dependencies.c

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/*-------------------------------------------------------------------------
 *
 * dependencies.c
 *    POSTGRES functional dependencies
 *
 * Portions Copyright (c) 1996-2023, PostgreSQL Global Development Group
 * Portions Copyright (c) 1994, Regents of the University of California
 *
 * IDENTIFICATION
 *    src/backend/statistics/dependencies.c
 *
 *-------------------------------------------------------------------------
 */
#include "postgres.h"
 
#include "access/htup_details.h"
#include "access/sysattr.h"
#include "catalog/pg_operator.h"
#include "catalog/pg_statistic_ext.h"
#include "catalog/pg_statistic_ext_data.h"
#include "lib/stringinfo.h"
#include "nodes/nodeFuncs.h"
#include "nodes/nodes.h"
#include "nodes/pathnodes.h"
#include "optimizer/clauses.h"
#include "optimizer/optimizer.h"
#include "parser/parsetree.h"
#include "statistics/extended_stats_internal.h"
#include "statistics/statistics.h"
#include "utils/bytea.h"
#include "utils/fmgroids.h"
#include "utils/fmgrprotos.h"
#include "utils/lsyscache.h"
#include "utils/memutils.h"
#include "utils/selfuncs.h"
#include "utils/syscache.h"
#include "utils/typcache.h"
 
/* size of the struct header fields (magic, type, ndeps) */
#define SizeOfHeader        (3 * sizeof(uint32))
 
/* size of a serialized dependency (degree, natts, atts) */
#define SizeOfItem(natts) \
    (sizeof(double) + sizeof(AttrNumber) * (1 + (natts)))
 
/* minimal size of a dependency (with two attributes) */
#define MinSizeOfItem   SizeOfItem(2)
 
/* minimal size of dependencies, when all deps are minimal */
#define MinSizeOfItems(ndeps) \
    (SizeOfHeader + (ndeps) * MinSizeOfItem)
 
/*
 * Internal state for DependencyGenerator of dependencies. Dependencies are similar to
 * k-permutations of n elements, except that the order does not matter for the
 * first (k-1) elements. That is, (a,b=>c) and (b,a=>c) are equivalent.
 */
typedef struct DependencyGeneratorData
{
    int         k;              /* size of the dependency */
    int         n;              /* number of possible attributes */
    int         current;        /* next dependency to return (index) */
    AttrNumber  ndependencies;  /* number of dependencies generated */
    AttrNumber *dependencies;   /* array of pre-generated dependencies  */
} DependencyGeneratorData;
 
typedef DependencyGeneratorData *DependencyGenerator;
 
static void generate_dependencies_recurse(DependencyGenerator state,
                                          int index, AttrNumber start, AttrNumber *current);
static void generate_dependencies(DependencyGenerator state);
static DependencyGenerator DependencyGenerator_init(int n, int k);
static void DependencyGenerator_free(DependencyGenerator state);
static AttrNumber *DependencyGenerator_next(DependencyGenerator state);
static double dependency_degree(StatsBuildData *data, int k, AttrNumber *dependency);
static bool dependency_is_fully_matched(MVDependency *dependency,
                                        Bitmapset *attnums);
static bool dependency_is_compatible_clause(Node *clause, Index relid,
                                            AttrNumber *attnum);
static bool dependency_is_compatible_expression(Node *clause, Index relid,
                                                List *statlist, Node **expr);
static MVDependency *find_strongest_dependency(MVDependencies **dependencies,
                                               int ndependencies, Bitmapset *attnums);
static Selectivity clauselist_apply_dependencies(PlannerInfo *root, List *clauses,
                                                 int varRelid, JoinType jointype,
                                                 SpecialJoinInfo *sjinfo,
                                                 MVDependency **dependencies,
                                                 int ndependencies,
                                                 AttrNumber *list_attnums,
                                                 Bitmapset **estimatedclauses);
 
static void
generate_dependencies_recurse(DependencyGenerator state, int index,
                              AttrNumber start, AttrNumber *current)
{
    /*
     * The generator handles the first (k-1) elements differently from the
     * last element.
     */
    if (index < (state->k - 1))
    {
        AttrNumber  i;
 
        /*
         * The first (k-1) values have to be in ascending order, which we
         * generate recursively.
         */
 
        for (i = start; i < state->n; i++)
        {
            current[index] = i;
            generate_dependencies_recurse(state, (index + 1), (i + 1), current);
        }
    }
    else
    {
        int         i;
 
        /*
         * the last element is the implied value, which does not respect the
         * ascending order. We just need to check that the value is not in the
         * first (k-1) elements.
         */
 
        for (i = 0; i < state->n; i++)
        {
            int         j;
            bool        match = false;
 
            current[index] = i;
 
            for (j = 0; j < index; j++)
            {
                if (current[j] == i)
                {
                    match = true;
                    break;
                }
            }
 
            /*
             * If the value is not found in the first part of the dependency,
             * we're done.
             */
            if (!match)
            {
                state->dependencies = (AttrNumber *) repalloc(state->dependencies,
                                                              state->k * (state->ndependencies + 1) * sizeof(AttrNumber));
                memcpy(&state->dependencies[(state->k * state->ndependencies)],
                       current, state->k * sizeof(AttrNumber));
                state->ndependencies++;
            }
        }
    }
}
 
/* generate all dependencies (k-permutations of n elements) */
static void
generate_dependencies(DependencyGenerator state)
{
    AttrNumber *current = (AttrNumber *) palloc0(sizeof(AttrNumber) * state->k);
 
    generate_dependencies_recurse(state, 0, 0, current);
 
    pfree(current);
}
 
/*
 * initialize the DependencyGenerator of variations, and prebuild the variations
 *
 * This pre-builds all the variations. We could also generate them in
 * DependencyGenerator_next(), but this seems simpler.
 */
static DependencyGenerator
DependencyGenerator_init(int n, int k)
{
    DependencyGenerator state;
 
    Assert((n >= k) && (k > 0));
 
    /* allocate the DependencyGenerator state */
    state = (DependencyGenerator) palloc0(sizeof(DependencyGeneratorData));
    state->dependencies = (AttrNumber *) palloc(k * sizeof(AttrNumber));
 
    state->ndependencies = 0;
    state->current = 0;
    state->k = k;
    state->n = n;
 
