pathkeys.c

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/*-------------------------------------------------------------------------
 *
 * pathkeys.c
 *    Utilities for matching and building path keys
 *
 * See src/backend/optimizer/README for a great deal of information about
 * the nature and use of path keys.
 *
 *
 * Portions Copyright (c) 1996-2021, PostgreSQL Global Development Group
 * Portions Copyright (c) 1994, Regents of the University of California
 *
 * IDENTIFICATION
 *    src/backend/optimizer/path/pathkeys.c
 *
 *-------------------------------------------------------------------------
 */
#include "postgres.h"

#include "access/stratnum.h"
#include "catalog/pg_opfamily.h"
#include "nodes/makefuncs.h"
#include "nodes/nodeFuncs.h"
#include "nodes/plannodes.h"
#include "optimizer/optimizer.h"
#include "optimizer/pathnode.h"
#include "optimizer/paths.h"
#include "partitioning/partbounds.h"
#include "utils/lsyscache.h"


static bool pathkey_is_redundant(PathKey *new_pathkey, List *pathkeys);
static bool matches_boolean_partition_clause(RestrictInfo *rinfo,
                                             RelOptInfo *partrel,
                                             int partkeycol);
static Var *find_var_for_subquery_tle(RelOptInfo *rel, TargetEntry *tle);
static bool right_merge_direction(PlannerInfo *root, PathKey *pathkey);


/****************************************************************************
 *      PATHKEY CONSTRUCTION AND REDUNDANCY TESTING
 ****************************************************************************/

/*
 * make_canonical_pathkey
 *    Given the parameters for a PathKey, find any pre-existing matching
 *    pathkey in the query's list of "canonical" pathkeys.  Make a new
 *    entry if there's not one already.
 *
 * Note that this function must not be used until after we have completed
 * merging EquivalenceClasses.
 */
PathKey *
make_canonical_pathkey(PlannerInfo *root,
                       EquivalenceClass *eclass, Oid opfamily,
                       int strategy, bool nulls_first)
{
    PathKey    *pk;
    ListCell   *lc;
    MemoryContext oldcontext;

    /* Can't make canonical pathkeys if the set of ECs might still change */
    if (!root->ec_merging_done)
        elog(ERROR, "too soon to build canonical pathkeys");

    /* The passed eclass might be non-canonical, so chase up to the top */
    while (eclass->ec_merged)
        eclass = eclass->ec_merged;

    foreach(lc, root->canon_pathkeys)
    {
        pk = (PathKey *) lfirst(lc);
        if (eclass == pk->pk_eclass &&
            opfamily == pk->pk_opfamily &&
            strategy == pk->pk_strategy &&
            nulls_first == pk->pk_nulls_first)
            return pk;
    }

    /*
     * Be sure canonical pathkeys are allocated in the main planning context.
     * Not an issue in normal planning, but it is for GEQO.
     */
    oldcontext = MemoryContextSwitchTo(root->planner_cxt);

    pk = makeNode(PathKey);
    pk->pk_eclass = eclass;
    pk->pk_opfamily = opfamily;
    pk->pk_strategy = strategy;
    pk->pk_nulls_first = nulls_first;

    root->canon_pathkeys = lappend(root->canon_pathkeys, pk);

    MemoryContextSwitchTo(oldcontext);

    return pk;
}

/*
 * pathkey_is_redundant
 *     Is a pathkey redundant with one already in the given list?
 *
 * We detect two cases:
 *
 * 1. If the new pathkey's equivalence class contains a constant, and isn't
 * below an outer join, then we can disregard it as a sort key.  An example:
 *          SELECT ... WHERE x = 42 ORDER BY x, y;
 * We may as well just sort by y.  Note that because of opfamily matching,
 * this is semantically correct: we know that the equality constraint is one
 * that actually binds the variable to a single value in the terms of any
 * ordering operator that might go with the eclass.  This rule not only lets
 * us simplify (or even skip) explicit sorts, but also allows matching index
 * sort orders to a query when there are don't-care index columns.
 *
 * 2. If the new pathkey's equivalence class is the same as that of any
 * existing member of the pathkey list, then it is redundant.  Some examples:
 *          SELECT ... ORDER BY x, x;
 *          SELECT ... ORDER BY x, x DESC;
 *          SELECT ... WHERE x = y ORDER BY x, y;
 * In all these cases the second sort key cannot distinguish values that are
 * considered equal by the first, and so there's no point in using it.
 * Note in particular that we need not compare opfamily (all the opfamilies
 * of the EC have the same notion of equality) nor sort direction.
 *
 * Both the given pathkey and the list members must be canonical for this
 * to work properly, but that's okay since we no longer ever construct any
 * non-canonical pathkeys.  (Note: the notion of a pathkey *list* being
 * canonical includes the additional requirement of no redundant entries,
 * which is exactly what we are checking for here.)
 *
 * Because the equivclass.c machinery forms only one copy of any EC per query,
 * pointer comparison is enough to decide whether canonical ECs are the same.
 */
static bool
pathkey_is_redundant(PathKey *new_pathkey, List *pathkeys)
{
    EquivalenceClass *new_ec = new_pathkey->pk_eclass;
    ListCell   *lc;

    /* Check for EC containing a constant --- unconditionally redundant */
    if (EC_MUST_BE_REDUNDANT(new_ec))
        return true;

    /* If same EC already used in list, then redundant */
    foreach(lc, pathkeys)
    {
        PathKey    *old_pathkey = (PathKey *) lfirst(lc);

        if (new_ec == old_pathkey->pk_eclass)
            return true;
    }

    return false;
}

/*
 * make_pathkey_from_sortinfo
 *    Given an expression and sort-order information, create a PathKey.
 *    The result is always a "canonical" PathKey, but it might be redundant.
 *
 * expr is the expression, and nullable_relids is the set of base relids
 * that are potentially nullable below it.
 *
 * If the PathKey is being generated from a SortGroupClause, sortref should be
 * the SortGroupClause's SortGroupRef; otherwise zero.
 *
 * If rel is not NULL, it identifies a specific relation we're considering
 * a path for, and indicates that child EC members for that relation can be
 * considered.  Otherwise child members are ignored.  (See the comments for
 * get_eclass_for_sort_expr.)
 *
 * create_it is true if we should create any missing EquivalenceClass
 * needed to represent the sort key.  If it's false, we return NULL if the
 * sort key isn't already present in any EquivalenceClass.
 */
static PathKey *
make_pathkey_from_sortinfo(PlannerInfo *root,
                           Expr *expr,
                           Relids nullable_relids,
                           Oid opfamily,
                           Oid opcintype,
                           Oid collation,
                           bool reverse_sort,
                           bool nulls_first,
                           Index sortref,
                           Relids rel,
                           bool create_it)
{
    int16       strategy;
    Oid         equality_op;
    List       *opfamilies;
    EquivalenceClass *eclass;

    strategy = reverse_sort ? BTGreaterStrategyNumber : BTLessStrategyNumber;

    /*
     * EquivalenceClasses need to contain opfamily lists based on the family
     * membership of mergejoinable equality operators, which could belong to
     * more than one opfamily.  So we have to look up the opfamily's equality
     * operator and get its membership.
     */
    equality_op = get_opfamily_member(opfamily,
                                      opcintype,
                                      opcintype,
                                      BTEqualStrategyNumber);
    if (!OidIsValid(equality_op))   /* shouldn't happen */
        elog(ERROR, "missing operator %d(%u,%u) in opfamily %u",
             BTEqualStrategyNumber, opcintype, opcintype, opfamily);
    opfamilies = get_mergejoin_opfamilies(equality_op);
    if (!opfamilies)            /* certainly should find some */
        elog(ERROR, "could not find opfamilies for equality operator %u",
             equality_op);

    /* Now find or (optionally) create a matching EquivalenceClass */
    eclass = get_eclass_for_sort_expr(root, expr, nullable_relids,
                                      opfamilies, opcintype, collation,
                                      sortref, rel, create_it);

