analyze.c

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/*-------------------------------------------------------------------------
 *
 * analyze.c
 *    the Postgres statistics generator
 *
 * Portions Copyright (c) 1996-2023, PostgreSQL Global Development Group
 * Portions Copyright (c) 1994, Regents of the University of California
 *
 *
 * IDENTIFICATION
 *    src/backend/commands/analyze.c
 *
 *-------------------------------------------------------------------------
 */
#include "postgres.h"
 
#include <math.h>
 
#include "access/detoast.h"
#include "access/genam.h"
#include "access/multixact.h"
#include "access/relation.h"
#include "access/sysattr.h"
#include "access/table.h"
#include "access/tableam.h"
#include "access/transam.h"
#include "access/tupconvert.h"
#include "access/visibilitymap.h"
#include "access/xact.h"
#include "catalog/catalog.h"
#include "catalog/index.h"
#include "catalog/indexing.h"
#include "catalog/pg_collation.h"
#include "catalog/pg_inherits.h"
#include "catalog/pg_namespace.h"
#include "catalog/pg_statistic_ext.h"
#include "commands/dbcommands.h"
#include "commands/progress.h"
#include "commands/tablecmds.h"
#include "commands/vacuum.h"
#include "common/pg_prng.h"
#include "executor/executor.h"
#include "foreign/fdwapi.h"
#include "miscadmin.h"
#include "nodes/nodeFuncs.h"
#include "parser/parse_oper.h"
#include "parser/parse_relation.h"
#include "pgstat.h"
#include "postmaster/autovacuum.h"
#include "statistics/extended_stats_internal.h"
#include "statistics/statistics.h"
#include "storage/bufmgr.h"
#include "storage/lmgr.h"
#include "storage/proc.h"
#include "storage/procarray.h"
#include "utils/acl.h"
#include "utils/attoptcache.h"
#include "utils/builtins.h"
#include "utils/datum.h"
#include "utils/fmgroids.h"
#include "utils/guc.h"
#include "utils/lsyscache.h"
#include "utils/memutils.h"
#include "utils/pg_rusage.h"
#include "utils/sampling.h"
#include "utils/sortsupport.h"
#include "utils/spccache.h"
#include "utils/syscache.h"
#include "utils/timestamp.h"
 
 
/* Per-index data for ANALYZE */
typedef struct AnlIndexData
{
    IndexInfo  *indexInfo;      /* BuildIndexInfo result */
    double      tupleFract;     /* fraction of rows for partial index */
    VacAttrStats **vacattrstats;    /* index attrs to analyze */
    int         attr_cnt;
} AnlIndexData;
 
 
/* Default statistics target (GUC parameter) */
int         default_statistics_target = 100;
 
/* A few variables that don't seem worth passing around as parameters */
static MemoryContext anl_context = NULL;
static BufferAccessStrategy vac_strategy;
 
 
static void do_analyze_rel(Relation onerel,
                           VacuumParams *params, List *va_cols,
                           AcquireSampleRowsFunc acquirefunc, BlockNumber relpages,
                           bool inh, bool in_outer_xact, int elevel);
static void compute_index_stats(Relation onerel, double totalrows,
                                AnlIndexData *indexdata, int nindexes,
                                HeapTuple *rows, int numrows,
                                MemoryContext col_context);
static VacAttrStats *examine_attribute(Relation onerel, int attnum,
                                       Node *index_expr);
static int  acquire_sample_rows(Relation onerel, int elevel,
                                HeapTuple *rows, int targrows,
                                double *totalrows, double *totaldeadrows);
static int  compare_rows(const void *a, const void *b, void *arg);
static int  acquire_inherited_sample_rows(Relation onerel, int elevel,
                                          HeapTuple *rows, int targrows,
                                          double *totalrows, double *totaldeadrows);
static void update_attstats(Oid relid, bool inh,
                            int natts, VacAttrStats **vacattrstats);
static Datum std_fetch_func(VacAttrStatsP stats, int rownum, bool *isNull);
static Datum ind_fetch_func(VacAttrStatsP stats, int rownum, bool *isNull);
 
 
/*
 *  analyze_rel() -- analyze one relation
 *
 * relid identifies the relation to analyze.  If relation is supplied, use
 * the name therein for reporting any failure to open/lock the rel; do not
 * use it once we've successfully opened the rel, since it might be stale.
 */
void
analyze_rel(Oid relid, RangeVar *relation,
            VacuumParams *params, List *va_cols, bool in_outer_xact,
            BufferAccessStrategy bstrategy)
{
    Relation    onerel;
    int         elevel;
    AcquireSampleRowsFunc acquirefunc = NULL;
    BlockNumber relpages = 0;
 
    /* Select logging level */
    if (params->options & VACOPT_VERBOSE)
        elevel = INFO;
    else
        elevel = DEBUG2;
 
    /* Set up static variables */
    vac_strategy = bstrategy;
 
    /*
     * Check for user-requested abort.
     */
    CHECK_FOR_INTERRUPTS();
 
    /*
     * Open the relation, getting ShareUpdateExclusiveLock to ensure that two
     * ANALYZEs don't run on it concurrently.  (This also locks out a
     * concurrent VACUUM, which doesn't matter much at the moment but might
     * matter if we ever try to accumulate stats on dead tuples.) If the rel
     * has been dropped since we last saw it, we don't need to process it.
     *
     * Make sure to generate only logs for ANALYZE in this case.
     */
    onerel = vacuum_open_relation(relid, relation, params->options & ~(VACOPT_VACUUM),
                                  params->log_min_duration >= 0,
                                  ShareUpdateExclusiveLock);
 
    /* leave if relation could not be opened or locked */
    if (!onerel)
        return;
 
    /*
     * Check if relation needs to be skipped based on privileges.  This check
     * happens also when building the relation list to analyze for a manual
     * operation, and needs to be done additionally here as ANALYZE could
     * happen across multiple transactions where privileges could have changed
     * in-between.  Make sure to generate only logs for ANALYZE in this case.
     */
    if (!vacuum_is_permitted_for_relation(RelationGetRelid(onerel),
                                          onerel->rd_rel,
                                          params->options & VACOPT_ANALYZE))
    {
        relation_close(onerel, ShareUpdateExclusiveLock);
        return;
    }
 
    /*
     * Silently ignore tables that are temp tables of other backends ---
     * trying to analyze these is rather pointless, since their contents are
     * probably not up-to-date on disk.  (We don't throw a warning here; it
     * would just lead to chatter during a database-wide ANALYZE.)
     */
    if (RELATION_IS_OTHER_TEMP(onerel))
    {
        relation_close(onerel, ShareUpdateExclusiveLock);
        return;
    }
 
    /*
     * We can ANALYZE any table except pg_statistic. See update_attstats
     */
    if (RelationGetRelid(onerel) == StatisticRelationId)
    {
        relation_close(onerel, ShareUpdateExclusiveLock);
        return;
    }
 
    /*
     * Check that it's of an analyzable relkind, and set up appropriately.
     */
    if (onerel->rd_rel->relkind == RELKIND_RELATION ||
        onerel->rd_rel->relkind == RELKIND_MATVIEW)
    {
        /* Regular table, so we'll use the regular row acquisition function */
        acquirefunc = acquire_sample_rows;
        /* Also get regular table's size */
        relpages = RelationGetNumberOfBlocks(onerel);
    }
    else if (onerel->rd_rel->relkind == RELKIND_FOREIGN_TABLE)
    {
        /*
         * For a foreign table, call the FDW's hook function to see whether it
         * supports analysis.
         */
        FdwRoutine *fdwroutine;
        bool        ok = false;
 
        fdwroutine = GetFdwRoutineForRelation(onerel, false);
 
        if (fdwroutine->AnalyzeForeignTable != NULL)
            ok = fdwroutine->AnalyzeForeignTable(onerel,
                                                 &acquirefunc,
                                                 &relpages);
 
        if (!ok)
        {
            ereport(WARNING,
                    (errmsg("skipping \"%s\" --- cannot analyze this foreign table",
                            RelationGetRelationName(onerel))));
            relation_close(onerel, ShareUpdateExclusiveLock);
            return;
        }
    }
    else if (onerel->rd_rel->relkind == RELKIND_PARTITIONED_TABLE)
    {
        /*
         * For partitioned tables, we want to do the recursive ANALYZE below.
         */
    }
    else
    {
        /* No need for a WARNING if we already complained during VACUUM */
        if (!(params->options & VACOPT_VACUUM))
            ereport(WARNING,
                    (errmsg("skipping \"%s\" --- cannot analyze non-tables or special system tables",
                            RelationGetRelationName(onerel))));
        relation_close(onerel, ShareUpdateExclusiveLock);
        return;
    }
 
    /*
     * OK, let's do it.  First, initialize progress reporting.
     */
    pgstat_progress_start_command(PROGRESS_COMMAND_ANALYZE,
                                  RelationGetRelid(onerel));
 
    /*
     * Do the normal non-recursive ANALYZE.  We can skip this for partitioned
     * tables, which don't contain any rows.
     */
    if (onerel->rd_rel->relkind != RELKIND_PARTITIONED_TABLE)
        do_analyze_rel(onerel, params, va_cols, acquirefunc,
                       relpages, false, in_outer_xact, elevel);
 
    /*
     * If there are child tables, do recursive ANALYZE.
     */
    if (onerel->rd_rel->relhassubclass)
        do_analyze_rel(onerel, params, va_cols, acquirefunc, relpages,
                       true, in_outer_xact, elevel);
 
    /*
     * Close source relation now, but keep lock so that no one deletes it
     * before we commit.  (If someone did, they'd fail to clean up the entries
     * we made in pg_statistic.  Also, releasing the lock before commit would
     * expose us to concurrent-update failures in update_attstats.)
     */
    relation_close(onerel, NoLock);
 
    pgstat_progress_end_command();
}
 
/*
 *  do_analyze_rel() -- analyze one relation, recursively or not
 *
 * Note that "acquirefunc" is only relevant for the non-inherited case.
 * For the inherited case, acquire_inherited_sample_rows() determines the
 * appropriate acquirefunc for each child table.
 */
static void
do_analyze_rel(Relation onerel, VacuumParams *params,
               List *va_cols, AcquireSampleRowsFunc acquirefunc,
               BlockNumber relpages, bool inh, bool in_outer_xact,
               int elevel)
{
    int         attr_cnt,
                tcnt,
                i,
                ind;
    Relation   *Irel;
    int         nindexes;
    bool        hasindex;
    VacAttrStats **vacattrstats;
    AnlIndexData *indexdata;
    int         targrows,
                numrows,
                minrows;
    double      totalrows,
                totaldeadrows;
    HeapTuple  *rows;
    PGRUsage    ru0;
    TimestampTz starttime = 0;
    MemoryContext caller_context;
    Oid         save_userid;
    int         save_sec_context;
    int         save_nestlevel;
    int64       AnalyzePageHit = VacuumPageHit;
    int64       AnalyzePageMiss = VacuumPageMiss;
    int64       AnalyzePageDirty = VacuumPageDirty;
    PgStat_Counter startreadtime = 0;
    PgStat_Counter startwritetime = 0;
 
    if (inh)
        ereport(elevel,
                (errmsg("analyzing \"%s.%s\" inheritance tree",
                        get_namespace_name(RelationGetNamespace(onerel)),
                        RelationGetRelationName(onerel))));
    else
        ereport(elevel,
                (errmsg("analyzing \"%s.%s\"",
                        get_namespace_name(RelationGetNamespace(onerel)),
                        RelationGetRelationName(onerel))));
 
    /*
     * Set up a working context so that we can easily free whatever junk gets
     * created.
     */
    anl_context = AllocSetContextCreate(CurrentMemoryContext,
                                        "Analyze",
                                        ALLOCSET_DEFAULT_SIZES);
    caller_context = MemoryContextSwitchTo(anl_context);
 
    /*
     * Switch to the table owner's userid, so that any index functions are run
     * as that user.  Also lock down security-restricted operations and
     * arrange to make GUC variable changes local to this command.
     */
    GetUserIdAndSecContext(&save_userid, &save_sec_context);
    SetUserIdAndSecContext(onerel->rd_rel->relowner,
                           save_sec_context | SECURITY_RESTRICTED_OPERATION);
    save_nestlevel = NewGUCNestLevel();
 