    /* now actually pre-generate all the variations */
    generate_dependencies(state);
 
    return state;
}
 
/* free the DependencyGenerator state */
static void
DependencyGenerator_free(DependencyGenerator state)
{
    pfree(state->dependencies);
    pfree(state);
}
 
/* generate next combination */
static AttrNumber *
DependencyGenerator_next(DependencyGenerator state)
{
    if (state->current == state->ndependencies)
        return NULL;
 
    return &state->dependencies[state->k * state->current++];
}
 
 
/*
 * validates functional dependency on the data
 *
 * An actual work horse of detecting functional dependencies. Given a variation
 * of k attributes, it checks that the first (k-1) are sufficient to determine
 * the last one.
 */
static double
dependency_degree(StatsBuildData *data, int k, AttrNumber *dependency)
{
    int         i,
                nitems;
    MultiSortSupport mss;
    SortItem   *items;
    AttrNumber *attnums_dep;
 
    /* counters valid within a group */
    int         group_size = 0;
    int         n_violations = 0;
 
    /* total number of rows supporting (consistent with) the dependency */
    int         n_supporting_rows = 0;
 
    /* Make sure we have at least two input attributes. */
    Assert(k >= 2);
 
    /* sort info for all attributes columns */
    mss = multi_sort_init(k);
 
    /*
     * Translate the array of indexes to regular attnums for the dependency
     * (we will need this to identify the columns in StatsBuildData).
     */
    attnums_dep = (AttrNumber *) palloc(k * sizeof(AttrNumber));
    for (i = 0; i < k; i++)
        attnums_dep[i] = data->attnums[dependency[i]];
 
    /*
     * Verify the dependency (a,b,...)->z, using a rather simple algorithm:
     *
     * (a) sort the data lexicographically
     *
     * (b) split the data into groups by first (k-1) columns
     *
     * (c) for each group count different values in the last column
     *
     * We use the column data types' default sort operators and collations;
     * perhaps at some point it'd be worth using column-specific collations?
     */
 
    /* prepare the sort function for the dimensions */
    for (i = 0; i < k; i++)
    {
        VacAttrStats *colstat = data->stats[dependency[i]];
        TypeCacheEntry *type;
 
        type = lookup_type_cache(colstat->attrtypid, TYPECACHE_LT_OPR);
        if (type->lt_opr == InvalidOid) /* shouldn't happen */
            elog(ERROR, "cache lookup failed for ordering operator for type %u",
                 colstat->attrtypid);
 
        /* prepare the sort function for this dimension */
        multi_sort_add_dimension(mss, i, type->lt_opr, colstat->attrcollid);
    }
 
    /*
     * build an array of SortItem(s) sorted using the multi-sort support
     *
     * XXX This relies on all stats entries pointing to the same tuple
     * descriptor.  For now that assumption holds, but it might change in the
     * future for example if we support statistics on multiple tables.
     */
    items = build_sorted_items(data, &nitems, mss, k, attnums_dep);
 
    /*
     * Walk through the sorted array, split it into rows according to the
     * first (k-1) columns. If there's a single value in the last column, we
     * count the group as 'supporting' the functional dependency. Otherwise we
     * count it as contradicting.
     */
 
    /* start with the first row forming a group */
    group_size = 1;
 
    /* loop 1 beyond the end of the array so that we count the final group */
    for (i = 1; i <= nitems; i++)
    {
        /*
         * Check if the group ended, which may be either because we processed
         * all the items (i==nitems), or because the i-th item is not equal to
         * the preceding one.
         */
        if (i == nitems ||
            multi_sort_compare_dims(0, k - 2, &items[i - 1], &items[i], mss) != 0)
        {
            /*
             * If no violations were found in the group then track the rows of
             * the group as supporting the functional dependency.
             */
            if (n_violations == 0)
                n_supporting_rows += group_size;
 
            /* Reset counters for the new group */
            n_violations = 0;
            group_size = 1;
            continue;
        }
        /* first columns match, but the last one does not (so contradicting) */
        else if (multi_sort_compare_dim(k - 1, &items[i - 1], &items[i], mss) != 0)
            n_violations++;
 
        group_size++;
    }
 
    /* Compute the 'degree of validity' as (supporting/total). */
    return (n_supporting_rows * 1.0 / data->numrows);
}
 
/*
 * detects functional dependencies between groups of columns
 *
 * Generates all possible subsets of columns (variations) and computes
 * the degree of validity for each one. For example when creating statistics
 * on three columns (a,b,c) there are 9 possible dependencies
 *
 *     two columns            three columns
 *     -----------            -------------
 *     (a) -> b               (a,b) -> c
 *     (a) -> c               (a,c) -> b
 *     (b) -> a               (b,c) -> a
 *     (b) -> c
 *     (c) -> a
 *     (c) -> b
 */
MVDependencies *
statext_dependencies_build(StatsBuildData *data)
{
    int         i,
                k;
 
    /* result */
    MVDependencies *dependencies = NULL;
    MemoryContext cxt;
 
    Assert(data->nattnums >= 2);
 
    /* tracks memory allocated by dependency_degree calls */
    cxt = AllocSetContextCreate(CurrentMemoryContext,
                                "dependency_degree cxt",
                                ALLOCSET_DEFAULT_SIZES);
 
    /*
     * We'll try build functional dependencies starting from the smallest ones
     * covering just 2 columns, to the largest ones, covering all columns
     * included in the statistics object.  We start from the smallest ones
     * because we want to be able to skip already implied ones.
     */
    for (k = 2; k <= data->nattnums; k++)
    {
        AttrNumber *dependency; /* array with k elements */
 
        /* prepare a DependencyGenerator of variation */
        DependencyGenerator DependencyGenerator = DependencyGenerator_init(data->nattnums, k);
 
        /* generate all possible variations of k values (out of n) */
        while ((dependency = DependencyGenerator_next(DependencyGenerator)))
        {
            double      degree;
            MVDependency *d;
            MemoryContext oldcxt;
 
            /* release memory used by dependency degree calculation */
            oldcxt = MemoryContextSwitchTo(cxt);
 
            /* compute how valid the dependency seems */
            degree = dependency_degree(data, k, dependency);
 
            MemoryContextSwitchTo(oldcxt);
            MemoryContextReset(cxt);
 