    /* Fail if no EC and !create_it */
    if (!eclass)
        return NULL;

    /* And finally we can find or create a PathKey node */
    return make_canonical_pathkey(root, eclass, opfamily,
                                  strategy, nulls_first);
}

/*
 * make_pathkey_from_sortop
 *    Like make_pathkey_from_sortinfo, but work from a sort operator.
 *
 * This should eventually go away, but we need to restructure SortGroupClause
 * first.
 */
static PathKey *
make_pathkey_from_sortop(PlannerInfo *root,
                         Expr *expr,
                         Relids nullable_relids,
                         Oid ordering_op,
                         bool nulls_first,
                         Index sortref,
                         bool create_it)
{
    Oid         opfamily,
                opcintype,
                collation;
    int16       strategy;

    /* Find the operator in pg_amop --- failure shouldn't happen */
    if (!get_ordering_op_properties(ordering_op,
                                    &opfamily, &opcintype, &strategy))
        elog(ERROR, "operator %u is not a valid ordering operator",
             ordering_op);

    /* Because SortGroupClause doesn't carry collation, consult the expr */
    collation = exprCollation((Node *) expr);

    return make_pathkey_from_sortinfo(root,
                                      expr,
                                      nullable_relids,
                                      opfamily,
                                      opcintype,
                                      collation,
                                      (strategy == BTGreaterStrategyNumber),
                                      nulls_first,
                                      sortref,
                                      NULL,
                                      create_it);
}


/****************************************************************************
 *      PATHKEY COMPARISONS
 ****************************************************************************/

/*
 * compare_pathkeys
 *    Compare two pathkeys to see if they are equivalent, and if not whether
 *    one is "better" than the other.
 *
 *    We assume the pathkeys are canonical, and so they can be checked for
 *    equality by simple pointer comparison.
 */
PathKeysComparison
compare_pathkeys(List *keys1, List *keys2)
{
    ListCell   *key1,
               *key2;

    /*
     * Fall out quickly if we are passed two identical lists.  This mostly
     * catches the case where both are NIL, but that's common enough to
     * warrant the test.
     */
    if (keys1 == keys2)
        return PATHKEYS_EQUAL;

    forboth(key1, keys1, key2, keys2)
    {
        PathKey    *pathkey1 = (PathKey *) lfirst(key1);
        PathKey    *pathkey2 = (PathKey *) lfirst(key2);

        if (pathkey1 != pathkey2)
            return PATHKEYS_DIFFERENT;  /* no need to keep looking */
    }

    /*
     * If we reached the end of only one list, the other is longer and
     * therefore not a subset.
     */
    if (key1 != NULL)
        return PATHKEYS_BETTER1;    /* key1 is longer */
    if (key2 != NULL)
        return PATHKEYS_BETTER2;    /* key2 is longer */
    return PATHKEYS_EQUAL;
}

/*
 * pathkeys_contained_in
 *    Common special case of compare_pathkeys: we just want to know
 *    if keys2 are at least as well sorted as keys1.
 */
bool
pathkeys_contained_in(List *keys1, List *keys2)
{
    switch (compare_pathkeys(keys1, keys2))
    {
        case PATHKEYS_EQUAL:
        case PATHKEYS_BETTER2:
            return true;
        default:
            break;
    }
    return false;
}

/*
 * pathkeys_count_contained_in
 *    Same as pathkeys_contained_in, but also sets length of longest
 *    common prefix of keys1 and keys2.
 */
bool
pathkeys_count_contained_in(List *keys1, List *keys2, int *n_common)
{
    int         n = 0;
    ListCell   *key1,
               *key2;

    /*
     * See if we can avoiding looping through both lists. This optimization
     * gains us several percent in planning time in a worst-case test.
     */
    if (keys1 == keys2)
    {
        *n_common = list_length(keys1);
        return true;
    }
    else if (keys1 == NIL)
    {
        *n_common = 0;
        return true;
    }
    else if (keys2 == NIL)
    {
        *n_common = 0;
        return false;
    }

    /*
     * If both lists are non-empty, iterate through both to find out how many
     * items are shared.
     */
    forboth(key1, keys1, key2, keys2)
    {
        PathKey    *pathkey1 = (PathKey *) lfirst(key1);
        PathKey    *pathkey2 = (PathKey *) lfirst(key2);

        if (pathkey1 != pathkey2)
        {
            *n_common = n;
            return false;
        }
        n++;
    }

    /* If we ended with a null value, then we've processed the whole list. */
    *n_common = n;
    return (key1 == NULL);
}

/*
 * get_cheapest_path_for_pathkeys
 *    Find the cheapest path (according to the specified criterion) that
 *    satisfies the given pathkeys and parameterization.
 *    Return NULL if no such path.
 *
 * 'paths' is a list of possible paths that all generate the same relation
 * 'pathkeys' represents a required ordering (in canonical form!)
 * 'required_outer' denotes allowable outer relations for parameterized paths
 * 'cost_criterion' is STARTUP_COST or TOTAL_COST
 * 'require_parallel_safe' causes us to consider only parallel-safe paths
 */
Path *
get_cheapest_path_for_pathkeys(List *paths, List *pathkeys,
                               Relids required_outer,
                               CostSelector cost_criterion,
                               bool require_parallel_safe)
{
    Path       *matched_path = NULL;
    ListCell   *l;

    foreach(l, paths)
    {
        Path       *path = (Path *) lfirst(l);

        /*
         * Since cost comparison is a lot cheaper than pathkey comparison, do
         * that first.  (XXX is that still true?)
         */
        if (matched_path != NULL &&
            compare_path_costs(matched_path, path, cost_criterion) <= 0)
            continue;

        if (require_parallel_safe && !path->parallel_safe)
            continue;

        if (pathkeys_contained_in(pathkeys, path->pathkeys) &&
            bms_is_subset(PATH_REQ_OUTER(path), required_outer))
            matched_path = path;
    }
    return matched_path;
}

/*
 * get_cheapest_fractional_path_for_pathkeys
 *    Find the cheapest path (for retrieving a specified fraction of all
 *    the tuples) that satisfies the given pathkeys and parameterization.
 *    Return NULL if no such path.
 *
 * See compare_fractional_path_costs() for the interpretation of the fraction
 * parameter.
 *
 * 'paths' is a list of possible paths that all generate the same relation
 * 'pathkeys' represents a required ordering (in canonical form!)
 * 'required_outer' denotes allowable outer relations for parameterized paths
 * 'fraction' is the fraction of the total tuples expected to be retrieved
 */
Path *
get_cheapest_fractional_path_for_pathkeys(List *paths,
                                          List *pathkeys,
                                          Relids required_outer,
                                          double fraction)
{
    Path       *matched_path = NULL;
    ListCell   *l;

    foreach(l, paths)
    {
        Path       *path = (Path *) lfirst(l);

        /*
         * Since cost comparison is a lot cheaper than pathkey comparison, do
         * that first.  (XXX is that still true?)
         */
        if (matched_path != NULL &&
            compare_fractional_path_costs(matched_path, path, fraction) <= 0)
            continue;

        if (pathkeys_contained_in(pathkeys, path->pathkeys) &&
            bms_is_subset(PATH_REQ_OUTER(path), required_outer))
            matched_path = path;
    }
    return matched_path;
}


/*
 * get_cheapest_parallel_safe_total_inner
 *    Find the unparameterized parallel-safe path with the least total cost.
 */
Path *
get_cheapest_parallel_safe_total_inner(List *paths)
{
    ListCell   *l;

    foreach(l, paths)
    {
        Path       *innerpath = (Path *) lfirst(l);

        if (innerpath->parallel_safe &&
            bms_is_empty(PATH_REQ_OUTER(innerpath)))
            return innerpath;
    }

    return NULL;
}

/****************************************************************************
 *      NEW PATHKEY FORMATION
 ****************************************************************************/

