array_typanalyze.c

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/*-------------------------------------------------------------------------
 *
 * array_typanalyze.c
 *    Functions for gathering statistics from array columns
 *
 * Portions Copyright (c) 1996-2018, PostgreSQL Global Development Group
 * Portions Copyright (c) 1994, Regents of the University of California
 *
 *
 * IDENTIFICATION
 *    src/backend/utils/adt/array_typanalyze.c
 *
 *-------------------------------------------------------------------------
 */
#include "postgres.h"

#include "access/tuptoaster.h"
#include "catalog/pg_collation.h"
#include "commands/vacuum.h"
#include "utils/array.h"
#include "utils/builtins.h"
#include "utils/datum.h"
#include "utils/lsyscache.h"
#include "utils/typcache.h"


/*
 * To avoid consuming too much memory, IO and CPU load during analysis, and/or
 * too much space in the resulting pg_statistic rows, we ignore arrays that
 * are wider than ARRAY_WIDTH_THRESHOLD (after detoasting!).  Note that this
 * number is considerably more than the similar WIDTH_THRESHOLD limit used
 * in analyze.c's standard typanalyze code.
 */
#define ARRAY_WIDTH_THRESHOLD 0x10000

/* Extra data for compute_array_stats function */
typedef struct
{
    /* Information about array element type */
    Oid         type_id;        /* element type's OID */
    Oid         eq_opr;         /* default equality operator's OID */
    bool        typbyval;       /* physical properties of element type */
    int16       typlen;
    char        typalign;

    /*
     * Lookup data for element type's comparison and hash functions (these are
     * in the type's typcache entry, which we expect to remain valid over the
     * lifespan of the ANALYZE run)
     */
    FmgrInfo   *cmp;
    FmgrInfo   *hash;

    /* Saved state from std_typanalyze() */
    AnalyzeAttrComputeStatsFunc std_compute_stats;
    void       *std_extra_data;
} ArrayAnalyzeExtraData;

/*
 * While compute_array_stats is running, we keep a pointer to the extra data
 * here for use by assorted subroutines.  compute_array_stats doesn't
 * currently need to be re-entrant, so avoiding this is not worth the extra
 * notational cruft that would be needed.
 */
static ArrayAnalyzeExtraData *array_extra_data;

/* A hash table entry for the Lossy Counting algorithm */
typedef struct
{
    Datum       key;            /* This is 'e' from the LC algorithm. */
    int         frequency;      /* This is 'f'. */
    int         delta;          /* And this is 'delta'. */
    int         last_container; /* For de-duplication of array elements. */
} TrackItem;

/* A hash table entry for distinct-elements counts */
typedef struct
{
    int         count;          /* Count of distinct elements in an array */
    int         frequency;      /* Number of arrays seen with this count */
} DECountItem;

static void compute_array_stats(VacAttrStats *stats,
                    AnalyzeAttrFetchFunc fetchfunc, int samplerows, double totalrows);
static void prune_element_hashtable(HTAB *elements_tab, int b_current);
static uint32 element_hash(const void *key, Size keysize);
static int  element_match(const void *key1, const void *key2, Size keysize);
static int  element_compare(const void *key1, const void *key2);
static int  trackitem_compare_frequencies_desc(const void *e1, const void *e2);
static int  trackitem_compare_element(const void *e1, const void *e2);
static int  countitem_compare_count(const void *e1, const void *e2);


/*
 * array_typanalyze -- typanalyze function for array columns
 */
Datum
array_typanalyze(PG_FUNCTION_ARGS)
{
    VacAttrStats *stats = (VacAttrStats *) PG_GETARG_POINTER(0);
    Oid         element_typeid;
    TypeCacheEntry *typentry;
    ArrayAnalyzeExtraData *extra_data;

    /*
     * Call the standard typanalyze function.  It may fail to find needed
     * operators, in which case we also can't do anything, so just fail.
     */
    if (!std_typanalyze(stats))
        PG_RETURN_BOOL(false);

