$Header: -> $PostgreSQL Changes ...

[postgresql] / src / backend / optimizer / path / costsize.c

/*-------------------------------------------------------------------------
 *
 * costsize.c
 *        Routines to compute (and set) relation sizes and path costs
 *
 * Path costs are measured in units of disk accesses: one sequential page
 * fetch has cost 1.  All else is scaled relative to a page fetch, using
 * the scaling parameters
 *
 *      random_page_cost        Cost of a non-sequential page fetch
 *      cpu_tuple_cost          Cost of typical CPU time to process a tuple
 *      cpu_index_tuple_cost  Cost of typical CPU time to process an index tuple
 *      cpu_operator_cost       Cost of CPU time to process a typical WHERE operator
 *
 * We also use a rough estimate "effective_cache_size" of the number of
 * disk pages in Postgres + OS-level disk cache.  (We can't simply use
 * NBuffers for this purpose because that would ignore the effects of
 * the kernel's disk cache.)
 *
 * Obviously, taking constants for these values is an oversimplification,
 * but it's tough enough to get any useful estimates even at this level of
 * detail.      Note that all of these parameters are user-settable, in case
 * the default values are drastically off for a particular platform.
 *
 * We compute two separate costs for each path:
 *              total_cost: total estimated cost to fetch all tuples
 *              startup_cost: cost that is expended before first tuple is fetched
 * In some scenarios, such as when there is a LIMIT or we are implementing
 * an EXISTS(...) sub-select, it is not necessary to fetch all tuples of the
 * path's result.  A caller can estimate the cost of fetching a partial
 * result by interpolating between startup_cost and total_cost.  In detail:
 *              actual_cost = startup_cost +
 *                      (total_cost - startup_cost) * tuples_to_fetch / path->parent->rows;
 * Note that a base relation's rows count (and, by extension, plan_rows for
 * plan nodes below the LIMIT node) are set without regard to any LIMIT, so
 * that this equation works properly.  (Also, these routines guarantee not to
 * set the rows count to zero, so there will be no zero divide.)  The LIMIT is
 * applied as a top-level plan node.
 *
 * For largely historical reasons, most of the routines in this module use
 * the passed result Path only to store their startup_cost and total_cost
 * results into.  All the input data they need is passed as separate
 * parameters, even though much of it could be extracted from the Path.
 * An exception is made for the cost_XXXjoin() routines, which expect all
 * the non-cost fields of the passed XXXPath to be filled in.
 *
 *
 * Portions Copyright (c) 1996-2003, PostgreSQL Global Development Group
 * Portions Copyright (c) 1994, Regents of the University of California
 *
 * IDENTIFICATION
 *        $PostgreSQL: pgsql/src/backend/optimizer/path/costsize.c,v 1.116 2003/11/29 19:51:50 pgsql Exp $
 *
 *-------------------------------------------------------------------------
 */

#include "postgres.h"

#include <math.h>

#include "catalog/pg_statistic.h"
#include "executor/nodeHash.h"
#include "miscadmin.h"
#include "optimizer/clauses.h"
#include "optimizer/cost.h"
#include "optimizer/pathnode.h"
#include "optimizer/plancat.h"
#include "parser/parsetree.h"
#include "utils/selfuncs.h"
#include "utils/lsyscache.h"
#include "utils/syscache.h"


#define LOG2(x)  (log(x) / 0.693147180559945)
#define LOG6(x)  (log(x) / 1.79175946922805)

/*
 * Some Paths return less than the nominal number of rows of their parent
 * relations; join nodes need to do this to get the correct input count:
 */
#define PATH_ROWS(path) \
        (IsA(path, UniquePath) ? \
         ((UniquePath *) (path))->rows : \
         (path)->parent->rows)


double          effective_cache_size = DEFAULT_EFFECTIVE_CACHE_SIZE;
double          random_page_cost = DEFAULT_RANDOM_PAGE_COST;
double          cpu_tuple_cost = DEFAULT_CPU_TUPLE_COST;
double          cpu_index_tuple_cost = DEFAULT_CPU_INDEX_TUPLE_COST;
double          cpu_operator_cost = DEFAULT_CPU_OPERATOR_COST;

Cost            disable_cost = 100000000.0;

bool            enable_seqscan = true;
bool            enable_indexscan = true;
bool            enable_tidscan = true;
bool            enable_sort = true;
bool            enable_hashagg = true;
bool            enable_nestloop = true;
bool            enable_mergejoin = true;
bool            enable_hashjoin = true;


static Selectivity estimate_hash_bucketsize(Query *root, Var *var,
                                                 int nbuckets);
static bool cost_qual_eval_walker(Node *node, QualCost *total);
static Selectivity approx_selectivity(Query *root, List *quals,
                                   JoinType jointype);
static void set_rel_width(Query *root, RelOptInfo *rel);
static double relation_byte_size(double tuples, int width);
static double page_size(double tuples, int width);


/*
 * cost_seqscan
 *        Determines and returns the cost of scanning a relation sequentially.
 */
void
cost_seqscan(Path *path, Query *root,
                         RelOptInfo *baserel)
{
        Cost            startup_cost = 0;
        Cost            run_cost = 0;
        Cost            cpu_per_tuple;

        /* Should only be applied to base relations */
        Assert(baserel->relid > 0);
        Assert(baserel->rtekind == RTE_RELATION);

        if (!enable_seqscan)
                startup_cost += disable_cost;

        /*
         * disk costs
         *
         * The cost of reading a page sequentially is 1.0, by definition. Note
         * that the Unix kernel will typically do some amount of read-ahead
         * optimization, so that this cost is less than the true cost of
         * reading a page from disk.  We ignore that issue here, but must take
         * it into account when estimating the cost of non-sequential
         * accesses!
         */
        run_cost += baserel->pages; /* sequential fetches with cost 1.0 */

        /* CPU costs */
        startup_cost += baserel->baserestrictcost.startup;
        cpu_per_tuple = cpu_tuple_cost + baserel->baserestrictcost.per_tuple;
        run_cost += cpu_per_tuple * baserel->tuples;

        path->startup_cost = startup_cost;
        path->total_cost = startup_cost + run_cost;
}

/*
 * cost_nonsequential_access
 *        Estimate the cost of accessing one page at random from a relation
 *        (or sort temp file) of the given size in pages.
 *
 * The simplistic model that the cost is random_page_cost is what we want
 * to use for large relations; but for small ones that is a serious
 * overestimate because of the effects of caching.      This routine tries to
 * account for that.
 *
 * Unfortunately we don't have any good way of estimating the effective cache
 * size we are working with --- we know that Postgres itself has NBuffers
 * internal buffers, but the size of the kernel's disk cache is uncertain,
 * and how much of it we get to use is even less certain.  We punt the problem
 * for now by assuming we are given an effective_cache_size parameter.
 *
 * Given a guesstimated cache size, we estimate the actual I/O cost per page
 * with the entirely ad-hoc equations:
 *      if relpages >= effective_cache_size:
 *              random_page_cost * (1 - (effective_cache_size/relpages)/2)
 *      if relpages < effective_cache_size:
 *              1 + (random_page_cost/2-1) * (relpages/effective_cache_size) ** 2
 * These give the right asymptotic behavior (=> 1.0 as relpages becomes
 * small, => random_page_cost as it becomes large) and meet in the middle
 * with the estimate that the cache is about 50% effective for a relation
 * of the same size as effective_cache_size.  (XXX this is probably all
 * wrong, but I haven't been able to find any theory about how effective
 * a disk cache should be presumed to be.)
 */
static Cost
cost_nonsequential_access(double relpages)
{
        double          relsize;

        /* don't crash on bad input data */
        if (relpages <= 0.0 || effective_cache_size <= 0.0)
                return random_page_cost;

        relsize = relpages / effective_cache_size;

        if (relsize >= 1.0)
                return random_page_cost * (1.0 - 0.5 / relsize);
        else
                return 1.0 + (random_page_cost * 0.5 - 1.0) * relsize * relsize;
}

