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

/*-------------------------------------------------------------------------
 *
 * costsize.c
 *        Routines to compute (and set) relation sizes and path costs
 *
 * Path costs are measured in arbitrary units established by these basic
 * parameters:
 *
 *      seq_page_cost           Cost of a sequential page fetch
 *      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 execute an operator or function
 *
 * We expect that the kernel will typically do some amount of read-ahead
 * optimization; this in conjunction with seek costs means that seq_page_cost
 * is normally considerably less than random_page_cost.  (However, if the
 * database is fully cached in RAM, it is reasonable to set them equal.)
 *
 * 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) 1996-2006, PostgreSQL Global Development Group
 * Portions Copyright (c) 1994, Regents of the University of California
 *
 * IDENTIFICATION
 *        $PostgreSQL: pgsql/src/backend/optimizer/path/costsize.c,v 1.161 2006/07/11 16:35:31 momjian 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/array.h"
#include "utils/lsyscache.h"
#include "utils/selfuncs.h"
#include "utils/syscache.h"
#include "utils/tuplesort.h"


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

/*
 * 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          seq_page_cost = DEFAULT_SEQ_PAGE_COST;
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;

double          effective_cache_size = DEFAULT_EFFECTIVE_CACHE_SIZE;

Cost            disable_cost = 100000000.0;

bool            enable_seqscan = true;
bool            enable_indexscan = true;
bool            enable_bitmapscan = 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 bool cost_qual_eval_walker(Node *node, QualCost *total);
static Selectivity approx_selectivity(PlannerInfo *root, List *quals,
                                   JoinType jointype);
static Selectivity join_in_selectivity(JoinPath *path, PlannerInfo *root);
static void set_rel_width(PlannerInfo *root, RelOptInfo *rel);
static double relation_byte_size(double tuples, int width);
static double page_size(double tuples, int width);


/*
 * clamp_row_est
 *              Force a row-count estimate to a sane value.
 */
double
clamp_row_est(double nrows)
{
        /*
         * Force estimate to be at least one row, to make explain output look
         * better and to avoid possible divide-by-zero when interpolating costs.
         * Make it an integer, too.
         */
        if (nrows <= 1.0)
                nrows = 1.0;
        else
                nrows = rint(nrows);

        return nrows;
}


/*
 * cost_seqscan
 *        Determines and returns the cost of scanning a relation sequentially.
 */
void
cost_seqscan(Path *path, PlannerInfo *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
         */
        run_cost += seq_page_cost * baserel->pages;

        /* 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_index
 *        Determines and returns the cost of scanning a relation using an index.
 *
 * 'index' is the index to be used
 * 'indexQuals' is the list of applicable qual clauses (implicit AND semantics)
 * 'outer_rel' is the outer relation when we are considering using the index
 *              scan as the inside of a nestloop join (hence, some of the indexQuals
 *              are join clauses, and we should expect repeated scans of the index);
 *              NULL for a plain index scan
 *
 * cost_index() takes an IndexPath not just a Path, because it sets a few
 * additional fields of the IndexPath besides startup_cost and total_cost.
 * These fields are needed if the IndexPath is used in a BitmapIndexScan.
 *
 * 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.
 *
 * NOTE: as of 8.0, indexQuals is a list of RestrictInfo nodes, where formerly
 * it was a list of bare clause expressions.
 */
void
cost_index(IndexPath *path, PlannerInfo *root,
                   IndexOptInfo *index,
                   List *indexQuals,
                   RelOptInfo *outer_rel)
{
        RelOptInfo *baserel = index->rel;
        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;

        /* 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(index),
                                         PointerGetDatum(indexQuals),
                                         PointerGetDatum(outer_rel),
                                         PointerGetDatum(&indexStartupCost),
                                         PointerGetDatum(&indexTotalCost),
                                         PointerGetDatum(&indexSelectivity),
                                         PointerGetDatum(&indexCorrelation));

        /*
         * Save amcostestimate's results for possible use in bitmap scan planning.
         * We don't bother to save indexStartupCost or indexCorrelation, because a
         * bitmap scan doesn't care about either.
         */
        path->indextotalcost = indexTotalCost;
        path->indexselectivity = indexSelectivity;

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

        /* estimate number of main-table tuples fetched */
        tuples_fetched = clamp_row_est(indexSelectivity * baserel->tuples);