    /*
     * Check attribute data type is a varlena array (or a domain over one).
     */
    element_typeid = get_base_element_type(stats->attrtypid);
    if (!OidIsValid(element_typeid))
        elog(ERROR, "array_typanalyze was invoked for non-array type %u",
             stats->attrtypid);

    /*
     * Gather information about the element type.  If we fail to find
     * something, return leaving the state from std_typanalyze() in place.
     */
    typentry = lookup_type_cache(element_typeid,
                                 TYPECACHE_EQ_OPR |
                                 TYPECACHE_CMP_PROC_FINFO |
                                 TYPECACHE_HASH_PROC_FINFO);

    if (!OidIsValid(typentry->eq_opr) ||
        !OidIsValid(typentry->cmp_proc_finfo.fn_oid) ||
        !OidIsValid(typentry->hash_proc_finfo.fn_oid))
        PG_RETURN_BOOL(true);

    /* Store our findings for use by compute_array_stats() */
    extra_data = (ArrayAnalyzeExtraData *) palloc(sizeof(ArrayAnalyzeExtraData));
    extra_data->type_id = typentry->type_id;
    extra_data->eq_opr = typentry->eq_opr;
    extra_data->typbyval = typentry->typbyval;
    extra_data->typlen = typentry->typlen;
    extra_data->typalign = typentry->typalign;
    extra_data->cmp = &typentry->cmp_proc_finfo;
    extra_data->hash = &typentry->hash_proc_finfo;

    /* Save old compute_stats and extra_data for scalar statistics ... */
    extra_data->std_compute_stats = stats->compute_stats;
    extra_data->std_extra_data = stats->extra_data;

    /* ... and replace with our info */
    stats->compute_stats = compute_array_stats;
    stats->extra_data = extra_data;

    /*
     * Note we leave stats->minrows set as std_typanalyze set it.  Should it
     * be increased for array analysis purposes?
     */

    PG_RETURN_BOOL(true);
}

/*
 * compute_array_stats() -- compute statistics for an array column
 *
 * This function computes statistics useful for determining selectivity of
 * the array operators <@, &&, and @>.  It is invoked by ANALYZE via the
 * compute_stats hook after sample rows have been collected.
 *
 * We also invoke the standard compute_stats function, which will compute
 * "scalar" statistics relevant to the btree-style array comparison operators.
 * However, exact duplicates of an entire array may be rare despite many
 * arrays sharing individual elements.  This especially afflicts long arrays,
 * which are also liable to lack all scalar statistics due to the low
 * WIDTH_THRESHOLD used in analyze.c.  So, in addition to the standard stats,
 * we find the most common array elements and compute a histogram of distinct
 * element counts.
 *
 * The algorithm used is Lossy Counting, as proposed in the paper "Approximate
 * frequency counts over data streams" by G. S. Manku and R. Motwani, in
 * Proceedings of the 28th International Conference on Very Large Data Bases,
 * Hong Kong, China, August 2002, section 4.2. The paper is available at
 * http://www.vldb.org/conf/2002/S10P03.pdf
 *
 * The Lossy Counting (aka LC) algorithm goes like this:
 * Let s be the threshold frequency for an item (the minimum frequency we
 * are interested in) and epsilon the error margin for the frequency. Let D
 * be a set of triples (e, f, delta), where e is an element value, f is that
 * element's frequency (actually, its current occurrence count) and delta is
 * the maximum error in f. We start with D empty and process the elements in
 * batches of size w. (The batch size is also known as "bucket size" and is
 * equal to 1/epsilon.) Let the current batch number be b_current, starting
 * with 1. For each element e we either increment its f count, if it's
 * already in D, or insert a new triple into D with values (e, 1, b_current
 * - 1). After processing each batch we prune D, by removing from it all
 * elements with f + delta <= b_current.  After the algorithm finishes we
 * suppress all elements from D that do not satisfy f >= (s - epsilon) * N,
 * where N is the total number of elements in the input.  We emit the
 * remaining elements with estimated frequency f/N.  The LC paper proves
 * that this algorithm finds all elements with true frequency at least s,
 * and that no frequency is overestimated or is underestimated by more than
 * epsilon.  Furthermore, given reasonable assumptions about the input
 * distribution, the required table size is no more than about 7 times w.
 *
 * In the absence of a principled basis for other particular values, we
 * follow ts_typanalyze() and use parameters s = 0.07/K, epsilon = s/10.
 * But we leave out the correction for stopwords, which do not apply to
 * arrays.  These parameters give bucket width w = K/0.007 and maximum
 * expected hashtable size of about 1000 * K.
 *
 * Elements may repeat within an array.  Since duplicates do not change the
 * behavior of <@, && or @>, we want to count each element only once per
 * array.  Therefore, we store in the finished pg_statistic entry each
 * element's frequency as the fraction of all non-null rows that contain it.
 * We divide the raw counts by nonnull_cnt to get those figures.
 */
static void
compute_array_stats(VacAttrStats *stats, AnalyzeAttrFetchFunc fetchfunc,
                    int samplerows, double totalrows)
{
    ArrayAnalyzeExtraData *extra_data;
    int         num_mcelem;
    int         null_cnt = 0;
    int         null_elem_cnt = 0;
    int         analyzed_rows = 0;