/*
 * cost_index
 *        Determines and returns the cost of scanning a relation using an index.
 *
 *        NOTE: an indexscan plan node can actually represent several passes,
 *        but here we consider the cost of just one pass.
 *
 * 'root' is the query root
 * 'baserel' is the base relation the index is for
 * 'index' is the index to be used
 * 'indexQuals' is the list of applicable qual clauses (implicit AND semantics)
 * 'is_injoin' is T if we are considering using the index scan as the inside
 *              of a nestloop join (hence, some of the indexQuals are join clauses)
 *
 * NOTE: 'indexQuals' must contain only clauses usable as index restrictions.
 * Any additional quals evaluated as qpquals may reduce the number of returned
 * tuples, but they won't reduce the number of tuples we have to fetch from
 * the table, so they don't reduce the scan cost.
 */
void
cost_index(Path *path, Query *root,
                   RelOptInfo *baserel,
                   IndexOptInfo *index,
                   List *indexQuals,
                   bool is_injoin)
{
        Cost            startup_cost = 0;
        Cost            run_cost = 0;
        Cost            indexStartupCost;
        Cost            indexTotalCost;
        Selectivity indexSelectivity;
        double          indexCorrelation,
                                csquared;
        Cost            min_IO_cost,
                                max_IO_cost;
        Cost            cpu_per_tuple;
        double          tuples_fetched;
        double          pages_fetched;
        double          T,
                                b;

        /* Should only be applied to base relations */
        Assert(IsA(baserel, RelOptInfo) &&
                   IsA(index, IndexOptInfo));
        Assert(baserel->relid > 0);
        Assert(baserel->rtekind == RTE_RELATION);

        if (!enable_indexscan)
                startup_cost += disable_cost;

        /*
         * Call index-access-method-specific code to estimate the processing
         * cost for scanning the index, as well as the selectivity of the
         * index (ie, the fraction of main-table tuples we will have to
         * retrieve) and its correlation to the main-table tuple order.
         */
        OidFunctionCall8(index->amcostestimate,
                                         PointerGetDatum(root),
                                         PointerGetDatum(baserel),
                                         PointerGetDatum(index),
                                         PointerGetDatum(indexQuals),
                                         PointerGetDatum(&indexStartupCost),
                                         PointerGetDatum(&indexTotalCost),
                                         PointerGetDatum(&indexSelectivity),
                                         PointerGetDatum(&indexCorrelation));

        /* all costs for touching index itself included here */
        startup_cost += indexStartupCost;
        run_cost += indexTotalCost - indexStartupCost;

        /*----------
         * Estimate number of main-table tuples and pages fetched.
         *
         * When the index ordering is uncorrelated with the table ordering,
         * we use an approximation proposed by Mackert and Lohman, "Index Scans
         * Using a Finite LRU Buffer: A Validated I/O Model", ACM Transactions
         * on Database Systems, Vol. 14, No. 3, September 1989, Pages 401-424.
         * The Mackert and Lohman approximation is that the number of pages
         * fetched is
         *      PF =
         *              min(2TNs/(2T+Ns), T)                    when T <= b
         *              2TNs/(2T+Ns)                                    when T > b and Ns <= 2Tb/(2T-b)
         *              b + (Ns - 2Tb/(2T-b))*(T-b)/T   when T > b and Ns > 2Tb/(2T-b)
         * where
         *              T = # pages in table
         *              N = # tuples in table
         *              s = selectivity = fraction of table to be scanned
         *              b = # buffer pages available (we include kernel space here)
         *
         * When the index ordering is exactly correlated with the table ordering
         * (just after a CLUSTER, for example), the number of pages fetched should
         * be just sT.  What's more, these will be sequential fetches, not the
         * random fetches that occur in the uncorrelated case.  So, depending on
         * the extent of correlation, we should estimate the actual I/O cost
         * somewhere between s * T * 1.0 and PF * random_cost.  We currently
         * interpolate linearly between these two endpoints based on the
         * correlation squared (XXX is that appropriate?).
         *
         * In any case the number of tuples fetched is Ns.
         *----------
         */

        tuples_fetched = indexSelectivity * baserel->tuples;
        /* Don't believe estimates less than 1... */
        if (tuples_fetched < 1.0)
                tuples_fetched = 1.0;

        /* This part is the Mackert and Lohman formula */

        T = (baserel->pages > 1) ? (double) baserel->pages : 1.0;
        b = (effective_cache_size > 1) ? effective_cache_size : 1.0;

        if (T <= b)
        {
                pages_fetched =
                        (2.0 * T * tuples_fetched) / (2.0 * T + tuples_fetched);
                if (pages_fetched > T)
                        pages_fetched = T;
        }
        else
        {
                double          lim;

                lim = (2.0 * T * b) / (2.0 * T - b);
                if (tuples_fetched <= lim)
                {
                        pages_fetched =
                                (2.0 * T * tuples_fetched) / (2.0 * T + tuples_fetched);
                }
                else
                {
                        pages_fetched =
                                b + (tuples_fetched - lim) * (T - b) / T;
                }
        }

        /*
         * min_IO_cost corresponds to the perfectly correlated case
         * (csquared=1), max_IO_cost to the perfectly uncorrelated case
         * (csquared=0).  Note that we just charge random_page_cost per page
         * in the uncorrelated case, rather than using
         * cost_nonsequential_access, since we've already accounted for
         * caching effects by using the Mackert model.
         */
        min_IO_cost = ceil(indexSelectivity * T);
        max_IO_cost = pages_fetched * random_page_cost;

        /*
         * Now interpolate based on estimated index order correlation to get
         * total disk I/O cost for main table accesses.
         */
        csquared = indexCorrelation * indexCorrelation;

        run_cost += max_IO_cost + csquared * (min_IO_cost - max_IO_cost);

        /*
         * Estimate CPU costs per tuple.
         *
         * Normally the indexquals will be removed from the list of restriction
         * clauses that we have to evaluate as qpquals, so we should subtract
         * their costs from baserestrictcost.  But if we are doing a join then
         * some of the indexquals are join clauses and shouldn't be
         * subtracted. Rather than work out exactly how much to subtract, we
         * don't subtract anything.
         *
         * XXX For a lossy index, not all the quals will be removed and so we
         * really shouldn't subtract their costs; but detecting that seems
         * more expensive than it's worth.
         */
        startup_cost += baserel->baserestrictcost.startup;
        cpu_per_tuple = cpu_tuple_cost + baserel->baserestrictcost.per_tuple;

        if (!is_injoin)
        {
                QualCost        index_qual_cost;

                cost_qual_eval(&index_qual_cost, indexQuals);
                cpu_per_tuple -= index_qual_cost.per_tuple;
        }

        run_cost += cpu_per_tuple * tuples_fetched;

        path->startup_cost = startup_cost;
        path->total_cost = startup_cost + run_cost;
}

/*
 * cost_tidscan
 *        Determines and returns the cost of scanning a relation using TIDs.
 */
void
cost_tidscan(Path *path, Query *root,
                         RelOptInfo *baserel, List *tideval)
{
        Cost            startup_cost = 0;
        Cost            run_cost = 0;
        Cost            cpu_per_tuple;
        int                     ntuples = length(tideval);

        /* Should only be applied to base relations */
        Assert(baserel->relid > 0);
        Assert(baserel->rtekind == RTE_RELATION);

        if (!enable_tidscan)
                startup_cost += disable_cost;

        /* disk costs --- assume each tuple on a different page */
        run_cost += random_page_cost * ntuples;

        /* CPU costs */
        startup_cost += baserel->baserestrictcost.startup;
        cpu_per_tuple = cpu_tuple_cost + baserel->baserestrictcost.per_tuple;
        run_cost += cpu_per_tuple * ntuples;

        path->startup_cost = startup_cost;
        path->total_cost = startup_cost + run_cost;
}

/*
 * cost_subqueryscan
 *        Determines and returns the cost of scanning a subquery RTE.
 */
void
cost_subqueryscan(Path *path, RelOptInfo *baserel)
{
        Cost            startup_cost;
        Cost            run_cost;
        Cost            cpu_per_tuple;

        /* Should only be applied to base relations that are subqueries */
        Assert(baserel->relid > 0);
        Assert(baserel->rtekind == RTE_SUBQUERY);

        /*
         * Cost of path is cost of evaluating the subplan, plus cost of
         * evaluating any restriction clauses that will be attached to the
         * SubqueryScan node, plus cpu_tuple_cost to account for selection and
         * projection overhead.
         */
        path->startup_cost = baserel->subplan->startup_cost;
        path->total_cost = baserel->subplan->total_cost;

        startup_cost = baserel->baserestrictcost.startup;
        cpu_per_tuple = cpu_tuple_cost + baserel->baserestrictcost.per_tuple;
        run_cost = cpu_per_tuple * baserel->tuples;

        path->startup_cost += startup_cost;
        path->total_cost += startup_cost + run_cost;
}

/*
 * cost_functionscan
 *        Determines and returns the cost of scanning a function RTE.
 */
void
cost_functionscan(Path *path, Query *root, RelOptInfo *baserel)
{
        Cost            startup_cost = 0;
        Cost            run_cost = 0;
        Cost            cpu_per_tuple;

        /* Should only be applied to base relations that are functions */
        Assert(baserel->relid > 0);
        Assert(baserel->rtekind == RTE_FUNCTION);