        /*----------
         * Estimate number of main-table pages fetched, and compute I/O cost.
         *
         * When the index ordering is uncorrelated with the table ordering,
         * we use an approximation proposed by Mackert and Lohman (see
         * index_pages_fetched() for details) to compute the number of pages
         * fetched, and then charge random_page_cost per page fetched.
         *
         * When the index ordering is exactly correlated with the table ordering
         * (just after a CLUSTER, for example), the number of pages fetched should
         * be exactly selectivity * table_size.  What's more, all but the first
         * will be sequential fetches, not the random fetches that occur in the
         * uncorrelated case.  So if the number of pages is more than 1, we
         * ought to charge
         *              random_page_cost + (pages_fetched - 1) * seq_page_cost
         * For partially-correlated indexes, we ought to charge somewhere between
         * these two estimates.  We currently interpolate linearly between the
         * estimates based on the correlation squared (XXX is that appropriate?).
         *----------
         */
        if (outer_rel != NULL && outer_rel->rows > 1)
        {
                /*
                 * For repeated indexscans, scale up the number of tuples fetched
                 * in the Mackert and Lohman formula by the number of scans, so
                 * that we estimate the number of pages fetched by all the scans.
                 * Then pro-rate the costs for one scan.  In this case we assume
                 * all the fetches are random accesses.  XXX it'd be good to
                 * include correlation in this model, but it's not clear how to do
                 * that without double-counting cache effects.
                 */
                double          num_scans = outer_rel->rows;

                pages_fetched = index_pages_fetched(tuples_fetched * num_scans,
                                                                                        baserel->pages,
                                                                                        index->pages);

                run_cost += (pages_fetched * random_page_cost) / num_scans;
        }
        else
        {
                /*
                 * Normal case: apply the Mackert and Lohman formula, and then
                 * interpolate between that and the correlation-derived result.
                 */
                pages_fetched = index_pages_fetched(tuples_fetched,
                                                                                        baserel->pages,
                                                                                        index->pages);

                /* max_IO_cost is for the perfectly uncorrelated case (csquared=0) */
                max_IO_cost = pages_fetched * random_page_cost;

                /* min_IO_cost is for the perfectly correlated case (csquared=1) */
                pages_fetched = ceil(indexSelectivity * (double) baserel->pages);
                min_IO_cost = random_page_cost;
                if (pages_fetched > 1)
                        min_IO_cost += (pages_fetched - 1) * seq_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.
         */
        startup_cost += baserel->baserestrictcost.startup;
        cpu_per_tuple = cpu_tuple_cost + baserel->baserestrictcost.per_tuple;

        if (outer_rel == NULL)
        {
                QualCost        index_qual_cost;

                cost_qual_eval(&index_qual_cost, indexQuals);
                /* any startup cost still has to be paid ... */
                cpu_per_tuple -= index_qual_cost.per_tuple;
        }

        run_cost += cpu_per_tuple * tuples_fetched;

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

/*
 * index_pages_fetched
 *        Estimate the number of pages actually fetched after accounting for
 *        cache effects.
 *
 * 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)
 *
 * We assume that effective_cache_size is the total number of buffer pages
 * available for both table and index, and pro-rate that space between the
 * table and index.  (Ideally other_pages should include all the other
 * tables and indexes used by the query too; but we don't have a good way
 * to get that number here.)
 *
 * The product Ns is the number of tuples fetched; we pass in that
 * product rather than calculating it here.
 *
 * Caller is expected to have ensured that tuples_fetched is greater than zero
 * and rounded to integer (see clamp_row_est).  The result will likewise be
 * greater than zero and integral.
 */
double
index_pages_fetched(double tuples_fetched, BlockNumber pages,
                                        BlockNumber other_pages)
{
        double          pages_fetched;
        double          T,
                                b;

        /* T is # pages in table, but don't allow it to be zero */
        T = (pages > 1) ? (double) pages : 1.0;

        /* b is pro-rated share of effective_cache_size */
        b = effective_cache_size * T / (T + (double) other_pages);
        /* force it positive and integral */
        if (b <= 1.0)
                b = 1.0;
        else
                b = ceil(b);

        /* This part is the Mackert and Lohman formula */
        if (T <= b)
        {
                pages_fetched =
                        (2.0 * T * tuples_fetched) / (2.0 * T + tuples_fetched);
                if (pages_fetched >= T)
                        pages_fetched = T;
                else
                        pages_fetched = ceil(pages_fetched);
        }
        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;
                }
                pages_fetched = ceil(pages_fetched);
        }
        return pages_fetched;
}

