Modify optimizer data structures so that IndexOptInfo lists built for

[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.
 *
 *
 * Portions Copyright (c) 1996-2001, PostgreSQL Global Development Group
 * Portions Copyright (c) 1994, Regents of the University of California
 *
 * IDENTIFICATION
 *        $Header: /cvsroot/pgsql/src/backend/optimizer/path/costsize.c,v 1.74 2001/05/20 20:28:18 tgl 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 "parser/parsetree.h"
#include "utils/lsyscache.h"
#include "utils/syscache.h"


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


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_nestloop = true;
bool            enable_mergejoin = true;
bool            enable_hashjoin = true;


static bool cost_qual_eval_walker(Node *node, Cost *total);
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.
 *
 * Note: for historical reasons, this routine and the others 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.
 */
void
cost_seqscan(Path *path, RelOptInfo *baserel)
{
        Cost            startup_cost = 0;
        Cost            run_cost = 0;
        Cost            cpu_per_tuple;

        /* Should only be applied to base relations */
        Assert(length(baserel->relids) == 1);
        Assert(!baserel->issubquery);

        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 */
        cpu_per_tuple = cpu_tuple_cost + baserel->baserestrictcost;
        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:
 *      for rel_size <= effective_cache_size:
 *              1 + (random_page_cost/2-1) * (rel_size/effective_cache_size) ** 2
 *      for rel_size >= effective_cache_size:
 *              random_page_cost * (1 - (effective_cache_size/rel_size)/2)
 * These give the right asymptotic behavior (=> 1.0 as rel_size 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(length(baserel->relids) == 1);
        Assert(!baserel->issubquery);

        if (!enable_indexscan && !is_injoin)
                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.  For a lossy
         * index, however, we will have to recheck all the quals and so
         * mustn't subtract anything. Also, 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 in that case either.
         */
        cpu_per_tuple = cpu_tuple_cost + baserel->baserestrictcost;

        if (!index->lossy && !is_injoin)
                cpu_per_tuple -= cost_qual_eval(indexQuals);

        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 tid-s.
 */
void
cost_tidscan(Path *path, RelOptInfo *baserel, List *tideval)
{
        Cost            startup_cost = 0;
        Cost            run_cost = 0;
        Cost            cpu_per_tuple;
        int                     ntuples = length(tideval);

        if (!enable_tidscan)
                startup_cost += disable_cost;

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

        /* CPU costs */
        cpu_per_tuple = cpu_tuple_cost + baserel->baserestrictcost;
        run_cost += cpu_per_tuple * ntuples;

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

/*
 * cost_sort
 *        Determines and returns the cost of sorting a relation.
 *
 * The cost of supplying the input data is NOT included; the caller should
 * add that cost to both startup and total costs returned from this routine!
 *
 * 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
 * '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.
 */
void
cost_sort(Path *path, List *pathkeys, double tuples, int width)
{
        Cost            startup_cost = 0;
        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;
        }

        /*
         * Note: should we bother to assign a nonzero run_cost to reflect the
         * overhead of extracting tuples from the sort result?  Probably not
         * worth worrying about.
         */
        path->startup_cost = startup_cost;
        path->total_cost = startup_cost + run_cost;
}


/*
 * cost_nestloop
 *        Determines and returns the cost of joining two relations using the
 *        nested loop algorithm.
 *
 * 'outer_path' is the path for the outer relation
 * 'inner_path' is the path for the inner relation
 * 'restrictlist' are the RestrictInfo nodes to be applied at the join
 */
void
cost_nestloop(Path *path,
                          Path *outer_path,
                          Path *inner_path,
                          List *restrictlist)
{
        Cost            startup_cost = 0;
        Cost            run_cost = 0;
        Cost            cpu_per_tuple;
        double          ntuples;

        if (!enable_nestloop)
                startup_cost += disable_cost;

        /* 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, but probably
         * is true otherwise.  Since we don't yet have clean handling of the
         * decision whether to materialize a path, we can't tell here which
         * will happen.  As a compromise, charge 50% of the inner startup cost
         * for each restart.
         */
        startup_cost += outer_path->startup_cost + inner_path->startup_cost;
        run_cost += outer_path->total_cost - outer_path->startup_cost;
        run_cost += outer_path->parent->rows *
                (inner_path->total_cost - inner_path->startup_cost);
        if (outer_path->parent->rows > 1)
                run_cost += (outer_path->parent->rows - 1) * inner_path->startup_cost;