        /*
         * For now, estimate function's cost at one operator eval per function
         * call.  Someday we should revive the function cost estimate columns
         * in pg_proc...
         */
        cpu_per_tuple = cpu_operator_cost;

        /* Add scanning CPU costs */
        startup_cost += baserel->baserestrictcost.startup;
        cpu_per_tuple += cpu_tuple_cost + baserel->baserestrictcost.per_tuple;
        run_cost += cpu_per_tuple * baserel->tuples;

        path->startup_cost = startup_cost;
        path->total_cost = startup_cost + run_cost;
}

/*
 * cost_sort
 *        Determines and returns the cost of sorting a relation, including
 *        the cost of reading the input data.
 *
 * If the total volume of data to sort is less than SortMem, we will do
 * an in-memory sort, which requires no I/O and about t*log2(t) tuple
 * comparisons for t tuples.
 *
 * If the total volume exceeds SortMem, we switch to a tape-style merge
 * algorithm.  There will still be about t*log2(t) tuple comparisons in
 * total, but we will also need to write and read each tuple once per
 * merge pass.  We expect about ceil(log6(r)) merge passes where r is the
 * number of initial runs formed (log6 because tuplesort.c uses six-tape
 * merging).  Since the average initial run should be about twice SortMem,
 * we have
 *              disk traffic = 2 * relsize * ceil(log6(p / (2*SortMem)))
 *              cpu = comparison_cost * t * log2(t)
 *
 * The disk traffic is assumed to be half sequential and half random
 * accesses (XXX can't we refine that guess?)
 *
 * We charge two operator evals per tuple comparison, which should be in
 * the right ballpark in most cases.
 *
 * 'pathkeys' is a list of sort keys
 * 'input_cost' is the total cost for reading the input data
 * 'tuples' is the number of tuples in the relation
 * 'width' is the average tuple width in bytes
 *
 * NOTE: some callers currently pass NIL for pathkeys because they
 * can't conveniently supply the sort keys.  Since this routine doesn't
 * currently do anything with pathkeys anyway, that doesn't matter...
 * but if it ever does, it should react gracefully to lack of key data.
 * (Actually, the thing we'd most likely be interested in is just the number
 * of sort keys, which all callers *could* supply.)
 */
void
cost_sort(Path *path, Query *root,
                  List *pathkeys, Cost input_cost, double tuples, int width)
{
        Cost            startup_cost = input_cost;
        Cost            run_cost = 0;
        double          nbytes = relation_byte_size(tuples, width);
        long            sortmembytes = SortMem * 1024L;

        if (!enable_sort)
                startup_cost += disable_cost;

        /*
         * We want to be sure the cost of a sort is never estimated as zero,
         * even if passed-in tuple count is zero.  Besides, mustn't do
         * log(0)...
         */
        if (tuples < 2.0)
                tuples = 2.0;

        /*
         * CPU costs
         *
         * Assume about two operator evals per tuple comparison and N log2 N
         * comparisons
         */
        startup_cost += 2.0 * cpu_operator_cost * tuples * LOG2(tuples);

        /* disk costs */
        if (nbytes > sortmembytes)
        {
                double          npages = ceil(nbytes / BLCKSZ);
                double          nruns = nbytes / (sortmembytes * 2);
                double          log_runs = ceil(LOG6(nruns));
                double          npageaccesses;

                if (log_runs < 1.0)
                        log_runs = 1.0;
                npageaccesses = 2.0 * npages * log_runs;
                /* Assume half are sequential (cost 1), half are not */
                startup_cost += npageaccesses *
                        (1.0 + cost_nonsequential_access(npages)) * 0.5;
        }

        /*
         * Also charge a small amount (arbitrarily set equal to operator cost)
         * per extracted tuple.
         */
        run_cost += cpu_operator_cost * tuples;

        path->startup_cost = startup_cost;
        path->total_cost = startup_cost + run_cost;
}

/*
 * cost_material
 *        Determines and returns the cost of materializing a relation, including
 *        the cost of reading the input data.
 *
 * If the total volume of data to materialize exceeds SortMem, we will need
 * to write it to disk, so the cost is much higher in that case.
 */
void
cost_material(Path *path,
                          Cost input_cost, double tuples, int width)
{
        Cost            startup_cost = input_cost;
        Cost            run_cost = 0;
        double          nbytes = relation_byte_size(tuples, width);
        long            sortmembytes = SortMem * 1024L;

        /* disk costs */
        if (nbytes > sortmembytes)
        {
                double          npages = ceil(nbytes / BLCKSZ);

                /* We'll write during startup and read during retrieval */
                startup_cost += npages;
                run_cost += npages;
        }

        /*
         * Also charge a small amount per extracted tuple.      We use
         * cpu_tuple_cost so that it doesn't appear worthwhile to materialize
         * a bare seqscan.
         */
        run_cost += cpu_tuple_cost * tuples;

        path->startup_cost = startup_cost;
        path->total_cost = startup_cost + run_cost;
}

/*
 * cost_agg
 *              Determines and returns the cost of performing an Agg plan node,
 *              including the cost of its input.
 *
 * Note: when aggstrategy == AGG_SORTED, caller must ensure that input costs
 * are for appropriately-sorted input.
 */
void
cost_agg(Path *path, Query *root,
                 AggStrategy aggstrategy, int numAggs,
                 int numGroupCols, double numGroups,
                 Cost input_startup_cost, Cost input_total_cost,
                 double input_tuples)
{
        Cost            startup_cost;
        Cost            total_cost;

        /*
         * We charge one cpu_operator_cost per aggregate function per input
         * tuple, and another one per output tuple (corresponding to transfn
         * and finalfn calls respectively).  If we are grouping, we charge an
         * additional cpu_operator_cost per grouping column per input tuple
         * for grouping comparisons.
         *
         * We will produce a single output tuple if not grouping, and a tuple per
         * group otherwise.
         *
         * Note: in this cost model, AGG_SORTED and AGG_HASHED have exactly the
         * same total CPU cost, but AGG_SORTED has lower startup cost.  If the
         * input path is already sorted appropriately, AGG_SORTED should be
         * preferred (since it has no risk of memory overflow).  This will
         * happen as long as the computed total costs are indeed exactly equal
         * --- but if there's roundoff error we might do the wrong thing.  So
         * be sure that the computations below form the same intermediate
         * values in the same order.
         */
        if (aggstrategy == AGG_PLAIN)
        {
                startup_cost = input_total_cost;
                startup_cost += cpu_operator_cost * (input_tuples + 1) * numAggs;
                /* we aren't grouping */
                total_cost = startup_cost;
        }
        else if (aggstrategy == AGG_SORTED)
        {
                /* Here we are able to deliver output on-the-fly */
                startup_cost = input_startup_cost;
                total_cost = input_total_cost;
                /* calcs phrased this way to match HASHED case, see note above */
                total_cost += cpu_operator_cost * input_tuples * numGroupCols;
                total_cost += cpu_operator_cost * input_tuples * numAggs;
                total_cost += cpu_operator_cost * numGroups * numAggs;
        }
        else
        {
                /* must be AGG_HASHED */
                startup_cost = input_total_cost;
                startup_cost += cpu_operator_cost * input_tuples * numGroupCols;
                startup_cost += cpu_operator_cost * input_tuples * numAggs;
                total_cost = startup_cost;
                total_cost += cpu_operator_cost * numGroups * numAggs;
        }

        path->startup_cost = startup_cost;
        path->total_cost = total_cost;
}

/*
 * cost_group
 *              Determines and returns the cost of performing a Group plan node,
 *              including the cost of its input.
 *
 * Note: caller must ensure that input costs are for appropriately-sorted
 * input.
 */
void
cost_group(Path *path, Query *root,
                   int numGroupCols, double numGroups,
                   Cost input_startup_cost, Cost input_total_cost,
                   double input_tuples)
{
        Cost            startup_cost;
        Cost            total_cost;

        startup_cost = input_startup_cost;
        total_cost = input_total_cost;

        /*
         * Charge one cpu_operator_cost per comparison per input tuple. We
         * assume all columns get compared at most of the tuples.
         */
        total_cost += cpu_operator_cost * input_tuples * numGroupCols;

        path->startup_cost = startup_cost;
        path->total_cost = total_cost;
}

/*
 * cost_nestloop
 *        Determines and returns the cost of joining two relations using the
 *        nested loop algorithm.
 *
 * 'path' is already filled in except for the cost fields
 */
void
cost_nestloop(NestPath *path, Query *root)
{
        Path       *outer_path = path->outerjoinpath;
        Path       *inner_path = path->innerjoinpath;
        List       *restrictlist = path->joinrestrictinfo;
        Cost            startup_cost = 0;
        Cost            run_cost = 0;
        Cost            cpu_per_tuple;
        QualCost        restrict_qual_cost;
        double          outer_path_rows = PATH_ROWS(outer_path);
        double          inner_path_rows = PATH_ROWS(inner_path);
        double          ntuples;
        Selectivity joininfactor;

        if (!enable_nestloop)
                startup_cost += disable_cost;