            /*
             * if the dependency seems entirely invalid, don't store it
             */
            if (degree == 0.0)
                continue;
 
            d = (MVDependency *) palloc0(offsetof(MVDependency, attributes)
                                         + k * sizeof(AttrNumber));
 
            /* copy the dependency (and keep the indexes into stxkeys) */
            d->degree = degree;
            d->nattributes = k;
            for (i = 0; i < k; i++)
                d->attributes[i] = data->attnums[dependency[i]];
 
            /* initialize the list of dependencies */
            if (dependencies == NULL)
            {
                dependencies
                    = (MVDependencies *) palloc0(sizeof(MVDependencies));
 
                dependencies->magic = STATS_DEPS_MAGIC;
                dependencies->type = STATS_DEPS_TYPE_BASIC;
                dependencies->ndeps = 0;
            }
 
            dependencies->ndeps++;
            dependencies = (MVDependencies *) repalloc(dependencies,
                                                       offsetof(MVDependencies, deps)
                                                       + dependencies->ndeps * sizeof(MVDependency *));
 
            dependencies->deps[dependencies->ndeps - 1] = d;
        }
 
        /*
         * we're done with variations of k elements, so free the
         * DependencyGenerator
         */
        DependencyGenerator_free(DependencyGenerator);
    }
 
    MemoryContextDelete(cxt);
 
    return dependencies;
}
 
 
/*
 * Serialize list of dependencies into a bytea value.
 */
bytea *
statext_dependencies_serialize(MVDependencies *dependencies)
{
    int         i;
    bytea      *output;
    char       *tmp;
    Size        len;
 
    /* we need to store ndeps, with a number of attributes for each one */
    len = VARHDRSZ + SizeOfHeader;
 
    /* and also include space for the actual attribute numbers and degrees */
    for (i = 0; i < dependencies->ndeps; i++)
        len += SizeOfItem(dependencies->deps[i]->nattributes);
 
    output = (bytea *) palloc0(len);
    SET_VARSIZE(output, len);
 
    tmp = VARDATA(output);
 
    /* Store the base struct values (magic, type, ndeps) */
    memcpy(tmp, &dependencies->magic, sizeof(uint32));
    tmp += sizeof(uint32);
    memcpy(tmp, &dependencies->type, sizeof(uint32));
    tmp += sizeof(uint32);
    memcpy(tmp, &dependencies->ndeps, sizeof(uint32));
    tmp += sizeof(uint32);
 
    /* store number of attributes and attribute numbers for each dependency */
    for (i = 0; i < dependencies->ndeps; i++)
    {
        MVDependency *d = dependencies->deps[i];
 
        memcpy(tmp, &d->degree, sizeof(double));
        tmp += sizeof(double);
 
        memcpy(tmp, &d->nattributes, sizeof(AttrNumber));
        tmp += sizeof(AttrNumber);
 
        memcpy(tmp, d->attributes, sizeof(AttrNumber) * d->nattributes);
        tmp += sizeof(AttrNumber) * d->nattributes;
 
        /* protect against overflow */
        Assert(tmp <= ((char *) output + len));
    }
 
    /* make sure we've produced exactly the right amount of data */
    Assert(tmp == ((char *) output + len));
 
    return output;
}
 
/*
 * Reads serialized dependencies into MVDependencies structure.
 */
MVDependencies *
statext_dependencies_deserialize(bytea *data)
{
    int         i;
    Size        min_expected_size;
    MVDependencies *dependencies;
    char       *tmp;
 
    if (data == NULL)
        return NULL;
 
    if (VARSIZE_ANY_EXHDR(data) < SizeOfHeader)
        elog(ERROR, "invalid MVDependencies size %zu (expected at least %zu)",
             VARSIZE_ANY_EXHDR(data), SizeOfHeader);
 
    /* read the MVDependencies header */
    dependencies = (MVDependencies *) palloc0(sizeof(MVDependencies));
 
    /* initialize pointer to the data part (skip the varlena header) */
    tmp = VARDATA_ANY(data);
 
    /* read the header fields and perform basic sanity checks */
    memcpy(&dependencies->magic, tmp, sizeof(uint32));
    tmp += sizeof(uint32);
    memcpy(&dependencies->type, tmp, sizeof(uint32));
    tmp += sizeof(uint32);
    memcpy(&dependencies->ndeps, tmp, sizeof(uint32));
    tmp += sizeof(uint32);
 
    if (dependencies->magic != STATS_DEPS_MAGIC)
        elog(ERROR, "invalid dependency magic %d (expected %d)",
             dependencies->magic, STATS_DEPS_MAGIC);
 
    if (dependencies->type != STATS_DEPS_TYPE_BASIC)
        elog(ERROR, "invalid dependency type %d (expected %d)",
             dependencies->type, STATS_DEPS_TYPE_BASIC);
 
    if (dependencies->ndeps == 0)
        elog(ERROR, "invalid zero-length item array in MVDependencies");
 
    /* what minimum bytea size do we expect for those parameters */
    min_expected_size = SizeOfItem(dependencies->ndeps);
 
    if (VARSIZE_ANY_EXHDR(data) < min_expected_size)
        elog(ERROR, "invalid dependencies size %zu (expected at least %zu)",
             VARSIZE_ANY_EXHDR(data), min_expected_size);
 
    /* allocate space for the MCV items */
    dependencies = repalloc(dependencies, offsetof(MVDependencies, deps)
                            + (dependencies->ndeps * sizeof(MVDependency *)));
 
    for (i = 0; i < dependencies->ndeps; i++)
    {
        double      degree;
        AttrNumber  k;
        MVDependency *d;
 
        /* degree of validity */
        memcpy(&degree, tmp, sizeof(double));
        tmp += sizeof(double);
 
        /* number of attributes */
        memcpy(&k, tmp, sizeof(AttrNumber));
        tmp += sizeof(AttrNumber);
 
        /* is the number of attributes valid? */
        Assert((k >= 2) && (k <= STATS_MAX_DIMENSIONS));
 
        /* now that we know the number of attributes, allocate the dependency */
        d = (MVDependency *) palloc0(offsetof(MVDependency, attributes)
                                     + (k * sizeof(AttrNumber)));
 
        d->degree = degree;
        d->nattributes = k;
 
        /* copy attribute numbers */
        memcpy(d->attributes, tmp, sizeof(AttrNumber) * d->nattributes);
        tmp += sizeof(AttrNumber) * d->nattributes;
 
        dependencies->deps[i] = d;
 