/*
 * cost_bitmap_heap_scan
 *        Determines and returns the cost of scanning a relation using a bitmap
 *        index-then-heap plan.
 *
 * 'baserel' is the relation to be scanned
 * 'bitmapqual' is a tree of IndexPaths, BitmapAndPaths, and BitmapOrPaths
 *
 * Note: we take no explicit notice here of whether this is a join inner path.
 * If it is, the component IndexPaths should have been costed accordingly.
 */
void
cost_bitmap_heap_scan(Path *path, PlannerInfo *root, RelOptInfo *baserel,
                                          Path *bitmapqual)
{
        Cost            startup_cost = 0;
        Cost            run_cost = 0;
        Cost            indexTotalCost;
        Selectivity indexSelectivity;
        Cost            cpu_per_tuple;
        Cost            cost_per_page;
        double          tuples_fetched;
        double          pages_fetched;
        double          T;

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

        if (!enable_bitmapscan)
                startup_cost += disable_cost;

        /*
         * Fetch total cost of obtaining the bitmap, as well as its total
         * selectivity.
         */
        cost_bitmap_tree_node(bitmapqual, &indexTotalCost, &indexSelectivity);

        startup_cost += indexTotalCost;

        /*
         * The number of heap pages that need to be fetched is the same as the
         * Mackert and Lohman formula for the case T <= b (ie, no re-reads
         * needed).
         */
        tuples_fetched = clamp_row_est(indexSelectivity * baserel->tuples);

        T = (baserel->pages > 1) ? (double) baserel->pages : 1.0;
        pages_fetched = (2.0 * T * tuples_fetched) / (2.0 * T + tuples_fetched);
        if (pages_fetched >= T)
                pages_fetched = T;
        else
                pages_fetched = ceil(pages_fetched);

        /*
         * For small numbers of pages we should charge random_page_cost apiece,
         * while if nearly all the table's pages are being read, it's more
         * appropriate to charge seq_page_cost apiece.  The effect is nonlinear,
         * too. For lack of a better idea, interpolate like this to determine the
         * cost per page.
         */
        if (pages_fetched >= 2.0)
                cost_per_page = random_page_cost -
                        (random_page_cost - seq_page_cost) * sqrt(pages_fetched / T);
        else
                cost_per_page = random_page_cost;

        run_cost += pages_fetched * cost_per_page;

        /*
         * Estimate CPU costs per tuple.
         *
         * Often the indexquals don't need to be rechecked at each tuple ... but
         * not always, especially not if there are enough tuples involved that the
         * bitmaps become lossy.  For the moment, just assume they will be
         * rechecked always.
         */
        startup_cost += baserel->baserestrictcost.startup;
        cpu_per_tuple = cpu_tuple_cost + baserel->baserestrictcost.per_tuple;

        run_cost += cpu_per_tuple * tuples_fetched;

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

/*
 * cost_bitmap_tree_node
 *              Extract cost and selectivity from a bitmap tree node (index/and/or)
 */
void
cost_bitmap_tree_node(Path *path, Cost *cost, Selectivity *selec)
{
        if (IsA(path, IndexPath))
        {
                *cost = ((IndexPath *) path)->indextotalcost;
                *selec = ((IndexPath *) path)->indexselectivity;
        }
        else if (IsA(path, BitmapAndPath))
        {
                *cost = path->total_cost;
                *selec = ((BitmapAndPath *) path)->bitmapselectivity;
        }
        else if (IsA(path, BitmapOrPath))
        {
                *cost = path->total_cost;
                *selec = ((BitmapOrPath *) path)->bitmapselectivity;
        }
        else
                elog(ERROR, "unrecognized node type: %d", nodeTag(path));
}

/*
 * cost_bitmap_and_node
 *              Estimate the cost of a BitmapAnd node
 *
 * Note that this considers only the costs of index scanning and bitmap
 * creation, not the eventual heap access.      In that sense the object isn't
 * truly a Path, but it has enough path-like properties (costs in particular)
 * to warrant treating it as one.
 */
void
cost_bitmap_and_node(BitmapAndPath *path, PlannerInfo *root)
{
        Cost            totalCost;
        Selectivity selec;
        ListCell   *l;