        /*
         * If we're doing JOIN_IN then we will stop scanning inner tuples for
         * an outer tuple as soon as we have one match.  Account for the
         * effects of this by scaling down the cost estimates in proportion to
         * the expected output size.  (This assumes that all the quals
         * attached to the join are IN quals, which should be true.)
         *
         * Note: it's probably bogus to use the normal selectivity calculation
         * here when either the outer or inner path is a UniquePath.
         */
        if (path->jointype == JOIN_IN)
        {
                Selectivity qual_selec = approx_selectivity(root, restrictlist,
                                                                                                        path->jointype);
                double          qptuples;

                qptuples = ceil(qual_selec * outer_path_rows * inner_path_rows);
                if (qptuples > path->path.parent->rows)
                        joininfactor = path->path.parent->rows / qptuples;
                else
                        joininfactor = 1.0;
        }
        else
                joininfactor = 1.0;

        /* cost of source data */

        /*
         * NOTE: clearly, we must pay both outer and inner paths' startup_cost
         * before we can start returning tuples, so the join's startup cost is
         * their sum.  What's not so clear is whether the inner path's
         * startup_cost must be paid again on each rescan of the inner path.
         * This is not true if the inner path is materialized or is a
         * hashjoin, but probably is true otherwise.
         */
        startup_cost += outer_path->startup_cost + inner_path->startup_cost;
        run_cost += outer_path->total_cost - outer_path->startup_cost;
        if (IsA(inner_path, MaterialPath) ||
                IsA(inner_path, HashPath))
        {
                /* charge only run cost for each iteration of inner path */
        }
        else
        {
                /*
                 * charge startup cost for each iteration of inner path, except we
                 * already charged the first startup_cost in our own startup
                 */
                run_cost += (outer_path_rows - 1) * inner_path->startup_cost;
        }
        run_cost += outer_path_rows *
                (inner_path->total_cost - inner_path->startup_cost) * joininfactor;

        /*
         * Compute number of tuples processed (not number emitted!). If inner
         * path is an indexscan, be sure to use its estimated output row
         * count, which may be lower than the restriction-clause-only row
         * count of its parent.  (We don't include this case in the PATH_ROWS
         * macro because it applies *only* to a nestloop's inner relation.)
         * Note: it is correct to use the unadjusted inner_path_rows in the
         * above calculation for joininfactor, since otherwise we'd be
         * double-counting the selectivity of the join clause being used for
         * the index.
         */
        if (IsA(inner_path, IndexPath))
                inner_path_rows = ((IndexPath *) inner_path)->rows;

        ntuples = inner_path_rows * outer_path_rows;

        /* CPU costs */
        cost_qual_eval(&restrict_qual_cost, restrictlist);
        startup_cost += restrict_qual_cost.startup;
        cpu_per_tuple = cpu_tuple_cost + restrict_qual_cost.per_tuple;
        run_cost += cpu_per_tuple * ntuples;

        path->path.startup_cost = startup_cost;
        path->path.total_cost = startup_cost + run_cost;
}

/*
 * cost_mergejoin
 *        Determines and returns the cost of joining two relations using the
 *        merge join algorithm.
 *
 * 'path' is already filled in except for the cost fields
 *
 * Notes: path's mergeclauses should be a subset of the joinrestrictinfo list;
 * outersortkeys and innersortkeys are lists of the keys to be used
 * to sort the outer and inner relations, or NIL if no explicit
 * sort is needed because the source path is already ordered.
 */
void
cost_mergejoin(MergePath *path, Query *root)
{
        Path       *outer_path = path->jpath.outerjoinpath;
        Path       *inner_path = path->jpath.innerjoinpath;
        List       *restrictlist = path->jpath.joinrestrictinfo;
        List       *mergeclauses = path->path_mergeclauses;
        List       *outersortkeys = path->outersortkeys;
        List       *innersortkeys = path->innersortkeys;
        Cost            startup_cost = 0;
        Cost            run_cost = 0;
        Cost            cpu_per_tuple;
        Selectivity merge_selec;
        Selectivity qp_selec;
        QualCost        merge_qual_cost;
        QualCost        qp_qual_cost;
        RestrictInfo *firstclause;
        List       *qpquals;
        double          outer_path_rows = PATH_ROWS(outer_path);
        double          inner_path_rows = PATH_ROWS(inner_path);
        double          outer_rows,
                                inner_rows;
        double          mergejointuples,
                                rescannedtuples;
        double          qptuples;
        double          rescanratio;
        Selectivity outerscansel,
                                innerscansel;
        Selectivity joininfactor;
        Path            sort_path;              /* dummy for result of cost_sort */

        if (!enable_mergejoin)
                startup_cost += disable_cost;

        /*
         * Compute cost and selectivity of the mergequals and qpquals (other
         * restriction clauses) separately.  We use approx_selectivity here
         * for speed --- in most cases, any errors won't affect the result
         * much.
         *
         * Note: it's probably bogus to use the normal selectivity calculation
         * here when either the outer or inner path is a UniquePath.
         */
        merge_selec = approx_selectivity(root, mergeclauses,
                                                                         path->jpath.jointype);
        cost_qual_eval(&merge_qual_cost, mergeclauses);
        qpquals = set_ptrDifference(restrictlist, mergeclauses);
        qp_selec = approx_selectivity(root, qpquals,
                                                                  path->jpath.jointype);
        cost_qual_eval(&qp_qual_cost, qpquals);
        freeList(qpquals);

        /* approx # tuples passing the merge quals */
        mergejointuples = ceil(merge_selec * outer_path_rows * inner_path_rows);
        /* approx # tuples passing qpquals as well */
        qptuples = ceil(mergejointuples * qp_selec);

        /*
         * When there are equal merge keys in the outer relation, the
         * mergejoin must rescan any matching tuples in the inner relation.
         * This means re-fetching inner tuples.  Our cost model for this is
         * that a re-fetch costs the same as an original fetch, which is
         * probably an overestimate; but on the other hand we ignore the
         * bookkeeping costs of mark/restore. Not clear if it's worth
         * developing a more refined model.
         *
         * The number of re-fetches can be estimated approximately as size of
         * merge join output minus size of inner relation.      Assume that the
         * distinct key values are 1, 2, ..., and denote the number of values
         * of each key in the outer relation as m1, m2, ...; in the inner
         * relation, n1, n2, ...  Then we have
         *
         * size of join = m1 * n1 + m2 * n2 + ...
         *
         * number of rescanned tuples = (m1 - 1) * n1 + (m2 - 1) * n2 + ... = m1 *
         * n1 + m2 * n2 + ... - (n1 + n2 + ...) = size of join - size of inner
         * relation
         *
         * This equation works correctly for outer tuples having no inner match
         * (nk = 0), but not for inner tuples having no outer match (mk = 0);
         * we are effectively subtracting those from the number of rescanned
         * tuples, when we should not.  Can we do better without expensive
         * selectivity computations?
         */
        if (IsA(outer_path, UniquePath))
                rescannedtuples = 0;
        else
        {
                rescannedtuples = mergejointuples - inner_path_rows;
                /* Must clamp because of possible underestimate */
                if (rescannedtuples < 0)
                        rescannedtuples = 0;
        }
        /* We'll inflate inner run cost this much to account for rescanning */
        rescanratio = 1.0 + (rescannedtuples / inner_path_rows);

        /*
         * A merge join will stop as soon as it exhausts either input stream.
         * Estimate fraction of the left and right inputs that will actually
         * need to be scanned.  We use only the first (most significant) merge
         * clause for this purpose.
         *
         * Since this calculation is somewhat expensive, and will be the same for
         * all mergejoin paths associated with the merge clause, we cache the
         * results in the RestrictInfo node.
         */
        firstclause = (RestrictInfo *) lfirst(mergeclauses);
        if (firstclause->left_mergescansel < 0)         /* not computed yet? */
                mergejoinscansel(root, (Node *) firstclause->clause,
                                                 &firstclause->left_mergescansel,
                                                 &firstclause->right_mergescansel);

        if (bms_is_subset(firstclause->left_relids, outer_path->parent->relids))
        {
                /* left side of clause is outer */
                outerscansel = firstclause->left_mergescansel;
                innerscansel = firstclause->right_mergescansel;
        }
        else
        {
                /* left side of clause is inner */
                outerscansel = firstclause->right_mergescansel;
                innerscansel = firstclause->left_mergescansel;
        }