        /* still within the bytea */
        Assert(tmp <= ((char *) data + VARSIZE_ANY(data)));
    }
 
    /* we should have consumed the whole bytea exactly */
    Assert(tmp == ((char *) data + VARSIZE_ANY(data)));
 
    return dependencies;
}
 
/*
 * dependency_is_fully_matched
 *      checks that a functional dependency is fully matched given clauses on
 *      attributes (assuming the clauses are suitable equality clauses)
 */
static bool
dependency_is_fully_matched(MVDependency *dependency, Bitmapset *attnums)
{
    int         j;
 
    /*
     * Check that the dependency actually is fully covered by clauses. We have
     * to translate all attribute numbers, as those are referenced
     */
    for (j = 0; j < dependency->nattributes; j++)
    {
        int         attnum = dependency->attributes[j];
 
        if (!bms_is_member(attnum, attnums))
            return false;
    }
 
    return true;
}
 
/*
 * statext_dependencies_load
 *      Load the functional dependencies for the indicated pg_statistic_ext tuple
 */
MVDependencies *
statext_dependencies_load(Oid mvoid, bool inh)
{
    MVDependencies *result;
    bool        isnull;
    Datum       deps;
    HeapTuple   htup;
 
    htup = SearchSysCache2(STATEXTDATASTXOID,
                           ObjectIdGetDatum(mvoid),
                           BoolGetDatum(inh));
    if (!HeapTupleIsValid(htup))
        elog(ERROR, "cache lookup failed for statistics object %u", mvoid);
 
    deps = SysCacheGetAttr(STATEXTDATASTXOID, htup,
                           Anum_pg_statistic_ext_data_stxddependencies, &isnull);
    if (isnull)
        elog(ERROR,
             "requested statistics kind \"%c\" is not yet built for statistics object %u",
             STATS_EXT_DEPENDENCIES, mvoid);
 
    result = statext_dependencies_deserialize(DatumGetByteaPP(deps));
 
    ReleaseSysCache(htup);
 
    return result;
}
 
/*
 * pg_dependencies_in       - input routine for type pg_dependencies.
 *
 * pg_dependencies is real enough to be a table column, but it has no operations
 * of its own, and disallows input too
 */
Datum
pg_dependencies_in(PG_FUNCTION_ARGS)
{
    /*
     * pg_node_list stores the data in binary form and parsing text input is
     * not needed, so disallow this.
     */
    ereport(ERROR,
            (errcode(ERRCODE_FEATURE_NOT_SUPPORTED),
             errmsg("cannot accept a value of type %s", "pg_dependencies")));
 
    PG_RETURN_VOID();           /* keep compiler quiet */
}
 
/*
 * pg_dependencies      - output routine for type pg_dependencies.
 */
Datum
pg_dependencies_out(PG_FUNCTION_ARGS)
{
    bytea      *data = PG_GETARG_BYTEA_PP(0);
    MVDependencies *dependencies = statext_dependencies_deserialize(data);
    int         i,
                j;
    StringInfoData str;
 
    initStringInfo(&str);
    appendStringInfoChar(&str, '{');
 
    for (i = 0; i < dependencies->ndeps; i++)
    {
        MVDependency *dependency = dependencies->deps[i];
 
        if (i > 0)
            appendStringInfoString(&str, ", ");
 
        appendStringInfoChar(&str, '"');
        for (j = 0; j < dependency->nattributes; j++)
        {
            if (j == dependency->nattributes - 1)
                appendStringInfoString(&str, " => ");
            else if (j > 0)
                appendStringInfoString(&str, ", ");
 
            appendStringInfo(&str, "%d", dependency->attributes[j]);
        }
        appendStringInfo(&str, "\": %f", dependency->degree);
    }
 
    appendStringInfoChar(&str, '}');
 
    PG_RETURN_CSTRING(str.data);
}
 
/*
 * pg_dependencies_recv     - binary input routine for type pg_dependencies.
 */
Datum
pg_dependencies_recv(PG_FUNCTION_ARGS)
{
    ereport(ERROR,
            (errcode(ERRCODE_FEATURE_NOT_SUPPORTED),
             errmsg("cannot accept a value of type %s", "pg_dependencies")));
 
    PG_RETURN_VOID();           /* keep compiler quiet */
}
 
/*
 * pg_dependencies_send     - binary output routine for type pg_dependencies.
 *
 * Functional dependencies are serialized in a bytea value (although the type
 * is named differently), so let's just send that.
 */
Datum
pg_dependencies_send(PG_FUNCTION_ARGS)
{
    return byteasend(fcinfo);
}
 
/*
 * dependency_is_compatible_clause
 *      Determines if the clause is compatible with functional dependencies
 *
 * Only clauses that have the form of equality to a pseudoconstant, or can be
 * interpreted that way, are currently accepted.  Furthermore the variable
 * part of the clause must be a simple Var belonging to the specified
 * relation, whose attribute number we return in *attnum on success.
 */
static bool
dependency_is_compatible_clause(Node *clause, Index relid, AttrNumber *attnum)
{
    Var        *var;
    Node       *clause_expr;
 
    if (IsA(clause, RestrictInfo))
    {
        RestrictInfo *rinfo = (RestrictInfo *) clause;
 
        /* Pseudoconstants are not interesting (they couldn't contain a Var) */
        if (rinfo->pseudoconstant)
            return false;
 
        /* Clauses referencing multiple, or no, varnos are incompatible */
        if (bms_membership(rinfo->clause_relids) != BMS_SINGLETON)
            return false;
 
        clause = (Node *) rinfo->clause;
    }
 
    if (is_opclause(clause))
    {
        /* If it's an opclause, check for Var = Const or Const = Var. */
        OpExpr     *expr = (OpExpr *) clause;
 
        /* Only expressions with two arguments are candidates. */
        if (list_length(expr->args) != 2)
            return false;
 
        /* Make sure non-selected argument is a pseudoconstant. */
        if (is_pseudo_constant_clause(lsecond(expr->args)))
            clause_expr = linitial(expr->args);
        else if (is_pseudo_constant_clause(linitial(expr->args)))
            clause_expr = lsecond(expr->args);
        else
            return false;
 