    /* This is D from the LC algorithm. */
    HTAB       *elements_tab;
    HASHCTL     elem_hash_ctl;
    HASH_SEQ_STATUS scan_status;

    /* This is the current bucket number from the LC algorithm */
    int         b_current;

    /* This is 'w' from the LC algorithm */
    int         bucket_width;
    int         array_no;
    int64       element_no;
    TrackItem  *item;
    int         slot_idx;
    HTAB       *count_tab;
    HASHCTL     count_hash_ctl;
    DECountItem *count_item;

    extra_data = (ArrayAnalyzeExtraData *) stats->extra_data;

    /*
     * Invoke analyze.c's standard analysis function to create scalar-style
     * stats for the column.  It will expect its own extra_data pointer, so
     * temporarily install that.
     */
    stats->extra_data = extra_data->std_extra_data;
    extra_data->std_compute_stats(stats, fetchfunc, samplerows, totalrows);
    stats->extra_data = extra_data;

    /*
     * Set up static pointer for use by subroutines.  We wait till here in
     * case std_compute_stats somehow recursively invokes us (probably not
     * possible, but ...)
     */
    array_extra_data = extra_data;

    /*
     * We want statistics_target * 10 elements in the MCELEM array. This
     * multiplier is pretty arbitrary, but is meant to reflect the fact that
     * the number of individual elements tracked in pg_statistic ought to be
     * more than the number of values for a simple scalar column.
     */
    num_mcelem = stats->attr->attstattarget * 10;

    /*
     * We set bucket width equal to num_mcelem / 0.007 as per the comment
     * above.
     */
    bucket_width = num_mcelem * 1000 / 7;

    /*
     * Create the hashtable. It will be in local memory, so we don't need to
     * worry about overflowing the initial size. Also we don't need to pay any
     * attention to locking and memory management.
     */
    MemSet(&elem_hash_ctl, 0, sizeof(elem_hash_ctl));
    elem_hash_ctl.keysize = sizeof(Datum);
    elem_hash_ctl.entrysize = sizeof(TrackItem);
    elem_hash_ctl.hash = element_hash;
    elem_hash_ctl.match = element_match;
    elem_hash_ctl.hcxt = CurrentMemoryContext;
    elements_tab = hash_create("Analyzed elements table",
                               num_mcelem,
                               &elem_hash_ctl,
                               HASH_ELEM | HASH_FUNCTION | HASH_COMPARE | HASH_CONTEXT);