        /* convert selectivity to row count; must scan at least one row */

        outer_rows = ceil(outer_path_rows * outerscansel);
        if (outer_rows < 1)
                outer_rows = 1;
        inner_rows = ceil(inner_path_rows * innerscansel);
        if (inner_rows < 1)
                inner_rows = 1;

        /*
         * Readjust scan selectivities to account for above rounding.  This is
         * normally an insignificant effect, but when there are only a few
         * rows in the inputs, failing to do this makes for a large percentage
         * error.
         */
        outerscansel = outer_rows / outer_path_rows;
        innerscansel = inner_rows / inner_path_rows;

        /* cost of source data */

        if (outersortkeys)                      /* do we need to sort outer? */
        {
                cost_sort(&sort_path,
                                  root,
                                  outersortkeys,
                                  outer_path->total_cost,
                                  outer_path_rows,
                                  outer_path->parent->width);
                startup_cost += sort_path.startup_cost;
                run_cost += (sort_path.total_cost - sort_path.startup_cost)
                        * outerscansel;
        }
        else
        {
                startup_cost += outer_path->startup_cost;
                run_cost += (outer_path->total_cost - outer_path->startup_cost)
                        * outerscansel;
        }

        if (innersortkeys)                      /* do we need to sort inner? */
        {
                cost_sort(&sort_path,
                                  root,
                                  innersortkeys,
                                  inner_path->total_cost,
                                  inner_path_rows,
                                  inner_path->parent->width);
                startup_cost += sort_path.startup_cost;
                run_cost += (sort_path.total_cost - sort_path.startup_cost)
                        * innerscansel * rescanratio;
        }
        else
        {
                startup_cost += inner_path->startup_cost;
                run_cost += (inner_path->total_cost - inner_path->startup_cost)
                        * innerscansel * rescanratio;
        }

        /* CPU costs */

        /*
         * If we're doing JOIN_IN then we will stop outputting inner tuples
         * for an outer tuple as soon as we have one match.  Account for the
         * effects of this by scaling down the cost estimates in proportion to
         * the expected output size.  (This assumes that all the quals
         * attached to the join are IN quals, which should be true.)
         */
        if (path->jpath.jointype == JOIN_IN &&
                qptuples > path->jpath.path.parent->rows)
                joininfactor = path->jpath.path.parent->rows / qptuples;
        else
                joininfactor = 1.0;

        /*
         * The number of tuple comparisons needed is approximately number of
         * outer rows plus number of inner rows plus number of rescanned
         * tuples (can we refine this?).  At each one, we need to evaluate the
         * mergejoin quals.  NOTE: JOIN_IN mode does not save any work here,
         * so do NOT include joininfactor.
         */
        startup_cost += merge_qual_cost.startup;
        run_cost += merge_qual_cost.per_tuple *
                (outer_rows + inner_rows * rescanratio);

        /*
         * For each tuple that gets through the mergejoin proper, we charge
         * cpu_tuple_cost plus the cost of evaluating additional restriction
         * clauses that are to be applied at the join.  (This is pessimistic
         * since not all of the quals may get evaluated at each tuple.)  This
         * work is skipped in JOIN_IN mode, so apply the factor.
         */
        startup_cost += qp_qual_cost.startup;
        cpu_per_tuple = cpu_tuple_cost + qp_qual_cost.per_tuple;
        run_cost += cpu_per_tuple * mergejointuples * joininfactor;

        path->jpath.path.startup_cost = startup_cost;
        path->jpath.path.total_cost = startup_cost + run_cost;
}

/*
 * cost_hashjoin
 *        Determines and returns the cost of joining two relations using the
 *        hash join algorithm.
 *
 * 'path' is already filled in except for the cost fields
 *
 * Note: path's hashclauses should be a subset of the joinrestrictinfo list
 */
void
cost_hashjoin(HashPath *path, Query *root)
{
        Path       *outer_path = path->jpath.outerjoinpath;
        Path       *inner_path = path->jpath.innerjoinpath;
        List       *restrictlist = path->jpath.joinrestrictinfo;
        List       *hashclauses = path->path_hashclauses;
        Cost            startup_cost = 0;
        Cost            run_cost = 0;
        Cost            cpu_per_tuple;
        Selectivity hash_selec;
        Selectivity qp_selec;
        QualCost        hash_qual_cost;
        QualCost        qp_qual_cost;
        double          hashjointuples;
        double          qptuples;
        double          outer_path_rows = PATH_ROWS(outer_path);
        double          inner_path_rows = PATH_ROWS(inner_path);
        double          outerbytes = relation_byte_size(outer_path_rows,
                                                                                          outer_path->parent->width);
        double          innerbytes = relation_byte_size(inner_path_rows,
                                                                                          inner_path->parent->width);
        int                     num_hashclauses = length(hashclauses);
        int                     virtualbuckets;
        int                     physicalbuckets;
        int                     numbatches;
        Selectivity innerbucketsize;
        Selectivity joininfactor;
        List       *hcl;
        List       *qpquals;

        if (!enable_hashjoin)
                startup_cost += disable_cost;

        /*
         * Compute cost and selectivity of the hashquals and qpquals (other
         * restriction clauses) separately.  We use approx_selectivity here
         * for speed --- in most cases, any errors won't affect the result
         * much.
         *
         * Note: it's probably bogus to use the normal selectivity calculation
         * here when either the outer or inner path is a UniquePath.
         */
        hash_selec = approx_selectivity(root, hashclauses,
                                                                        path->jpath.jointype);
        cost_qual_eval(&hash_qual_cost, hashclauses);
        qpquals = set_ptrDifference(restrictlist, hashclauses);
        qp_selec = approx_selectivity(root, qpquals,
                                                                  path->jpath.jointype);
        cost_qual_eval(&qp_qual_cost, qpquals);
        freeList(qpquals);

        /* approx # tuples passing the hash quals */
        hashjointuples = ceil(hash_selec * outer_path_rows * inner_path_rows);
        /* approx # tuples passing qpquals as well */
        qptuples = ceil(hashjointuples * qp_selec);

        /* cost of source data */
        startup_cost += outer_path->startup_cost;
        run_cost += outer_path->total_cost - outer_path->startup_cost;
        startup_cost += inner_path->total_cost;

        /*
         * Cost of computing hash function: must do it once per input tuple.
         * We charge one cpu_operator_cost for each column's hash function.
         *
         * XXX when a hashclause is more complex than a single operator, we
         * really should charge the extra eval costs of the left or right
         * side, as appropriate, here.  This seems more work than it's worth
         * at the moment.
         */
        startup_cost += cpu_operator_cost * num_hashclauses * inner_path_rows;
        run_cost += cpu_operator_cost * num_hashclauses * outer_path_rows;

        /* Get hash table size that executor would use for inner relation */
        ExecChooseHashTableSize(inner_path_rows,
                                                        inner_path->parent->width,
                                                        &virtualbuckets,
                                                        &physicalbuckets,
                                                        &numbatches);

        /*
         * Determine bucketsize fraction for inner relation.  We use the
         * smallest bucketsize estimated for any individual hashclause; this
         * is undoubtedly conservative.
         *
         * BUT: if inner relation has been unique-ified, we can assume it's good
         * for hashing.  This is important both because it's the right answer,
         * and because we avoid contaminating the cache with a value that's
         * wrong for non-unique-ified paths.
         */
        if (IsA(inner_path, UniquePath))
                innerbucketsize = 1.0 / virtualbuckets;
        else
        {
                innerbucketsize = 1.0;
                foreach(hcl, hashclauses)
                {
                        RestrictInfo *restrictinfo = (RestrictInfo *) lfirst(hcl);
                        Selectivity thisbucketsize;

                        Assert(IsA(restrictinfo, RestrictInfo));

                        /*
                         * First we have to figure out which side of the hashjoin
                         * clause is the inner side.
                         *
                         * Since we tend to visit the same clauses over and over when
                         * planning a large query, we cache the bucketsize estimate in
                         * the RestrictInfo node to avoid repeated lookups of
                         * statistics.
                         */
                        if (bms_is_subset(restrictinfo->right_relids,
                                                          inner_path->parent->relids))
                        {
                                /* righthand side is inner */
                                thisbucketsize = restrictinfo->right_bucketsize;
                                if (thisbucketsize < 0)
                                {
                                        /* not cached yet */
                                        thisbucketsize =
                                                estimate_hash_bucketsize(root,
                                                           (Var *) get_rightop(restrictinfo->clause),
                                                                                                 virtualbuckets);
                                        restrictinfo->right_bucketsize = thisbucketsize;
                                }
                        }
                        else
                        {
                                Assert(bms_is_subset(restrictinfo->left_relids,
                                                                         inner_path->parent->relids));
                                /* lefthand side is inner */
                                thisbucketsize = restrictinfo->left_bucketsize;
                                if (thisbucketsize < 0)
                                {
                                        /* not cached yet */
                                        thisbucketsize =
                                                estimate_hash_bucketsize(root,
                                                                (Var *) get_leftop(restrictinfo->clause),
                                                                                                 virtualbuckets);
                                        restrictinfo->left_bucketsize = thisbucketsize;
                                }
                        }