        /*
         * 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.
         */
        if (IsA(inner_path, IndexPath))
                ntuples = ((IndexPath *) inner_path)->rows;
        else
                ntuples = inner_path->parent->rows;
        ntuples *= outer_path->parent->rows;

        /* CPU costs */
        cpu_per_tuple = cpu_tuple_cost + cost_qual_eval(restrictlist);
        run_cost += cpu_per_tuple * ntuples;

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

/*
 * cost_mergejoin
 *        Determines and returns the cost of joining two relations using the
 *        merge join algorithm.
 *
 * 'outer_path' is the path for the outer relation
 * 'inner_path' is the path for the inner relation
 * 'restrictlist' are the RestrictInfo nodes to be applied at the join
 * '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(Path *path,
                           Path *outer_path,
                           Path *inner_path,
                           List *restrictlist,
                           List *outersortkeys,
                           List *innersortkeys)
{
        Cost            startup_cost = 0;
        Cost            run_cost = 0;
        Cost            cpu_per_tuple;
        double          ntuples;
        Path            sort_path;              /* dummy for result of cost_sort */

        if (!enable_mergejoin)
                startup_cost += disable_cost;

        /* cost of source data */

        /*
         * Note we are assuming that each source tuple is fetched just once,
         * which is not right in the presence of equal keys.  If we had a way
         * of estimating the proportion of equal keys, we could apply a
         * correction factor...
         */
        if (outersortkeys)                      /* do we need to sort outer? */
        {
                startup_cost += outer_path->total_cost;
                cost_sort(&sort_path,
                                  outersortkeys,
                                  outer_path->parent->rows,
                                  outer_path->parent->width);
                startup_cost += sort_path.startup_cost;
                run_cost += sort_path.total_cost - sort_path.startup_cost;
        }
        else
        {
                startup_cost += outer_path->startup_cost;
                run_cost += outer_path->total_cost - outer_path->startup_cost;
        }

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

        /*
         * Estimate the number of tuples to be processed in the mergejoin
         * itself as one per tuple in the two source relations.  This could be
         * a drastic underestimate if there are many equal-keyed tuples in
         * either relation, but we have no good way of estimating that...
         */
        ntuples = outer_path->parent->rows + inner_path->parent->rows;

        /* CPU costs */
        cpu_per_tuple = cpu_tuple_cost + cost_qual_eval(restrictlist);
        run_cost += cpu_per_tuple * ntuples;

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

/*
 * cost_hashjoin
 *        Determines and returns the cost of joining two relations using the
 *        hash join algorithm.
 *
 * 'outer_path' is the path for the outer relation
 * 'inner_path' is the path for the inner relation
 * 'restrictlist' are the RestrictInfo nodes to be applied at the join
 * 'innerbucketsize' is an estimate of the bucketsize statistic
 *                              for the inner hash key.
 */
void
cost_hashjoin(Path *path,
                          Path *outer_path,
                          Path *inner_path,
                          List *restrictlist,
                          Selectivity innerbucketsize)
{
        Cost            startup_cost = 0;
        Cost            run_cost = 0;
        Cost            cpu_per_tuple;
        double          ntuples;
        double          outerbytes = relation_byte_size(outer_path->parent->rows,
                                                                                          outer_path->parent->width);
        double          innerbytes = relation_byte_size(inner_path->parent->rows,
                                                                                          inner_path->parent->width);
        long            hashtablebytes = SortMem * 1024L;

        if (!enable_hashjoin)
                startup_cost += disable_cost;

        /* 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 */
        startup_cost += cpu_operator_cost * inner_path->parent->rows;
        run_cost += cpu_operator_cost * outer_path->parent->rows;

        /*
         * 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.
         * We charge one cpu_operator_cost per tuple comparison.
         */
        run_cost += cpu_operator_cost * outer_path->parent->rows *
                ceil(inner_path->parent->rows * innerbucketsize);

        /*
         * Estimate the number of tuples that get through the hashing filter
         * as one per tuple in the two source relations.  This could be a
         * drastic underestimate if there are many equal-keyed tuples in
         * either relation, but we have no simple way of estimating that;
         * and since this is only a second-order parameter, it's probably
         * not worth expending a lot of effort on the estimate.
         */
        ntuples = outer_path->parent->rows + inner_path->parent->rows;

        /* CPU costs */
        cpu_per_tuple = cpu_tuple_cost + cost_qual_eval(restrictlist);
        run_cost += cpu_per_tuple * ntuples;

