fn^on * (1 - fn)^(1 - on), o1, o2, ..., on) | o1 + o2 + .. on = m
 */
static Selectivity
mcelem_array_contained_selec(Datum *mcelem, int nmcelem,
                             float4 *numbers, int nnumbers,
                             Datum *array_data, int nitems,
                             float4 *hist, int nhist,
                             Oid operator, FmgrInfo *cmpfunc)
{
    int         mcelem_index,
                i,
                unique_nitems = 0;
    float       selec,
                minfreq,
                nullelem_freq;
    float      *dist,
               *mcelem_dist,
               *hist_part;
    float       avg_count,
                mult,
                rest;
    float      *elem_selec;

    /*
     * There should be three more Numbers than Values in the MCELEM slot,
     * because the last three cells should hold minimal and maximal frequency
     * among the non-null elements, and then the frequency of null elements.
     * Punt if not right, because we can't do much without the element freqs.
     */
    if (numbers == NULL || nnumbers != nmcelem + 3)
        return DEFAULT_CONTAIN_SEL;

    /* Can't do much without a count histogram, either */
    if (hist == NULL || nhist < 3)
        return DEFAULT_CONTAIN_SEL;

    /*
     * Grab some of the summary statistics that compute_array_stats() stores:
     * lowest frequency, frequency of null elements, and average distinct
     * element count.
     */
    minfreq = numbers[nmcelem];
    nullelem_freq = numbers[nmcelem + 2];
    avg_count = hist[nhist - 1];

    /*
     * "rest" will be the sum of the frequencies of all elements not
     * represented in MCELEM.  The average distinct element count is the sum
     * of the frequencies of *all* elements.  Begin with that; we will proceed
     * to subtract the MCELEM frequencies.
     */
    rest = avg_count;

    /*
     * mult is a multiplier representing estimate of probability that each
     * mcelem that is not present in constant doesn't occur.
     */
    mult = 1.0f;

    /*
     * elem_selec is array of estimated frequencies for elements in the
     * constant.
     */
    elem_selec = (float *) palloc(sizeof(float) * nitems);

    /* Scan mcelem and array in parallel. */
    mcelem_index = 0;
    for (i = 0; i < nitems; i++)
    {
        bool        match = false;

        /* Ignore any duplicates in the array data. */
        if (i > 0 &&
            element_compare(&array_data[i - 1], &array_data[i], cmpfunc) == 0)
            continue;

        /*
         * Iterate over MCELEM until we find an entry greater than or equal to
         * this element of the constant.  Update "rest" and "mult" for mcelem
         * entries skipped over.
         */
        while (mcelem_index < nmcelem)
        {
            int         cmp = element_compare(&mcelem[mcelem_index],
                                              &array_data[i],
                                              cmpfunc);

            if (cmp < 0)
            {
                mult *= (1.0f - numbers[mcelem_index]);
                rest -= numbers[mcelem_index];
                mcelem_index++;
            }
            else
            {
                if (cmp == 0)
                    match = true;       /* mcelem is found */
                break;
            }
        }

        if (match)
        {
            /* MCELEM matches the array item. */
            elem_selec[unique_nitems] = numbers[mcelem_index];
            /* "rest" is decremented for all mcelems, matched or not */
            rest -= numbers[mcelem_index];
            mcelem_index++;
        }
        else
        {
            /*
             * The element is not in MCELEM.  Punt, but assume that the
             * selectivity cannot be more than minfreq / 2.
             */
            elem_selec[unique_nitems] = Min(DEFAULT_CONTAIN_SEL,
                                            minfreq / 2);
        }

        unique_nitems++;
    }

    /*
     * If we handled all constant elements without exhausting the MCELEM
     * array, finish walking it to complete calculation of "rest" and "mult".
     */
    while (mcelem_index < nmcelem)
    {
        mult *= (1.0f - numbers[mcelem_index]);
        rest -= numbers[mcelem_index];
        mcelem_index++;
    }

    /*
     * The presence of many distinct rare elements materially decreases
     * selectivity.  Use the Poisson distribution to estimate the probability
     * of a column value having zero occurrences of such elements.  See above
     * for the definition of "rest".
     */
    mult *= exp(-rest);