                        if (innerbucketsize > thisbucketsize)
                                innerbucketsize = thisbucketsize;
                }
        }

        /*
         * if inner relation is too big then we will need to "batch" the join,
         * which implies writing and reading most of the tuples to disk an
         * extra time.  Charge one cost unit per page of I/O (correct since it
         * should be nice and sequential...).  Writing the inner rel counts as
         * startup cost, all the rest as run cost.
         */
        if (numbatches)
        {
                double          outerpages = page_size(outer_path_rows,
                                                                                   outer_path->parent->width);
                double          innerpages = page_size(inner_path_rows,
                                                                                   inner_path->parent->width);

                startup_cost += innerpages;
                run_cost += innerpages + 2 * outerpages;
        }

        /* CPU costs */

        /*
         * If we're doing JOIN_IN then we will stop comparing inner tuples to
         * an outer tuple as soon as we have one match.  Account for the
         * effects of this by scaling down the cost estimates in proportion to
         * the expected output size.  (This assumes that all the quals
         * attached to the join are IN quals, which should be true.)
         */
        if (path->jpath.jointype == JOIN_IN &&
                qptuples > path->jpath.path.parent->rows)
                joininfactor = path->jpath.path.parent->rows / qptuples;
        else
                joininfactor = 1.0;

        /*
         * The number of tuple comparisons needed is the number of outer
         * tuples times the typical number of tuples in a hash bucket, which
         * is the inner relation size times its bucketsize fraction.  At each
         * one, we need to evaluate the hashjoin quals.
         */
        startup_cost += hash_qual_cost.startup;
        run_cost += hash_qual_cost.per_tuple *
                outer_path_rows * ceil(inner_path_rows * innerbucketsize) *
                joininfactor;

        /*
         * For each tuple that gets through the hashjoin proper, we charge
         * cpu_tuple_cost plus the cost of evaluating additional restriction
         * clauses that are to be applied at the join.  (This is pessimistic
         * since not all of the quals may get evaluated at each tuple.)
         */
        startup_cost += qp_qual_cost.startup;
        cpu_per_tuple = cpu_tuple_cost + qp_qual_cost.per_tuple;
        run_cost += cpu_per_tuple * hashjointuples * joininfactor;

        /*
         * Bias against putting larger relation on inside.      We don't want an
         * absolute prohibition, though, since larger relation might have
         * better bucketsize --- and we can't trust the size estimates
         * unreservedly, anyway.  Instead, inflate the run cost by the square
         * root of the size ratio.      (Why square root?  No real good reason,
         * but it seems reasonable...)
         *
         * Note: before 7.4 we implemented this by inflating startup cost; but if
         * there's a disable_cost component in the input paths' startup cost,
         * that unfairly penalizes the hash.  Probably it'd be better to keep
         * track of disable penalty separately from cost.
         */
        if (innerbytes > outerbytes && outerbytes > 0)
                run_cost *= sqrt(innerbytes / outerbytes);

        path->jpath.path.startup_cost = startup_cost;
        path->jpath.path.total_cost = startup_cost + run_cost;
}

/*
 * Estimate hash bucketsize fraction (ie, number of entries in a bucket
 * divided by total tuples in relation) if the specified Var is used
 * as a hash key.
 *
 * XXX This is really pretty bogus since we're effectively assuming that the
 * distribution of hash keys will be the same after applying restriction
 * clauses as it was in the underlying relation.  However, we are not nearly
 * smart enough to figure out how the restrict clauses might change the
 * distribution, so this will have to do for now.
 *
 * We are passed the number of buckets the executor will use for the given
 * input relation.      If the data were perfectly distributed, with the same
 * number of tuples going into each available bucket, then the bucketsize
 * fraction would be 1/nbuckets.  But this happy state of affairs will occur
 * only if (a) there are at least nbuckets distinct data values, and (b)
 * we have a not-too-skewed data distribution.  Otherwise the buckets will
 * be nonuniformly occupied.  If the other relation in the join has a key
 * distribution similar to this one's, then the most-loaded buckets are
 * exactly those that will be probed most often.  Therefore, the "average"
 * bucket size for costing purposes should really be taken as something close
 * to the "worst case" bucket size.  We try to estimate this by adjusting the
 * fraction if there are too few distinct data values, and then scaling up
 * by the ratio of the most common value's frequency to the average frequency.
 *
 * If no statistics are available, use a default estimate of 0.1.  This will
 * discourage use of a hash rather strongly if the inner relation is large,
 * which is what we want.  We do not want to hash unless we know that the
 * inner rel is well-dispersed (or the alternatives seem much worse).
 */
static Selectivity
estimate_hash_bucketsize(Query *root, Var *var, int nbuckets)
{
        Oid                     relid;
        RelOptInfo *rel;
        HeapTuple       tuple;
        Form_pg_statistic stats;
        double          estfract,
                                ndistinct,
                                mcvfreq,
                                avgfreq;
        float4     *numbers;
        int                     nnumbers;

        /*
         * Lookup info about var's relation and attribute; if none available,
         * return default estimate.
         */
        if (var == NULL || !IsA(var, Var))
                return 0.1;

        relid = getrelid(var->varno, root->rtable);
        if (relid == InvalidOid)
                return 0.1;

        rel = find_base_rel(root, var->varno);

        if (rel->tuples <= 0.0 || rel->rows <= 0.0)
                return 0.1;                             /* ensure we can divide below */

        tuple = SearchSysCache(STATRELATT,
                                                   ObjectIdGetDatum(relid),
                                                   Int16GetDatum(var->varattno),
                                                   0, 0);
        if (!HeapTupleIsValid(tuple))
        {
                /*
                 * If the attribute is known unique because of an index,
                 * we can treat it as well-distributed.
                 */
                if (has_unique_index(rel, var->varattno))
                        return 1.0 / (double) nbuckets;

                /*
                 * Perhaps the Var is a system attribute; if so, it will have no
                 * entry in pg_statistic, but we may be able to guess something
                 * about its distribution anyway.
                 */
                switch (var->varattno)
                {
                        case ObjectIdAttributeNumber:
                        case SelfItemPointerAttributeNumber:
                                /* these are unique, so buckets should be well-distributed */
                                return 1.0 / (double) nbuckets;
                        case TableOidAttributeNumber:
                                /* hashing this is a terrible idea... */
                                return 1.0;
                }
                return 0.1;
        }
        stats = (Form_pg_statistic) GETSTRUCT(tuple);

        /*
         * Obtain number of distinct data values in raw relation.
         */
        ndistinct = stats->stadistinct;
        if (ndistinct < 0.0)
                ndistinct = -ndistinct * rel->tuples;

        if (ndistinct <= 0.0)           /* ensure we can divide */
        {
                ReleaseSysCache(tuple);
                return 0.1;
        }

        /* Also compute avg freq of all distinct data values in raw relation */
        avgfreq = (1.0 - stats->stanullfrac) / ndistinct;

        /*
         * Adjust ndistinct to account for restriction clauses.  Observe we
         * are assuming that the data distribution is affected uniformly by
         * the restriction clauses!
         *
         * XXX Possibly better way, but much more expensive: multiply by
         * selectivity of rel's restriction clauses that mention the target
         * Var.
         */
        ndistinct *= rel->rows / rel->tuples;

        /*
         * Initial estimate of bucketsize fraction is 1/nbuckets as long as
         * the number of buckets is less than the expected number of distinct
         * values; otherwise it is 1/ndistinct.
         */
        if (ndistinct > (double) nbuckets)
                estfract = 1.0 / (double) nbuckets;
        else
                estfract = 1.0 / ndistinct;

        /*
         * Look up the frequency of the most common value, if available.
         */
        mcvfreq = 0.0;

        if (get_attstatsslot(tuple, var->vartype, var->vartypmod,
                                                 STATISTIC_KIND_MCV, InvalidOid,
                                                 NULL, NULL, &numbers, &nnumbers))
        {
                /*
                 * The first MCV stat is for the most common value.
                 */
                if (nnumbers > 0)
                        mcvfreq = numbers[0];
                free_attstatsslot(var->vartype, NULL, 0,
                                                  numbers, nnumbers);
        }