    /* measure elapsed time iff autovacuum logging requires it */
    if (IsAutoVacuumWorkerProcess() && params->log_min_duration >= 0)
    {
        if (track_io_timing)
        {
            startreadtime = pgStatBlockReadTime;
            startwritetime = pgStatBlockWriteTime;
        }
 
        pg_rusage_init(&ru0);
        starttime = GetCurrentTimestamp();
    }
 
    /*
     * Determine which columns to analyze
     *
     * Note that system attributes are never analyzed, so we just reject them
     * at the lookup stage.  We also reject duplicate column mentions.  (We
     * could alternatively ignore duplicates, but analyzing a column twice
     * won't work; we'd end up making a conflicting update in pg_statistic.)
     */
    if (va_cols != NIL)
    {
        Bitmapset  *unique_cols = NULL;
        ListCell   *le;
 
        vacattrstats = (VacAttrStats **) palloc(list_length(va_cols) *
                                                sizeof(VacAttrStats *));
        tcnt = 0;
        foreach(le, va_cols)
        {
            char       *col = strVal(lfirst(le));
 
            i = attnameAttNum(onerel, col, false);
            if (i == InvalidAttrNumber)
                ereport(ERROR,
                        (errcode(ERRCODE_UNDEFINED_COLUMN),
                         errmsg("column \"%s\" of relation \"%s\" does not exist",
                                col, RelationGetRelationName(onerel))));
            if (bms_is_member(i, unique_cols))
                ereport(ERROR,
                        (errcode(ERRCODE_DUPLICATE_COLUMN),
                         errmsg("column \"%s\" of relation \"%s\" appears more than once",
                                col, RelationGetRelationName(onerel))));
            unique_cols = bms_add_member(unique_cols, i);
 
            vacattrstats[tcnt] = examine_attribute(onerel, i, NULL);
            if (vacattrstats[tcnt] != NULL)
                tcnt++;
        }
        attr_cnt = tcnt;
    }
    else
    {
        attr_cnt = onerel->rd_att->natts;
        vacattrstats = (VacAttrStats **)
            palloc(attr_cnt * sizeof(VacAttrStats *));
        tcnt = 0;
        for (i = 1; i <= attr_cnt; i++)
        {
            vacattrstats[tcnt] = examine_attribute(onerel, i, NULL);
            if (vacattrstats[tcnt] != NULL)
                tcnt++;
        }
        attr_cnt = tcnt;
    }
 
    /*
     * Open all indexes of the relation, and see if there are any analyzable
     * columns in the indexes.  We do not analyze index columns if there was
     * an explicit column list in the ANALYZE command, however.
     *
     * If we are doing a recursive scan, we don't want to touch the parent's
     * indexes at all.  If we're processing a partitioned table, we need to
     * know if there are any indexes, but we don't want to process them.
     */
    if (onerel->rd_rel->relkind == RELKIND_PARTITIONED_TABLE)
    {
        List       *idxs = RelationGetIndexList(onerel);
 
        Irel = NULL;
        nindexes = 0;
        hasindex = idxs != NIL;
        list_free(idxs);
    }
    else if (!inh)
    {
        vac_open_indexes(onerel, AccessShareLock, &nindexes, &Irel);
        hasindex = nindexes > 0;
    }
    else
    {
        Irel = NULL;
        nindexes = 0;
        hasindex = false;
    }
    indexdata = NULL;
    if (nindexes > 0)
    {
        indexdata = (AnlIndexData *) palloc0(nindexes * sizeof(AnlIndexData));
        for (ind = 0; ind < nindexes; ind++)
        {
            AnlIndexData *thisdata = &indexdata[ind];
            IndexInfo  *indexInfo;
 
            thisdata->indexInfo = indexInfo = BuildIndexInfo(Irel[ind]);
            thisdata->tupleFract = 1.0; /* fix later if partial */
            if (indexInfo->ii_Expressions != NIL && va_cols == NIL)
            {
                ListCell   *indexpr_item = list_head(indexInfo->ii_Expressions);
 
                thisdata->vacattrstats = (VacAttrStats **)
                    palloc(indexInfo->ii_NumIndexAttrs * sizeof(VacAttrStats *));
                tcnt = 0;
                for (i = 0; i < indexInfo->ii_NumIndexAttrs; i++)
                {
                    int         keycol = indexInfo->ii_IndexAttrNumbers[i];
 
                    if (keycol == 0)
                    {
                        /* Found an index expression */
                        Node       *indexkey;
 
                        if (indexpr_item == NULL)   /* shouldn't happen */
                            elog(ERROR, "too few entries in indexprs list");
                        indexkey = (Node *) lfirst(indexpr_item);
                        indexpr_item = lnext(indexInfo->ii_Expressions,
                                             indexpr_item);
                        thisdata->vacattrstats[tcnt] =
                            examine_attribute(Irel[ind], i + 1, indexkey);
                        if (thisdata->vacattrstats[tcnt] != NULL)
                            tcnt++;
                    }
                }
                thisdata->attr_cnt = tcnt;
            }
        }
    }
 
    /*
     * Determine how many rows we need to sample, using the worst case from
     * all analyzable columns.  We use a lower bound of 100 rows to avoid
     * possible overflow in Vitter's algorithm.  (Note: that will also be the
     * target in the corner case where there are no analyzable columns.)
     */
    targrows = 100;
    for (i = 0; i < attr_cnt; i++)
    {
        if (targrows < vacattrstats[i]->minrows)
            targrows = vacattrstats[i]->minrows;
    }
    for (ind = 0; ind < nindexes; ind++)
    {
        AnlIndexData *thisdata = &indexdata[ind];
 
        for (i = 0; i < thisdata->attr_cnt; i++)
        {
            if (targrows < thisdata->vacattrstats[i]->minrows)
                targrows = thisdata->vacattrstats[i]->minrows;
        }
    }
 
    /*
     * Look at extended statistics objects too, as those may define custom
     * statistics target. So we may need to sample more rows and then build
     * the statistics with enough detail.
     */
    minrows = ComputeExtStatisticsRows(onerel, attr_cnt, vacattrstats);
 
    if (targrows < minrows)
        targrows = minrows;
 
    /*
     * Acquire the sample rows
     */
    rows = (HeapTuple *) palloc(targrows * sizeof(HeapTuple));
    pgstat_progress_update_param(PROGRESS_ANALYZE_PHASE,
                                 inh ? PROGRESS_ANALYZE_PHASE_ACQUIRE_SAMPLE_ROWS_INH :
                                 PROGRESS_ANALYZE_PHASE_ACQUIRE_SAMPLE_ROWS);
    if (inh)
        numrows = acquire_inherited_sample_rows(onerel, elevel,
                                                rows, targrows,
                                                &totalrows, &totaldeadrows);
    else
        numrows = (*acquirefunc) (onerel, elevel,
                                  rows, targrows,
                                  &totalrows, &totaldeadrows);
 
    /*
     * Compute the statistics.  Temporary results during the calculations for
     * each column are stored in a child context.  The calc routines are
     * responsible to make sure that whatever they store into the VacAttrStats
     * structure is allocated in anl_context.
     */
    if (numrows > 0)
    {
        MemoryContext col_context,
                    old_context;
 
        pgstat_progress_update_param(PROGRESS_ANALYZE_PHASE,
                                     PROGRESS_ANALYZE_PHASE_COMPUTE_STATS);
 
        col_context = AllocSetContextCreate(anl_context,
                                            "Analyze Column",
                                            ALLOCSET_DEFAULT_SIZES);
        old_context = MemoryContextSwitchTo(col_context);
 
        for (i = 0; i < attr_cnt; i++)
        {
            VacAttrStats *stats = vacattrstats[i];
            AttributeOpts *aopt;
 
            stats->rows = rows;
            stats->tupDesc = onerel->rd_att;
            stats->compute_stats(stats,
                                 std_fetch_func,
                                 numrows,
                                 totalrows);
 
            /*
             * If the appropriate flavor of the n_distinct option is
             * specified, override with the corresponding value.
             */
            aopt = get_attribute_options(onerel->rd_id, stats->attr->attnum);
            if (aopt != NULL)
            {
                float8      n_distinct;
 
                n_distinct = inh ? aopt->n_distinct_inherited : aopt->n_distinct;
                if (n_distinct != 0.0)
                    stats->stadistinct = n_distinct;
            }
 
            MemoryContextResetAndDeleteChildren(col_context);
        }
 
        if (nindexes > 0)
            compute_index_stats(onerel, totalrows,
                                indexdata, nindexes,
                                rows, numrows,
                                col_context);
 
        MemoryContextSwitchTo(old_context);
        MemoryContextDelete(col_context);
 
        /*
         * Emit the completed stats rows into pg_statistic, replacing any
         * previous statistics for the target columns.  (If there are stats in
         * pg_statistic for columns we didn't process, we leave them alone.)
         */
        update_attstats(RelationGetRelid(onerel), inh,
                        attr_cnt, vacattrstats);
 
        for (ind = 0; ind < nindexes; ind++)
        {
            AnlIndexData *thisdata = &indexdata[ind];
 
            update_attstats(RelationGetRelid(Irel[ind]), false,
                            thisdata->attr_cnt, thisdata->vacattrstats);
        }
 
        /* Build extended statistics (if there are any). */
        BuildRelationExtStatistics(onerel, inh, totalrows, numrows, rows,
                                   attr_cnt, vacattrstats);
    }
 
    pgstat_progress_update_param(PROGRESS_ANALYZE_PHASE,
                                 PROGRESS_ANALYZE_PHASE_FINALIZE_ANALYZE);
 
    /*
     * Update pages/tuples stats in pg_class ... but not if we're doing
     * inherited stats.
     *
     * We assume that VACUUM hasn't set pg_class.reltuples already, even
     * during a VACUUM ANALYZE.  Although VACUUM often updates pg_class,
     * exceptions exist.  A "VACUUM (ANALYZE, INDEX_CLEANUP OFF)" command will
     * never update pg_class entries for index relations.  It's also possible
     * that an individual index's pg_class entry won't be updated during
     * VACUUM if the index AM returns NULL from its amvacuumcleanup() routine.
     */
    if (!inh)
    {
        BlockNumber relallvisible;
 
        visibilitymap_count(onerel, &relallvisible, NULL);
 
        /* Update pg_class for table relation */
        vac_update_relstats(onerel,
                            relpages,
                            totalrows,
                            relallvisible,
                            hasindex,
                            InvalidTransactionId,
                            InvalidMultiXactId,
                            NULL, NULL,
                            in_outer_xact);
 
        /* Same for indexes */
        for (ind = 0; ind < nindexes; ind++)
        {
            AnlIndexData *thisdata = &indexdata[ind];
            double      totalindexrows;
 
            totalindexrows = ceil(thisdata->tupleFract * totalrows);
            vac_update_relstats(Irel[ind],
                                RelationGetNumberOfBlocks(Irel[ind]),
                                totalindexrows,
                                0,
                                false,
                                InvalidTransactionId,
                                InvalidMultiXactId,
                                NULL, NULL,
                                in_outer_xact);
        }
    }
    else if (onerel->rd_rel->relkind == RELKIND_PARTITIONED_TABLE)
    {
        /*
         * Partitioned tables don't have storage, so we don't set any fields
         * in their pg_class entries except for reltuples and relhasindex.
         */
        vac_update_relstats(onerel, -1, totalrows,
                            0, hasindex, InvalidTransactionId,
                            InvalidMultiXactId,
                            NULL, NULL,
                            in_outer_xact);
    }
 
    /*
     * Now report ANALYZE to the cumulative stats system.  For regular tables,
     * we do it only if not doing inherited stats.  For partitioned tables, we
     * only do it for inherited stats. (We're never called for not-inherited
     * stats on partitioned tables anyway.)
     *
     * Reset the changes_since_analyze counter only if we analyzed all
     * columns; otherwise, there is still work for auto-analyze to do.
     */
    if (!inh)
        pgstat_report_analyze(onerel, totalrows, totaldeadrows,
                              (va_cols == NIL));
    else if (onerel->rd_rel->relkind == RELKIND_PARTITIONED_TABLE)
        pgstat_report_analyze(onerel, 0, 0, (va_cols == NIL));
 
    /*
     * If this isn't part of VACUUM ANALYZE, let index AMs do cleanup.
     *
     * Note that most index AMs perform a no-op as a matter of policy for
     * amvacuumcleanup() when called in ANALYZE-only mode.  The only exception
     * among core index AMs is GIN/ginvacuumcleanup().
     */
    if (!(params->options & VACOPT_VACUUM))
    {
        for (ind = 0; ind < nindexes; ind++)
        {
            IndexBulkDeleteResult *stats;
            IndexVacuumInfo ivinfo;
 
            ivinfo.index = Irel[ind];
            ivinfo.analyze_only = true;
            ivinfo.estimated_count = true;
            ivinfo.message_level = elevel;
            ivinfo.num_heap_tuples = onerel->rd_rel->reltuples;
            ivinfo.strategy = vac_strategy;
 
            stats = index_vacuum_cleanup(&ivinfo, NULL);
 
            if (stats)
                pfree(stats);
        }
    }
 
    /* Done with indexes */
    vac_close_indexes(nindexes, Irel, NoLock);
 
    /* Log the action if appropriate */
    if (IsAutoVacuumWorkerProcess() && params->log_min_duration >= 0)
    {
        TimestampTz endtime = GetCurrentTimestamp();
 
        if (params->log_min_duration == 0 ||
            TimestampDifferenceExceeds(starttime, endtime,
                                       params->log_min_duration))
        {
            long        delay_in_ms;
            double      read_rate = 0;
            double      write_rate = 0;
            StringInfoData buf;
 