        /*
         * If it's not an "=" operator, just ignore the clause, as it's not
         * compatible with functional dependencies.
         *
         * This uses the function for estimating selectivity, not the operator
         * directly (a bit awkward, but well ...).
         *
         * XXX this is pretty dubious; probably it'd be better to check btree
         * or hash opclass membership, so as not to be fooled by custom
         * selectivity functions, and to be more consistent with decisions
         * elsewhere in the planner.
         */
        if (get_oprrest(expr->opno) != F_EQSEL)
            return false;
 
        /* OK to proceed with checking "var" */
    }
    else if (IsA(clause, ScalarArrayOpExpr))
    {
        /* If it's an scalar array operator, check for Var IN Const. */
        ScalarArrayOpExpr *expr = (ScalarArrayOpExpr *) clause;
 
        /*
         * Reject ALL() variant, we only care about ANY/IN.
         *
         * XXX Maybe we should check if all the values are the same, and allow
         * ALL in that case? Doesn't seem very practical, though.
         */
        if (!expr->useOr)
            return false;
 
        /* Only expressions with two arguments are candidates. */
        if (list_length(expr->args) != 2)
            return false;
 
        /*
         * We know it's always (Var IN Const), so we assume the var is the
         * first argument, and pseudoconstant is the second one.
         */
        if (!is_pseudo_constant_clause(lsecond(expr->args)))
            return false;
 
        clause_expr = linitial(expr->args);
 
        /*
         * If it's not an "=" operator, just ignore the clause, as it's not
         * compatible with functional dependencies. The operator is identified
         * simply by looking at which function it uses to estimate
         * selectivity. That's a bit strange, but it's what other similar
         * places do.
         */
        if (get_oprrest(expr->opno) != F_EQSEL)
            return false;
 
        /* OK to proceed with checking "var" */
    }
    else if (is_orclause(clause))
    {
        BoolExpr   *bool_expr = (BoolExpr *) clause;
        ListCell   *lc;
 
        /* start with no attribute number */
        *attnum = InvalidAttrNumber;
 
        foreach(lc, bool_expr->args)
        {
            AttrNumber  clause_attnum;
 
            /*
             * Had we found incompatible clause in the arguments, treat the
             * whole clause as incompatible.
             */
            if (!dependency_is_compatible_clause((Node *) lfirst(lc),
                                                 relid, &clause_attnum))
                return false;
 
            if (*attnum == InvalidAttrNumber)
                *attnum = clause_attnum;
 
            /* ensure all the variables are the same (same attnum) */
            if (*attnum != clause_attnum)
                return false;
        }
 
        /* the Var is already checked by the recursive call */
        return true;
    }
    else if (is_notclause(clause))
    {
        /*
         * "NOT x" can be interpreted as "x = false", so get the argument and
         * proceed with seeing if it's a suitable Var.
         */
        clause_expr = (Node *) get_notclausearg(clause);
    }
    else
    {
        /*
         * A boolean expression "x" can be interpreted as "x = true", so
         * proceed with seeing if it's a suitable Var.
         */
        clause_expr = (Node *) clause;
    }
 
    /*
     * We may ignore any RelabelType node above the operand.  (There won't be
     * more than one, since eval_const_expressions has been applied already.)
     */
    if (IsA(clause_expr, RelabelType))
        clause_expr = (Node *) ((RelabelType *) clause_expr)->arg;
 
    /* We only support plain Vars for now */
    if (!IsA(clause_expr, Var))
        return false;
 
    /* OK, we know we have a Var */
    var = (Var *) clause_expr;
 
    /* Ensure Var is from the correct relation */
    if (var->varno != relid)
        return false;
 
    /* We also better ensure the Var is from the current level */
    if (var->varlevelsup != 0)
        return false;
 
    /* Also ignore system attributes (we don't allow stats on those) */
    if (!AttrNumberIsForUserDefinedAttr(var->varattno))
        return false;
 
    *attnum = var->varattno;
    return true;
}
 
/*
 * find_strongest_dependency
 *      find the strongest dependency on the attributes
 *
 * When applying functional dependencies, we start with the strongest
 * dependencies. That is, we select the dependency that:
 *
 * (a) has all attributes covered by equality clauses
 *
 * (b) has the most attributes
 *
 * (c) has the highest degree of validity
 *
 * This guarantees that we eliminate the most redundant conditions first
 * (see the comment in dependencies_clauselist_selectivity).
 */
static MVDependency *
find_strongest_dependency(MVDependencies **dependencies, int ndependencies,
                          Bitmapset *attnums)
{
    int         i,
                j;
    MVDependency *strongest = NULL;
 
    /* number of attnums in clauses */
    int         nattnums = bms_num_members(attnums);
 
    /*
     * Iterate over the MVDependency items and find the strongest one from the
     * fully-matched dependencies. We do the cheap checks first, before
     * matching it against the attnums.
     */
    for (i = 0; i < ndependencies; i++)
    {
        for (j = 0; j < dependencies[i]->ndeps; j++)
        {
            MVDependency *dependency = dependencies[i]->deps[j];
 
            /*
             * Skip dependencies referencing more attributes than available
             * clauses, as those can't be fully matched.
             */
            if (dependency->nattributes > nattnums)
                continue;
 
            if (strongest)
            {
                /* skip dependencies on fewer attributes than the strongest. */
                if (dependency->nattributes < strongest->nattributes)
                    continue;
 
                /* also skip weaker dependencies when attribute count matches */
                if (strongest->nattributes == dependency->nattributes &&
                    strongest->degree > dependency->degree)
                    continue;
            }
 