        /*
         * We estimate AND selectivity on the assumption that the inputs are
         * independent.  This is probably often wrong, but we don't have the info
         * to do better.
         *
         * The runtime cost of the BitmapAnd itself is estimated at 100x
         * cpu_operator_cost for each tbm_intersect needed.  Probably too small,
         * definitely too simplistic?
         */
        totalCost = 0.0;
        selec = 1.0;
        foreach(l, path->bitmapquals)
        {
                Path       *subpath = (Path *) lfirst(l);
                Cost            subCost;
                Selectivity subselec;

                cost_bitmap_tree_node(subpath, &subCost, &subselec);

                selec *= subselec;

                totalCost += subCost;
                if (l != list_head(path->bitmapquals))
                        totalCost += 100.0 * cpu_operator_cost;
        }
        path->bitmapselectivity = selec;
        path->path.startup_cost = totalCost;
        path->path.total_cost = totalCost;
}

/*
 * cost_bitmap_or_node
 *              Estimate the cost of a BitmapOr node
 *
 * See comments for cost_bitmap_and_node.
 */
void
cost_bitmap_or_node(BitmapOrPath *path, PlannerInfo *root)
{
        Cost            totalCost;
        Selectivity selec;
        ListCell   *l;

        /*
         * We estimate OR selectivity on the assumption that the inputs are
         * non-overlapping, since that's often the case in "x IN (list)" type
         * situations.  Of course, we clamp to 1.0 at the end.
         *
         * The runtime cost of the BitmapOr itself is estimated at 100x
         * cpu_operator_cost for each tbm_union needed.  Probably too small,
         * definitely too simplistic?  We are aware that the tbm_unions are
         * optimized out when the inputs are BitmapIndexScans.
         */
        totalCost = 0.0;
        selec = 0.0;
        foreach(l, path->bitmapquals)
        {
                Path       *subpath = (Path *) lfirst(l);
                Cost            subCost;
                Selectivity subselec;

                cost_bitmap_tree_node(subpath, &subCost, &subselec);

                selec += subselec;

                totalCost += subCost;
                if (l != list_head(path->bitmapquals) &&
                        !IsA(subpath, IndexPath))
                        totalCost += 100.0 * cpu_operator_cost;
        }
        path->bitmapselectivity = Min(selec, 1.0);
        path->path.startup_cost = totalCost;
        path->path.total_cost = totalCost;
}

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

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

        if (!enable_tidscan)
                startup_cost += disable_cost;

        /* Count how many tuples we expect to retrieve */
        ntuples = 0;
        foreach(l, tidquals)
        {
                if (IsA(lfirst(l), ScalarArrayOpExpr))
                {
                        /* Each element of the array yields 1 tuple */
                        ScalarArrayOpExpr *saop = (ScalarArrayOpExpr *) lfirst(l);
                        Node *arraynode = (Node *) lsecond(saop->args);

                        ntuples += estimate_array_length(arraynode);
                }
                else
                {
                        /* It's just CTID = something, count 1 tuple */
                        ntuples++;
                }
        }

        /* 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, PlannerInfo *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 work_mem, 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 work_mem, 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(logM(r)) merge passes where r is the
 * number of initial runs formed and M is the merge order used by tuplesort.c.
 * Since the average initial run should be about twice work_mem, we have
 *              disk traffic = 2 * relsize * ceil(logM(p / (2*work_mem)))
 *              cpu = comparison_cost * t * log2(t)
 *
 * The disk traffic is assumed to be 3/4ths sequential and 1/4th 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, PlannerInfo *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            work_mem_bytes = work_mem * 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 > work_mem_bytes)
        {
                double          npages = ceil(nbytes / BLCKSZ);
                double          nruns = (nbytes / work_mem_bytes) * 0.5;
                double          mergeorder = tuplesort_merge_order(work_mem_bytes);
                double          log_runs;
                double          npageaccesses;

                /* Compute logM(r) as log(r) / log(M) */
                if (nruns > mergeorder)
                        log_runs = ceil(log(nruns) / log(mergeorder));
                else
                        log_runs = 1.0;
                npageaccesses = 2.0 * npages * log_runs;
                /* Assume 3/4ths of accesses are sequential, 1/4th are not */
                startup_cost += npageaccesses *
                        (seq_page_cost * 0.75 + random_page_cost * 0.25);
        }

        /*
         * 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 work_mem, 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            work_mem_bytes = work_mem * 1024L;