            /*
             * Calculate the difference in the Page Hit/Miss/Dirty that
             * happened as part of the analyze by subtracting out the
             * pre-analyze values which we saved above.
             */
            AnalyzePageHit = VacuumPageHit - AnalyzePageHit;
            AnalyzePageMiss = VacuumPageMiss - AnalyzePageMiss;
            AnalyzePageDirty = VacuumPageDirty - AnalyzePageDirty;
 
            /*
             * We do not expect an analyze to take > 25 days and it simplifies
             * things a bit to use TimestampDifferenceMilliseconds.
             */
            delay_in_ms = TimestampDifferenceMilliseconds(starttime, endtime);
 
            /*
             * Note that we are reporting these read/write rates in the same
             * manner as VACUUM does, which means that while the 'average read
             * rate' here actually corresponds to page misses and resulting
             * reads which are also picked up by track_io_timing, if enabled,
             * the 'average write rate' is actually talking about the rate of
             * pages being dirtied, not being written out, so it's typical to
             * have a non-zero 'avg write rate' while I/O timings only reports
             * reads.
             *
             * It's not clear that an ANALYZE will ever result in
             * FlushBuffer() being called, but we track and support reporting
             * on I/O write time in case that changes as it's practically free
             * to do so anyway.
             */
 
            if (delay_in_ms > 0)
            {
                read_rate = (double) BLCKSZ * AnalyzePageMiss / (1024 * 1024) /
                    (delay_in_ms / 1000.0);
                write_rate = (double) BLCKSZ * AnalyzePageDirty / (1024 * 1024) /
                    (delay_in_ms / 1000.0);
            }
 
            /*
             * We split this up so we don't emit empty I/O timing values when
             * track_io_timing isn't enabled.
             */
 
            initStringInfo(&buf);
            appendStringInfo(&buf, _("automatic analyze of table \"%s.%s.%s\"\n"),
                             get_database_name(MyDatabaseId),
                             get_namespace_name(RelationGetNamespace(onerel)),
                             RelationGetRelationName(onerel));
            if (track_io_timing)
            {
                double      read_ms = (double) (pgStatBlockReadTime - startreadtime) / 1000;
                double      write_ms = (double) (pgStatBlockWriteTime - startwritetime) / 1000;
 
                appendStringInfo(&buf, _("I/O timings: read: %.3f ms, write: %.3f ms\n"),
                                 read_ms, write_ms);
            }
            appendStringInfo(&buf, _("avg read rate: %.3f MB/s, avg write rate: %.3f MB/s\n"),
                             read_rate, write_rate);
            appendStringInfo(&buf, _("buffer usage: %lld hits, %lld misses, %lld dirtied\n"),
                             (long long) AnalyzePageHit,
                             (long long) AnalyzePageMiss,
                             (long long) AnalyzePageDirty);
            appendStringInfo(&buf, _("system usage: %s"), pg_rusage_show(&ru0));
 
            ereport(LOG,
                    (errmsg_internal("%s", buf.data)));
 
            pfree(buf.data);
        }
    }
 
    /* Roll back any GUC changes executed by index functions */
    AtEOXact_GUC(false, save_nestlevel);
 
    /* Restore userid and security context */
    SetUserIdAndSecContext(save_userid, save_sec_context);
 
    /* Restore current context and release memory */
    MemoryContextSwitchTo(caller_context);
    MemoryContextDelete(anl_context);
    anl_context = NULL;
}
 
/*
 * Compute statistics about indexes of a relation
 */
static void
compute_index_stats(Relation onerel, double totalrows,
                    AnlIndexData *indexdata, int nindexes,
                    HeapTuple *rows, int numrows,
                    MemoryContext col_context)
{
    MemoryContext ind_context,
                old_context;
    Datum       values[INDEX_MAX_KEYS];
    bool        isnull[INDEX_MAX_KEYS];
    int         ind,
                i;
 
    ind_context = AllocSetContextCreate(anl_context,
                                        "Analyze Index",
                                        ALLOCSET_DEFAULT_SIZES);
    old_context = MemoryContextSwitchTo(ind_context);
 
    for (ind = 0; ind < nindexes; ind++)
    {
        AnlIndexData *thisdata = &indexdata[ind];
        IndexInfo  *indexInfo = thisdata->indexInfo;
        int         attr_cnt = thisdata->attr_cnt;
        TupleTableSlot *slot;
        EState     *estate;
        ExprContext *econtext;
        ExprState  *predicate;
        Datum      *exprvals;
        bool       *exprnulls;
        int         numindexrows,
                    tcnt,
                    rowno;
        double      totalindexrows;
 
        /* Ignore index if no columns to analyze and not partial */
        if (attr_cnt == 0 && indexInfo->ii_Predicate == NIL)
            continue;
 
        /*
         * Need an EState for evaluation of index expressions and
         * partial-index predicates.  Create it in the per-index context to be
         * sure it gets cleaned up at the bottom of the loop.
         */
        estate = CreateExecutorState();
        econtext = GetPerTupleExprContext(estate);
        /* Need a slot to hold the current heap tuple, too */
        slot = MakeSingleTupleTableSlot(RelationGetDescr(onerel),
                                        &TTSOpsHeapTuple);
 
        /* Arrange for econtext's scan tuple to be the tuple under test */
        econtext->ecxt_scantuple = slot;
 
        /* Set up execution state for predicate. */
        predicate = ExecPrepareQual(indexInfo->ii_Predicate, estate);
 
        /* Compute and save index expression values */
        exprvals = (Datum *) palloc(numrows * attr_cnt * sizeof(Datum));
        exprnulls = (bool *) palloc(numrows * attr_cnt * sizeof(bool));
        numindexrows = 0;
        tcnt = 0;
        for (rowno = 0; rowno < numrows; rowno++)
        {
            HeapTuple   heapTuple = rows[rowno];
 
            vacuum_delay_point();
 
            /*
             * Reset the per-tuple context each time, to reclaim any cruft
             * left behind by evaluating the predicate or index expressions.
             */
            ResetExprContext(econtext);
 
            /* Set up for predicate or expression evaluation */
            ExecStoreHeapTuple(heapTuple, slot, false);
 
            /* If index is partial, check predicate */
            if (predicate != NULL)
            {
                if (!ExecQual(predicate, econtext))
                    continue;
            }
            numindexrows++;
 
            if (attr_cnt > 0)
            {
                /*
                 * Evaluate the index row to compute expression values. We
                 * could do this by hand, but FormIndexDatum is convenient.
                 */
                FormIndexDatum(indexInfo,
                               slot,
                               estate,
                               values,
                               isnull);
 
                /*
                 * Save just the columns we care about.  We copy the values
                 * into ind_context from the estate's per-tuple context.
                 */
                for (i = 0; i < attr_cnt; i++)
                {
                    VacAttrStats *stats = thisdata->vacattrstats[i];
                    int         attnum = stats->attr->attnum;
 
                    if (isnull[attnum - 1])
                    {
                        exprvals[tcnt] = (Datum) 0;
                        exprnulls[tcnt] = true;
                    }
                    else
                    {
                        exprvals[tcnt] = datumCopy(values[attnum - 1],
                                                   stats->attrtype->typbyval,
                                                   stats->attrtype->typlen);
                        exprnulls[tcnt] = false;
                    }
                    tcnt++;
                }
            }
        }
 
        /*
         * Having counted the number of rows that pass the predicate in the
         * sample, we can estimate the total number of rows in the index.
         */
        thisdata->tupleFract = (double) numindexrows / (double) numrows;
        totalindexrows = ceil(thisdata->tupleFract * totalrows);
 
        /*
         * Now we can compute the statistics for the expression columns.
         */
        if (numindexrows > 0)
        {
            MemoryContextSwitchTo(col_context);
            for (i = 0; i < attr_cnt; i++)
            {
                VacAttrStats *stats = thisdata->vacattrstats[i];
 
                stats->exprvals = exprvals + i;
                stats->exprnulls = exprnulls + i;
                stats->rowstride = attr_cnt;
                stats->compute_stats(stats,
                                     ind_fetch_func,
                                     numindexrows,
                                     totalindexrows);
 
                MemoryContextResetAndDeleteChildren(col_context);
            }
        }
 
        /* And clean up */
        MemoryContextSwitchTo(ind_context);
 
        ExecDropSingleTupleTableSlot(slot);
        FreeExecutorState(estate);
        MemoryContextResetAndDeleteChildren(ind_context);
    }
 
    MemoryContextSwitchTo(old_context);
    MemoryContextDelete(ind_context);
}
 
/*
 * examine_attribute -- pre-analysis of a single column
 *
 * Determine whether the column is analyzable; if so, create and initialize
 * a VacAttrStats struct for it.  If not, return NULL.
 *
 * If index_expr isn't NULL, then we're trying to analyze an expression index,
 * and index_expr is the expression tree representing the column's data.
 */
static VacAttrStats *
examine_attribute(Relation onerel, int attnum, Node *index_expr)
{
    Form_pg_attribute attr = TupleDescAttr(onerel->rd_att, attnum - 1);
    HeapTuple   typtuple;
    VacAttrStats *stats;
    int         i;
    bool        ok;
 
    /* Never analyze dropped columns */
    if (attr->attisdropped)
        return NULL;
 
    /* Don't analyze column if user has specified not to */
    if (attr->attstattarget == 0)
        return NULL;
 
    /*
     * Create the VacAttrStats struct.  Note that we only have a copy of the
     * fixed fields of the pg_attribute tuple.
     */
    stats = (VacAttrStats *) palloc0(sizeof(VacAttrStats));
    stats->attr = (Form_pg_attribute) palloc(ATTRIBUTE_FIXED_PART_SIZE);
    memcpy(stats->attr, attr, ATTRIBUTE_FIXED_PART_SIZE);
 
    /*
     * When analyzing an expression index, believe the expression tree's type
     * not the column datatype --- the latter might be the opckeytype storage
     * type of the opclass, which is not interesting for our purposes.  (Note:
     * if we did anything with non-expression index columns, we'd need to
     * figure out where to get the correct type info from, but for now that's
     * not a problem.)  It's not clear whether anyone will care about the
     * typmod, but we store that too just in case.
     */
    if (index_expr)
    {
        stats->attrtypid = exprType(index_expr);
        stats->attrtypmod = exprTypmod(index_expr);
 
        /*
         * If a collation has been specified for the index column, use that in
         * preference to anything else; but if not, fall back to whatever we
         * can get from the expression.
         */
        if (OidIsValid(onerel->rd_indcollation[attnum - 1]))
            stats->attrcollid = onerel->rd_indcollation[attnum - 1];
        else
            stats->attrcollid = exprCollation(index_expr);
    }
    else
    {
        stats->attrtypid = attr->atttypid;
        stats->attrtypmod = attr->atttypmod;
        stats->attrcollid = attr->attcollation;
    }
 
    typtuple = SearchSysCacheCopy1(TYPEOID,
                                   ObjectIdGetDatum(stats->attrtypid));
    if (!HeapTupleIsValid(typtuple))
        elog(ERROR, "cache lookup failed for type %u", stats->attrtypid);
    stats->attrtype = (Form_pg_type) GETSTRUCT(typtuple);
    stats->anl_context = anl_context;
    stats->tupattnum = attnum;
 
    /*
     * The fields describing the stats->stavalues[n] element types default to
     * the type of the data being analyzed, but the type-specific typanalyze
     * function can change them if it wants to store something else.
     */
    for (i = 0; i < STATISTIC_NUM_SLOTS; i++)
    {
        stats->statypid[i] = stats->attrtypid;
        stats->statyplen[i] = stats->attrtype->typlen;
        stats->statypbyval[i] = stats->attrtype->typbyval;
        stats->statypalign[i] = stats->attrtype->typalign;
    }
 