            /*
             * this dependency is stronger, but we must still check that it's
             * fully matched to these attnums. We perform this check last as
             * it's slightly more expensive than the previous checks.
             */
            if (dependency_is_fully_matched(dependency, attnums))
                strongest = dependency; /* save new best match */
        }
    }
 
    return strongest;
}
 
/*
 * clauselist_apply_dependencies
 *      Apply the specified functional dependencies to a list of clauses and
 *      return the estimated selectivity of the clauses that are compatible
 *      with any of the given dependencies.
 *
 * This will estimate all not-already-estimated clauses that are compatible
 * with functional dependencies, and which have an attribute mentioned by any
 * of the given dependencies (either as an implying or implied attribute).
 *
 * Given (lists of) clauses on attributes (a,b) and a functional dependency
 * (a=>b), the per-column selectivities P(a) and P(b) are notionally combined
 * using the formula
 *
 *      P(a,b) = f * P(a) + (1-f) * P(a) * P(b)
 *
 * where 'f' is the degree of dependency.  This reflects the fact that we
 * expect a fraction f of all rows to be consistent with the dependency
 * (a=>b), and so have a selectivity of P(a), while the remaining rows are
 * treated as independent.
 *
 * In practice, we use a slightly modified version of this formula, which uses
 * a selectivity of Min(P(a), P(b)) for the dependent rows, since the result
 * should obviously not exceed either column's individual selectivity.  I.e.,
 * we actually combine selectivities using the formula
 *
 *      P(a,b) = f * Min(P(a), P(b)) + (1-f) * P(a) * P(b)
 *
 * This can make quite a difference if the specific values matching the
 * clauses are not consistent with the functional dependency.
 */
static Selectivity
clauselist_apply_dependencies(PlannerInfo *root, List *clauses,
                              int varRelid, JoinType jointype,
                              SpecialJoinInfo *sjinfo,
                              MVDependency **dependencies, int ndependencies,
                              AttrNumber *list_attnums,
                              Bitmapset **estimatedclauses)
{
    Bitmapset  *attnums;
    int         i;
    int         j;
    int         nattrs;
    Selectivity *attr_sel;
    int         attidx;
    int         listidx;
    ListCell   *l;
    Selectivity s1;
 
    /*
     * Extract the attnums of all implying and implied attributes from all the
     * given dependencies.  Each of these attributes is expected to have at
     * least 1 not-already-estimated compatible clause that we will estimate
     * here.
     */
    attnums = NULL;
    for (i = 0; i < ndependencies; i++)
    {
        for (j = 0; j < dependencies[i]->nattributes; j++)
        {
            AttrNumber  attnum = dependencies[i]->attributes[j];
 
            attnums = bms_add_member(attnums, attnum);
        }
    }
 
    /*
     * Compute per-column selectivity estimates for each of these attributes,
     * and mark all the corresponding clauses as estimated.
     */
    nattrs = bms_num_members(attnums);
    attr_sel = (Selectivity *) palloc(sizeof(Selectivity) * nattrs);
 
    attidx = 0;
    i = -1;
    while ((i = bms_next_member(attnums, i)) >= 0)
    {
        List       *attr_clauses = NIL;
        Selectivity simple_sel;
 
        listidx = -1;
        foreach(l, clauses)
        {
            Node       *clause = (Node *) lfirst(l);
 
            listidx++;
            if (list_attnums[listidx] == i)
            {
                attr_clauses = lappend(attr_clauses, clause);
                *estimatedclauses = bms_add_member(*estimatedclauses, listidx);
            }
        }
 
        simple_sel = clauselist_selectivity_ext(root, attr_clauses, varRelid,
                                                jointype, sjinfo, false);
        attr_sel[attidx++] = simple_sel;
    }
 
    /*
     * Now combine these selectivities using the dependency information.  For
     * chains of dependencies such as a -> b -> c, the b -> c dependency will
     * come before the a -> b dependency in the array, so we traverse the
     * array backwards to ensure such chains are computed in the right order.
     *
     * As explained above, pairs of selectivities are combined using the
     * formula
     *
     * P(a,b) = f * Min(P(a), P(b)) + (1-f) * P(a) * P(b)
     *
     * to ensure that the combined selectivity is never greater than either
     * individual selectivity.
     *
     * Where multiple dependencies apply (e.g., a -> b -> c), we use
     * conditional probabilities to compute the overall result as follows:
     *
     * P(a,b,c) = P(c|a,b) * P(a,b) = P(c|a,b) * P(b|a) * P(a)
     *
     * so we replace the selectivities of all implied attributes with
     * conditional probabilities, that are conditional on all their implying
     * attributes.  The selectivities of all other non-implied attributes are
     * left as they are.
     */
    for (i = ndependencies - 1; i >= 0; i--)
    {
        MVDependency *dependency = dependencies[i];
        AttrNumber  attnum;
        Selectivity s2;
        double      f;
 
        /* Selectivity of all the implying attributes */
        s1 = 1.0;
        for (j = 0; j < dependency->nattributes - 1; j++)
        {
            attnum = dependency->attributes[j];
            attidx = bms_member_index(attnums, attnum);
            s1 *= attr_sel[attidx];
        }
 
        /* Original selectivity of the implied attribute */
        attnum = dependency->attributes[j];
        attidx = bms_member_index(attnums, attnum);
        s2 = attr_sel[attidx];
 
        /*
         * Replace s2 with the conditional probability s2 given s1, computed
         * using the formula P(b|a) = P(a,b) / P(a), which simplifies to
         *
         * P(b|a) = f * Min(P(a), P(b)) / P(a) + (1-f) * P(b)
         *
         * where P(a) = s1, the selectivity of the implying attributes, and
         * P(b) = s2, the selectivity of the implied attribute.
         */
        f = dependency->degree;
 
        if (s1 <= s2)
            attr_sel[attidx] = f + (1 - f) * s2;
        else
            attr_sel[attidx] = f * s2 / s1 + (1 - f) * s2;
    }
 
    /*
     * The overall selectivity of all the clauses on all these attributes is
     * then the product of all the original (non-implied) probabilities and
     * the new conditional (implied) probabilities.
     */
    s1 = 1.0;
    for (i = 0; i < nattrs; i++)
        s1 *= attr_sel[i];
 
    CLAMP_PROBABILITY(s1);
 
    pfree(attr_sel);
    bms_free(attnums);
 
    return s1;
}
 
/*
 * dependency_is_compatible_expression
 *      Determines if the expression is compatible with functional dependencies
 *
 * Similar to dependency_is_compatible_clause, but doesn't enforce that the
 * expression is a simple Var.  On success, return the matching statistics
 * expression into *expr.
 */
static bool
dependency_is_compatible_expression(Node *clause, Index relid, List *statlist, Node **expr)
{
    ListCell   *lc,
               *lc2;
    Node       *clause_expr;
 
    if (IsA(clause, RestrictInfo))
    {
        RestrictInfo *rinfo = (RestrictInfo *) clause;
 
        /* Pseudoconstants are not interesting (they couldn't contain a Var) */
        if (rinfo->pseudoconstant)
            return false;
 