        /*
         * Adjust estimated bucketsize upward to account for skewed
         * distribution.
         */
        if (avgfreq > 0.0 && mcvfreq > avgfreq)
                estfract *= mcvfreq / avgfreq;

        /*
         * Clamp bucketsize to sane range (the above adjustment could easily
         * produce an out-of-range result).  We set the lower bound a little
         * above zero, since zero isn't a very sane result.
         */
        if (estfract < 1.0e-6)
                estfract = 1.0e-6;
        else if (estfract > 1.0)
                estfract = 1.0;

        ReleaseSysCache(tuple);

        return (Selectivity) estfract;
}


/*
 * cost_qual_eval
 *              Estimate the CPU costs of evaluating a WHERE clause.
 *              The input can be either an implicitly-ANDed list of boolean
 *              expressions, or a list of RestrictInfo nodes.
 *              The result includes both a one-time (startup) component,
 *              and a per-evaluation component.
 */
void
cost_qual_eval(QualCost *cost, List *quals)
{
        List       *l;

        cost->startup = 0;
        cost->per_tuple = 0;

        /* We don't charge any cost for the implicit ANDing at top level ... */

        foreach(l, quals)
        {
                Node       *qual = (Node *) lfirst(l);

                /*
                 * RestrictInfo nodes contain an eval_cost field reserved for this
                 * routine's use, so that it's not necessary to evaluate the qual
                 * clause's cost more than once.  If the clause's cost hasn't been
                 * computed yet, the field's startup value will contain -1.
                 */
                if (qual && IsA(qual, RestrictInfo))
                {
                        RestrictInfo *restrictinfo = (RestrictInfo *) qual;

                        if (restrictinfo->eval_cost.startup < 0)
                        {
                                restrictinfo->eval_cost.startup = 0;
                                restrictinfo->eval_cost.per_tuple = 0;
                                cost_qual_eval_walker((Node *) restrictinfo->clause,
                                                                          &restrictinfo->eval_cost);
                        }
                        cost->startup += restrictinfo->eval_cost.startup;
                        cost->per_tuple += restrictinfo->eval_cost.per_tuple;
                }
                else
                {
                        /* If it's a bare expression, must always do it the hard way */
                        cost_qual_eval_walker(qual, cost);
                }
        }
}

static bool
cost_qual_eval_walker(Node *node, QualCost *total)
{
        if (node == NULL)
                return false;

        /*
         * Our basic strategy is to charge one cpu_operator_cost for each
         * operator or function node in the given tree.  Vars and Consts are
         * charged zero, and so are boolean operators (AND, OR, NOT).
         * Simplistic, but a lot better than no model at all.
         *
         * Should we try to account for the possibility of short-circuit
         * evaluation of AND/OR?
         */
        if (IsA(node, FuncExpr) ||
                IsA(node, OpExpr) ||
                IsA(node, DistinctExpr) ||
                IsA(node, NullIfExpr))
                total->per_tuple += cpu_operator_cost;
        else if (IsA(node, ScalarArrayOpExpr))
        {
                /* should charge more than 1 op cost, but how many? */
                total->per_tuple += cpu_operator_cost * 10;
        }
        else if (IsA(node, SubLink))
        {
                /* This routine should not be applied to un-planned expressions */
                elog(ERROR, "cannot handle unplanned sub-select");
        }
        else if (IsA(node, SubPlan))
        {
                /*
                 * A subplan node in an expression typically indicates that the
                 * subplan will be executed on each evaluation, so charge
                 * accordingly. (Sub-selects that can be executed as InitPlans
                 * have already been removed from the expression.)
                 *
                 * An exception occurs when we have decided we can implement the
                 * subplan by hashing.
                 *
                 */
                SubPlan    *subplan = (SubPlan *) node;
                Plan       *plan = subplan->plan;

                if (subplan->useHashTable)
                {
                        /*
                         * If we are using a hash table for the subquery outputs, then
                         * the cost of evaluating the query is a one-time cost. We
                         * charge one cpu_operator_cost per tuple for the work of
                         * loading the hashtable, too.
                         */
                        total->startup += plan->total_cost +
                                cpu_operator_cost * plan->plan_rows;

                        /*
                         * The per-tuple costs include the cost of evaluating the
                         * lefthand expressions, plus the cost of probing the
                         * hashtable. Recursion into the exprs list will handle the
                         * lefthand expressions properly, and will count one
                         * cpu_operator_cost for each comparison operator.      That is
                         * probably too low for the probing cost, but it's hard to
                         * make a better estimate, so live with it for now.
                         */
                }
                else
                {
                        /*
                         * Otherwise we will be rescanning the subplan output on each
                         * evaluation.  We need to estimate how much of the output we
                         * will actually need to scan.  NOTE: this logic should agree
                         * with the estimates used by make_subplan() in
                         * plan/subselect.c.
                         */
                        Cost            plan_run_cost = plan->total_cost - plan->startup_cost;

                        if (subplan->subLinkType == EXISTS_SUBLINK)
                        {
                                /* we only need to fetch 1 tuple */
                                total->per_tuple += plan_run_cost / plan->plan_rows;
                        }
                        else if (subplan->subLinkType == ALL_SUBLINK ||
                                         subplan->subLinkType == ANY_SUBLINK)
                        {
                                /* assume we need 50% of the tuples */
                                total->per_tuple += 0.50 * plan_run_cost;
                                /* also charge a cpu_operator_cost per row examined */
                                total->per_tuple += 0.50 * plan->plan_rows * cpu_operator_cost;
                        }
                        else
                        {
                                /* assume we need all tuples */
                                total->per_tuple += plan_run_cost;
                        }

                        /*
                         * Also account for subplan's startup cost. If the subplan is
                         * uncorrelated or undirect correlated, AND its topmost node
                         * is a Sort or Material node, assume that we'll only need to
                         * pay its startup cost once; otherwise assume we pay the
                         * startup cost every time.
                         */
                        if (subplan->parParam == NIL &&
                                (IsA(plan, Sort) ||
                                 IsA(plan, Material)))
                                total->startup += plan->startup_cost;
                        else
                                total->per_tuple += plan->startup_cost;
                }
        }

        return expression_tree_walker(node, cost_qual_eval_walker,
                                                                  (void *) total);
}


/*
 * approx_selectivity
 *              Quick-and-dirty estimation of clause selectivities.
 *              The input can be either an implicitly-ANDed list of boolean
 *              expressions, or a list of RestrictInfo nodes (typically the latter).
 *
 * The "quick" part comes from caching the selectivity estimates so we can
 * avoid recomputing them later.  (Since the same clauses are typically
 * examined over and over in different possible join trees, this makes a
 * big difference.)
 *
 * The "dirty" part comes from the fact that the selectivities of multiple
 * clauses are estimated independently and multiplied together.  Now
 * clauselist_selectivity often can't do any better than that anyhow, but
 * for some situations (such as range constraints) it is smarter.
 *
 * Since we are only using the results to estimate how many potential
 * output tuples are generated and passed through qpqual checking, it
 * seems OK to live with the approximation.
 */
static Selectivity
approx_selectivity(Query *root, List *quals, JoinType jointype)
{
        Selectivity total = 1.0;
        List       *l;

        foreach(l, quals)
        {
                Node       *qual = (Node *) lfirst(l);
                Selectivity selec;

                /*
                 * RestrictInfo nodes contain a this_selec field reserved for this
                 * routine's use, so that it's not necessary to evaluate the qual
                 * clause's selectivity more than once.  If the clause's
                 * selectivity hasn't been computed yet, the field will contain
                 * -1.
                 */
                if (qual && IsA(qual, RestrictInfo))
                {
                        RestrictInfo *restrictinfo = (RestrictInfo *) qual;

                        if (restrictinfo->this_selec < 0)
                                restrictinfo->this_selec =
                                        clause_selectivity(root,
                                                                           (Node *) restrictinfo->clause,
                                                                           0,
                                                                           jointype);
                        selec = restrictinfo->this_selec;
                }
                else
                {
                        /* If it's a bare expression, must always do it the hard way */
                        selec = clause_selectivity(root, qual, 0, jointype);
                }
                total *= selec;
        }
        return total;
}


/*
 * set_baserel_size_estimates
 *              Set the size estimates for the given base relation.
 *
 * The rel's targetlist and restrictinfo list must have been constructed
 * already.
 *
 * We set the following fields of the rel node:
 *      rows: the estimated number of output tuples (after applying
 *                restriction clauses).
 *      width: the estimated average output tuple width in bytes.
 *      baserestrictcost: estimated cost of evaluating baserestrictinfo clauses.
 */
void
set_baserel_size_estimates(Query *root, RelOptInfo *rel)
{
        double          temp;