    /*
     * Call the type-specific typanalyze function.  If none is specified, use
     * std_typanalyze().
     */
    if (OidIsValid(stats->attrtype->typanalyze))
        ok = DatumGetBool(OidFunctionCall1(stats->attrtype->typanalyze,
                                           PointerGetDatum(stats)));
    else
        ok = std_typanalyze(stats);
 
    if (!ok || stats->compute_stats == NULL || stats->minrows <= 0)
    {
        heap_freetuple(typtuple);
        pfree(stats->attr);
        pfree(stats);
        return NULL;
    }
 
    return stats;
}
 
/*
 * acquire_sample_rows -- acquire a random sample of rows from the table
 *
 * Selected rows are returned in the caller-allocated array rows[], which
 * must have at least targrows entries.
 * The actual number of rows selected is returned as the function result.
 * We also estimate the total numbers of live and dead rows in the table,
 * and return them into *totalrows and *totaldeadrows, respectively.
 *
 * The returned list of tuples is in order by physical position in the table.
 * (We will rely on this later to derive correlation estimates.)
 *
 * As of May 2004 we use a new two-stage method:  Stage one selects up
 * to targrows random blocks (or all blocks, if there aren't so many).
 * Stage two scans these blocks and uses the Vitter algorithm to create
 * a random sample of targrows rows (or less, if there are less in the
 * sample of blocks).  The two stages are executed simultaneously: each
 * block is processed as soon as stage one returns its number and while
 * the rows are read stage two controls which ones are to be inserted
 * into the sample.
 *
 * Although every row has an equal chance of ending up in the final
 * sample, this sampling method is not perfect: not every possible
 * sample has an equal chance of being selected.  For large relations
 * the number of different blocks represented by the sample tends to be
 * too small.  We can live with that for now.  Improvements are welcome.
 *
 * An important property of this sampling method is that because we do
 * look at a statistically unbiased set of blocks, we should get
 * unbiased estimates of the average numbers of live and dead rows per
 * block.  The previous sampling method put too much credence in the row
 * density near the start of the table.
 */
static int
acquire_sample_rows(Relation onerel, int elevel,
                    HeapTuple *rows, int targrows,
                    double *totalrows, double *totaldeadrows)
{
    int         numrows = 0;    /* # rows now in reservoir */
    double      samplerows = 0; /* total # rows collected */
    double      liverows = 0;   /* # live rows seen */
    double      deadrows = 0;   /* # dead rows seen */
    double      rowstoskip = -1;    /* -1 means not set yet */
    uint32      randseed;       /* Seed for block sampler(s) */
    BlockNumber totalblocks;
    TransactionId OldestXmin;
    BlockSamplerData bs;
    ReservoirStateData rstate;
    TupleTableSlot *slot;
    TableScanDesc scan;
    BlockNumber nblocks;
    BlockNumber blksdone = 0;
#ifdef USE_PREFETCH
    int         prefetch_maximum = 0;   /* blocks to prefetch if enabled */
    BlockSamplerData prefetch_bs;
#endif
 
    Assert(targrows > 0);
 
    totalblocks = RelationGetNumberOfBlocks(onerel);
 
    /* Need a cutoff xmin for HeapTupleSatisfiesVacuum */
    OldestXmin = GetOldestNonRemovableTransactionId(onerel);
 
    /* Prepare for sampling block numbers */
    randseed = pg_prng_uint32(&pg_global_prng_state);
    nblocks = BlockSampler_Init(&bs, totalblocks, targrows, randseed);
 
#ifdef USE_PREFETCH
    prefetch_maximum = get_tablespace_maintenance_io_concurrency(onerel->rd_rel->reltablespace);
    /* Create another BlockSampler, using the same seed, for prefetching */
    if (prefetch_maximum)
        (void) BlockSampler_Init(&prefetch_bs, totalblocks, targrows, randseed);
#endif
 
    /* Report sampling block numbers */
    pgstat_progress_update_param(PROGRESS_ANALYZE_BLOCKS_TOTAL,
                                 nblocks);
 
    /* Prepare for sampling rows */
    reservoir_init_selection_state(&rstate, targrows);
 
    scan = table_beginscan_analyze(onerel);
    slot = table_slot_create(onerel, NULL);
 
#ifdef USE_PREFETCH
 
    /*
     * If we are doing prefetching, then go ahead and tell the kernel about
     * the first set of pages we are going to want.  This also moves our
     * iterator out ahead of the main one being used, where we will keep it so
     * that we're always pre-fetching out prefetch_maximum number of blocks
     * ahead.
     */
    if (prefetch_maximum)
    {
        for (int i = 0; i < prefetch_maximum; i++)
        {
            BlockNumber prefetch_block;
 
            if (!BlockSampler_HasMore(&prefetch_bs))
                break;
 
            prefetch_block = BlockSampler_Next(&prefetch_bs);
            PrefetchBuffer(scan->rs_rd, MAIN_FORKNUM, prefetch_block);
        }
    }
#endif
 
    /* Outer loop over blocks to sample */
    while (BlockSampler_HasMore(&bs))
    {
        bool        block_accepted;
        BlockNumber targblock = BlockSampler_Next(&bs);
#ifdef USE_PREFETCH
        BlockNumber prefetch_targblock = InvalidBlockNumber;
 
        /*
         * Make sure that every time the main BlockSampler is moved forward
         * that our prefetch BlockSampler also gets moved forward, so that we
         * always stay out ahead.
         */
        if (prefetch_maximum && BlockSampler_HasMore(&prefetch_bs))
            prefetch_targblock = BlockSampler_Next(&prefetch_bs);
#endif
 
        vacuum_delay_point();
 
        block_accepted = table_scan_analyze_next_block(scan, targblock, vac_strategy);
 
#ifdef USE_PREFETCH
 
        /*
         * When pre-fetching, after we get a block, tell the kernel about the
         * next one we will want, if there's any left.
         *
         * We want to do this even if the table_scan_analyze_next_block() call
         * above decides against analyzing the block it picked.
         */
        if (prefetch_maximum && prefetch_targblock != InvalidBlockNumber)
            PrefetchBuffer(scan->rs_rd, MAIN_FORKNUM, prefetch_targblock);
#endif
 
        /*
         * Don't analyze if table_scan_analyze_next_block() indicated this
         * block is unsuitable for analyzing.
         */
        if (!block_accepted)
            continue;
 
        while (table_scan_analyze_next_tuple(scan, OldestXmin, &liverows, &deadrows, slot))
        {
            /*
             * The first targrows sample rows are simply copied into the
             * reservoir. Then we start replacing tuples in the sample until
             * we reach the end of the relation.  This algorithm is from Jeff
             * Vitter's paper (see full citation in utils/misc/sampling.c). It
             * works by repeatedly computing the number of tuples to skip
             * before selecting a tuple, which replaces a randomly chosen
             * element of the reservoir (current set of tuples).  At all times
             * the reservoir is a true random sample of the tuples we've
             * passed over so far, so when we fall off the end of the relation
             * we're done.
             */
            if (numrows < targrows)
                rows[numrows++] = ExecCopySlotHeapTuple(slot);
            else
            {
                /*
                 * t in Vitter's paper is the number of records already
                 * processed.  If we need to compute a new S value, we must
                 * use the not-yet-incremented value of samplerows as t.
                 */
                if (rowstoskip < 0)
                    rowstoskip = reservoir_get_next_S(&rstate, samplerows, targrows);
 
                if (rowstoskip <= 0)
                {
                    /*
                     * Found a suitable tuple, so save it, replacing one old
                     * tuple at random
                     */
                    int         k = (int) (targrows * sampler_random_fract(&rstate.randstate));
 
                    Assert(k >= 0 && k < targrows);
                    heap_freetuple(rows[k]);
                    rows[k] = ExecCopySlotHeapTuple(slot);
                }
 
                rowstoskip -= 1;
            }
 
            samplerows += 1;
        }
 
        pgstat_progress_update_param(PROGRESS_ANALYZE_BLOCKS_DONE,
                                     ++blksdone);
    }
 
    ExecDropSingleTupleTableSlot(slot);
    table_endscan(scan);
 
    /*
     * If we didn't find as many tuples as we wanted then we're done. No sort
     * is needed, since they're already in order.
     *
     * Otherwise we need to sort the collected tuples by position
     * (itempointer). It's not worth worrying about corner cases where the
     * tuples are already sorted.
     */
    if (numrows == targrows)
        qsort_interruptible(rows, numrows, sizeof(HeapTuple),
                            compare_rows, NULL);
 
    /*
     * Estimate total numbers of live and dead rows in relation, extrapolating
     * on the assumption that the average tuple density in pages we didn't
     * scan is the same as in the pages we did scan.  Since what we scanned is
     * a random sample of the pages in the relation, this should be a good
     * assumption.
     */
    if (bs.m > 0)
    {
        *totalrows = floor((liverows / bs.m) * totalblocks + 0.5);
        *totaldeadrows = floor((deadrows / bs.m) * totalblocks + 0.5);
    }
    else
    {
        *totalrows = 0.0;
        *totaldeadrows = 0.0;
    }
 
    /*
     * Emit some interesting relation info
     */
    ereport(elevel,
            (errmsg("\"%s\": scanned %d of %u pages, "
                    "containing %.0f live rows and %.0f dead rows; "
                    "%d rows in sample, %.0f estimated total rows",
                    RelationGetRelationName(onerel),
                    bs.m, totalblocks,
                    liverows, deadrows,
                    numrows, *totalrows)));
 
    return numrows;
}
 
/*
 * Comparator for sorting rows[] array
 */
static int
compare_rows(const void *a, const void *b, void *arg)
{
    HeapTuple   ha = *(const HeapTuple *) a;
    HeapTuple   hb = *(const HeapTuple *) b;
    BlockNumber ba = ItemPointerGetBlockNumber(&ha->t_self);
    OffsetNumber oa = ItemPointerGetOffsetNumber(&ha->t_self);
    BlockNumber bb = ItemPointerGetBlockNumber(&hb->t_self);
    OffsetNumber ob = ItemPointerGetOffsetNumber(&hb->t_self);
 
    if (ba < bb)
        return -1;
    if (ba > bb)
        return 1;
    if (oa < ob)
        return -1;
    if (oa > ob)
        return 1;
    return 0;
}
 
 
/*
 * acquire_inherited_sample_rows -- acquire sample rows from inheritance tree
 *
 * This has the same API as acquire_sample_rows, except that rows are
 * collected from all inheritance children as well as the specified table.
 * We fail and return zero if there are no inheritance children, or if all
 * children are foreign tables that don't support ANALYZE.
 */
static int
acquire_inherited_sample_rows(Relation onerel, int elevel,
                              HeapTuple *rows, int targrows,
                              double *totalrows, double *totaldeadrows)
{
    List       *tableOIDs;
    Relation   *rels;
    AcquireSampleRowsFunc *acquirefuncs;
    double     *relblocks;
    double      totalblocks;
    int         numrows,
                nrels,
                i;
    ListCell   *lc;
    bool        has_child;
 
    /*
     * Find all members of inheritance set.  We only need AccessShareLock on
     * the children.
     */
    tableOIDs =
        find_all_inheritors(RelationGetRelid(onerel), AccessShareLock, NULL);
 
    /*
     * Check that there's at least one descendant, else fail.  This could
     * happen despite analyze_rel's relhassubclass check, if table once had a
     * child but no longer does.  In that case, we can clear the
     * relhassubclass field so as not to make the same mistake again later.
     * (This is safe because we hold ShareUpdateExclusiveLock.)
     */
    if (list_length(tableOIDs) < 2)
    {
        /* CCI because we already updated the pg_class row in this command */
        CommandCounterIncrement();
        SetRelationHasSubclass(RelationGetRelid(onerel), false);
        ereport(elevel,
                (errmsg("skipping analyze of \"%s.%s\" inheritance tree --- this inheritance tree contains no child tables",
                        get_namespace_name(RelationGetNamespace(onerel)),
                        RelationGetRelationName(onerel))));
        return 0;
    }
 
    /*
     * Identify acquirefuncs to use, and count blocks in all the relations.
     * The result could overflow BlockNumber, so we use double arithmetic.
     */
    rels = (Relation *) palloc(list_length(tableOIDs) * sizeof(Relation));
    acquirefuncs = (AcquireSampleRowsFunc *)
        palloc(list_length(tableOIDs) * sizeof(AcquireSampleRowsFunc));
    relblocks = (double *) palloc(list_length(tableOIDs) * sizeof(double));
    totalblocks = 0;
    nrels = 0;
    has_child = false;
    foreach(lc, tableOIDs)
    {
        Oid         childOID = lfirst_oid(lc);
        Relation    childrel;
        AcquireSampleRowsFunc acquirefunc = NULL;
        BlockNumber relpages = 0;
 
        /* We already got the needed lock */
        childrel = table_open(childOID, NoLock);
 
        /* Ignore if temp table of another backend */
        if (RELATION_IS_OTHER_TEMP(childrel))
        {
            /* ... but release the lock on it */
            Assert(childrel != onerel);
            table_close(childrel, AccessShareLock);
            continue;
        }
 
        /* Check table type (MATVIEW can't happen, but might as well allow) */
        if (childrel->rd_rel->relkind == RELKIND_RELATION ||
            childrel->rd_rel->relkind == RELKIND_MATVIEW)
        {
            /* Regular table, so use the regular row acquisition function */
            acquirefunc = acquire_sample_rows;
            relpages = RelationGetNumberOfBlocks(childrel);
        }
        else if (childrel->rd_rel->relkind == RELKIND_FOREIGN_TABLE)
        {
            /*
             * For a foreign table, call the FDW's hook function to see
             * whether it supports analysis.
             */
            FdwRoutine *fdwroutine;
            bool        ok = false;
 
            fdwroutine = GetFdwRoutineForRelation(childrel, false);
 
            if (fdwroutine->AnalyzeForeignTable != NULL)
                ok = fdwroutine->AnalyzeForeignTable(childrel,
                                                     &acquirefunc,
                                                     &relpages);
 
            if (!ok)
            {
                /* ignore, but release the lock on it */
                Assert(childrel != onerel);
                table_close(childrel, AccessShareLock);
                continue;
            }
        }
        else
        {
            /*
             * ignore, but release the lock on it.  don't try to unlock the
             * passed-in relation
             */
            Assert(childrel->rd_rel->relkind == RELKIND_PARTITIONED_TABLE);
            if (childrel != onerel)
                table_close(childrel, AccessShareLock);
            else
                table_close(childrel, NoLock);
            continue;
        }
 