    /* hashtable for array distinct elements counts */
    MemSet(&count_hash_ctl, 0, sizeof(count_hash_ctl));
    count_hash_ctl.keysize = sizeof(int);
    count_hash_ctl.entrysize = sizeof(DECountItem);
    count_hash_ctl.hcxt = CurrentMemoryContext;
    count_tab = hash_create("Array distinct element count table",
                            64,
                            &count_hash_ctl,
                            HASH_ELEM | HASH_BLOBS | HASH_CONTEXT);

    /* Initialize counters. */
    b_current = 1;
    element_no = 0;

    /* Loop over the arrays. */
    for (array_no = 0; array_no < samplerows; array_no++)
    {
        Datum       value;
        bool        isnull;
        ArrayType  *array;
        int         num_elems;
        Datum      *elem_values;
        bool       *elem_nulls;
        bool        null_present;
        int         j;
        int64       prev_element_no = element_no;
        int         distinct_count;
        bool        count_item_found;

        vacuum_delay_point();

        value = fetchfunc(stats, array_no, &isnull);
        if (isnull)
        {
            /* array is null, just count that */
            null_cnt++;
            continue;
        }

        /* Skip too-large values. */
        if (toast_raw_datum_size(value) > ARRAY_WIDTH_THRESHOLD)
            continue;
        else
            analyzed_rows++;

        /*
         * Now detoast the array if needed, and deconstruct into datums.
         */
        array = DatumGetArrayTypeP(value);

        Assert(ARR_ELEMTYPE(array) == extra_data->type_id);
        deconstruct_array(array,
                          extra_data->type_id,
                          extra_data->typlen,
                          extra_data->typbyval,
                          extra_data->typalign,
                          &elem_values, &elem_nulls, &num_elems);

        /*
         * We loop through the elements in the array and add them to our
         * tracking hashtable.
         */
        null_present = false;
        for (j = 0; j < num_elems; j++)
        {
            Datum       elem_value;
            bool        found;

            /* No null element processing other than flag setting here */
            if (elem_nulls[j])
            {
                null_present = true;
                continue;
            }

            /* Lookup current element in hashtable, adding it if new */
            elem_value = elem_values[j];
            item = (TrackItem *) hash_search(elements_tab,
                                             (const void *) &elem_value,
                                             HASH_ENTER, &found);

            if (found)
            {
                /* The element value is already on the tracking list */

                /*
                 * The operators we assist ignore duplicate array elements, so
                 * count a given distinct element only once per array.
                 */
                if (item->last_container == array_no)
                    continue;

                item->frequency++;
                item->last_container = array_no;
            }
            else
            {
                /* Initialize new tracking list element */

                /*
                 * If element type is pass-by-reference, we must copy it into
                 * palloc'd space, so that we can release the array below. (We
                 * do this so that the space needed for element values is
                 * limited by the size of the hashtable; if we kept all the
                 * array values around, it could be much more.)
                 */
                item->key = datumCopy(elem_value,
                                      extra_data->typbyval,
                                      extra_data->typlen);

                item->frequency = 1;
                item->delta = b_current - 1;
                item->last_container = array_no;
            }

            /* element_no is the number of elements processed (ie N) */
            element_no++;

            /* We prune the D structure after processing each bucket */
            if (element_no % bucket_width == 0)
            {
                prune_element_hashtable(elements_tab, b_current);
                b_current++;
            }
        }

        /* Count null element presence once per array. */
        if (null_present)
            null_elem_cnt++;

        /* Update frequency of the particular array distinct element count. */
        distinct_count = (int) (element_no - prev_element_no);
        count_item = (DECountItem *) hash_search(count_tab, &distinct_count,
                                                 HASH_ENTER,
                                                 &count_item_found);

        if (count_item_found)
            count_item->frequency++;
        else
            count_item->frequency = 1;

        /* Free memory allocated while detoasting. */
        if (PointerGetDatum(array) != value)
            pfree(array);
        pfree(elem_values);
        pfree(elem_nulls);
    }

    /* Skip pg_statistic slots occupied by standard statistics */
    slot_idx = 0;
    while (slot_idx < STATISTIC_NUM_SLOTS && stats->stakind[slot_idx] != 0)
        slot_idx++;
    if (slot_idx > STATISTIC_NUM_SLOTS - 2)
        elog(ERROR, "insufficient pg_statistic slots for array stats");