        /* Should only be applied to base relations */
        Assert(rel->relid > 0);

        temp = rel->tuples *
                restrictlist_selectivity(root,
                                                                 rel->baserestrictinfo,
                                                                 rel->relid,
                                                                 JOIN_INNER);

        /*
         * Force estimate to be at least one row, to make explain output look
         * better and to avoid possible divide-by-zero when interpolating
         * cost.  Make it an integer, too.
         */
        if (temp < 1.0)
                temp = 1.0;
        else
                temp = ceil(temp);

        rel->rows = temp;

        cost_qual_eval(&rel->baserestrictcost, rel->baserestrictinfo);

        set_rel_width(root, rel);
}

/*
 * set_joinrel_size_estimates
 *              Set the size estimates for the given join relation.
 *
 * The rel's targetlist must have been constructed already, and a
 * restriction clause list that matches the given component rels must
 * be provided.
 *
 * Since there is more than one way to make a joinrel for more than two
 * base relations, the results we get here could depend on which component
 * rel pair is provided.  In theory we should get the same answers no matter
 * which pair is provided; in practice, since the selectivity estimation
 * routines don't handle all cases equally well, we might not.  But there's
 * not much to be done about it.  (Would it make sense to repeat the
 * calculations for each pair of input rels that's encountered, and somehow
 * average the results?  Probably way more trouble than it's worth.)
 *
 * It's important that the results for symmetric JoinTypes be symmetric,
 * eg, (rel1, rel2, JOIN_LEFT) should produce the same result as (rel2,
 * rel1, JOIN_RIGHT).  Also, JOIN_IN should produce the same result as
 * JOIN_UNIQUE_INNER, likewise JOIN_REVERSE_IN == JOIN_UNIQUE_OUTER.
 *
 * We set the same relnode fields as set_baserel_size_estimates() does.
 */
void
set_joinrel_size_estimates(Query *root, RelOptInfo *rel,
                                                   RelOptInfo *outer_rel,
                                                   RelOptInfo *inner_rel,
                                                   JoinType jointype,
                                                   List *restrictlist)
{
        Selectivity selec;
        double          temp;
        UniquePath *upath;

        /*
         * Compute joinclause selectivity.      Note that we are only considering
         * clauses that become restriction clauses at this join level; we are
         * not double-counting them because they were not considered in
         * estimating the sizes of the component rels.
         */
        selec = restrictlist_selectivity(root,
                                                                         restrictlist,
                                                                         0,
                                                                         jointype);

        /*
         * Basically, we multiply size of Cartesian product by selectivity.
         *
         * If we are doing an outer join, take that into account: the output must
         * be at least as large as the non-nullable input.      (Is there any
         * chance of being even smarter?)
         *
         * For JOIN_IN and variants, the Cartesian product is figured with
         * respect to a unique-ified input, and then we can clamp to the size
         * of the other input.
         */
        switch (jointype)
        {
                case JOIN_INNER:
                        temp = outer_rel->rows * inner_rel->rows * selec;
                        break;
                case JOIN_LEFT:
                        temp = outer_rel->rows * inner_rel->rows * selec;
                        if (temp < outer_rel->rows)
                                temp = outer_rel->rows;
                        break;
                case JOIN_RIGHT:
                        temp = outer_rel->rows * inner_rel->rows * selec;
                        if (temp < inner_rel->rows)
                                temp = inner_rel->rows;
                        break;
                case JOIN_FULL:
                        temp = outer_rel->rows * inner_rel->rows * selec;
                        if (temp < outer_rel->rows)
                                temp = outer_rel->rows;
                        if (temp < inner_rel->rows)
                                temp = inner_rel->rows;
                        break;
                case JOIN_IN:
                case JOIN_UNIQUE_INNER:
                        upath = create_unique_path(root, inner_rel,
                                                                           inner_rel->cheapest_total_path);
                        temp = outer_rel->rows * upath->rows * selec;
                        if (temp > outer_rel->rows)
                                temp = outer_rel->rows;
                        break;
                case JOIN_REVERSE_IN:
                case JOIN_UNIQUE_OUTER:
                        upath = create_unique_path(root, outer_rel,
                                                                           outer_rel->cheapest_total_path);
                        temp = upath->rows * inner_rel->rows * selec;
                        if (temp > inner_rel->rows)
                                temp = inner_rel->rows;
                        break;
                default:
                        elog(ERROR, "unrecognized join type: %d", (int) jointype);
                        temp = 0;                       /* keep compiler quiet */
                        break;
        }

        /*
         * Force estimate to be at least one row, to make explain output look
         * better and to avoid possible divide-by-zero when interpolating
         * cost.  Make it an integer, too.
         */
        if (temp < 1.0)
                temp = 1.0;
        else
                temp = ceil(temp);

        rel->rows = temp;

        /*
         * We need not compute the output width here, because
         * build_joinrel_tlist already did.
         */
}

/*
 * set_function_size_estimates
 *              Set the size estimates for a base relation that is a function call.
 *
 * The rel's targetlist and restrictinfo list must have been constructed
 * already.
 *
 * We set the following fields of the rel node:
 *      rows: the estimated number of output tuples (after applying
 *                restriction clauses).
 *      width: the estimated average output tuple width in bytes.
 *      baserestrictcost: estimated cost of evaluating baserestrictinfo clauses.
 */
void
set_function_size_estimates(Query *root, RelOptInfo *rel)
{
        double          temp;

        /* Should only be applied to base relations that are functions */
        Assert(rel->relid > 0);
        Assert(rel->rtekind == RTE_FUNCTION);

        /*
         * Estimate number of rows the function itself will return.
         *
         * XXX no idea how to do this yet; but should at least check whether
         * function returns set or not...
         */
        rel->tuples = 1000;

        /* Now estimate number of output rows */
        temp = rel->tuples *
                restrictlist_selectivity(root,
                                                                 rel->baserestrictinfo,
                                                                 rel->relid,
                                                                 JOIN_INNER);

        /*
         * Force estimate to be at least one row, to make explain output look
         * better and to avoid possible divide-by-zero when interpolating
         * cost.  Make it an integer, too.
         */
        if (temp < 1.0)
                temp = 1.0;
        else
                temp = ceil(temp);

        rel->rows = temp;

        cost_qual_eval(&rel->baserestrictcost, rel->baserestrictinfo);

        set_rel_width(root, rel);
}


/*
 * set_rel_width
 *              Set the estimated output width of a base relation.
 *
 * NB: this works best on plain relations because it prefers to look at
 * real Vars.  It will fail to make use of pg_statistic info when applied
 * to a subquery relation, even if the subquery outputs are simple vars
 * that we could have gotten info for.  Is it worth trying to be smarter
 * about subqueries?
 *
 * The per-attribute width estimates are cached for possible re-use while
 * building join relations.
 */
static void
set_rel_width(Query *root, RelOptInfo *rel)
{
        int32           tuple_width = 0;
        List       *tllist;

        foreach(tllist, FastListValue(&rel->reltargetlist))
        {
                Var                *var = (Var *) lfirst(tllist);
                int                     ndx = var->varattno - rel->min_attr;
                Oid                     relid;
                int32           item_width;

                Assert(IsA(var, Var));

                /*
                 * The width probably hasn't been cached yet, but may as well
                 * check
                 */
                if (rel->attr_widths[ndx] > 0)
                {
                        tuple_width += rel->attr_widths[ndx];
                        continue;
                }

                relid = getrelid(var->varno, root->rtable);
                if (relid != InvalidOid)
                {
                        item_width = get_attavgwidth(relid, var->varattno);
                        if (item_width > 0)
                        {
                                rel->attr_widths[ndx] = item_width;
                                tuple_width += item_width;
                                continue;
                        }
                }

                /*
                 * Not a plain relation, or can't find statistics for it. Estimate
                 * using just the type info.
                 */
                item_width = get_typavgwidth(var->vartype, var->vartypmod);
                Assert(item_width > 0);
                rel->attr_widths[ndx] = item_width;
                tuple_width += item_width;
        }
        Assert(tuple_width >= 0);
        rel->width = tuple_width;
}

/*
 * relation_byte_size
 *        Estimate the storage space in bytes for a given number of tuples
 *        of a given width (size in bytes).
 */
static double
relation_byte_size(double tuples, int width)
{
        return tuples * (MAXALIGN(width) + MAXALIGN(sizeof(HeapTupleData)));
}

/*
 * page_size
 *        Returns an estimate of the number of pages covered by a given
 *        number of tuples of a given width (size in bytes).
 */
static double
page_size(double tuples, int width)
{
        return ceil(relation_byte_size(tuples, width) / BLCKSZ);
}