        /*
         * 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 (innerbytes > hashtablebytes)
        {
                double          outerpages = page_size(outer_path->parent->rows,
                                                                                   outer_path->parent->width);
                double          innerpages = page_size(inner_path->parent->rows,
                                                                                   inner_path->parent->width);

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

        /*
         * 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 startup cost by the
         * square root of the size ratio.  (Why square root?  No real good
         * reason, but it seems reasonable...)
         */
        if (innerbytes > outerbytes && outerbytes > 0)
                startup_cost *= sqrt(innerbytes / outerbytes);

        path->startup_cost = startup_cost;
        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.
 *
 * This statistic is used by cost_hashjoin.  We split out the calculation
 * because it's useful to cache the result for re-use across multiple path
 * cost calculations.
 *
 * 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.
 *
 * The executor tries for average bucket loading of NTUP_PER_BUCKET by setting
 * number of buckets equal to ntuples / NTUP_PER_BUCKET, which would yield
 * a bucketsize fraction of NTUP_PER_BUCKET / ntuples.  But that goal will
 * be reached only if the data values are uniformly distributed among the
 * buckets, which requires (a) at least ntuples / NTUP_PER_BUCKET distinct
 * data values, and (b) a not-too-skewed data distribution.  Otherwise the
 * buckets will be nonuniformly occupied.  If the other relation in the join
 * has a similar distribution, 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 first scaling up if there
 * are too few distinct data values, and then scaling up again 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).
 */
Selectivity
estimate_hash_bucketsize(Query *root, Var *var)
{
        Oid                     relid;
        RelOptInfo *rel;
        HeapTuple       tuple;
        Form_pg_statistic stats;
        double          estfract,
                                ndistinct,
                                needdistinct,
                                mcvfreq,
                                avgfreq;
        float4     *numbers;
        int                     nnumbers;

        /*
         * Lookup info about var's relation and attribute;
         * if none available, return default estimate.
         */
        if (!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))
        {
                /*
                 * 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 (double) NTUP_PER_BUCKET / rel->rows;
                        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;

        /*
         * 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;

        /*
         * Discourage use of hash join if there seem not to be very many distinct
         * data values.  The threshold here is somewhat arbitrary, as is the
         * fraction used to "discourage" the choice.
         */
        if (ndistinct < 50.0)
        {
                ReleaseSysCache(tuple);
                return 0.5;
        }

        /*
         * Form initial estimate of bucketsize fraction.  Here we use rel->rows,
         * ie the number of rows after applying restriction clauses, because
         * that's what the fraction will eventually be multiplied by in
         * cost_heapjoin.
         */
        estfract = (double) NTUP_PER_BUCKET / rel->rows;

        /*
         * Adjust estimated bucketsize if too few distinct values to fill
         * all the buckets.
         */
        needdistinct = rel->rows / (double) NTUP_PER_BUCKET;
        if (ndistinct < needdistinct)
                estfract *= needdistinct / 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.
         */
        avgfreq = (1.0 - stats->stanullfrac) / ndistinct;

        if (avgfreq > 0.0 && mcvfreq > avgfreq)
                estfract *= mcvfreq / avgfreq;

        ReleaseSysCache(tuple);

        return (Selectivity) estfract;
}


/*
 * cost_qual_eval
 *              Estimate the CPU cost of evaluating a WHERE clause (once).
 *              The input can be either an implicitly-ANDed list of boolean
 *              expressions, or a list of RestrictInfo nodes.
 */
Cost
cost_qual_eval(List *quals)
{
        Cost            total = 0;
        List       *l;

        /* 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 will contain -1.
                 */
                if (qual && IsA(qual, RestrictInfo))
                {
                        RestrictInfo *restrictinfo = (RestrictInfo *) qual;

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

static bool
cost_qual_eval_walker(Node *node, Cost *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, Expr))
        {
                Expr       *expr = (Expr *) node;

                switch (expr->opType)
                {
                        case OP_EXPR:
                        case FUNC_EXPR:
                                *total += cpu_operator_cost;
                                break;
                        case OR_EXPR:
                        case AND_EXPR:
                        case NOT_EXPR:
                                break;
                        case SUBPLAN_EXPR:

                                /*
                                 * A subplan node in an expression indicates that the
                                 * subplan will be executed on each evaluation, so charge
                                 * accordingly. (We assume that sub-selects that can be
                                 * executed as InitPlans have already been removed from
                                 * the expression.)
                                 *
                                 * NOTE: this logic should agree with the estimates used by
                                 * make_subplan() in plan/subselect.c.
                                 */
                                {
                                        SubPlan    *subplan = (SubPlan *) expr->oper;
                                        Plan       *plan = subplan->plan;
                                        Cost            subcost;