        /* OK, we'll process this child */
        has_child = true;
        rels[nrels] = childrel;
        acquirefuncs[nrels] = acquirefunc;
        relblocks[nrels] = (double) relpages;
        totalblocks += (double) relpages;
        nrels++;
    }
 
    /*
     * If we don't have at least one child table to consider, fail.  If the
     * relation is a partitioned table, it's not counted as a child table.
     */
    if (!has_child)
    {
        ereport(elevel,
                (errmsg("skipping analyze of \"%s.%s\" inheritance tree --- this inheritance tree contains no analyzable child tables",
                        get_namespace_name(RelationGetNamespace(onerel)),
                        RelationGetRelationName(onerel))));
        return 0;
    }
 
    /*
     * Now sample rows from each relation, proportionally to its fraction of
     * the total block count.  (This might be less than desirable if the child
     * rels have radically different free-space percentages, but it's not
     * clear that it's worth working harder.)
     */
    pgstat_progress_update_param(PROGRESS_ANALYZE_CHILD_TABLES_TOTAL,
                                 nrels);
    numrows = 0;
    *totalrows = 0;
    *totaldeadrows = 0;
    for (i = 0; i < nrels; i++)
    {
        Relation    childrel = rels[i];
        AcquireSampleRowsFunc acquirefunc = acquirefuncs[i];
        double      childblocks = relblocks[i];
 
        pgstat_progress_update_param(PROGRESS_ANALYZE_CURRENT_CHILD_TABLE_RELID,
                                     RelationGetRelid(childrel));
 
        if (childblocks > 0)
        {
            int         childtargrows;
 
            childtargrows = (int) rint(targrows * childblocks / totalblocks);
            /* Make sure we don't overrun due to roundoff error */
            childtargrows = Min(childtargrows, targrows - numrows);
            if (childtargrows > 0)
            {
                int         childrows;
                double      trows,
                            tdrows;
 
                /* Fetch a random sample of the child's rows */
                childrows = (*acquirefunc) (childrel, elevel,
                                            rows + numrows, childtargrows,
                                            &trows, &tdrows);
 
                /* We may need to convert from child's rowtype to parent's */
                if (childrows > 0 &&
                    !equalTupleDescs(RelationGetDescr(childrel),
                                     RelationGetDescr(onerel)))
                {
                    TupleConversionMap *map;
 
                    map = convert_tuples_by_name(RelationGetDescr(childrel),
                                                 RelationGetDescr(onerel));
                    if (map != NULL)
                    {
                        int         j;
 
                        for (j = 0; j < childrows; j++)
                        {
                            HeapTuple   newtup;
 
                            newtup = execute_attr_map_tuple(rows[numrows + j], map);
                            heap_freetuple(rows[numrows + j]);
                            rows[numrows + j] = newtup;
                        }
                        free_conversion_map(map);
                    }
                }
 
                /* And add to counts */
                numrows += childrows;
                *totalrows += trows;
                *totaldeadrows += tdrows;
            }
        }
 
        /*
         * Note: we cannot release the child-table locks, since we may have
         * pointers to their TOAST tables in the sampled rows.
         */
        table_close(childrel, NoLock);
        pgstat_progress_update_param(PROGRESS_ANALYZE_CHILD_TABLES_DONE,
                                     i + 1);
    }
 
    return numrows;
}
 
 
/*
 *  update_attstats() -- update attribute statistics for one relation
 *
 *      Statistics are stored in several places: the pg_class row for the
 *      relation has stats about the whole relation, and there is a
 *      pg_statistic row for each (non-system) attribute that has ever
 *      been analyzed.  The pg_class values are updated by VACUUM, not here.
 *
 *      pg_statistic rows are just added or updated normally.  This means
 *      that pg_statistic will probably contain some deleted rows at the
 *      completion of a vacuum cycle, unless it happens to get vacuumed last.
 *
 *      To keep things simple, we punt for pg_statistic, and don't try
 *      to compute or store rows for pg_statistic itself in pg_statistic.
 *      This could possibly be made to work, but it's not worth the trouble.
 *      Note analyze_rel() has seen to it that we won't come here when
 *      vacuuming pg_statistic itself.
 *
 *      Note: there would be a race condition here if two backends could
 *      ANALYZE the same table concurrently.  Presently, we lock that out
 *      by taking a self-exclusive lock on the relation in analyze_rel().
 */
static void
update_attstats(Oid relid, bool inh, int natts, VacAttrStats **vacattrstats)
{
    Relation    sd;
    int         attno;
    CatalogIndexState indstate = NULL;
 
    if (natts <= 0)
        return;                 /* nothing to do */
 
    sd = table_open(StatisticRelationId, RowExclusiveLock);
 
    for (attno = 0; attno < natts; attno++)
    {
        VacAttrStats *stats = vacattrstats[attno];
        HeapTuple   stup,
                    oldtup;
        int         i,
                    k,
                    n;
        Datum       values[Natts_pg_statistic];
        bool        nulls[Natts_pg_statistic];
        bool        replaces[Natts_pg_statistic];
 
        /* Ignore attr if we weren't able to collect stats */
        if (!stats->stats_valid)
            continue;
 
        /*
         * Construct a new pg_statistic tuple
         */
        for (i = 0; i < Natts_pg_statistic; ++i)
        {
            nulls[i] = false;
            replaces[i] = true;
        }
 
        values[Anum_pg_statistic_starelid - 1] = ObjectIdGetDatum(relid);
        values[Anum_pg_statistic_staattnum - 1] = Int16GetDatum(stats->attr->attnum);
        values[Anum_pg_statistic_stainherit - 1] = BoolGetDatum(inh);
        values[Anum_pg_statistic_stanullfrac - 1] = Float4GetDatum(stats->stanullfrac);
        values[Anum_pg_statistic_stawidth - 1] = Int32GetDatum(stats->stawidth);
        values[Anum_pg_statistic_stadistinct - 1] = Float4GetDatum(stats->stadistinct);
        i = Anum_pg_statistic_stakind1 - 1;
        for (k = 0; k < STATISTIC_NUM_SLOTS; k++)
        {
            values[i++] = Int16GetDatum(stats->stakind[k]); /* stakindN */
        }
        i = Anum_pg_statistic_staop1 - 1;
        for (k = 0; k < STATISTIC_NUM_SLOTS; k++)
        {
            values[i++] = ObjectIdGetDatum(stats->staop[k]);    /* staopN */
        }
        i = Anum_pg_statistic_stacoll1 - 1;
        for (k = 0; k < STATISTIC_NUM_SLOTS; k++)
        {
            values[i++] = ObjectIdGetDatum(stats->stacoll[k]);  /* stacollN */
        }
        i = Anum_pg_statistic_stanumbers1 - 1;
        for (k = 0; k < STATISTIC_NUM_SLOTS; k++)
        {
            int         nnum = stats->numnumbers[k];
 
            if (nnum > 0)
            {
                Datum      *numdatums = (Datum *) palloc(nnum * sizeof(Datum));
                ArrayType  *arry;
 
                for (n = 0; n < nnum; n++)
                    numdatums[n] = Float4GetDatum(stats->stanumbers[k][n]);
                arry = construct_array_builtin(numdatums, nnum, FLOAT4OID);
                values[i++] = PointerGetDatum(arry);    /* stanumbersN */
            }
            else
            {
                nulls[i] = true;
                values[i++] = (Datum) 0;
            }
        }
        i = Anum_pg_statistic_stavalues1 - 1;
        for (k = 0; k < STATISTIC_NUM_SLOTS; k++)
        {
            if (stats->numvalues[k] > 0)
            {
                ArrayType  *arry;
 
                arry = construct_array(stats->stavalues[k],
                                       stats->numvalues[k],
                                       stats->statypid[k],
                                       stats->statyplen[k],
                                       stats->statypbyval[k],
                                       stats->statypalign[k]);
                values[i++] = PointerGetDatum(arry);    /* stavaluesN */
            }
            else
            {
                nulls[i] = true;
                values[i++] = (Datum) 0;
            }
        }
 
        /* Is there already a pg_statistic tuple for this attribute? */
        oldtup = SearchSysCache3(STATRELATTINH,
                                 ObjectIdGetDatum(relid),
                                 Int16GetDatum(stats->attr->attnum),
                                 BoolGetDatum(inh));
 
        /* Open index information when we know we need it */
        if (indstate == NULL)
            indstate = CatalogOpenIndexes(sd);
 
        if (HeapTupleIsValid(oldtup))
        {
            /* Yes, replace it */
            stup = heap_modify_tuple(oldtup,
                                     RelationGetDescr(sd),
                                     values,
                                     nulls,
                                     replaces);
            ReleaseSysCache(oldtup);
            CatalogTupleUpdateWithInfo(sd, &stup->t_self, stup, indstate);
        }
        else
        {
            /* No, insert new tuple */
            stup = heap_form_tuple(RelationGetDescr(sd), values, nulls);
            CatalogTupleInsertWithInfo(sd, stup, indstate);
        }
 
        heap_freetuple(stup);
    }
 
    if (indstate != NULL)
        CatalogCloseIndexes(indstate);
    table_close(sd, RowExclusiveLock);
}
 
/*
 * Standard fetch function for use by compute_stats subroutines.
 *
 * This exists to provide some insulation between compute_stats routines
 * and the actual storage of the sample data.
 */
static Datum
std_fetch_func(VacAttrStatsP stats, int rownum, bool *isNull)
{
    int         attnum = stats->tupattnum;
    HeapTuple   tuple = stats->rows[rownum];
    TupleDesc   tupDesc = stats->tupDesc;
 
    return heap_getattr(tuple, attnum, tupDesc, isNull);
}
 
/*
 * Fetch function for analyzing index expressions.
 *
 * We have not bothered to construct index tuples, instead the data is
 * just in Datum arrays.
 */
static Datum
ind_fetch_func(VacAttrStatsP stats, int rownum, bool *isNull)
{
    int         i;
 
    /* exprvals and exprnulls are already offset for proper column */
    i = rownum * stats->rowstride;
    *isNull = stats->exprnulls[i];
    return stats->exprvals[i];
}
 
 
/*==========================================================================
 *
 * Code below this point represents the "standard" type-specific statistics
 * analysis algorithms.  This code can be replaced on a per-data-type basis
 * by setting a nonzero value in pg_type.typanalyze.
 *
 *==========================================================================
 */
 
 
/*
 * To avoid consuming too much memory during analysis and/or too much space
 * in the resulting pg_statistic rows, we ignore varlena datums that are wider
 * than WIDTH_THRESHOLD (after detoasting!).  This is legitimate for MCV
 * and distinct-value calculations since a wide value is unlikely to be
 * duplicated at all, much less be a most-common value.  For the same reason,
 * ignoring wide values will not affect our estimates of histogram bin
 * boundaries very much.
 */
#define WIDTH_THRESHOLD  1024
 
#define swapInt(a,b)    do {int _tmp; _tmp=a; a=b; b=_tmp;} while(0)
#define swapDatum(a,b)  do {Datum _tmp; _tmp=a; a=b; b=_tmp;} while(0)
 
/*
 * Extra information used by the default analysis routines
 */
typedef struct
{
    int         count;          /* # of duplicates */
    int         first;          /* values[] index of first occurrence */
} ScalarMCVItem;
 
typedef struct
{
    SortSupport ssup;
    int        *tupnoLink;
} CompareScalarsContext;
 
 
static void compute_trivial_stats(VacAttrStatsP stats,
                                  AnalyzeAttrFetchFunc fetchfunc,
                                  int samplerows,
                                  double totalrows);
static void compute_distinct_stats(VacAttrStatsP stats,
                                   AnalyzeAttrFetchFunc fetchfunc,
                                   int samplerows,
                                   double totalrows);
static void compute_scalar_stats(VacAttrStatsP stats,
                                 AnalyzeAttrFetchFunc fetchfunc,
                                 int samplerows,
                                 double totalrows);
static int  compare_scalars(const void *a, const void *b, void *arg);
static int  compare_mcvs(const void *a, const void *b, void *arg);
static int  analyze_mcv_list(int *mcv_counts,
                             int num_mcv,
                             double stadistinct,
                             double stanullfrac,
                             int samplerows,
                             double totalrows);
 
 
/*
 * std_typanalyze -- the default type-specific typanalyze function
 */
bool
std_typanalyze(VacAttrStats *stats)
{
    Form_pg_attribute attr = stats->attr;
    Oid         ltopr;
    Oid         eqopr;
    StdAnalyzeData *mystats;
 
    /* If the attstattarget column is negative, use the default value */
    /* NB: it is okay to scribble on stats->attr since it's a copy */
    if (attr->attstattarget < 0)
        attr->attstattarget = default_statistics_target;
 
    /* Look for default "<" and "=" operators for column's type */
    get_sort_group_operators(stats->attrtypid,
                             false, false, false,
                             &ltopr, &eqopr, NULL,
                             NULL);
 