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

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

        /*
         * Charge a very small amount per inserted tuple, to reflect bookkeeping
         * costs.  We use cpu_tuple_cost/10 for this.  This is needed to break the
         * tie that would otherwise exist between nestloop with A outer,
         * materialized B inner and nestloop with B outer, materialized A inner.
         * The extra cost ensures we'll prefer materializing the smaller rel.
         */
        startup_cost += cpu_tuple_cost * 0.1 * tuples;

        /*
         * 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, PlannerInfo *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.  We charge cpu_tuple_cost for each output tuple.
         *
         * 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 + cpu_tuple_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;
                total_cost += cpu_tuple_cost * numGroups;
        }
        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;
                total_cost += cpu_tuple_cost * numGroups;
        }

        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, PlannerInfo *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;
}

/*
 * If a nestloop's 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.)  We have to
 * be prepared to recurse through Append nodes in case of an appendrel.
 */
static double
nestloop_inner_path_rows(Path *path)
{
        double          result;

        if (IsA(path, IndexPath))
                result = ((IndexPath *) path)->rows;
        else if (IsA(path, BitmapHeapPath))
                result = ((BitmapHeapPath *) path)->rows;
        else if (IsA(path, AppendPath))
        {
                ListCell   *l;

                result = 0;
                foreach(l, ((AppendPath *) path)->subpaths)
                {
                        result += nestloop_inner_path_rows((Path *) lfirst(l));
                }
        }
        else
                result = PATH_ROWS(path);

        return result;
}

/*
 * 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, PlannerInfo *root)
{
        Path       *outer_path = path->outerjoinpath;
        Path       *inner_path = path->innerjoinpath;
        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 = nestloop_inner_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 JOIN_IN
         * selectivity.  (This assumes that all the quals attached to the join are
         * IN quals, which should be true.)
         */
        joininfactor = join_in_selectivity(path, root);

        /* 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!)
         */
        ntuples = outer_path_rows * inner_path_rows * joininfactor;

        /* CPU costs */
        cost_qual_eval(&restrict_qual_cost, path->joinrestrictinfo);
        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, PlannerInfo *root)
{
        Path       *outer_path = path->jpath.outerjoinpath;
        Path       *inner_path = path->jpath.innerjoinpath;
        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;
        QualCost        merge_qual_cost;
        QualCost        qp_qual_cost;
        RestrictInfo *firstclause;
        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          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);
        cost_qual_eval(&qp_qual_cost, path->jpath.joinrestrictinfo);
        qp_qual_cost.startup -= merge_qual_cost.startup;
        qp_qual_cost.per_tuple -= merge_qual_cost.per_tuple;

        /* approx # tuples passing the merge quals */
        mergejointuples = clamp_row_est(merge_selec * outer_path_rows * inner_path_rows);

        /*
         * 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
         * (unless it's an outer join, in which case the outer side has to be
         * scanned all the way anyway).  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.
         */
        if (mergeclauses && path->jpath.jointype != JOIN_FULL)
        {
                firstclause = (RestrictInfo *) linitial(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;
                }
                if (path->jpath.jointype == JOIN_LEFT)
                        outerscansel = 1.0;
                else if (path->jpath.jointype == JOIN_RIGHT)
                        innerscansel = 1.0;
        }
        else
        {
                /* cope with clauseless or full mergejoin */
                outerscansel = innerscansel = 1.0;
        }

        /* convert selectivity to row count; must scan at least one row */
        outer_rows = clamp_row_est(outer_path_rows * outerscansel);
        inner_rows = clamp_row_est(inner_path_rows * innerscansel);

        /*
         * 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.)
         */
        joininfactor = join_in_selectivity(&path->jpath, root);

        /*
         * 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, PlannerInfo *root)
{
        Path       *outer_path = path->jpath.outerjoinpath;
        Path       *inner_path = path->jpath.innerjoinpath;
        List       *hashclauses = path->path_hashclauses;
        Cost            startup_cost = 0;
        Cost            run_cost = 0;
        Cost            cpu_per_tuple;
        Selectivity hash_selec;
        QualCost        hash_qual_cost;
        QualCost        qp_qual_cost;
        double          hashjointuples;
        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 = list_length(hashclauses);
        int                     numbuckets;
        int                     numbatches;
        double          virtualbuckets;
        Selectivity innerbucketsize;
        Selectivity joininfactor;
        ListCell   *hcl;