    /* We can only compute real stats if we found some non-null values. */
    if (analyzed_rows > 0)
    {
        int         nonnull_cnt = analyzed_rows;
        int         count_items_count;
        int         i;
        TrackItem **sort_table;
        int         track_len;
        int64       cutoff_freq;
        int64       minfreq,
                    maxfreq;

        /*
         * We assume the standard stats code already took care of setting
         * stats_valid, stanullfrac, stawidth, stadistinct.  We'd have to
         * re-compute those values if we wanted to not store the standard
         * stats.
         */

        /*
         * Construct an array of the interesting hashtable items, that is,
         * those meeting the cutoff frequency (s - epsilon)*N.  Also identify
         * the minimum and maximum frequencies among these items.
         *
         * Since epsilon = s/10 and bucket_width = 1/epsilon, the cutoff
         * frequency is 9*N / bucket_width.
         */
        cutoff_freq = 9 * element_no / bucket_width;

        i = hash_get_num_entries(elements_tab); /* surely enough space */
        sort_table = (TrackItem **) palloc(sizeof(TrackItem *) * i);

        hash_seq_init(&scan_status, elements_tab);
        track_len = 0;
        minfreq = element_no;
        maxfreq = 0;
        while ((item = (TrackItem *) hash_seq_search(&scan_status)) != NULL)
        {
            if (item->frequency > cutoff_freq)
            {
                sort_table[track_len++] = item;
                minfreq = Min(minfreq, item->frequency);
                maxfreq = Max(maxfreq, item->frequency);
            }
        }
        Assert(track_len <= i);

        /* emit some statistics for debug purposes */
        elog(DEBUG3, "compute_array_stats: target # mces = %d, "
             "bucket width = %d, "
             "# elements = " INT64_FORMAT ", hashtable size = %d, "
             "usable entries = %d",
             num_mcelem, bucket_width, element_no, i, track_len);

        /*
         * If we obtained more elements than we really want, get rid of those
         * with least frequencies.  The easiest way is to qsort the array into
         * descending frequency order and truncate the array.
         */
        if (num_mcelem < track_len)
        {
            qsort(sort_table, track_len, sizeof(TrackItem *),
                  trackitem_compare_frequencies_desc);
            /* reset minfreq to the smallest frequency we're keeping */
            minfreq = sort_table[num_mcelem - 1]->frequency;
        }
        else
            num_mcelem = track_len;

        /* Generate MCELEM slot entry */
        if (num_mcelem > 0)
        {
            MemoryContext old_context;
            Datum      *mcelem_values;
            float4     *mcelem_freqs;

            /*
             * We want to store statistics sorted on the element value using
             * the element type's default comparison function.  This permits
             * fast binary searches in selectivity estimation functions.
             */
            qsort(sort_table, num_mcelem, sizeof(TrackItem *),
                  trackitem_compare_element);

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

            /*
             * We sorted statistics on the element value, but we want to be
             * able to find the minimal and maximal frequencies without going
             * through all the values.  We also want the frequency of null
             * elements.  Store these three values at the end of mcelem_freqs.
             */
            mcelem_values = (Datum *) palloc(num_mcelem * sizeof(Datum));
            mcelem_freqs = (float4 *) palloc((num_mcelem + 3) * sizeof(float4));

            /*
             * See comments above about use of nonnull_cnt as the divisor for
             * the final frequency estimates.
             */
            for (i = 0; i < num_mcelem; i++)
            {
                TrackItem  *item = sort_table[i];

                mcelem_values[i] = datumCopy(item->key,
                                             extra_data->typbyval,
                                             extra_data->typlen);
                mcelem_freqs[i] = (double) item->frequency /
                    (double) nonnull_cnt;
            }
            mcelem_freqs[i++] = (double) minfreq / (double) nonnull_cnt;
            mcelem_freqs[i++] = (double) maxfreq / (double) nonnull_cnt;
            mcelem_freqs[i++] = (double) null_elem_cnt / (double) nonnull_cnt;