/*
 * build_index_pathkeys
 *    Build a pathkeys list that describes the ordering induced by an index
 *    scan using the given index.  (Note that an unordered index doesn't
 *    induce any ordering, so we return NIL.)
 *
 * If 'scandir' is BackwardScanDirection, build pathkeys representing a
 * backwards scan of the index.
 *
 * We iterate only key columns of covering indexes, since non-key columns
 * don't influence index ordering.  The result is canonical, meaning that
 * redundant pathkeys are removed; it may therefore have fewer entries than
 * there are key columns in the index.
 *
 * Another reason for stopping early is that we may be able to tell that
 * an index column's sort order is uninteresting for this query.  However,
 * that test is just based on the existence of an EquivalenceClass and not
 * on position in pathkey lists, so it's not complete.  Caller should call
 * truncate_useless_pathkeys() to possibly remove more pathkeys.
 */
List *
build_index_pathkeys(PlannerInfo *root,
                     IndexOptInfo *index,
                     ScanDirection scandir)
{
    List       *retval = NIL;
    ListCell   *lc;
    int         i;

    if (index->sortopfamily == NULL)
        return NIL;             /* non-orderable index */

    i = 0;
    foreach(lc, index->indextlist)
    {
        TargetEntry *indextle = (TargetEntry *) lfirst(lc);
        Expr       *indexkey;
        bool        reverse_sort;
        bool        nulls_first;
        PathKey    *cpathkey;

        /*
         * INCLUDE columns are stored in index unordered, so they don't
         * support ordered index scan.
         */
        if (i >= index->nkeycolumns)
            break;

        /* We assume we don't need to make a copy of the tlist item */
        indexkey = indextle->expr;

        if (ScanDirectionIsBackward(scandir))
        {
            reverse_sort = !index->reverse_sort[i];
            nulls_first = !index->nulls_first[i];
        }
        else
        {
            reverse_sort = index->reverse_sort[i];
            nulls_first = index->nulls_first[i];
        }

        /*
         * OK, try to make a canonical pathkey for this sort key.  Note we're
         * underneath any outer joins, so nullable_relids should be NULL.
         */
        cpathkey = make_pathkey_from_sortinfo(root,
                                              indexkey,
                                              NULL,
                                              index->sortopfamily[i],
                                              index->opcintype[i],
                                              index->indexcollations[i],
                                              reverse_sort,
                                              nulls_first,
                                              0,
                                              index->rel->relids,
                                              false);

        if (cpathkey)
        {
            /*
             * We found the sort key in an EquivalenceClass, so it's relevant
             * for this query.  Add it to list, unless it's redundant.
             */
            if (!pathkey_is_redundant(cpathkey, retval))
                retval = lappend(retval, cpathkey);
        }
        else
        {
            /*
             * Boolean index keys might be redundant even if they do not
             * appear in an EquivalenceClass, because of our special treatment
             * of boolean equality conditions --- see the comment for
             * indexcol_is_bool_constant_for_query().  If that applies, we can
             * continue to examine lower-order index columns.  Otherwise, the
             * sort key is not an interesting sort order for this query, so we
             * should stop considering index columns; any lower-order sort
             * keys won't be useful either.
             */
            if (!indexcol_is_bool_constant_for_query(root, index, i))
                break;
        }

        i++;
    }

    return retval;
}

/*
 * partkey_is_bool_constant_for_query
 *
 * If a partition key column is constrained to have a constant value by the
 * query's WHERE conditions, then it's irrelevant for sort-order
 * considerations.  Usually that means we have a restriction clause
 * WHERE partkeycol = constant, which gets turned into an EquivalenceClass
 * containing a constant, which is recognized as redundant by
 * build_partition_pathkeys().  But if the partition key column is a
 * boolean variable (or expression), then we are not going to see such a
 * WHERE clause, because expression preprocessing will have simplified it
 * to "WHERE partkeycol" or "WHERE NOT partkeycol".  So we are not going
 * to have a matching EquivalenceClass (unless the query also contains
 * "ORDER BY partkeycol").  To allow such cases to work the same as they would
 * for non-boolean values, this function is provided to detect whether the
 * specified partition key column matches a boolean restriction clause.
 */
static bool
partkey_is_bool_constant_for_query(RelOptInfo *partrel, int partkeycol)
{
    PartitionScheme partscheme = partrel->part_scheme;
    ListCell   *lc;

    /* If the partkey isn't boolean, we can't possibly get a match */
    if (!IsBooleanOpfamily(partscheme->partopfamily[partkeycol]))
        return false;

    /* Check each restriction clause for the partitioned rel */
    foreach(lc, partrel->baserestrictinfo)
    {
        RestrictInfo *rinfo = (RestrictInfo *) lfirst(lc);

        /* Ignore pseudoconstant quals, they won't match */
        if (rinfo->pseudoconstant)
            continue;

        /* See if we can match the clause's expression to the partkey column */
        if (matches_boolean_partition_clause(rinfo, partrel, partkeycol))
            return true;
    }

    return false;
}

/*
 * matches_boolean_partition_clause
 *      Determine if the boolean clause described by rinfo matches
 *      partrel's partkeycol-th partition key column.
 *
 * "Matches" can be either an exact match (equivalent to partkey = true),
 * or a NOT above an exact match (equivalent to partkey = false).
 */
static bool
matches_boolean_partition_clause(RestrictInfo *rinfo,
                                 RelOptInfo *partrel, int partkeycol)
{
    Node       *clause = (Node *) rinfo->clause;
    Node       *partexpr = (Node *) linitial(partrel->partexprs[partkeycol]);

    /* Direct match? */
    if (equal(partexpr, clause))
        return true;
    /* NOT clause? */
    else if (is_notclause(clause))
    {
        Node       *arg = (Node *) get_notclausearg((Expr *) clause);

        if (equal(partexpr, arg))
            return true;
    }

    return false;
}

/*
 * build_partition_pathkeys
 *    Build a pathkeys list that describes the ordering induced by the
 *    partitions of partrel, under either forward or backward scan
 *    as per scandir.
 *
 * Caller must have checked that the partitions are properly ordered,
 * as detected by partitions_are_ordered().
 *
 * Sets *partialkeys to true if pathkeys were only built for a prefix of the
 * partition key, or false if the pathkeys include all columns of the
 * partition key.
 */
List *
build_partition_pathkeys(PlannerInfo *root, RelOptInfo *partrel,
                         ScanDirection scandir, bool *partialkeys)
{
    List       *retval = NIL;
    PartitionScheme partscheme = partrel->part_scheme;
    int         i;

    Assert(partscheme != NULL);
    Assert(partitions_are_ordered(partrel->boundinfo, partrel->nparts));
    /* For now, we can only cope with baserels */
    Assert(IS_SIMPLE_REL(partrel));

    for (i = 0; i < partscheme->partnatts; i++)
    {
        PathKey    *cpathkey;
        Expr       *keyCol = (Expr *) linitial(partrel->partexprs[i]);

        /*
         * Try to make a canonical pathkey for this partkey.
         *
         * We're considering a baserel scan, so nullable_relids should be
         * NULL.  Also, we assume the PartitionDesc lists any NULL partition
         * last, so we treat the scan like a NULLS LAST index: we have
         * nulls_first for backwards scan only.
         */
        cpathkey = make_pathkey_from_sortinfo(root,
                                              keyCol,
                                              NULL,
                                              partscheme->partopfamily[i],
                                              partscheme->partopcintype[i],
                                              partscheme->partcollation[i],
                                              ScanDirectionIsBackward(scandir),
                                              ScanDirectionIsBackward(scandir),
                                              0,
                                              partrel->relids,
                                              false);


        if (cpathkey)
        {
            /*
             * We found the sort key in an EquivalenceClass, so it's relevant
             * for this query.  Add it to list, unless it's redundant.
             */
            if (!pathkey_is_redundant(cpathkey, retval))
                retval = lappend(retval, cpathkey);
        }
        else
        {
            /*
             * Boolean partition keys might be redundant even if they do not
             * appear in an EquivalenceClass, because of our special treatment
             * of boolean equality conditions --- see the comment for
             * partkey_is_bool_constant_for_query().  If that applies, we can
             * continue to examine lower-order partition keys.  Otherwise, the
             * sort key is not an interesting sort order for this query, so we
             * should stop considering partition columns; any lower-order sort
             * keys won't be useful either.
             */
            if (!partkey_is_bool_constant_for_query(partrel, i))
            {
                *partialkeys = true;
                return retval;
            }
        }
    }