        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);
        cost_qual_eval(&qp_qual_cost, path->jpath.joinrestrictinfo);
        qp_qual_cost.startup -= hash_qual_cost.startup;
        qp_qual_cost.per_tuple -= hash_qual_cost.per_tuple;

        /* approx # tuples passing the hash quals */
        hashjointuples = clamp_row_est(hash_selec * outer_path_rows * inner_path_rows);

        /* 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,
                                                        &numbuckets,
                                                        &numbatches);
        virtualbuckets = (double) numbuckets *(double) 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,
                                                                                   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,
                                                                                        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 > 1)
        {
                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.)
         */
        joininfactor = join_in_selectivity(&path->jpath, root);

        /*
         * 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.  (Note: charging the full qual eval cost
         * at each tuple is pessimistic, since we don't evaluate the quals unless
         * the hash values match exactly.)
         */
        startup_cost += hash_qual_cost.startup;
        run_cost += hash_qual_cost.per_tuple *
                outer_path_rows * clamp_row_est(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;
}


/*
 * 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)
{
        ListCell   *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 the RestrictInfo is marked pseudoconstant, it will be tested
                 * only once, so treat its cost as all startup cost.
                 */
                if (qual && IsA(qual, RestrictInfo))
                {
                        RestrictInfo *rinfo = (RestrictInfo *) qual;

                        if (rinfo->eval_cost.startup < 0)
                        {
                                rinfo->eval_cost.startup = 0;
                                rinfo->eval_cost.per_tuple = 0;
                                cost_qual_eval_walker((Node *) rinfo->clause,
                                                                          &rinfo->eval_cost);
                                if (rinfo->pseudoconstant)
                                {
                                        /* count one execution during startup */
                                        rinfo->eval_cost.startup += rinfo->eval_cost.per_tuple;
                                        rinfo->eval_cost.per_tuple = 0;
                                }
                        }
                        cost->startup += rinfo->eval_cost.startup;
                        cost->per_tuple += rinfo->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))
        {
                /*
                 * Estimate that the operator will be applied to about half of the
                 * array elements before the answer is determined.
                 */
                ScalarArrayOpExpr *saop = (ScalarArrayOpExpr *) node;
                Node *arraynode = (Node *) lsecond(saop->args);

                total->per_tuple +=
                        cpu_operator_cost * estimate_array_length(arraynode) * 0.5;
        }
        else if (IsA(node, RowCompareExpr))
        {
                /* Conservatively assume we will check all the columns */
                RowCompareExpr *rcexpr = (RowCompareExpr *) node;

                total->per_tuple += cpu_operator_cost * list_length(rcexpr->opnos);
        }
        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 testexpr 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).
 *
 * This is quick-and-dirty because we bypass clauselist_selectivity, and
 * simply multiply the independent clause selectivities together.  Now
 * clauselist_selectivity often can't do any better than that anyhow, but
 * for some situations (such as range constraints) it is smarter.  However,
 * we can't effectively cache the results of clauselist_selectivity, whereas
 * the individual clause selectivities can be and are cached.
 *
 * 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(PlannerInfo *root, List *quals, JoinType jointype)
{
        Selectivity total = 1.0;
        ListCell   *l;

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

                /* Note that clause_selectivity will be able to cache its result */
                total *= clause_selectivity(root, qual, 0, jointype);
        }
        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(PlannerInfo *root, RelOptInfo *rel)
{
        double          nrows;

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

        nrows = rel->tuples *
                clauselist_selectivity(root,
                                                           rel->baserestrictinfo,
                                                           0,
                                                           JOIN_INNER);

        rel->rows = clamp_row_est(nrows);