            MemoryContextSwitchTo(old_context);

            stats->stakind[slot_idx] = STATISTIC_KIND_MCELEM;
            stats->staop[slot_idx] = extra_data->eq_opr;
            stats->stanumbers[slot_idx] = mcelem_freqs;
            /* See above comment about extra stanumber entries */
            stats->numnumbers[slot_idx] = num_mcelem + 3;
            stats->stavalues[slot_idx] = mcelem_values;
            stats->numvalues[slot_idx] = num_mcelem;
            /* We are storing values of element type */
            stats->statypid[slot_idx] = extra_data->type_id;
            stats->statyplen[slot_idx] = extra_data->typlen;
            stats->statypbyval[slot_idx] = extra_data->typbyval;
            stats->statypalign[slot_idx] = extra_data->typalign;
            slot_idx++;
        }

        /* Generate DECHIST slot entry */
        count_items_count = hash_get_num_entries(count_tab);
        if (count_items_count > 0)
        {
            int         num_hist = stats->attr->attstattarget;
            DECountItem **sorted_count_items;
            int         j;
            int         delta;
            int64       frac;
            float4     *hist;

            /* num_hist must be at least 2 for the loop below to work */
            num_hist = Max(num_hist, 2);

            /*
             * Create an array of DECountItem pointers, and sort them into
             * increasing count order.
             */
            sorted_count_items = (DECountItem **)
                palloc(sizeof(DECountItem *) * count_items_count);
            hash_seq_init(&scan_status, count_tab);
            j = 0;
            while ((count_item = (DECountItem *) hash_seq_search(&scan_status)) != NULL)
            {
                sorted_count_items[j++] = count_item;
            }
            qsort(sorted_count_items, count_items_count,
                  sizeof(DECountItem *), countitem_compare_count);

            /*
             * Prepare to fill stanumbers with the histogram, followed by the
             * average count.  This array must be stored in anl_context.
             */
            hist = (float4 *)
                MemoryContextAlloc(stats->anl_context,
                                   sizeof(float4) * (num_hist + 1));
            hist[num_hist] = (double) element_no / (double) nonnull_cnt;

            /*----------
             * Construct the histogram of distinct-element counts (DECs).
             *
             * The object of this loop is to copy the min and max DECs to
             * hist[0] and hist[num_hist - 1], along with evenly-spaced DECs
             * in between (where "evenly-spaced" is with reference to the
             * whole input population of arrays).  If we had a complete sorted
             * array of DECs, one per analyzed row, the i'th hist value would
             * come from DECs[i * (analyzed_rows - 1) / (num_hist - 1)]
             * (compare the histogram-making loop in compute_scalar_stats()).
             * But instead of that we have the sorted_count_items[] array,
             * which holds unique DEC values with their frequencies (that is,
             * a run-length-compressed version of the full array).  So we
             * control advancing through sorted_count_items[] with the
             * variable "frac", which is defined as (x - y) * (num_hist - 1),
             * where x is the index in the notional DECs array corresponding
             * to the start of the next sorted_count_items[] element's run,
             * and y is the index in DECs from which we should take the next
             * histogram value.  We have to advance whenever x <= y, that is
             * frac <= 0.  The x component is the sum of the frequencies seen
             * so far (up through the current sorted_count_items[] element),
             * and of course y * (num_hist - 1) = i * (analyzed_rows - 1),
             * per the subscript calculation above.  (The subscript calculation
             * implies dropping any fractional part of y; in this formulation
             * that's handled by not advancing until frac reaches 1.)
             *
             * Even though frac has a bounded range, it could overflow int32
             * when working with very large statistics targets, so we do that
             * math in int64.
             *----------
             */
            delta = analyzed_rows - 1;
            j = 0;              /* current index in sorted_count_items */
            /* Initialize frac for sorted_count_items[0]; y is initially 0 */
            frac = (int64) sorted_count_items[0]->frequency * (num_hist - 1);
            for (i = 0; i < num_hist; i++)
            {
                while (frac <= 0)
                {
                    /* Advance, and update x component of frac */
                    j++;
                    frac += (int64) sorted_count_items[j]->frequency * (num_hist - 1);
                }
                hist[i] = sorted_count_items[j]->count;
                frac -= delta;  /* update y for upcoming i increment */
            }
            Assert(j == count_items_count - 1);