    /* Save the operator info for compute_stats routines */
    mystats = (StdAnalyzeData *) palloc(sizeof(StdAnalyzeData));
    mystats->eqopr = eqopr;
    mystats->eqfunc = OidIsValid(eqopr) ? get_opcode(eqopr) : InvalidOid;
    mystats->ltopr = ltopr;
    stats->extra_data = mystats;
 
    /*
     * Determine which standard statistics algorithm to use
     */
    if (OidIsValid(eqopr) && OidIsValid(ltopr))
    {
        /* Seems to be a scalar datatype */
        stats->compute_stats = compute_scalar_stats;
        /*--------------------
         * The following choice of minrows is based on the paper
         * "Random sampling for histogram construction: how much is enough?"
         * by Surajit Chaudhuri, Rajeev Motwani and Vivek Narasayya, in
         * Proceedings of ACM SIGMOD International Conference on Management
         * of Data, 1998, Pages 436-447.  Their Corollary 1 to Theorem 5
         * says that for table size n, histogram size k, maximum relative
         * error in bin size f, and error probability gamma, the minimum
         * random sample size is
         *      r = 4 * k * ln(2*n/gamma) / f^2
         * Taking f = 0.5, gamma = 0.01, n = 10^6 rows, we obtain
         *      r = 305.82 * k
         * Note that because of the log function, the dependence on n is
         * quite weak; even at n = 10^12, a 300*k sample gives <= 0.66
         * bin size error with probability 0.99.  So there's no real need to
         * scale for n, which is a good thing because we don't necessarily
         * know it at this point.
         *--------------------
         */
        stats->minrows = 300 * attr->attstattarget;
    }
    else if (OidIsValid(eqopr))
    {
        /* We can still recognize distinct values */
        stats->compute_stats = compute_distinct_stats;
        /* Might as well use the same minrows as above */
        stats->minrows = 300 * attr->attstattarget;
    }
    else
    {
        /* Can't do much but the trivial stuff */
        stats->compute_stats = compute_trivial_stats;
        /* Might as well use the same minrows as above */
        stats->minrows = 300 * attr->attstattarget;
    }
 
    return true;
}
 
 
/*
 *  compute_trivial_stats() -- compute very basic column statistics
 *
 *  We use this when we cannot find a hash "=" operator for the datatype.
 *
 *  We determine the fraction of non-null rows and the average datum width.
 */
static void
compute_trivial_stats(VacAttrStatsP stats,
                      AnalyzeAttrFetchFunc fetchfunc,
                      int samplerows,
                      double totalrows)
{
    int         i;
    int         null_cnt = 0;
    int         nonnull_cnt = 0;
    double      total_width = 0;
    bool        is_varlena = (!stats->attrtype->typbyval &&
                              stats->attrtype->typlen == -1);
    bool        is_varwidth = (!stats->attrtype->typbyval &&
                               stats->attrtype->typlen < 0);
 
    for (i = 0; i < samplerows; i++)
    {
        Datum       value;
        bool        isnull;
 
        vacuum_delay_point();
 
        value = fetchfunc(stats, i, &isnull);
 
        /* Check for null/nonnull */
        if (isnull)
        {
            null_cnt++;
            continue;
        }
        nonnull_cnt++;
 
        /*
         * If it's a variable-width field, add up widths for average width
         * calculation.  Note that if the value is toasted, we use the toasted
         * width.  We don't bother with this calculation if it's a fixed-width
         * type.
         */
        if (is_varlena)
        {
            total_width += VARSIZE_ANY(DatumGetPointer(value));
        }
        else if (is_varwidth)
        {
            /* must be cstring */
            total_width += strlen(DatumGetCString(value)) + 1;
        }
    }
 
    /* We can only compute average width if we found some non-null values. */
    if (nonnull_cnt > 0)
    {
        stats->stats_valid = true;
        /* Do the simple null-frac and width stats */
        stats->stanullfrac = (double) null_cnt / (double) samplerows;
        if (is_varwidth)
            stats->stawidth = total_width / (double) nonnull_cnt;
        else
            stats->stawidth = stats->attrtype->typlen;
        stats->stadistinct = 0.0;   /* "unknown" */
    }
    else if (null_cnt > 0)
    {
        /* We found only nulls; assume the column is entirely null */
        stats->stats_valid = true;
        stats->stanullfrac = 1.0;
        if (is_varwidth)
            stats->stawidth = 0;    /* "unknown" */
        else
            stats->stawidth = stats->attrtype->typlen;
        stats->stadistinct = 0.0;   /* "unknown" */
    }
}
 
 
/*
 *  compute_distinct_stats() -- compute column statistics including ndistinct
 *
 *  We use this when we can find only an "=" operator for the datatype.
 *
 *  We determine the fraction of non-null rows, the average width, the
 *  most common values, and the (estimated) number of distinct values.
 *
 *  The most common values are determined by brute force: we keep a list
 *  of previously seen values, ordered by number of times seen, as we scan
 *  the samples.  A newly seen value is inserted just after the last
 *  multiply-seen value, causing the bottommost (oldest) singly-seen value
 *  to drop off the list.  The accuracy of this method, and also its cost,
 *  depend mainly on the length of the list we are willing to keep.
 */
static void
compute_distinct_stats(VacAttrStatsP stats,
                       AnalyzeAttrFetchFunc fetchfunc,
                       int samplerows,
                       double totalrows)
{
    int         i;
    int         null_cnt = 0;
    int         nonnull_cnt = 0;
    int         toowide_cnt = 0;
    double      total_width = 0;
    bool        is_varlena = (!stats->attrtype->typbyval &&
                              stats->attrtype->typlen == -1);
    bool        is_varwidth = (!stats->attrtype->typbyval &&
                               stats->attrtype->typlen < 0);
    FmgrInfo    f_cmpeq;
    typedef struct
    {
        Datum       value;
        int         count;
    } TrackItem;
    TrackItem  *track;
    int         track_cnt,
                track_max;
    int         num_mcv = stats->attr->attstattarget;
    StdAnalyzeData *mystats = (StdAnalyzeData *) stats->extra_data;
 
    /*
     * We track up to 2*n values for an n-element MCV list; but at least 10
     */
    track_max = 2 * num_mcv;
    if (track_max < 10)
        track_max = 10;
    track = (TrackItem *) palloc(track_max * sizeof(TrackItem));
    track_cnt = 0;
 
    fmgr_info(mystats->eqfunc, &f_cmpeq);
 
    for (i = 0; i < samplerows; i++)
    {
        Datum       value;
        bool        isnull;
        bool        match;
        int         firstcount1,
                    j;
 
        vacuum_delay_point();
 
        value = fetchfunc(stats, i, &isnull);
 
        /* Check for null/nonnull */
        if (isnull)
        {
            null_cnt++;
            continue;
        }
        nonnull_cnt++;
 
        /*
         * If it's a variable-width field, add up widths for average width
         * calculation.  Note that if the value is toasted, we use the toasted
         * width.  We don't bother with this calculation if it's a fixed-width
         * type.
         */
        if (is_varlena)
        {
            total_width += VARSIZE_ANY(DatumGetPointer(value));
 
            /*
             * If the value is toasted, we want to detoast it just once to
             * avoid repeated detoastings and resultant excess memory usage
             * during the comparisons.  Also, check to see if the value is
             * excessively wide, and if so don't detoast at all --- just
             * ignore the value.
             */
            if (toast_raw_datum_size(value) > WIDTH_THRESHOLD)
            {
                toowide_cnt++;
                continue;
            }
            value = PointerGetDatum(PG_DETOAST_DATUM(value));
        }
        else if (is_varwidth)
        {
            /* must be cstring */
            total_width += strlen(DatumGetCString(value)) + 1;
        }
 
        /*
         * See if the value matches anything we're already tracking.
         */
        match = false;
        firstcount1 = track_cnt;
        for (j = 0; j < track_cnt; j++)
        {
            if (DatumGetBool(FunctionCall2Coll(&f_cmpeq,
                                               stats->attrcollid,
                                               value, track[j].value)))
            {
                match = true;
                break;
            }
            if (j < firstcount1 && track[j].count == 1)
                firstcount1 = j;
        }
 
        if (match)
        {
            /* Found a match */
            track[j].count++;
            /* This value may now need to "bubble up" in the track list */
            while (j > 0 && track[j].count > track[j - 1].count)
            {
                swapDatum(track[j].value, track[j - 1].value);
                swapInt(track[j].count, track[j - 1].count);
                j--;
            }
        }
        else
        {
            /* No match.  Insert at head of count-1 list */
            if (track_cnt < track_max)
                track_cnt++;
            for (j = track_cnt - 1; j > firstcount1; j--)
            {
                track[j].value = track[j - 1].value;
                track[j].count = track[j - 1].count;
            }
            if (firstcount1 < track_cnt)
            {
                track[firstcount1].value = value;
                track[firstcount1].count = 1;
            }
        }
    }
 
    /* We can only compute real stats if we found some non-null values. */
    if (nonnull_cnt > 0)
    {
        int         nmultiple,
                    summultiple;
 
        stats->stats_valid = true;
        /* Do the simple null-frac and width stats */
        stats->stanullfrac = (double) null_cnt / (double) samplerows;
        if (is_varwidth)
            stats->stawidth = total_width / (double) nonnull_cnt;
        else
            stats->stawidth = stats->attrtype->typlen;
 
        /* Count the number of values we found multiple times */
        summultiple = 0;
        for (nmultiple = 0; nmultiple < track_cnt; nmultiple++)
        {
            if (track[nmultiple].count == 1)
                break;
            summultiple += track[nmultiple].count;
        }
 
        if (nmultiple == 0)
        {
            /*
             * If we found no repeated non-null values, assume it's a unique
             * column; but be sure to discount for any nulls we found.
             */
            stats->stadistinct = -1.0 * (1.0 - stats->stanullfrac);
        }
        else if (track_cnt < track_max && toowide_cnt == 0 &&
                 nmultiple == track_cnt)
        {
            /*
             * Our track list includes every value in the sample, and every
             * value appeared more than once.  Assume the column has just
             * these values.  (This case is meant to address columns with
             * small, fixed sets of possible values, such as boolean or enum
             * columns.  If there are any values that appear just once in the
             * sample, including too-wide values, we should assume that that's
             * not what we're dealing with.)
             */
            stats->stadistinct = track_cnt;
        }
        else
        {
            /*----------
             * Estimate the number of distinct values using the estimator
             * proposed by Haas and Stokes in IBM Research Report RJ 10025:
             *      n*d / (n - f1 + f1*n/N)
             * where f1 is the number of distinct values that occurred
             * exactly once in our sample of n rows (from a total of N),
             * and d is the total number of distinct values in the sample.
             * This is their Duj1 estimator; the other estimators they
             * recommend are considerably more complex, and are numerically
             * very unstable when n is much smaller than N.
             *
             * In this calculation, we consider only non-nulls.  We used to
             * include rows with null values in the n and N counts, but that
             * leads to inaccurate answers in columns with many nulls, and
             * it's intuitively bogus anyway considering the desired result is
             * the number of distinct non-null values.
             *
             * We assume (not very reliably!) that all the multiply-occurring
             * values are reflected in the final track[] list, and the other
             * nonnull values all appeared but once.  (XXX this usually
             * results in a drastic overestimate of ndistinct.  Can we do
             * any better?)
             *----------
             */
            int         f1 = nonnull_cnt - summultiple;
            int         d = f1 + nmultiple;
            double      n = samplerows - null_cnt;
            double      N = totalrows * (1.0 - stats->stanullfrac);
            double      stadistinct;
 
            /* N == 0 shouldn't happen, but just in case ... */
            if (N > 0)
                stadistinct = (n * d) / ((n - f1) + f1 * n / N);
            else
                stadistinct = 0;
 
            /* Clamp to sane range in case of roundoff error */
            if (stadistinct < d)
                stadistinct = d;
            if (stadistinct > N)
                stadistinct = N;
            /* And round to integer */
            stats->stadistinct = floor(stadistinct + 0.5);
        }
 
        /*
         * If we estimated the number of distinct values at more than 10% of
         * the total row count (a very arbitrary limit), then assume that
         * stadistinct should scale with the row count rather than be a fixed
         * value.
         */
        if (stats->stadistinct > 0.1 * totalrows)
            stats->stadistinct = -(stats->stadistinct / totalrows);
 
        /*
         * Decide how many values are worth storing as most-common values. If
         * we are able to generate a complete MCV list (all the values in the
         * sample will fit, and we think these are all the ones in the table),
         * then do so.  Otherwise, store only those values that are
         * significantly more common than the values not in the list.
         *
         * Note: the first of these cases is meant to address columns with
         * small, fixed sets of possible values, such as boolean or enum
         * columns.  If we can *completely* represent the column population by
         * an MCV list that will fit into the stats target, then we should do
         * so and thus provide the planner with complete information.  But if
         * the MCV list is not complete, it's generally worth being more
         * selective, and not just filling it all the way up to the stats
         * target.
         */
        if (track_cnt < track_max && toowide_cnt == 0 &&
            stats->stadistinct > 0 &&
            track_cnt <= num_mcv)
        {
            /* Track list includes all values seen, and all will fit */
            num_mcv = track_cnt;
        }
        else
        {
            int        *mcv_counts;
 