    /*----------
     * Using the distinct element count histogram requires
     *      O(unique_nitems * (nmcelem + unique_nitems))
     * operations.  Beyond a certain computational cost threshold, it's
     * reasonable to sacrifice accuracy for decreased planning time.  We limit
     * the number of operations to EFFORT * nmcelem; since nmcelem is limited
     * by the column's statistics target, the work done is user-controllable.
     *
     * If the number of operations would be too large, we can reduce it
     * without losing all accuracy by reducing unique_nitems and considering
     * only the most-common elements of the constant array.  To make the
     * results exactly match what we would have gotten with only those
     * elements to start with, we'd have to remove any discarded elements'
     * frequencies from "mult", but since this is only an approximation
     * anyway, we don't bother with that.  Therefore it's sufficient to qsort
     * elem_selec[] and take the largest elements.  (They will no longer match
     * up with the elements of array_data[], but we don't care.)
     *----------
     */
#define EFFORT 100

    if ((nmcelem + unique_nitems) > 0 &&
        unique_nitems > EFFORT * nmcelem / (nmcelem + unique_nitems))
    {
        /*
         * Use the quadratic formula to solve for largest allowable N.  We
         * have A = 1, B = nmcelem, C = - EFFORT * nmcelem.
         */
        double      b = (double) nmcelem;
        int         n;

        n = (int) ((sqrt(b * b + 4 * EFFORT * b) - b) / 2);

        /* Sort, then take just the first n elements */
        qsort(elem_selec, unique_nitems, sizeof(float),
              float_compare_desc);
        unique_nitems = n;
    }

    /*
     * Calculate probabilities of each distinct element count for both mcelems
     * and constant elements.  At this point, assume independent element
     * occurrence.
     */
    dist = calc_distr(elem_selec, unique_nitems, unique_nitems, 0.0f);
    mcelem_dist = calc_distr(numbers, nmcelem, unique_nitems, rest);

    /* ignore hist[nhist-1], which is the average not a histogram member */
    hist_part = calc_hist(hist, nhist - 1, unique_nitems);

    selec = 0.0f;
    for (i = 0; i <= unique_nitems; i++)
    {
        /*
         * mult * dist[i] / mcelem_dist[i] gives us probability of qual
         * matching from assumption of independent element occurrence with the
         * condition that distinct element count = i.
         */
        if (mcelem_dist[i] > 0)
            selec += hist_part[i] * mult * dist[i] / mcelem_dist[i];
    }

    pfree(dist);
    pfree(mcelem_dist);
    pfree(hist_part);
    pfree(elem_selec);

    /* Take into account occurrence of NULL element. */
    selec *= (1.0f - nullelem_freq);

    CLAMP_PROBABILITY(selec);

    return selec;
}

/*
 * Calculate the first n distinct element count probabilities from a
 * histogram of distinct element counts.
 *
 * Returns a palloc'd array of n+1 entries, with array[k] being the
 * probability of element count k, k in [0..n].
 *
 * We assume that a histogram box with bounds a and b gives 1 / ((b - a + 1) *
 * (nhist - 1)) probability to each value in (a,b) and an additional half of
 * that to a and b themselves.
 */
static float *
calc_hist(const float4 *hist, int nhist, int n)
{
    float      *hist_part;
    int         k,
                i = 0;
    float       prev_interval = 0,
                next_interval;
    float       frac;

    hist_part = (float *) palloc((n + 1) * sizeof(float));

    /*
     * frac is a probability contribution for each interval between histogram
     * values.  We have nhist - 1 intervals, so contribution of each one will
     * be 1 / (nhist - 1).
     */
    frac = 1.0f / ((float) (nhist - 1));

    for (k = 0; k <= n; k++)
    {
        int         count = 0;

        /*
         * Count the histogram boundaries equal to k.  (Although the histogram
         * should theoretically contain only exact integers, entries are
         * floats so there could be roundoff error in large values.  Treat any
         * fractional value as equal to the next larger k.)
         */
        while (i < nhist && hist[i] <= k)
        {
            count++;
            i++;
        }

        if (count > 0)
        {
            /* k is an exact bound for at least one histogram box. */
            float       val;

            /* Find length between current histogram value and the next one */
            if (i < nhist)
                next_interval = hist[i] - hist[i - 1];
            else
                next_interval = 0;

            /*
             * count - 1 histogram boxes contain k exclusively.  They
             * contribute a total of (count - 1) * frac probability.  Also
             * factor in the partial histogram boxes on either side.
             */
            val = (float) (count - 1);
            if (next_interval > 0)
                val += 0.5f / next_interval;
            if (prev_interval > 0)
                val += 0.5f / prev_interval;
            hist_part[k] = frac * val;

            prev_interval = next_interval;
        }
        else
        {
            /* k does not appear as an exact histogram bound. */
            if (prev_interval > 0)
                hist_part[k] = frac / prev_interval;
            else
                hist_part[k] = 0.0f;
        }
    }

    return hist_part;
}

/*
 * Consider n independent events with probabilities p[].  This function
 * calculates probabilities of exact k of events occurrence for k in [0..m].
 * Returns a palloc'd array of size m+1.
 *
 * "rest" is the sum of the probabilities of all low-probability events not
 * included in p.
 *
 * Imagine matrix M of size (n + 1) x (m + 1).  Element M[i,j] denotes the
 * probability that exactly j of first i events occur.  Obviously M[0,0] = 1.
 * For any constant j, each increment of i increases the probability iff the
 * event occurs.  So, by the law of total probability:
 *  M[i,j] = M[i - 1, j] * (1 - p[i]) + M[i - 1, j - 1] * p[i]
 *      for i > 0, j > 0.
 *  M[i,0] = M[i - 1, 0] * (1 - p[i]) for i > 0.
 */
static float *
calc_distr(const float *p, int n, int m, float rest)
{
    float      *row,
               *prev_row,
               *tmp;
    int         i,
                j;