    *partialkeys = false;
    return retval;
}

/*
 * build_expression_pathkey
 *    Build a pathkeys list that describes an ordering by a single expression
 *    using the given sort operator.
 *
 * expr, nullable_relids, and rel are as for make_pathkey_from_sortinfo.
 * We induce the other arguments assuming default sort order for the operator.
 *
 * Similarly to make_pathkey_from_sortinfo, the result is NIL if create_it
 * is false and the expression isn't already in some EquivalenceClass.
 */
List *
build_expression_pathkey(PlannerInfo *root,
                         Expr *expr,
                         Relids nullable_relids,
                         Oid opno,
                         Relids rel,
                         bool create_it)
{
    List       *pathkeys;
    Oid         opfamily,
                opcintype;
    int16       strategy;
    PathKey    *cpathkey;

    /* Find the operator in pg_amop --- failure shouldn't happen */
    if (!get_ordering_op_properties(opno,
                                    &opfamily, &opcintype, &strategy))
        elog(ERROR, "operator %u is not a valid ordering operator",
             opno);

    cpathkey = make_pathkey_from_sortinfo(root,
                                          expr,
                                          nullable_relids,
                                          opfamily,
                                          opcintype,
                                          exprCollation((Node *) expr),
                                          (strategy == BTGreaterStrategyNumber),
                                          (strategy == BTGreaterStrategyNumber),
                                          0,
                                          rel,
                                          create_it);

    if (cpathkey)
        pathkeys = list_make1(cpathkey);
    else
        pathkeys = NIL;

    return pathkeys;
}

/*
 * convert_subquery_pathkeys
 *    Build a pathkeys list that describes the ordering of a subquery's
 *    result, in the terms of the outer query.  This is essentially a
 *    task of conversion.
 *
 * 'rel': outer query's RelOptInfo for the subquery relation.
 * 'subquery_pathkeys': the subquery's output pathkeys, in its terms.
 * 'subquery_tlist': the subquery's output targetlist, in its terms.
 *
 * We intentionally don't do truncate_useless_pathkeys() here, because there
 * are situations where seeing the raw ordering of the subquery is helpful.
 * For example, if it returns ORDER BY x DESC, that may prompt us to
 * construct a mergejoin using DESC order rather than ASC order; but the
 * right_merge_direction heuristic would have us throw the knowledge away.
 */
List *
convert_subquery_pathkeys(PlannerInfo *root, RelOptInfo *rel,
                          List *subquery_pathkeys,
                          List *subquery_tlist)
{
    List       *retval = NIL;
    int         retvallen = 0;
    int         outer_query_keys = list_length(root->query_pathkeys);
    ListCell   *i;

    foreach(i, subquery_pathkeys)
    {
        PathKey    *sub_pathkey = (PathKey *) lfirst(i);
        EquivalenceClass *sub_eclass = sub_pathkey->pk_eclass;
        PathKey    *best_pathkey = NULL;

        if (sub_eclass->ec_has_volatile)
        {
            /*
             * If the sub_pathkey's EquivalenceClass is volatile, then it must
             * have come from an ORDER BY clause, and we have to match it to
             * that same targetlist entry.
             */
            TargetEntry *tle;
            Var        *outer_var;

            if (sub_eclass->ec_sortref == 0)    /* can't happen */
                elog(ERROR, "volatile EquivalenceClass has no sortref");
            tle = get_sortgroupref_tle(sub_eclass->ec_sortref, subquery_tlist);
            Assert(tle);
            /* Is TLE actually available to the outer query? */
            outer_var = find_var_for_subquery_tle(rel, tle);
            if (outer_var)
            {
                /* We can represent this sub_pathkey */
                EquivalenceMember *sub_member;
                EquivalenceClass *outer_ec;

                Assert(list_length(sub_eclass->ec_members) == 1);
                sub_member = (EquivalenceMember *) linitial(sub_eclass->ec_members);

                /*
                 * Note: it might look funny to be setting sortref = 0 for a
                 * reference to a volatile sub_eclass.  However, the
                 * expression is *not* volatile in the outer query: it's just
                 * a Var referencing whatever the subquery emitted. (IOW, the
                 * outer query isn't going to re-execute the volatile
                 * expression itself.)  So this is okay.  Likewise, it's
                 * correct to pass nullable_relids = NULL, because we're
                 * underneath any outer joins appearing in the outer query.
                 */
                outer_ec =
                    get_eclass_for_sort_expr(root,
                                             (Expr *) outer_var,
                                             NULL,
                                             sub_eclass->ec_opfamilies,
                                             sub_member->em_datatype,
                                             sub_eclass->ec_collation,
                                             0,
                                             rel->relids,
                                             false);

                /*
                 * If we don't find a matching EC, sub-pathkey isn't
                 * interesting to the outer query
                 */
                if (outer_ec)
                    best_pathkey =
                        make_canonical_pathkey(root,
                                               outer_ec,
                                               sub_pathkey->pk_opfamily,
                                               sub_pathkey->pk_strategy,
                                               sub_pathkey->pk_nulls_first);
            }
        }
        else
        {
            /*
             * Otherwise, the sub_pathkey's EquivalenceClass could contain
             * multiple elements (representing knowledge that multiple items
             * are effectively equal).  Each element might match none, one, or
             * more of the output columns that are visible to the outer query.
             * This means we may have multiple possible representations of the
             * sub_pathkey in the context of the outer query.  Ideally we
             * would generate them all and put them all into an EC of the
             * outer query, thereby propagating equality knowledge up to the
             * outer query.  Right now we cannot do so, because the outer
             * query's EquivalenceClasses are already frozen when this is
             * called. Instead we prefer the one that has the highest "score"
             * (number of EC peers, plus one if it matches the outer
             * query_pathkeys). This is the most likely to be useful in the
             * outer query.
             */
            int         best_score = -1;
            ListCell   *j;

            foreach(j, sub_eclass->ec_members)
            {
                EquivalenceMember *sub_member = (EquivalenceMember *) lfirst(j);
                Expr       *sub_expr = sub_member->em_expr;
                Oid         sub_expr_type = sub_member->em_datatype;
                Oid         sub_expr_coll = sub_eclass->ec_collation;
                ListCell   *k;

                if (sub_member->em_is_child)
                    continue;   /* ignore children here */

                foreach(k, subquery_tlist)
                {
                    TargetEntry *tle = (TargetEntry *) lfirst(k);
                    Var        *outer_var;
                    Expr       *tle_expr;
                    EquivalenceClass *outer_ec;
                    PathKey    *outer_pk;
                    int         score;

                    /* Is TLE actually available to the outer query? */
                    outer_var = find_var_for_subquery_tle(rel, tle);
                    if (!outer_var)
                        continue;

                    /*
                     * The targetlist entry is considered to match if it
                     * matches after sort-key canonicalization.  That is
                     * needed since the sub_expr has been through the same
                     * process.
                     */
                    tle_expr = canonicalize_ec_expression(tle->expr,
                                                          sub_expr_type,
                                                          sub_expr_coll);
                    if (!equal(tle_expr, sub_expr))
                        continue;