        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 only the rows field here.  The width field was already set by
 * build_joinrel_tlist, and baserestrictcost is not used for join rels.
 */
void
set_joinrel_size_estimates(PlannerInfo *root, RelOptInfo *rel,
                                                   RelOptInfo *outer_rel,
                                                   RelOptInfo *inner_rel,
                                                   JoinType jointype,
                                                   List *restrictlist)
{
        Selectivity selec;
        double          nrows;
        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 = clauselist_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?)  (XXX this is not really right, because it
         * assumes all the restriction clauses are join clauses; we should figure
         * pushed-down clauses separately.)
         *
         * 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:
                        nrows = outer_rel->rows * inner_rel->rows * selec;
                        break;
                case JOIN_LEFT:
                        nrows = outer_rel->rows * inner_rel->rows * selec;
                        if (nrows < outer_rel->rows)
                                nrows = outer_rel->rows;
                        break;
                case JOIN_RIGHT:
                        nrows = outer_rel->rows * inner_rel->rows * selec;
                        if (nrows < inner_rel->rows)
                                nrows = inner_rel->rows;
                        break;
                case JOIN_FULL:
                        nrows = outer_rel->rows * inner_rel->rows * selec;
                        if (nrows < outer_rel->rows)
                                nrows = outer_rel->rows;
                        if (nrows < inner_rel->rows)
                                nrows = inner_rel->rows;
                        break;
                case JOIN_IN:
                case JOIN_UNIQUE_INNER:
                        upath = create_unique_path(root, inner_rel,
                                                                           inner_rel->cheapest_total_path);
                        nrows = outer_rel->rows * upath->rows * selec;
                        if (nrows > outer_rel->rows)
                                nrows = outer_rel->rows;
                        break;
                case JOIN_REVERSE_IN:
                case JOIN_UNIQUE_OUTER:
                        upath = create_unique_path(root, outer_rel,
                                                                           outer_rel->cheapest_total_path);
                        nrows = upath->rows * inner_rel->rows * selec;
                        if (nrows > inner_rel->rows)
                                nrows = inner_rel->rows;
                        break;
                default:
                        elog(ERROR, "unrecognized join type: %d", (int) jointype);
                        nrows = 0;                      /* keep compiler quiet */
                        break;
        }

        rel->rows = clamp_row_est(nrows);
}

/*
 * join_in_selectivity
 *        Determines the factor by which a JOIN_IN join's result is expected
 *        to be smaller than an ordinary inner join.
 *
 * 'path' is already filled in except for the cost fields
 */
static Selectivity
join_in_selectivity(JoinPath *path, PlannerInfo *root)
{
        RelOptInfo *innerrel;
        UniquePath *innerunique;
        Selectivity selec;
        double          nrows;

        /* Return 1.0 whenever it's not JOIN_IN */
        if (path->jointype != JOIN_IN)
                return 1.0;

        /*
         * Return 1.0 if the inner side is already known unique.  The case where
         * the inner path is already a UniquePath probably cannot happen in
         * current usage, but check it anyway for completeness.  The interesting
         * case is where we've determined the inner relation itself is unique,
         * which we can check by looking at the rows estimate for its UniquePath.
         */
        if (IsA(path->innerjoinpath, UniquePath))
                return 1.0;
        innerrel = path->innerjoinpath->parent;
        innerunique = create_unique_path(root,
                                                                         innerrel,
                                                                         innerrel->cheapest_total_path);
        if (innerunique->rows >= innerrel->rows)
                return 1.0;

        /*
         * Compute same result set_joinrel_size_estimates would compute for
         * JOIN_INNER.  Note that we use the input rels' absolute size estimates,
         * not PATH_ROWS() which might be less; if we used PATH_ROWS() we'd be
         * double-counting the effects of any join clauses used in input scans.
         */
        selec = clauselist_selectivity(root,
                                                                   path->joinrestrictinfo,
                                                                   0,
                                                                   JOIN_INNER);
        nrows = path->outerjoinpath->parent->rows * innerrel->rows * selec;

        nrows = clamp_row_est(nrows);

        /* See if it's larger than the actual JOIN_IN size estimate */
        if (nrows > path->path.parent->rows)
                return path->path.parent->rows / nrows;
        else
                return 1.0;
}

/*
 * 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 same fields as set_baserel_size_estimates.
 */
void
set_function_size_estimates(PlannerInfo *root, RelOptInfo *rel)
{
        RangeTblEntry *rte;

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

        /*
         * Estimate number of rows the function itself will return.
         *
         * XXX no idea how to do this yet; but we can at least check whether
         * function returns set or not...
         */
        if (expression_returns_set(rte->funcexpr))
                rel->tuples = 1000;
        else
                rel->tuples = 1;

        /* Now estimate number of output rows, etc */
        set_baserel_size_estimates(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(PlannerInfo *root, RelOptInfo *rel)
{
        int32           tuple_width = 0;
        ListCell   *tllist;

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

                /* For now, punt on whole-row child Vars */
                if (!IsA(var, Var))
                {
                        tuple_width += 32;      /* arbitrary */
                        continue;
                }

                ndx = var->varattno - rel->min_attr;

                /*
                 * 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->parse->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(HeapTupleHeaderData)));
}

/*
 * 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);
}