        /* Clauses referencing multiple, or no, varnos are incompatible */
        if (bms_membership(rinfo->clause_relids) != BMS_SINGLETON)
            return false;
 
        clause = (Node *) rinfo->clause;
    }
 
    if (is_opclause(clause))
    {
        /* If it's an opclause, check for Var = Const or Const = Var. */
        OpExpr     *expr = (OpExpr *) clause;
 
        /* Only expressions with two arguments are candidates. */
        if (list_length(expr->args) != 2)
            return false;
 
        /* Make sure non-selected argument is a pseudoconstant. */
        if (is_pseudo_constant_clause(lsecond(expr->args)))
            clause_expr = linitial(expr->args);
        else if (is_pseudo_constant_clause(linitial(expr->args)))
            clause_expr = lsecond(expr->args);
        else
            return false;
 
        /*
         * If it's not an "=" operator, just ignore the clause, as it's not
         * compatible with functional dependencies.
         *
         * This uses the function for estimating selectivity, not the operator
         * directly (a bit awkward, but well ...).
         *
         * XXX this is pretty dubious; probably it'd be better to check btree
         * or hash opclass membership, so as not to be fooled by custom
         * selectivity functions, and to be more consistent with decisions
         * elsewhere in the planner.
         */
        if (get_oprrest(expr->opno) != F_EQSEL)
            return false;
 
        /* OK to proceed with checking "var" */
    }
    else if (IsA(clause, ScalarArrayOpExpr))
    {
        /* If it's an scalar array operator, check for Var IN Const. */
        ScalarArrayOpExpr *expr = (ScalarArrayOpExpr *) clause;
 
        /*
         * Reject ALL() variant, we only care about ANY/IN.
         *
         * FIXME Maybe we should check if all the values are the same, and
         * allow ALL in that case? Doesn't seem very practical, though.
         */
        if (!expr->useOr)
            return false;
 
        /* Only expressions with two arguments are candidates. */
        if (list_length(expr->args) != 2)
            return false;
 
        /*
         * We know it's always (Var IN Const), so we assume the var is the
         * first argument, and pseudoconstant is the second one.
         */
        if (!is_pseudo_constant_clause(lsecond(expr->args)))
            return false;
 
        clause_expr = linitial(expr->args);
 
        /*
         * If it's not an "=" operator, just ignore the clause, as it's not
         * compatible with functional dependencies. The operator is identified
         * simply by looking at which function it uses to estimate
         * selectivity. That's a bit strange, but it's what other similar
         * places do.
         */
        if (get_oprrest(expr->opno) != F_EQSEL)
            return false;
 
        /* OK to proceed with checking "var" */
    }
    else if (is_orclause(clause))
    {
        BoolExpr   *bool_expr = (BoolExpr *) clause;
 
        /* start with no expression (we'll use the first match) */
        *expr = NULL;
 
        foreach(lc, bool_expr->args)
        {
            Node       *or_expr = NULL;
 
            /*
             * Had we found incompatible expression in the arguments, treat
             * the whole expression as incompatible.
             */
            if (!dependency_is_compatible_expression((Node *) lfirst(lc), relid,
                                                     statlist, &or_expr))
                return false;
 
            if (*expr == NULL)
                *expr = or_expr;
 
            /* ensure all the expressions are the same */
            if (!equal(or_expr, *expr))
                return false;
        }
 
        /* the expression is already checked by the recursive call */
        return true;
    }
    else if (is_notclause(clause))
    {
        /*
         * "NOT x" can be interpreted as "x = false", so get the argument and
         * proceed with seeing if it's a suitable Var.
         */
        clause_expr = (Node *) get_notclausearg(clause);
    }
    else
    {
        /*
         * A boolean expression "x" can be interpreted as "x = true", so
         * proceed with seeing if it's a suitable Var.
         */
        clause_expr = (Node *) clause;
    }
 
    /*
     * We may ignore any RelabelType node above the operand.  (There won't be
     * more than one, since eval_const_expressions has been applied already.)
     */
    if (IsA(clause_expr, RelabelType))
        clause_expr = (Node *) ((RelabelType *) clause_expr)->arg;
 
    /*
     * Search for a matching statistics expression.
     */
    foreach(lc, statlist)
    {
        StatisticExtInfo *info = (StatisticExtInfo *) lfirst(lc);
 
        /* ignore stats without dependencies */
        if (info->kind != STATS_EXT_DEPENDENCIES)
            continue;
 
        foreach(lc2, info->exprs)
        {
            Node       *stat_expr = (Node *) lfirst(lc2);
 
            if (equal(clause_expr, stat_expr))
            {
                *expr = stat_expr;
                return true;
            }
        }
    }
 
    return false;
}
 
/*
 * dependencies_clauselist_selectivity
 *      Return the estimated selectivity of (a subset of) the given clauses
 *      using functional dependency statistics, or 1.0 if no useful functional
 *      dependency statistic exists.
 *
 * 'estimatedclauses' is an input/output argument that gets a bit set
 * corresponding to the (zero-based) list index of each clause that is included
 * in the estimated selectivity.
 *
 * Given equality clauses on attributes (a,b) we find the strongest dependency
 * between them, i.e. either (a=>b) or (b=>a). Assuming (a=>b) is the selected
 * dependency, we then combine the per-clause selectivities using the formula
 *
 *     P(a,b) = f * P(a) + (1-f) * P(a) * P(b)
 *
 * where 'f' is the degree of the dependency.  (Actually we use a slightly
 * modified version of this formula -- see clauselist_apply_dependencies()).
 *
 * With clauses on more than two attributes, the dependencies are applied
 * recursively, starting with the widest/strongest dependencies. For example
 * P(a,b,c) is first split like this:
 *
 *     P(a,b,c) = f * P(a,b) + (1-f) * P(a,b) * P(c)
 *
 * assuming (a,b=>c) is the strongest dependency.
 */
Selectivity
dependencies_clauselist_selectivity(PlannerInfo *root,
                                    List *clauses,
                                    int varRelid,
                                    JoinType jointype,
                                    SpecialJoinInfo *sjinfo,
                                    RelOptInfo *rel,
                                    Bitmapset **estimatedclauses)
{
    Selectivity s1 = 1.0;
    ListCell   *l;
    Bitmapset  *clauses_attnums = NULL;
    AttrNumber *list_attnums;
    int         listidx;
    MVDependencies **func_dependencies;
    int         nfunc_dependencies;
    int         total_ndeps;
    MVDependency **dependencies;
    int         ndependencies;
    int         i;
    AttrNumber  attnum_offset;
    RangeTblEntry *rte = planner_rt_fetch(rel->relid, root);
 