                    /* See if we have a matching EC for the TLE */
                    outer_ec = get_eclass_for_sort_expr(root,
                                                        (Expr *) outer_var,
                                                        NULL,
                                                        sub_eclass->ec_opfamilies,
                                                        sub_expr_type,
                                                        sub_expr_coll,
                                                        0,
                                                        rel->relids,
                                                        false);

                    /*
                     * If we don't find a matching EC, this sub-pathkey isn't
                     * interesting to the outer query
                     */
                    if (!outer_ec)
                        continue;

                    outer_pk = make_canonical_pathkey(root,
                                                      outer_ec,
                                                      sub_pathkey->pk_opfamily,
                                                      sub_pathkey->pk_strategy,
                                                      sub_pathkey->pk_nulls_first);
                    /* score = # of equivalence peers */
                    score = list_length(outer_ec->ec_members) - 1;
                    /* +1 if it matches the proper query_pathkeys item */
                    if (retvallen < outer_query_keys &&
                        list_nth(root->query_pathkeys, retvallen) == outer_pk)
                        score++;
                    if (score > best_score)
                    {
                        best_pathkey = outer_pk;
                        best_score = score;
                    }
                }
            }
        }

        /*
         * If we couldn't find a representation of this sub_pathkey, we're
         * done (we can't use the ones to its right, either).
         */
        if (!best_pathkey)
            break;

        /*
         * Eliminate redundant ordering info; could happen if outer query
         * equivalences subquery keys...
         */
        if (!pathkey_is_redundant(best_pathkey, retval))
        {
            retval = lappend(retval, best_pathkey);
            retvallen++;
        }
    }

    return retval;
}

/*
 * find_var_for_subquery_tle
 *
 * If the given subquery tlist entry is due to be emitted by the subquery's
 * scan node, return a Var for it, else return NULL.
 *
 * We need this to ensure that we don't return pathkeys describing values
 * that are unavailable above the level of the subquery scan.
 */
static Var *
find_var_for_subquery_tle(RelOptInfo *rel, TargetEntry *tle)
{
    ListCell   *lc;

    /* If the TLE is resjunk, it's certainly not visible to the outer query */
    if (tle->resjunk)
        return NULL;

    /* Search the rel's targetlist to see what it will return */
    foreach(lc, rel->reltarget->exprs)
    {
        Var        *var = (Var *) lfirst(lc);

        /* Ignore placeholders */
        if (!IsA(var, Var))
            continue;
        Assert(var->varno == rel->relid);

        /* If we find a Var referencing this TLE, we're good */
        if (var->varattno == tle->resno)
            return copyObject(var); /* Make a copy for safety */
    }
    return NULL;
}

/*
 * build_join_pathkeys
 *    Build the path keys for a join relation constructed by mergejoin or
 *    nestloop join.  This is normally the same as the outer path's keys.
 *
 *    EXCEPTION: in a FULL or RIGHT join, we cannot treat the result as
 *    having the outer path's path keys, because null lefthand rows may be
 *    inserted at random points.  It must be treated as unsorted.
 *
 *    We truncate away any pathkeys that are uninteresting for higher joins.
 *
 * 'joinrel' is the join relation that paths are being formed for
 * 'jointype' is the join type (inner, left, full, etc)
 * 'outer_pathkeys' is the list of the current outer path's path keys
 *
 * Returns the list of new path keys.
 */
List *
build_join_pathkeys(PlannerInfo *root,
                    RelOptInfo *joinrel,
                    JoinType jointype,
                    List *outer_pathkeys)
{
    if (jointype == JOIN_FULL || jointype == JOIN_RIGHT)
        return NIL;

    /*
     * This used to be quite a complex bit of code, but now that all pathkey
     * sublists start out life canonicalized, we don't have to do a darn thing
     * here!
     *
     * We do, however, need to truncate the pathkeys list, since it may
     * contain pathkeys that were useful for forming this joinrel but are
     * uninteresting to higher levels.
     */
    return truncate_useless_pathkeys(root, joinrel, outer_pathkeys);
}

/****************************************************************************
 *      PATHKEYS AND SORT CLAUSES
 ****************************************************************************/

/*
 * make_pathkeys_for_sortclauses
 *      Generate a pathkeys list that represents the sort order specified
 *      by a list of SortGroupClauses
 *
 * The resulting PathKeys are always in canonical form.  (Actually, there
 * is no longer any code anywhere that creates non-canonical PathKeys.)
 *
 * We assume that root->nullable_baserels is the set of base relids that could
 * have gone to NULL below the SortGroupClause expressions.  This is okay if
 * the expressions came from the query's top level (ORDER BY, DISTINCT, etc)
 * and if this function is only invoked after deconstruct_jointree.  In the
 * future we might have to make callers pass in the appropriate
 * nullable-relids set, but for now it seems unnecessary.
 *
 * 'sortclauses' is a list of SortGroupClause nodes
 * 'tlist' is the targetlist to find the referenced tlist entries in
 */
List *
make_pathkeys_for_sortclauses(PlannerInfo *root,
                              List *sortclauses,
                              List *tlist)
{
    List       *pathkeys = NIL;
    ListCell   *l;

    foreach(l, sortclauses)
    {
        SortGroupClause *sortcl = (SortGroupClause *) lfirst(l);
        Expr       *sortkey;
        PathKey    *pathkey;

        sortkey = (Expr *) get_sortgroupclause_expr(sortcl, tlist);
        Assert(OidIsValid(sortcl->sortop));
        pathkey = make_pathkey_from_sortop(root,
                                           sortkey,
                                           root->nullable_baserels,
                                           sortcl->sortop,
                                           sortcl->nulls_first,
                                           sortcl->tleSortGroupRef,
                                           true);

        /* Canonical form eliminates redundant ordering keys */
        if (!pathkey_is_redundant(pathkey, pathkeys))
            pathkeys = lappend(pathkeys, pathkey);
    }
    return pathkeys;
}

/****************************************************************************
 *      PATHKEYS AND MERGECLAUSES
 ****************************************************************************/

/*
 * initialize_mergeclause_eclasses
 *      Set the EquivalenceClass links in a mergeclause restrictinfo.
 *
 * RestrictInfo contains fields in which we may cache pointers to
 * EquivalenceClasses for the left and right inputs of the mergeclause.
 * (If the mergeclause is a true equivalence clause these will be the
 * same EquivalenceClass, otherwise not.)  If the mergeclause is either
 * used to generate an EquivalenceClass, or derived from an EquivalenceClass,
 * then it's easy to set up the left_ec and right_ec members --- otherwise,
 * this function should be called to set them up.  We will generate new
 * EquivalenceClauses if necessary to represent the mergeclause's left and
 * right sides.
 *
 * Note this is called before EC merging is complete, so the links won't
 * necessarily point to canonical ECs.  Before they are actually used for
 * anything, update_mergeclause_eclasses must be called to ensure that
 * they've been updated to point to canonical ECs.
 */
void
initialize_mergeclause_eclasses(PlannerInfo *root, RestrictInfo *restrictinfo)
{
    Expr       *clause = restrictinfo->clause;
    Oid         lefttype,
                righttype;

    /* Should be a mergeclause ... */
    Assert(restrictinfo->mergeopfamilies != NIL);
    /* ... with links not yet set */
    Assert(restrictinfo->left_ec == NULL);
    Assert(restrictinfo->right_ec == NULL);

    /* Need the declared input types of the operator */
    op_input_types(((OpExpr *) clause)->opno, &lefttype, &righttype);

    /* Find or create a matching EquivalenceClass for each side */
    restrictinfo->left_ec =
        get_eclass_for_sort_expr(root,
                                 (Expr *) get_leftop(clause),
                                 restrictinfo->nullable_relids,
                                 restrictinfo->mergeopfamilies,
                                 lefttype,
                                 ((OpExpr *) clause)->inputcollid,
                                 0,
                                 NULL,
                                 true);
    restrictinfo->right_ec =
        get_eclass_for_sort_expr(root,
                                 (Expr *) get_rightop(clause),
                                 restrictinfo->nullable_relids,
                                 restrictinfo->mergeopfamilies,
                                 righttype,
                                 ((OpExpr *) clause)->inputcollid,
                                 0,
                                 NULL,
                                 true);
}