    /*
     * Since we return only the last row of the matrix and need only the
     * current and previous row for calculations, allocate two rows.
     */
    row = (float *) palloc((m + 1) * sizeof(float));
    prev_row = (float *) palloc((m + 1) * sizeof(float));

    /* M[0,0] = 1 */
    row[0] = 1.0f;
    for (i = 1; i <= n; i++)
    {
        float       t = p[i - 1];

        /* Swap rows */
        tmp = row;
        row = prev_row;
        prev_row = tmp;

        /* Calculate next row */
        for (j = 0; j <= i && j <= m; j++)
        {
            float       val = 0.0f;

            if (j < i)
                val += prev_row[j] * (1.0f - t);
            if (j > 0)
                val += prev_row[j - 1] * t;
            row[j] = val;
        }
    }

    /*
     * The presence of many distinct rare (not in "p") elements materially
     * decreases selectivity.  Model their collective occurrence with the
     * Poisson distribution.
     */
    if (rest > DEFAULT_CONTAIN_SEL)
    {
        float       t;

        /* Swap rows */
        tmp = row;
        row = prev_row;
        prev_row = tmp;

        for (i = 0; i <= m; i++)
            row[i] = 0.0f;

        /* Value of Poisson distribution for 0 occurrences */
        t = exp(-rest);

        /*
         * Calculate convolution of previously computed distribution and the
         * Poisson distribution.
         */
        for (i = 0; i <= m; i++)
        {
            for (j = 0; j <= m - i; j++)
                row[j + i] += prev_row[j] * t;

            /* Get Poisson distribution value for (i + 1) occurrences */
            t *= rest / (float) (i + 1);
        }
    }

    pfree(prev_row);
    return row;
}

/* Fast function for floor value of 2 based logarithm calculation. */
static int
floor_log2(uint32 n)
{
    int         logval = 0;

    if (n == 0)
        return -1;
    if (n >= (1 << 16))
    {
        n >>= 16;
        logval += 16;
    }
    if (n >= (1 << 8))
    {
        n >>= 8;
        logval += 8;
    }
    if (n >= (1 << 4))
    {
        n >>= 4;
        logval += 4;
    }
    if (n >= (1 << 2))
    {
        n >>= 2;
        logval += 2;
    }
    if (n >= (1 << 1))
    {
        logval += 1;
    }
    return logval;
}

/*
 * find_next_mcelem binary-searches a most common elements array, starting
 * from *index, for the first member >= value.  It saves the position of the
 * match into *index and returns true if it's an exact match.  (Note: we
 * assume the mcelem elements are distinct so there can't be more than one
 * exact match.)
 */
static bool
find_next_mcelem(Datum *mcelem, int nmcelem, Datum value, int *index,
                 FmgrInfo *cmpfunc)
{
    int         l = *index,
                r = nmcelem - 1,
                i,
                res;

    while (l <= r)
    {
        i = (l + r) / 2;
        res = element_compare(&mcelem[i], &value, cmpfunc);
        if (res == 0)
        {
            *index = i;
            return true;
        }
        else if (res < 0)
            l = i + 1;
        else
            r = i - 1;
    }
    *index = l;
    return false;
}

/*
 * Comparison function for elements.
 *
 * We use the element type's default btree opclass, and the default collation
 * if the type is collation-sensitive.
 *
 * XXX consider using SortSupport infrastructure
 */
static int
element_compare(const void *key1, const void *key2, void *arg)
{
    Datum       d1 = *((const Datum *) key1);
    Datum       d2 = *((const Datum *) key2);
    FmgrInfo   *cmpfunc = (FmgrInfo *) arg;
    Datum       c;

    c = FunctionCall2Coll(cmpfunc, DEFAULT_COLLATION_OID, d1, d2);
    return DatumGetInt32(c);
}

/*
 * Comparison function for sorting floats into descending order.
 */
static int
float_compare_desc(const void *key1, const void *key2)
{
    float       d1 = *((const float *) key1);
    float       d2 = *((const float *) key2);

    if (d1 > d2)
        return -1;
    else if (d1 < d2)
        return 1;
    else
        return 0;
}