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Sourcecode: postgresql-8.4 version File versions

analyze.c

/*-------------------------------------------------------------------------
 *
 * analyze.c
 *      the Postgres statistics generator
 *
 * Portions Copyright (c) 1996-2009, PostgreSQL Global Development Group
 * Portions Copyright (c) 1994, Regents of the University of California
 *
 *
 * IDENTIFICATION
 *      $PostgreSQL: pgsql/src/backend/commands/analyze.c,v 1.139 2009/06/11 14:48:55 momjian Exp $
 *
 *-------------------------------------------------------------------------
 */
#include "postgres.h"

#include <math.h>

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


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

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


/* Default statistics target (GUC parameter) */
int               default_statistics_target = 100;

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

static MemoryContext anl_context = NULL;

static BufferAccessStrategy vac_strategy;


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

static bool std_typanalyze(VacAttrStats *stats);


/*
 *    analyze_rel() -- analyze one relation
 *
 * If update_reltuples is true, we update reltuples and relpages columns
 * in pg_class.  Caller should pass false if we're part of VACUUM ANALYZE,
 * and the VACUUM didn't skip any pages.  We only have an approximate count,
 * so we don't want to overwrite the accurate values already inserted by the
 * VACUUM in that case.  VACUUM always scans all indexes, however, so the
 * pg_class entries for indexes are never updated if we're part of VACUUM
 * ANALYZE.
 */
void
analyze_rel(Oid relid, VacuumStmt *vacstmt,
                  BufferAccessStrategy bstrategy, bool update_reltuples)
{
      Relation    onerel;
      int               attr_cnt,
                        tcnt,
                        i,
                        ind;
      Relation   *Irel;
      int               nindexes;
      bool        hasindex;
      bool        analyzableindex;
      VacAttrStats **vacattrstats;
      AnlIndexData *indexdata;
      int               targrows,
                        numrows;
      double            totalrows,
                        totaldeadrows;
      HeapTuple  *rows;
      PGRUsage    ru0;
      TimestampTz starttime = 0;
      Oid               save_userid;
      bool        save_secdefcxt;

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

      vac_strategy = bstrategy;

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      /*
       * Quit if no analyzable columns and no pg_class update needed.
       */
      if (attr_cnt <= 0 && !analyzableindex && !update_reltuples)
            goto cleanup;

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

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

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

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

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

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

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

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

            MemoryContextSwitchTo(old_context);
            MemoryContextDelete(col_context);

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

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

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

      /*
       * Update pages/tuples stats in pg_class.
       */
      if (update_reltuples)
      {
            vac_update_relstats(onerel,
                                          RelationGetNumberOfBlocks(onerel),
                                          totalrows, hasindex, InvalidTransactionId);
            /* report results to the stats collector, too */
            pgstat_report_analyze(onerel, totalrows, totaldeadrows);
      }

      /*
       * Same for indexes. Vacuum always scans all indexes, so if we're part of
       * VACUUM ANALYZE, don't overwrite the accurate count already inserted by
       * VACUUM.
       */
      if (!vacstmt->vacuum)
      {
            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, false, InvalidTransactionId);
            }
      }

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

      /* If this isn't part of VACUUM ANALYZE, let index AMs do cleanup */
      if (!vacstmt->vacuum)
      {
            for (ind = 0; ind < nindexes; ind++)
            {
                  IndexBulkDeleteResult *stats;
                  IndexVacuumInfo ivinfo;

                  ivinfo.index = Irel[ind];
                  ivinfo.vacuum_full = false;
                  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);

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      MemoryContextSwitchTo(old_context);
      MemoryContextDelete(ind_context);
}

/*
 * examine_attribute -- pre-analysis of a single column
 *
 * Determine whether the column is analyzable; if so, create and initialize
 * a VacAttrStats struct for it.  If not, return NULL.
 */
static VacAttrStats *
examine_attribute(Relation onerel, int attnum)
{
      Form_pg_attribute attr = onerel->rd_att->attrs[attnum - 1];
      HeapTuple   typtuple;
      VacAttrStats *stats;
      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);
      typtuple = SearchSysCache(TYPEOID,
                                            ObjectIdGetDatum(attr->atttypid),
                                            0, 0, 0);
      if (!HeapTupleIsValid(typtuple))
            elog(ERROR, "cache lookup failed for type %u", attr->atttypid);
      stats->attrtype = (Form_pg_type) palloc(sizeof(FormData_pg_type));
      memcpy(stats->attrtype, GETSTRUCT(typtuple), sizeof(FormData_pg_type));
      ReleaseSysCache(typtuple);
      stats->anl_context = anl_context;
      stats->tupattnum = attnum;