/*
 * update_mergeclause_eclasses
 *      Make the cached EquivalenceClass links valid in a mergeclause
 *      restrictinfo.
 *
 * These pointers should have been set by process_equivalence or
 * initialize_mergeclause_eclasses, but they might have been set to
 * non-canonical ECs that got merged later.  Chase up to the canonical
 * merged parent if so.
 */
void
update_mergeclause_eclasses(PlannerInfo *root, RestrictInfo *restrictinfo)
{
    /* Should be a merge clause ... */
    Assert(restrictinfo->mergeopfamilies != NIL);
    /* ... with pointers already set */
    Assert(restrictinfo->left_ec != NULL);
    Assert(restrictinfo->right_ec != NULL);

    /* Chase up to the top as needed */
    while (restrictinfo->left_ec->ec_merged)
        restrictinfo->left_ec = restrictinfo->left_ec->ec_merged;
    while (restrictinfo->right_ec->ec_merged)
        restrictinfo->right_ec = restrictinfo->right_ec->ec_merged;
}

/*
 * find_mergeclauses_for_outer_pathkeys
 *    This routine attempts to find a list of mergeclauses that can be
 *    used with a specified ordering for the join's outer relation.
 *    If successful, it returns a list of mergeclauses.
 *
 * 'pathkeys' is a pathkeys list showing the ordering of an outer-rel path.
 * 'restrictinfos' is a list of mergejoinable restriction clauses for the
 *          join relation being formed, in no particular order.
 *
 * The restrictinfos must be marked (via outer_is_left) to show which side
 * of each clause is associated with the current outer path.  (See
 * select_mergejoin_clauses())
 *
 * The result is NIL if no merge can be done, else a maximal list of
 * usable mergeclauses (represented as a list of their restrictinfo nodes).
 * The list is ordered to match the pathkeys, as required for execution.
 */
List *
find_mergeclauses_for_outer_pathkeys(PlannerInfo *root,
                                     List *pathkeys,
                                     List *restrictinfos)
{
    List       *mergeclauses = NIL;
    ListCell   *i;

    /* make sure we have eclasses cached in the clauses */
    foreach(i, restrictinfos)
    {
        RestrictInfo *rinfo = (RestrictInfo *) lfirst(i);

        update_mergeclause_eclasses(root, rinfo);
    }

    foreach(i, pathkeys)
    {
        PathKey    *pathkey = (PathKey *) lfirst(i);
        EquivalenceClass *pathkey_ec = pathkey->pk_eclass;
        List       *matched_restrictinfos = NIL;
        ListCell   *j;

        /*----------
         * A mergejoin clause matches a pathkey if it has the same EC.
         * If there are multiple matching clauses, take them all.  In plain
         * inner-join scenarios we expect only one match, because
         * equivalence-class processing will have removed any redundant
         * mergeclauses.  However, in outer-join scenarios there might be
         * multiple matches.  An example is
         *
         *  select * from a full join b
         *      on a.v1 = b.v1 and a.v2 = b.v2 and a.v1 = b.v2;
         *
         * Given the pathkeys ({a.v1}, {a.v2}) it is okay to return all three
         * clauses (in the order a.v1=b.v1, a.v1=b.v2, a.v2=b.v2) and indeed
         * we *must* do so or we will be unable to form a valid plan.
         *
         * We expect that the given pathkeys list is canonical, which means
         * no two members have the same EC, so it's not possible for this
         * code to enter the same mergeclause into the result list twice.
         *
         * It's possible that multiple matching clauses might have different
         * ECs on the other side, in which case the order we put them into our
         * result makes a difference in the pathkeys required for the inner
         * input rel.  However this routine hasn't got any info about which
         * order would be best, so we don't worry about that.
         *
         * It's also possible that the selected mergejoin clauses produce
         * a noncanonical ordering of pathkeys for the inner side, ie, we
         * might select clauses that reference b.v1, b.v2, b.v1 in that
         * order.  This is not harmful in itself, though it suggests that
         * the clauses are partially redundant.  Since the alternative is
         * to omit mergejoin clauses and thereby possibly fail to generate a
         * plan altogether, we live with it.  make_inner_pathkeys_for_merge()
         * has to delete duplicates when it constructs the inner pathkeys
         * list, and we also have to deal with such cases specially in
         * create_mergejoin_plan().
         *----------
         */
        foreach(j, restrictinfos)
        {
            RestrictInfo *rinfo = (RestrictInfo *) lfirst(j);
            EquivalenceClass *clause_ec;

            clause_ec = rinfo->outer_is_left ?
                rinfo->left_ec : rinfo->right_ec;
            if (clause_ec == pathkey_ec)
                matched_restrictinfos = lappend(matched_restrictinfos, rinfo);
        }

        /*
         * If we didn't find a mergeclause, we're done --- any additional
         * sort-key positions in the pathkeys are useless.  (But we can still
         * mergejoin if we found at least one mergeclause.)
         */
        if (matched_restrictinfos == NIL)
            break;

        /*
         * If we did find usable mergeclause(s) for this sort-key position,
         * add them to result list.
         */
        mergeclauses = list_concat(mergeclauses, matched_restrictinfos);
    }

    return mergeclauses;
}

/*
 * select_outer_pathkeys_for_merge
 *    Builds a pathkey list representing a possible sort ordering
 *    that can be used with the given mergeclauses.
 *
 * 'mergeclauses' is a list of RestrictInfos for mergejoin clauses
 *          that will be used in a merge join.
 * 'joinrel' is the join relation we are trying to construct.
 *
 * The restrictinfos must be marked (via outer_is_left) to show which side
 * of each clause is associated with the current outer path.  (See
 * select_mergejoin_clauses())
 *
 * Returns a pathkeys list that can be applied to the outer relation.
 *
 * Since we assume here that a sort is required, there is no particular use
 * in matching any available ordering of the outerrel.  (joinpath.c has an
 * entirely separate code path for considering sort-free mergejoins.)  Rather,
 * it's interesting to try to match the requested query_pathkeys so that a
 * second output sort may be avoided; and failing that, we try to list "more
 * popular" keys (those with the most unmatched EquivalenceClass peers)
 * earlier, in hopes of making the resulting ordering useful for as many
 * higher-level mergejoins as possible.
 */
List *
select_outer_pathkeys_for_merge(PlannerInfo *root,
                                List *mergeclauses,
                                RelOptInfo *joinrel)
{
    List       *pathkeys = NIL;
    int         nClauses = list_length(mergeclauses);
    EquivalenceClass **ecs;
    int        *scores;
    int         necs;
    ListCell   *lc;
    int         j;

    /* Might have no mergeclauses */
    if (nClauses == 0)
        return NIL;

    /*
     * Make arrays of the ECs used by the mergeclauses (dropping any
     * duplicates) and their "popularity" scores.
     */
    ecs = (EquivalenceClass **) palloc(nClauses * sizeof(EquivalenceClass *));
    scores = (int *) palloc(nClauses * sizeof(int));
    necs = 0;

    foreach(lc, mergeclauses)
    {
        RestrictInfo *rinfo = (RestrictInfo *) lfirst(lc);
        EquivalenceClass *oeclass;
        int         score;
        ListCell   *lc2;

        /* get the outer eclass */
        update_mergeclause_eclasses(root, rinfo);

        if (rinfo->outer_is_left)
            oeclass = rinfo->left_ec;
        else
            oeclass = rinfo->right_ec;

        /* reject duplicates */
        for (j = 0; j < necs; j++)
        {
            if (ecs[j] == oeclass)
                break;
        }
        if (j < necs)
            continue;

        /* compute score */
        score = 0;
        foreach(lc2, oeclass->ec_members)
        {
            EquivalenceMember *em = (EquivalenceMember *) lfirst(lc2);