            /* Incomplete list; decide how many values are worth keeping */
            if (num_mcv > track_cnt)
                num_mcv = track_cnt;
 
            if (num_mcv > 0)
            {
                mcv_counts = (int *) palloc(num_mcv * sizeof(int));
                for (i = 0; i < num_mcv; i++)
                    mcv_counts[i] = track[i].count;
 
                num_mcv = analyze_mcv_list(mcv_counts, num_mcv,
                                           stats->stadistinct,
                                           stats->stanullfrac,
                                           samplerows, totalrows);
            }
        }
 
        /* Generate MCV slot entry */
        if (num_mcv > 0)
        {
            MemoryContext old_context;
            Datum      *mcv_values;
            float4     *mcv_freqs;
 
            /* Must copy the target values into anl_context */
            old_context = MemoryContextSwitchTo(stats->anl_context);
            mcv_values = (Datum *) palloc(num_mcv * sizeof(Datum));
            mcv_freqs = (float4 *) palloc(num_mcv * sizeof(float4));
            for (i = 0; i < num_mcv; i++)
            {
                mcv_values[i] = datumCopy(track[i].value,
                                          stats->attrtype->typbyval,
                                          stats->attrtype->typlen);
                mcv_freqs[i] = (double) track[i].count / (double) samplerows;
            }
            MemoryContextSwitchTo(old_context);
 
            stats->stakind[0] = STATISTIC_KIND_MCV;
            stats->staop[0] = mystats->eqopr;
            stats->stacoll[0] = stats->attrcollid;
            stats->stanumbers[0] = mcv_freqs;
            stats->numnumbers[0] = num_mcv;
            stats->stavalues[0] = mcv_values;
            stats->numvalues[0] = num_mcv;
 
            /*
             * Accept the defaults for stats->statypid and others. They have
             * been set before we were called (see vacuum.h)
             */
        }
    }
    else if (null_cnt > 0)
    {
        /* We found only nulls; assume the column is entirely null */
        stats->stats_valid = true;
        stats->stanullfrac = 1.0;
        if (is_varwidth)
            stats->stawidth = 0;    /* "unknown" */
        else
            stats->stawidth = stats->attrtype->typlen;
        stats->stadistinct = 0.0;   /* "unknown" */
    }
 
    /* We don't need to bother cleaning up any of our temporary palloc's */
}
 
 
/*
 *  compute_scalar_stats() -- compute column statistics
 *
 *  We use this when we can find "=" and "<" operators for the datatype.
 *
 *  We determine the fraction of non-null rows, the average width, the
 *  most common values, the (estimated) number of distinct values, the
 *  distribution histogram, and the correlation of physical to logical order.
 *
 *  The desired stats can be determined fairly easily after sorting the
 *  data values into order.
 */
static void
compute_scalar_stats(VacAttrStatsP stats,
                     AnalyzeAttrFetchFunc fetchfunc,
                     int samplerows,
                     double totalrows)
{
    int         i;
    int         null_cnt = 0;
    int         nonnull_cnt = 0;
    int         toowide_cnt = 0;
    double      total_width = 0;
    bool        is_varlena = (!stats->attrtype->typbyval &&
                              stats->attrtype->typlen == -1);
    bool        is_varwidth = (!stats->attrtype->typbyval &&
                               stats->attrtype->typlen < 0);
    double      corr_xysum;
    SortSupportData ssup;
    ScalarItem *values;
    int         values_cnt = 0;
    int        *tupnoLink;
    ScalarMCVItem *track;
    int         track_cnt = 0;
    int         num_mcv = stats->attr->attstattarget;
    int         num_bins = stats->attr->attstattarget;
    StdAnalyzeData *mystats = (StdAnalyzeData *) stats->extra_data;
 
    values = (ScalarItem *) palloc(samplerows * sizeof(ScalarItem));
    tupnoLink = (int *) palloc(samplerows * sizeof(int));
    track = (ScalarMCVItem *) palloc(num_mcv * sizeof(ScalarMCVItem));
 
    memset(&ssup, 0, sizeof(ssup));
    ssup.ssup_cxt = CurrentMemoryContext;
    ssup.ssup_collation = stats->attrcollid;
    ssup.ssup_nulls_first = false;
 
    /*
     * For now, don't perform abbreviated key conversion, because full values
     * are required for MCV slot generation.  Supporting that optimization
     * would necessitate teaching compare_scalars() to call a tie-breaker.
     */
    ssup.abbreviate = false;
 
    PrepareSortSupportFromOrderingOp(mystats->ltopr, &ssup);
 
    /* Initial scan to find sortable values */
    for (i = 0; i < samplerows; i++)
    {
        Datum       value;
        bool        isnull;
 
        vacuum_delay_point();
 
        value = fetchfunc(stats, i, &isnull);
 
        /* Check for null/nonnull */
        if (isnull)
        {
            null_cnt++;
            continue;
        }
        nonnull_cnt++;
 
        /*
         * If it's a variable-width field, add up widths for average width
         * calculation.  Note that if the value is toasted, we use the toasted
         * width.  We don't bother with this calculation if it's a fixed-width
         * type.
         */
        if (is_varlena)
        {
            total_width += VARSIZE_ANY(DatumGetPointer(value));
 
            /*
             * If the value is toasted, we want to detoast it just once to
             * avoid repeated detoastings and resultant excess memory usage
             * during the comparisons.  Also, check to see if the value is
             * excessively wide, and if so don't detoast at all --- just
             * ignore the value.
             */
            if (toast_raw_datum_size(value) > WIDTH_THRESHOLD)
            {
                toowide_cnt++;
                continue;
            }
            value = PointerGetDatum(PG_DETOAST_DATUM(value));
        }
        else if (is_varwidth)
        {
            /* must be cstring */
            total_width += strlen(DatumGetCString(value)) + 1;
        }
 
        /* Add it to the list to be sorted */
        values[values_cnt].value = value;
        values[values_cnt].tupno = values_cnt;
        tupnoLink[values_cnt] = values_cnt;
        values_cnt++;
    }
 
    /* We can only compute real stats if we found some sortable values. */
    if (values_cnt > 0)
    {
        int         ndistinct,  /* # distinct values in sample */
                    nmultiple,  /* # that appear multiple times */
                    num_hist,
                    dups_cnt;
        int         slot_idx = 0;
        CompareScalarsContext cxt;
 
        /* Sort the collected values */
        cxt.ssup = &ssup;
        cxt.tupnoLink = tupnoLink;
        qsort_interruptible(values, values_cnt, sizeof(ScalarItem),
                            compare_scalars, &cxt);
 
        /*
         * Now scan the values in order, find the most common ones, and also
         * accumulate ordering-correlation statistics.
         *
         * To determine which are most common, we first have to count the
         * number of duplicates of each value.  The duplicates are adjacent in
         * the sorted list, so a brute-force approach is to compare successive
         * datum values until we find two that are not equal. However, that
         * requires N-1 invocations of the datum comparison routine, which are
         * completely redundant with work that was done during the sort.  (The
         * sort algorithm must at some point have compared each pair of items
         * that are adjacent in the sorted order; otherwise it could not know
         * that it's ordered the pair correctly.) We exploit this by having
         * compare_scalars remember the highest tupno index that each
         * ScalarItem has been found equal to.  At the end of the sort, a
         * ScalarItem's tupnoLink will still point to itself if and only if it
         * is the last item of its group of duplicates (since the group will
         * be ordered by tupno).
         */
        corr_xysum = 0;
        ndistinct = 0;
        nmultiple = 0;
        dups_cnt = 0;
        for (i = 0; i < values_cnt; i++)
        {
            int         tupno = values[i].tupno;
 
            corr_xysum += ((double) i) * ((double) tupno);
            dups_cnt++;
            if (tupnoLink[tupno] == tupno)
            {
                /* Reached end of duplicates of this value */
                ndistinct++;
                if (dups_cnt > 1)
                {
                    nmultiple++;
                    if (track_cnt < num_mcv ||
                        dups_cnt > track[track_cnt - 1].count)
                    {
                        /*
                         * Found a new item for the mcv list; find its
                         * position, bubbling down old items if needed. Loop
                         * invariant is that j points at an empty/ replaceable
                         * slot.
                         */
                        int         j;
 
                        if (track_cnt < num_mcv)
                            track_cnt++;
                        for (j = track_cnt - 1; j > 0; j--)
                        {
                            if (dups_cnt <= track[j - 1].count)
                                break;
                            track[j].count = track[j - 1].count;
                            track[j].first = track[j - 1].first;
                        }
                        track[j].count = dups_cnt;
                        track[j].first = i + 1 - dups_cnt;
                    }
                }
                dups_cnt = 0;
            }
        }
 
        stats->stats_valid = true;
        /* Do the simple null-frac and width stats */
        stats->stanullfrac = (double) null_cnt / (double) samplerows;
        if (is_varwidth)
            stats->stawidth = total_width / (double) nonnull_cnt;
        else
            stats->stawidth = stats->attrtype->typlen;
 
        if (nmultiple == 0)
        {
            /*
             * If we found no repeated non-null values, assume it's a unique
             * column; but be sure to discount for any nulls we found.
             */
            stats->stadistinct = -1.0 * (1.0 - stats->stanullfrac);
        }
        else if (toowide_cnt == 0 && nmultiple == ndistinct)
        {
            /*
             * Every value in the sample appeared more than once.  Assume the
             * column has just these values.  (This case is meant to address
             * columns with small, fixed sets of possible values, such as
             * boolean or enum columns.  If there are any values that appear
             * just once in the sample, including too-wide values, we should
             * assume that that's not what we're dealing with.)
             */
            stats->stadistinct = ndistinct;
        }
        else
        {
            /*----------
             * Estimate the number of distinct values using the estimator
             * proposed by Haas and Stokes in IBM Research Report RJ 10025:
             *      n*d / (n - f1 + f1*n/N)
             * where f1 is the number of distinct values that occurred
             * exactly once in our sample of n rows (from a total of N),
             * and d is the total number of distinct values in the sample.
             * This is their Duj1 estimator; the other estimators they
             * recommend are considerably more complex, and are numerically
             * very unstable when n is much smaller than N.
             *
             * In this calculation, we consider only non-nulls.  We used to
             * include rows with null values in the n and N counts, but that
             * leads to inaccurate answers in columns with many nulls, and
             * it's intuitively bogus anyway considering the desired result is
             * the number of distinct non-null values.
             *
             * Overwidth values are assumed to have been distinct.
             *----------
             */
            int         f1 = ndistinct - nmultiple + toowide_cnt;
            int         d = f1 + nmultiple;
            double      n = samplerows - null_cnt;
            double      N = totalrows * (1.0 - stats->stanullfrac);
            double      stadistinct;
 
            /* N == 0 shouldn't happen, but just in case ... */
            if (N > 0)
                stadistinct = (n * d) / ((n - f1) + f1 * n / N);
            else
                stadistinct = 0;
 
            /* Clamp to sane range in case of roundoff error */
            if (stadistinct < d)
                stadistinct = d;
            if (stadistinct > N)
                stadistinct = N;
            /* And round to integer */
            stats->stadistinct = floor(stadistinct + 0.5);
        }
 
        /*
         * If we estimated the number of distinct values at more than 10% of
         * the total row count (a very arbitrary limit), then assume that
         * stadistinct should scale with the row count rather than be a fixed
         * value.
         */
        if (stats->stadistinct > 0.1 * totalrows)
            stats->stadistinct = -(stats->stadistinct / totalrows);
 
        /*
         * Decide how many values are worth storing as most-common values. If
         * we are able to generate a complete MCV list (all the values in the
         * sample will fit, and we think these are all the ones in the table),
         * then do so.  Otherwise, store only those values that are
         * significantly more common than the values not in the list.
         *
         * Note: the first of these cases is meant to address columns with
         * small, fixed sets of possible values, such as boolean or enum
         * columns.  If we can *completely* represent the column population by
         * an MCV list that will fit into the stats target, then we should do
         * so and thus provide the planner with complete information.  But if
         * the MCV list is not complete, it's generally worth being more
         * selective, and not just filling it all the way up to the stats
         * target.
         */
        if (track_cnt == ndistinct && toowide_cnt == 0 &&
            stats->stadistinct > 0 &&
            track_cnt <= num_mcv)
        {
            /* Track list includes all values seen, and all will fit */
            num_mcv = track_cnt;
        }
        else
        {
            int        *mcv_counts;
 
            /* Incomplete list; decide how many values are worth keeping */
            if (num_mcv > track_cnt)
                num_mcv = track_cnt;
 
            if (num_mcv > 0)
            {
                mcv_counts = (int *) palloc(num_mcv * sizeof(int));
                for (i = 0; i < num_mcv; i++)
                    mcv_counts[i] = track[i].count;
 
                num_mcv = analyze_mcv_list(mcv_counts, num_mcv,
                                           stats->stadistinct,
                                           stats->stanullfrac,
                                           samplerows, totalrows);
            }
        }
 