      /*
       * The fields describing the stats->stavalues[n] element types default to
       * the type of the field 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->attr->atttypid;
            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)
      {
            pfree(stats->attrtype);
            pfree(stats->attr);
            pfree(stats);
            return NULL;
      }

      return stats;
}

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

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

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

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

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

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

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

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

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

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

      Assert(targrows > 1);

      totalblocks = RelationGetNumberOfBlocks(onerel);

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

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

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

            vacuum_delay_point();

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

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

                  itemid = PageGetItemId(targpage, targoffset);

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

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

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

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

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

                        case HEAPTUPLE_INSERT_IN_PROGRESS:

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

                        case HEAPTUPLE_DELETE_IN_PROGRESS:

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

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

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

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

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

                              rowstoskip -= 1;
                        }

                        samplerows += 1;
                  }
            }

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

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

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

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

      return numrows;
}

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

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

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

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

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

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

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

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

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


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

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

      sd = heap_open(StatisticRelationId, RowExclusiveLock);

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

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

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

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

                        arry = construct_array(stats->stavalues[k],
                                                         stats->numvalues[k],
                                                         stats->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 = SearchSysCache(STATRELATT,
                                                ObjectIdGetDatum(relid),
                                                Int16GetDatum(stats->attr->attnum),
                                                0, 0);

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

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

            heap_freetuple(stup);
      }

      heap_close(sd, RowExclusiveLock);
}

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

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

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

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


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


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

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

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

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

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

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


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


/*
 * std_typanalyze -- the default type-specific typanalyze function
 */
static bool
std_typanalyze(VacAttrStats *stats)
{
      Form_pg_attribute attr = stats->attr;
      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(attr->atttypid,
                                           false, false, false,
                                           &ltopr, &eqopr, NULL);

      /* If column has no "=" operator, we can't do much of anything */
      if (!OidIsValid(eqopr))
            return false;

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

      /*
       * Determine which standard statistics algorithm to use
       */
      if (OidIsValid(ltopr))
      {
            /* Seems to be a scalar datatype */
            stats->compute_stats = compute_scalar_stats;
            /*--------------------
             * The following choice of minrows is based on the paper
             * "Random sampling for histogram construction: how much is enough?"
             * by Surajit Chaudhuri, Rajeev Motwani and Vivek Narasayya, in
             * Proceedings of ACM SIGMOD International Conference on Management
             * of Data, 1998, Pages 436-447.  Their Corollary 1 to Theorem 5
             * says that for table size n, histogram size k, maximum relative
             * error in bin size f, and error probability gamma, the minimum
             * random sample size is
             *          r = 4 * k * ln(2*n/gamma) / f^2
             * Taking f = 0.5, gamma = 0.01, n = 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
      {
            /* Can't do much but the minimal stuff */
            stats->compute_stats = compute_minimal_stats;
            /* Might as well use the same minrows as above */
            stats->minrows = 300 * attr->attstattarget;
      }

      return true;
}

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

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

      fmgr_info(mystats->eqfunc, &f_cmpeq);

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

            vacuum_delay_point();

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

                  stats->stakind[0] = STATISTIC_KIND_MCV;
                  stats->staop[0] = mystats->eqopr;
                  stats->stanumbers[0] = mcv_freqs;
                  stats->numnumbers[0] = num_mcv;
                  stats->stavalues[0] = mcv_values;
                  stats->numvalues[0] = num_mcv;

                  /*
                   * 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->attr->attbyval &&
                                            stats->attr->attlen == -1);
      bool        is_varwidth = (!stats->attr->attbyval &&
                                             stats->attr->attlen < 0);
      double            corr_xysum;
      Oid               cmpFn;
      int               cmpFlags;
      FmgrInfo    f_cmpfn;
      ScalarItem *values;
      int               values_cnt = 0;
      int            *tupnoLink;
      ScalarMCVItem *track;
      int               track_cnt = 0;
      int               num_mcv = stats->attr->attstattarget;
      int               num_bins = stats->attr->attstattarget;
      StdAnalyzeData *mystats = (StdAnalyzeData *) stats->extra_data;

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

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

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

            vacuum_delay_point();

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

                  /*
                   * 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((void *) track, num_mcv,
                          sizeof(ScalarMCVItem), compare_mcvs);

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

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

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

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

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

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

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

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

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

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

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

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

      return da - db;
}

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