            /* Potential future join partner? */
            if (!em->em_is_const && !em->em_is_child &&
                !bms_overlap(em->em_relids, joinrel->relids))
                score++;
        }

        ecs[necs] = oeclass;
        scores[necs] = score;
        necs++;
    }

    /*
     * Find out if we have all the ECs mentioned in query_pathkeys; if so we
     * can generate a sort order that's also useful for final output. There is
     * no percentage in a partial match, though, so we have to have 'em all.
     */
    if (root->query_pathkeys)
    {
        foreach(lc, root->query_pathkeys)
        {
            PathKey    *query_pathkey = (PathKey *) lfirst(lc);
            EquivalenceClass *query_ec = query_pathkey->pk_eclass;

            for (j = 0; j < necs; j++)
            {
                if (ecs[j] == query_ec)
                    break;      /* found match */
            }
            if (j >= necs)
                break;          /* didn't find match */
        }
        /* if we got to the end of the list, we have them all */
        if (lc == NULL)
        {
            /* copy query_pathkeys as starting point for our output */
            pathkeys = list_copy(root->query_pathkeys);
            /* mark their ECs as already-emitted */
            foreach(lc, root->query_pathkeys)
            {
                PathKey    *query_pathkey = (PathKey *) lfirst(lc);
                EquivalenceClass *query_ec = query_pathkey->pk_eclass;

                for (j = 0; j < necs; j++)
                {
                    if (ecs[j] == query_ec)
                    {
                        scores[j] = -1;
                        break;
                    }
                }
            }
        }
    }

    /*
     * Add remaining ECs to the list in popularity order, using a default sort
     * ordering.  (We could use qsort() here, but the list length is usually
     * so small it's not worth it.)
     */
    for (;;)
    {
        int         best_j;
        int         best_score;
        EquivalenceClass *ec;
        PathKey    *pathkey;

        best_j = 0;
        best_score = scores[0];
        for (j = 1; j < necs; j++)
        {
            if (scores[j] > best_score)
            {
                best_j = j;
                best_score = scores[j];
            }
        }
        if (best_score < 0)
            break;              /* all done */
        ec = ecs[best_j];
        scores[best_j] = -1;
        pathkey = make_canonical_pathkey(root,
                                         ec,
                                         linitial_oid(ec->ec_opfamilies),
                                         BTLessStrategyNumber,
                                         false);
        /* can't be redundant because no duplicate ECs */
        Assert(!pathkey_is_redundant(pathkey, pathkeys));
        pathkeys = lappend(pathkeys, pathkey);
    }

    pfree(ecs);
    pfree(scores);

    return pathkeys;
}

/*
 * make_inner_pathkeys_for_merge
 *    Builds a pathkey list representing the explicit sort order that
 *    must be applied to an inner path to make it usable with the
 *    given mergeclauses.
 *
 * 'mergeclauses' is a list of RestrictInfos for the mergejoin clauses
 *          that will be used in a merge join, in order.
 * 'outer_pathkeys' are the already-known canonical pathkeys for the outer
 *          side of the join.
 *
 * The restrictinfos must be marked (via outer_is_left) to show which side
 * of each clause is associated with the current outer path.  (See
 * select_mergejoin_clauses())
 *
 * Returns a pathkeys list that can be applied to the inner relation.
 *
 * Note that it is not this routine's job to decide whether sorting is
 * actually needed for a particular input path.  Assume a sort is necessary;
 * just make the keys, eh?
 */
List *
make_inner_pathkeys_for_merge(PlannerInfo *root,
                              List *mergeclauses,
                              List *outer_pathkeys)
{
    List       *pathkeys = NIL;
    EquivalenceClass *lastoeclass;
    PathKey    *opathkey;
    ListCell   *lc;
    ListCell   *lop;

    lastoeclass = NULL;
    opathkey = NULL;
    lop = list_head(outer_pathkeys);

    foreach(lc, mergeclauses)
    {
        RestrictInfo *rinfo = (RestrictInfo *) lfirst(lc);
        EquivalenceClass *oeclass;
        EquivalenceClass *ieclass;
        PathKey    *pathkey;

        update_mergeclause_eclasses(root, rinfo);

        if (rinfo->outer_is_left)
        {
            oeclass = rinfo->left_ec;
            ieclass = rinfo->right_ec;
        }
        else
        {
            oeclass = rinfo->right_ec;
            ieclass = rinfo->left_ec;
        }

        /* outer eclass should match current or next pathkeys */
        /* we check this carefully for debugging reasons */
        if (oeclass != lastoeclass)
        {
            if (!lop)
                elog(ERROR, "too few pathkeys for mergeclauses");
            opathkey = (PathKey *) lfirst(lop);
            lop = lnext(outer_pathkeys, lop);
            lastoeclass = opathkey->pk_eclass;
            if (oeclass != lastoeclass)
                elog(ERROR, "outer pathkeys do not match mergeclause");
        }

        /*
         * Often, we'll have same EC on both sides, in which case the outer
         * pathkey is also canonical for the inner side, and we can skip a
         * useless search.
         */
        if (ieclass == oeclass)
            pathkey = opathkey;
        else
            pathkey = make_canonical_pathkey(root,
                                             ieclass,
                                             opathkey->pk_opfamily,
                                             opathkey->pk_strategy,
                                             opathkey->pk_nulls_first);

        /*
         * Don't generate redundant pathkeys (which can happen if multiple
         * mergeclauses refer to the same EC).  Because we do this, the output
         * pathkey list isn't necessarily ordered like the mergeclauses, which
         * complicates life for create_mergejoin_plan().  But if we didn't,
         * we'd have a noncanonical sort key list, which would be bad; for one
         * reason, it certainly wouldn't match any available sort order for
         * the input relation.
         */
        if (!pathkey_is_redundant(pathkey, pathkeys))
            pathkeys = lappend(pathkeys, pathkey);
    }

    return pathkeys;
}

/*
 * trim_mergeclauses_for_inner_pathkeys
 *    This routine trims a list of mergeclauses to include just those that
 *    work with a specified ordering for the join's inner relation.
 *
 * 'mergeclauses' is a list of RestrictInfos for mergejoin clauses for the
 *          join relation being formed, in an order known to work for the
 *          currently-considered sort ordering of the join's outer rel.
 * 'pathkeys' is a pathkeys list showing the ordering of an inner-rel path;
 *          it should be equal to, or a truncation of, the result of
 *          make_inner_pathkeys_for_merge for these mergeclauses.
 *
 * What we return will be a prefix of the given mergeclauses list.
 *
 * We need this logic because make_inner_pathkeys_for_merge's result isn't
 * necessarily in the same order as the mergeclauses.  That means that if we
 * consider an inner-rel pathkey list that is a truncation of that result,
 * we might need to drop mergeclauses even though they match a surviving inner
 * pathkey.  This happens when they are to the right of a mergeclause that
 * matches a removed inner pathkey.
 *
 * The mergeclauses must be marked (via outer_is_left) to show which side
 * of each clause is associated with the current outer path.  (See
 * select_mergejoin_clauses())
 */
List *
trim_mergeclauses_for_inner_pathkeys(PlannerInfo *root,
                                     List *mergeclauses,
                                     List *pathkeys)
{
    List       *new_mergeclauses = NIL;
    PathKey    *pathkey;
    EquivalenceClass *pathkey_ec;
    bool        matched_pathkey;
    ListCell   *lip;
    ListCell   *i;

    /* No pathkeys => no mergeclauses (though we don't expect this case) */
    if (pathkeys == NIL)
        return NIL;
    /* Initialize to consider first pathkey */
    lip = list_head(pathkeys);
    pathkey = (PathKey *) lfirst(lip);
    pathkey_ec = pathkey->pk_eclass;
    lip = lnext(pathkeys, lip);
    matched_pathkey = false;