                                        if (subplan->sublink->subLinkType == EXISTS_SUBLINK)
                                        {
                                                /* we only need to fetch 1 tuple */
                                                subcost = plan->startup_cost +
                                                        (plan->total_cost - plan->startup_cost) / plan->plan_rows;
                                        }
                                        else if (subplan->sublink->subLinkType == ALL_SUBLINK ||
                                                         subplan->sublink->subLinkType == ANY_SUBLINK)
                                        {
                                                /* assume we need 50% of the tuples */
                                                subcost = plan->startup_cost +
                                                        0.50 * (plan->total_cost - plan->startup_cost);
                                                /* XXX what if subplan has been materialized? */
                                        }
                                        else
                                        {
                                                /* assume we need all tuples */
                                                subcost = plan->total_cost;
                                        }
                                        *total += subcost;
                                }
                                break;
                }
                /* fall through to examine args of Expr node */
        }
        return expression_tree_walker(node, cost_qual_eval_walker,
                                                                  (void *) 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)
{
        /* Should only be applied to base relations */
        Assert(length(rel->relids) == 1);

        rel->rows = rel->tuples *
                restrictlist_selectivity(root,
                                                                 rel->baserestrictinfo,
                                                                 lfirsti(rel->relids));

        /*
         * Force estimate to be at least one row, to make explain output look
         * better and to avoid possible divide-by-zero when interpolating
         * cost.
         */
        if (rel->rows < 1.0)
                rel->rows = 1.0;

        rel->baserestrictcost = cost_qual_eval(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.)
 *
 * 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)
{
        double          temp;

        /* Start with the Cartesian product */
        temp = outer_rel->rows * inner_rel->rows;

        /*
         * Apply join restrictivity.  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.
         */
        temp *= restrictlist_selectivity(root,
                                                                         restrictlist,
                                                                         0);

        /*
         * 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?)
         */
        switch (jointype)
        {
                case JOIN_INNER:
                        break;
                case JOIN_LEFT:
                        if (temp < outer_rel->rows)
                                temp = outer_rel->rows;
                        break;
                case JOIN_RIGHT:
                        if (temp < inner_rel->rows)
                                temp = inner_rel->rows;
                        break;
                case JOIN_FULL:
                        if (temp < outer_rel->rows)
                                temp = outer_rel->rows;
                        if (temp < inner_rel->rows)
                                temp = inner_rel->rows;
                        break;
                default:
                        elog(ERROR, "set_joinrel_size_estimates: unsupported join type %d",
                                 (int) jointype);
                        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.
         */
        if (temp < 1.0)
                temp = 1.0;

        rel->rows = temp;

        /*
         * We could apply set_rel_width() to compute the output tuple width
         * from scratch, but at present it's always just the sum of the input
         * widths, so why work harder than necessary?  If relnode.c is ever
         * taught to remove unneeded columns from join targetlists, go back to
         * using set_rel_width here.
         */
        rel->width = outer_rel->width + inner_rel->width;
}

/*
 * set_rel_width
 *              Set the estimated output width of the relation.
 *
 * NB: this works best on base 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?
 */
static void
set_rel_width(Query *root, RelOptInfo *rel)
{
        int32           tuple_width = 0;
        List       *tllist;

        foreach(tllist, rel->targetlist)
        {
                TargetEntry *tle = (TargetEntry *) lfirst(tllist);
                int32   item_width;

                /*
                 * If it's a Var, try to get statistical info from pg_statistic.
                 */
                if (tle->expr && IsA(tle->expr, Var))
                {
                        Var        *var = (Var *) tle->expr;
                        Oid             relid;

                        relid = getrelid(var->varno, root->rtable);
                        if (relid != InvalidOid)
                        {
                                item_width = get_attavgwidth(relid, var->varattno);
                                if (item_width > 0)
                                {
                                        tuple_width += item_width;
                                        continue;
                                }
                        }
                }
                /*
                 * Not a Var, or can't find statistics for it.  Estimate using
                 * just the type info.
                 */
                item_width = get_typavgwidth(tle->resdom->restype,
                                                                         tle->resdom->restypmod);
                Assert(item_width > 0);
                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 * ((double) MAXALIGN(width + 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);
}