            stats->stakind[slot_idx] = STATISTIC_KIND_DECHIST;
            stats->staop[slot_idx] = extra_data->eq_opr;
            stats->stanumbers[slot_idx] = hist;
            stats->numnumbers[slot_idx] = num_hist + 1;
            slot_idx++;
        }
    }

    /*
     * We don't need to bother cleaning up any of our temporary palloc's. The
     * hashtable should also go away, as it used a child memory context.
     */
}

/*
 * A function to prune the D structure from the Lossy Counting algorithm.
 * Consult compute_tsvector_stats() for wider explanation.
 */
static void
prune_element_hashtable(HTAB *elements_tab, int b_current)
{
    HASH_SEQ_STATUS scan_status;
    TrackItem  *item;

    hash_seq_init(&scan_status, elements_tab);
    while ((item = (TrackItem *) hash_seq_search(&scan_status)) != NULL)
    {
        if (item->frequency + item->delta <= b_current)
        {
            Datum       value = item->key;

            if (hash_search(elements_tab, (const void *) &item->key,
                            HASH_REMOVE, NULL) == NULL)
                elog(ERROR, "hash table corrupted");
            /* We should free memory if element is not passed by value */
            if (!array_extra_data->typbyval)
                pfree(DatumGetPointer(value));
        }
    }
}

/*
 * Hash function for elements.
 *
 * We use the element type's default hash opclass, and the default collation
 * if the type is collation-sensitive.
 */
static uint32
element_hash(const void *key, Size keysize)
{
    Datum       d = *((const Datum *) key);
    Datum       h;

    h = FunctionCall1Coll(array_extra_data->hash, DEFAULT_COLLATION_OID, d);
    return DatumGetUInt32(h);
}

/*
 * Matching function for elements, to be used in hashtable lookups.
 */
static int
element_match(const void *key1, const void *key2, Size keysize)
{
    /* The keysize parameter is superfluous here */
    return element_compare(key1, key2);
}

/*
 * 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)
{
    Datum       d1 = *((const Datum *) key1);
    Datum       d2 = *((const Datum *) key2);
    Datum       c;

    c = FunctionCall2Coll(array_extra_data->cmp, DEFAULT_COLLATION_OID, d1, d2);
    return DatumGetInt32(c);
}

/*
 * qsort() comparator for sorting TrackItems by frequencies (descending sort)
 */
static int
trackitem_compare_frequencies_desc(const void *e1, const void *e2)
{
    const TrackItem *const *t1 = (const TrackItem *const *) e1;
    const TrackItem *const *t2 = (const TrackItem *const *) e2;

    return (*t2)->frequency - (*t1)->frequency;
}

/*
 * qsort() comparator for sorting TrackItems by element values
 */
static int
trackitem_compare_element(const void *e1, const void *e2)
{
    const TrackItem *const *t1 = (const TrackItem *const *) e1;
    const TrackItem *const *t2 = (const TrackItem *const *) e2;

    return element_compare(&(*t1)->key, &(*t2)->key);
}

/*
 * qsort() comparator for sorting DECountItems by count
 */
static int
countitem_compare_count(const void *e1, const void *e2)
{
    const DECountItem *const *t1 = (const DECountItem *const *) e1;
    const DECountItem *const *t2 = (const DECountItem *const *) e2;

    if ((*t1)->count < (*t2)->count)
        return -1;
    else if ((*t1)->count == (*t2)->count)
        return 0;
    else
        return 1;
}