        /* Generate MCV slot entry */
        if (num_mcv > 0)
        {
            MemoryContext old_context;
            Datum      *mcv_values;
            float4     *mcv_freqs;
 
            /* Must copy the target values into anl_context */
            old_context = MemoryContextSwitchTo(stats->anl_context);
            mcv_values = (Datum *) palloc(num_mcv * sizeof(Datum));
            mcv_freqs = (float4 *) palloc(num_mcv * sizeof(float4));
            for (i = 0; i < num_mcv; i++)
            {
                mcv_values[i] = datumCopy(values[track[i].first].value,
                                          stats->attrtype->typbyval,
                                          stats->attrtype->typlen);
                mcv_freqs[i] = (double) track[i].count / (double) samplerows;
            }
            MemoryContextSwitchTo(old_context);
 
            stats->stakind[slot_idx] = STATISTIC_KIND_MCV;
            stats->staop[slot_idx] = mystats->eqopr;
            stats->stacoll[slot_idx] = stats->attrcollid;
            stats->stanumbers[slot_idx] = mcv_freqs;
            stats->numnumbers[slot_idx] = num_mcv;
            stats->stavalues[slot_idx] = mcv_values;
            stats->numvalues[slot_idx] = num_mcv;
 
            /*
             * Accept the defaults for stats->statypid and others. They have
             * been set before we were called (see vacuum.h)
             */
            slot_idx++;
        }
 
        /*
         * Generate a histogram slot entry if there are at least two distinct
         * values not accounted for in the MCV list.  (This ensures the
         * histogram won't collapse to empty or a singleton.)
         */
        num_hist = ndistinct - num_mcv;
        if (num_hist > num_bins)
            num_hist = num_bins + 1;
        if (num_hist >= 2)
        {
            MemoryContext old_context;
            Datum      *hist_values;
            int         nvals;
            int         pos,
                        posfrac,
                        delta,
                        deltafrac;
 
            /* Sort the MCV items into position order to speed next loop */
            qsort_interruptible(track, num_mcv, sizeof(ScalarMCVItem),
                                compare_mcvs, NULL);
 
            /*
             * Collapse out the MCV items from the values[] array.
             *
             * Note we destroy the values[] array here... but we don't need it
             * for anything more.  We do, however, still need values_cnt.
             * nvals will be the number of remaining entries in values[].
             */
            if (num_mcv > 0)
            {
                int         src,
                            dest;
                int         j;
 
                src = dest = 0;
                j = 0;          /* index of next interesting MCV item */
                while (src < values_cnt)
                {
                    int         ncopy;
 
                    if (j < num_mcv)
                    {
                        int         first = track[j].first;
 
                        if (src >= first)
                        {
                            /* advance past this MCV item */
                            src = first + track[j].count;
                            j++;
                            continue;
                        }
                        ncopy = first - src;
                    }
                    else
                        ncopy = values_cnt - src;
                    memmove(&values[dest], &values[src],
                            ncopy * sizeof(ScalarItem));
                    src += ncopy;
                    dest += ncopy;
                }
                nvals = dest;
            }
            else
                nvals = values_cnt;
            Assert(nvals >= num_hist);
 
            /* Must copy the target values into anl_context */
            old_context = MemoryContextSwitchTo(stats->anl_context);
            hist_values = (Datum *) palloc(num_hist * sizeof(Datum));
 
            /*
             * The object of this loop is to copy the first and last values[]
             * entries along with evenly-spaced values in between.  So the
             * i'th value is values[(i * (nvals - 1)) / (num_hist - 1)].  But
             * computing that subscript directly risks integer overflow when
             * the stats target is more than a couple thousand.  Instead we
             * add (nvals - 1) / (num_hist - 1) to pos at each step, tracking
             * the integral and fractional parts of the sum separately.
             */
            delta = (nvals - 1) / (num_hist - 1);
            deltafrac = (nvals - 1) % (num_hist - 1);
            pos = posfrac = 0;
 
            for (i = 0; i < num_hist; i++)
            {
                hist_values[i] = datumCopy(values[pos].value,
                                           stats->attrtype->typbyval,
                                           stats->attrtype->typlen);
                pos += delta;
                posfrac += deltafrac;
                if (posfrac >= (num_hist - 1))
                {
                    /* fractional part exceeds 1, carry to integer part */
                    pos++;
                    posfrac -= (num_hist - 1);
                }
            }
 
            MemoryContextSwitchTo(old_context);
 
            stats->stakind[slot_idx] = STATISTIC_KIND_HISTOGRAM;
            stats->staop[slot_idx] = mystats->ltopr;
            stats->stacoll[slot_idx] = stats->attrcollid;
            stats->stavalues[slot_idx] = hist_values;
            stats->numvalues[slot_idx] = num_hist;
 
            /*
             * Accept the defaults for stats->statypid and others. They have
             * been set before we were called (see vacuum.h)
             */
            slot_idx++;
        }
 
        /* Generate a correlation entry if there are multiple values */
        if (values_cnt > 1)
        {
            MemoryContext old_context;
            float4     *corrs;
            double      corr_xsum,
                        corr_x2sum;
 
            /* Must copy the target values into anl_context */
            old_context = MemoryContextSwitchTo(stats->anl_context);
            corrs = (float4 *) palloc(sizeof(float4));
            MemoryContextSwitchTo(old_context);
 
            /*----------
             * Since we know the x and y value sets are both
             *      0, 1, ..., values_cnt-1
             * we have sum(x) = sum(y) =
             *      (values_cnt-1)*values_cnt / 2
             * and sum(x^2) = sum(y^2) =
             *      (values_cnt-1)*values_cnt*(2*values_cnt-1) / 6.
             *----------
             */
            corr_xsum = ((double) (values_cnt - 1)) *
                ((double) values_cnt) / 2.0;
            corr_x2sum = ((double) (values_cnt - 1)) *
                ((double) values_cnt) * (double) (2 * values_cnt - 1) / 6.0;
 
            /* And the correlation coefficient reduces to */
            corrs[0] = (values_cnt * corr_xysum - corr_xsum * corr_xsum) /
                (values_cnt * corr_x2sum - corr_xsum * corr_xsum);
 
            stats->stakind[slot_idx] = STATISTIC_KIND_CORRELATION;
            stats->staop[slot_idx] = mystats->ltopr;
            stats->stacoll[slot_idx] = stats->attrcollid;
            stats->stanumbers[slot_idx] = corrs;
            stats->numnumbers[slot_idx] = 1;
            slot_idx++;
        }
    }
    else if (nonnull_cnt > 0)
    {
        /* We found some non-null values, but they were all too wide */
        Assert(nonnull_cnt == toowide_cnt);
        stats->stats_valid = true;
        /* Do the simple null-frac and width stats */
        stats->stanullfrac = (double) null_cnt / (double) samplerows;
        if (is_varwidth)
            stats->stawidth = total_width / (double) nonnull_cnt;
        else
            stats->stawidth = stats->attrtype->typlen;
        /* Assume all too-wide values are distinct, so it's a unique column */
        stats->stadistinct = -1.0 * (1.0 - stats->stanullfrac);
    }
    else if (null_cnt > 0)
    {
        /* We found only nulls; assume the column is entirely null */
        stats->stats_valid = true;
        stats->stanullfrac = 1.0;
        if (is_varwidth)
            stats->stawidth = 0;    /* "unknown" */
        else
            stats->stawidth = stats->attrtype->typlen;
        stats->stadistinct = 0.0;   /* "unknown" */
    }
 
    /* We don't need to bother cleaning up any of our temporary palloc's */
}
 
/*
 * Comparator for sorting ScalarItems
 *
 * Aside from sorting the items, we update the tupnoLink[] array
 * whenever two ScalarItems are found to contain equal datums.  The array
 * is indexed by tupno; for each ScalarItem, it contains the highest
 * tupno that that item's datum has been found to be equal to.  This allows
 * us to avoid additional comparisons in compute_scalar_stats().
 */
static int
compare_scalars(const void *a, const void *b, void *arg)
{
    Datum       da = ((const ScalarItem *) a)->value;
    int         ta = ((const ScalarItem *) a)->tupno;
    Datum       db = ((const ScalarItem *) b)->value;
    int         tb = ((const ScalarItem *) b)->tupno;
    CompareScalarsContext *cxt = (CompareScalarsContext *) arg;
    int         compare;
 
    compare = ApplySortComparator(da, false, db, false, cxt->ssup);
    if (compare != 0)
        return compare;
 
    /*
     * The two datums are equal, so update cxt->tupnoLink[].
     */
    if (cxt->tupnoLink[ta] < tb)
        cxt->tupnoLink[ta] = tb;
    if (cxt->tupnoLink[tb] < ta)
        cxt->tupnoLink[tb] = ta;
 
    /*
     * For equal datums, sort by tupno
     */
    return ta - tb;
}
 
/*
 * Comparator for sorting ScalarMCVItems by position
 */
static int
compare_mcvs(const void *a, const void *b, void *arg)
{
    int         da = ((const ScalarMCVItem *) a)->first;
    int         db = ((const ScalarMCVItem *) b)->first;
 
    return da - db;
}
 
/*
 * Analyze the list of common values in the sample and decide how many are
 * worth storing in the table's MCV list.
 *
 * mcv_counts is assumed to be a list of the counts of the most common values
 * seen in the sample, starting with the most common.  The return value is the
 * number that are significantly more common than the values not in the list,
 * and which are therefore deemed worth storing in the table's MCV list.
 */
static int
analyze_mcv_list(int *mcv_counts,
                 int num_mcv,
                 double stadistinct,
                 double stanullfrac,
                 int samplerows,
                 double totalrows)
{
    double      ndistinct_table;
    double      sumcount;
    int         i;
 
    /*
     * If the entire table was sampled, keep the whole list.  This also
     * protects us against division by zero in the code below.
     */
    if (samplerows == totalrows || totalrows <= 1.0)
        return num_mcv;
 
    /* Re-extract the estimated number of distinct nonnull values in table */
    ndistinct_table = stadistinct;
    if (ndistinct_table < 0)
        ndistinct_table = -ndistinct_table * totalrows;
 
    /*
     * Exclude the least common values from the MCV list, if they are not
     * significantly more common than the estimated selectivity they would
     * have if they weren't in the list.  All non-MCV values are assumed to be
     * equally common, after taking into account the frequencies of all the
     * values in the MCV list and the number of nulls (c.f. eqsel()).
     *
     * Here sumcount tracks the total count of all but the last (least common)
     * value in the MCV list, allowing us to determine the effect of excluding
     * that value from the list.
     *
     * Note that we deliberately do this by removing values from the full
     * list, rather than starting with an empty list and adding values,
     * because the latter approach can fail to add any values if all the most
     * common values have around the same frequency and make up the majority
     * of the table, so that the overall average frequency of all values is
     * roughly the same as that of the common values.  This would lead to any
     * uncommon values being significantly overestimated.
     */
    sumcount = 0.0;
    for (i = 0; i < num_mcv - 1; i++)
        sumcount += mcv_counts[i];
 
    while (num_mcv > 0)
    {
        double      selec,
                    otherdistinct,
                    N,
                    n,
                    K,
                    variance,
                    stddev;
 
        /*
         * Estimated selectivity the least common value would have if it
         * wasn't in the MCV list (c.f. eqsel()).
         */
        selec = 1.0 - sumcount / samplerows - stanullfrac;
        if (selec < 0.0)
            selec = 0.0;
        if (selec > 1.0)
            selec = 1.0;
        otherdistinct = ndistinct_table - (num_mcv - 1);
        if (otherdistinct > 1)
            selec /= otherdistinct;
 
        /*
         * If the value is kept in the MCV list, its population frequency is
         * assumed to equal its sample frequency.  We use the lower end of a
         * textbook continuity-corrected Wald-type confidence interval to
         * determine if that is significantly more common than the non-MCV
         * frequency --- specifically we assume the population frequency is
         * highly likely to be within around 2 standard errors of the sample
         * frequency, which equates to an interval of 2 standard deviations
         * either side of the sample count, plus an additional 0.5 for the
         * continuity correction.  Since we are sampling without replacement,
         * this is a hypergeometric distribution.
         *
         * XXX: Empirically, this approach seems to work quite well, but it
         * may be worth considering more advanced techniques for estimating
         * the confidence interval of the hypergeometric distribution.
         */
        N = totalrows;
        n = samplerows;
        K = N * mcv_counts[num_mcv - 1] / n;
        variance = n * K * (N - K) * (N - n) / (N * N * (N - 1));
        stddev = sqrt(variance);
 
        if (mcv_counts[num_mcv - 1] > selec * samplerows + 2 * stddev + 0.5)
        {
            /*
             * The value is significantly more common than the non-MCV
             * selectivity would suggest.  Keep it, and all the other more
             * common values in the list.
             */
            break;
        }
        else
        {
            /* Discard this value and consider the next least common value */
            num_mcv--;
            if (num_mcv == 0)
                break;
            sumcount -= mcv_counts[num_mcv - 1];
        }
    }
    return num_mcv;
}