    /* Scan mergeclauses to see how many we can use */
    foreach(i, mergeclauses)
    {
        RestrictInfo *rinfo = (RestrictInfo *) lfirst(i);
        EquivalenceClass *clause_ec;

        /* Assume we needn't do update_mergeclause_eclasses again here */

        /* Check clause's inner-rel EC against current pathkey */
        clause_ec = rinfo->outer_is_left ?
            rinfo->right_ec : rinfo->left_ec;

        /* If we don't have a match, attempt to advance to next pathkey */
        if (clause_ec != pathkey_ec)
        {
            /* If we had no clauses matching this inner pathkey, must stop */
            if (!matched_pathkey)
                break;

            /* Advance to next inner pathkey, if any */
            if (lip == NULL)
                break;
            pathkey = (PathKey *) lfirst(lip);
            pathkey_ec = pathkey->pk_eclass;
            lip = lnext(pathkeys, lip);
            matched_pathkey = false;
        }

        /* If mergeclause matches current inner pathkey, we can use it */
        if (clause_ec == pathkey_ec)
        {
            new_mergeclauses = lappend(new_mergeclauses, rinfo);
            matched_pathkey = true;
        }
        else
        {
            /* Else, no hope of adding any more mergeclauses */
            break;
        }
    }

    return new_mergeclauses;
}


/****************************************************************************
 *      PATHKEY USEFULNESS CHECKS
 *
 * We only want to remember as many of the pathkeys of a path as have some
 * potential use, either for subsequent mergejoins or for meeting the query's
 * requested output ordering.  This ensures that add_path() won't consider
 * a path to have a usefully different ordering unless it really is useful.
 * These routines check for usefulness of given pathkeys.
 ****************************************************************************/

/*
 * pathkeys_useful_for_merging
 *      Count the number of pathkeys that may be useful for mergejoins
 *      above the given relation.
 *
 * We consider a pathkey potentially useful if it corresponds to the merge
 * ordering of either side of any joinclause for the rel.  This might be
 * overoptimistic, since joinclauses that require different other relations
 * might never be usable at the same time, but trying to be exact is likely
 * to be more trouble than it's worth.
 *
 * To avoid doubling the number of mergejoin paths considered, we would like
 * to consider only one of the two scan directions (ASC or DESC) as useful
 * for merging for any given target column.  The choice is arbitrary unless
 * one of the directions happens to match an ORDER BY key, in which case
 * that direction should be preferred, in hopes of avoiding a final sort step.
 * right_merge_direction() implements this heuristic.
 */
static int
pathkeys_useful_for_merging(PlannerInfo *root, RelOptInfo *rel, List *pathkeys)
{
    int         useful = 0;
    ListCell   *i;

    foreach(i, pathkeys)
    {
        PathKey    *pathkey = (PathKey *) lfirst(i);
        bool        matched = false;
        ListCell   *j;

        /* If "wrong" direction, not useful for merging */
        if (!right_merge_direction(root, pathkey))
            break;

        /*
         * First look into the EquivalenceClass of the pathkey, to see if
         * there are any members not yet joined to the rel.  If so, it's
         * surely possible to generate a mergejoin clause using them.
         */
        if (rel->has_eclass_joins &&
            eclass_useful_for_merging(root, pathkey->pk_eclass, rel))
            matched = true;
        else
        {
            /*
             * Otherwise search the rel's joininfo list, which contains
             * non-EquivalenceClass-derivable join clauses that might
             * nonetheless be mergejoinable.
             */
            foreach(j, rel->joininfo)
            {
                RestrictInfo *restrictinfo = (RestrictInfo *) lfirst(j);

                if (restrictinfo->mergeopfamilies == NIL)
                    continue;
                update_mergeclause_eclasses(root, restrictinfo);

                if (pathkey->pk_eclass == restrictinfo->left_ec ||
                    pathkey->pk_eclass == restrictinfo->right_ec)
                {
                    matched = true;
                    break;
                }
            }
        }

        /*
         * If we didn't find a mergeclause, we're done --- any additional
         * sort-key positions in the pathkeys are useless.  (But we can still
         * mergejoin if we found at least one mergeclause.)
         */
        if (matched)
            useful++;
        else
            break;
    }

    return useful;
}

/*
 * right_merge_direction
 *      Check whether the pathkey embodies the preferred sort direction
 *      for merging its target column.
 */
static bool
right_merge_direction(PlannerInfo *root, PathKey *pathkey)
{
    ListCell   *l;

    foreach(l, root->query_pathkeys)
    {
        PathKey    *query_pathkey = (PathKey *) lfirst(l);

        if (pathkey->pk_eclass == query_pathkey->pk_eclass &&
            pathkey->pk_opfamily == query_pathkey->pk_opfamily)
        {
            /*
             * Found a matching query sort column.  Prefer this pathkey's
             * direction iff it matches.  Note that we ignore pk_nulls_first,
             * which means that a sort might be needed anyway ... but we still
             * want to prefer only one of the two possible directions, and we
             * might as well use this one.
             */
            return (pathkey->pk_strategy == query_pathkey->pk_strategy);
        }
    }

    /* If no matching ORDER BY request, prefer the ASC direction */
    return (pathkey->pk_strategy == BTLessStrategyNumber);
}

/*
 * pathkeys_useful_for_ordering
 *      Count the number of pathkeys that are useful for meeting the
 *      query's requested output ordering.
 *
 * Because we the have the possibility of incremental sort, a prefix list of
 * keys is potentially useful for improving the performance of the requested
 * ordering. Thus we return 0, if no valuable keys are found, or the number
 * of leading keys shared by the list and the requested ordering..
 */
static int
pathkeys_useful_for_ordering(PlannerInfo *root, List *pathkeys)
{
    int         n_common_pathkeys;

    if (root->query_pathkeys == NIL)
        return 0;               /* no special ordering requested */

    if (pathkeys == NIL)
        return 0;               /* unordered path */

    (void) pathkeys_count_contained_in(root->query_pathkeys, pathkeys,
                                       &n_common_pathkeys);

    return n_common_pathkeys;
}

/*
 * truncate_useless_pathkeys
 *      Shorten the given pathkey list to just the useful pathkeys.
 */
List *
truncate_useless_pathkeys(PlannerInfo *root,
                          RelOptInfo *rel,
                          List *pathkeys)
{
    int         nuseful;
    int         nuseful2;

    nuseful = pathkeys_useful_for_merging(root, rel, pathkeys);
    nuseful2 = pathkeys_useful_for_ordering(root, pathkeys);
    if (nuseful2 > nuseful)
        nuseful = nuseful2;

    /*
     * Note: not safe to modify input list destructively, but we can avoid
     * copying the list if we're not actually going to change it
     */
    if (nuseful == 0)
        return NIL;
    else if (nuseful == list_length(pathkeys))
        return pathkeys;
    else
        return list_truncate(list_copy(pathkeys), nuseful);
}

/*
 * has_useful_pathkeys
 *      Detect whether the specified rel could have any pathkeys that are
 *      useful according to truncate_useless_pathkeys().
 *
 * This is a cheap test that lets us skip building pathkeys at all in very
 * simple queries.  It's OK to err in the direction of returning "true" when
 * there really aren't any usable pathkeys, but erring in the other direction
 * is bad --- so keep this in sync with the routines above!
 *
 * We could make the test more complex, for example checking to see if any of
 * the joinclauses are really mergejoinable, but that likely wouldn't win
 * often enough to repay the extra cycles.  Queries with neither a join nor
 * a sort are reasonably common, though, so this much work seems worthwhile.
 */
bool
has_useful_pathkeys(PlannerInfo *root, RelOptInfo *rel)
{
    if (rel->joininfo != NIL || rel->has_eclass_joins)
        return true;            /* might be able to use pathkeys for merging */
    if (root->query_pathkeys != NIL)
        return true;            /* might be able to use them for ordering */
    return false;               /* definitely useless */
}