    /* unique expressions */
    Node      **unique_exprs;
    int         unique_exprs_cnt;
 
    /* check if there's any stats that might be useful for us. */
    if (!has_stats_of_kind(rel->statlist, STATS_EXT_DEPENDENCIES))
        return 1.0;
 
    list_attnums = (AttrNumber *) palloc(sizeof(AttrNumber) *
                                         list_length(clauses));
 
    /*
     * We allocate space as if every clause was a unique expression, although
     * that's probably overkill. Some will be simple column references that
     * we'll translate to attnums, and there might be duplicates. But it's
     * easier and cheaper to just do one allocation than repalloc later.
     */
    unique_exprs = (Node **) palloc(sizeof(Node *) * list_length(clauses));
    unique_exprs_cnt = 0;
 
    /*
     * Pre-process the clauses list to extract the attnums seen in each item.
     * We need to determine if there's any clauses which will be useful for
     * dependency selectivity estimations. Along the way we'll record all of
     * the attnums for each clause in a list which we'll reference later so we
     * don't need to repeat the same work again. We'll also keep track of all
     * attnums seen.
     *
     * We also skip clauses that we already estimated using different types of
     * statistics (we treat them as incompatible).
     *
     * To handle expressions, we assign them negative attnums, as if it was a
     * system attribute (this is fine, as we only allow extended stats on user
     * attributes). And then we offset everything by the number of
     * expressions, so that we can store the values in a bitmapset.
     */
    listidx = 0;
    foreach(l, clauses)
    {
        Node       *clause = (Node *) lfirst(l);
        AttrNumber  attnum;
        Node       *expr = NULL;
 
        /* ignore clause by default */
        list_attnums[listidx] = InvalidAttrNumber;
 
        if (!bms_is_member(listidx, *estimatedclauses))
        {
            /*
             * If it's a simple column reference, just extract the attnum. If
             * it's an expression, assign a negative attnum as if it was a
             * system attribute.
             */
            if (dependency_is_compatible_clause(clause, rel->relid, &attnum))
            {
                list_attnums[listidx] = attnum;
            }
            else if (dependency_is_compatible_expression(clause, rel->relid,
                                                         rel->statlist,
                                                         &expr))
            {
                /* special attnum assigned to this expression */
                attnum = InvalidAttrNumber;
 
                Assert(expr != NULL);
 
                /* If the expression is duplicate, use the same attnum. */
                for (i = 0; i < unique_exprs_cnt; i++)
                {
                    if (equal(unique_exprs[i], expr))
                    {
                        /* negative attribute number to expression */
                        attnum = -(i + 1);
                        break;
                    }
                }
 
                /* not found in the list, so add it */
                if (attnum == InvalidAttrNumber)
                {
                    unique_exprs[unique_exprs_cnt++] = expr;
 
                    /* after incrementing the value, to get -1, -2, ... */
                    attnum = (-unique_exprs_cnt);
                }
 
                /* remember which attnum was assigned to this clause */
                list_attnums[listidx] = attnum;
            }
        }
 
        listidx++;
    }
 
    Assert(listidx == list_length(clauses));
 
    /*
     * How much we need to offset the attnums? If there are no expressions,
     * then no offset is needed. Otherwise we need to offset enough for the
     * lowest value (-unique_exprs_cnt) to become 1.
     */
    if (unique_exprs_cnt > 0)
        attnum_offset = (unique_exprs_cnt + 1);
    else
        attnum_offset = 0;
 
    /*
     * Now that we know how many expressions there are, we can offset the
     * values just enough to build the bitmapset.
     */
    for (i = 0; i < list_length(clauses); i++)
    {
        AttrNumber  attnum;
 
        /* ignore incompatible or already estimated clauses */
        if (list_attnums[i] == InvalidAttrNumber)
            continue;
 
        /* make sure the attnum is in the expected range */
        Assert(list_attnums[i] >= (-unique_exprs_cnt));
        Assert(list_attnums[i] <= MaxHeapAttributeNumber);
 
        /* make sure the attnum is positive (valid AttrNumber) */
        attnum = list_attnums[i] + attnum_offset;
 
        /*
         * Either it's a regular attribute, or it's an expression, in which
         * case we must not have seen it before (expressions are unique).
         *
         * XXX Check whether it's a regular attribute has to be done using the
         * original attnum, while the second check has to use the value with
         * an offset.
         */
        Assert(AttrNumberIsForUserDefinedAttr(list_attnums[i]) ||
               !bms_is_member(attnum, clauses_attnums));
 
        /*
         * Remember the offset attnum, both for attributes and expressions.
         * We'll pass list_attnums to clauselist_apply_dependencies, which
         * uses it to identify clauses in a bitmap. We could also pass the
         * offset, but this is more convenient.
         */
        list_attnums[i] = attnum;
 
        clauses_attnums = bms_add_member(clauses_attnums, attnum);
    }
 
    /*
     * If there's not at least two distinct attnums and expressions, then
     * reject the whole list of clauses. We must return 1.0 so the calling
     * function's selectivity is unaffected.
     */
    if (bms_membership(clauses_attnums) != BMS_MULTIPLE)
    {
        bms_free(clauses_attnums);
        pfree(list_attnums);
        return 1.0;
    }
 
    /*
     * Load all functional dependencies matching at least two parameters. We
     * can simply consider all dependencies at once, without having to search
     * for the best statistics object.
     *
     * To not waste cycles and memory, we deserialize dependencies only for
     * statistics that match at least two attributes. The array is allocated
     * with the assumption that all objects match - we could grow the array to
     * make it just the right size, but it's likely wasteful anyway thanks to
     * moving the freed chunks to freelists etc.
     */
    func_dependencies = (MVDependencies **) palloc(sizeof(MVDependencies *) *
                                                   list_length(rel->statlist));
    nfunc_dependencies = 0;
    total_ndeps = 0;
 
    foreach(l, rel->statlist)
    {
        StatisticExtInfo *stat = (StatisticExtInfo *) lfirst(l);
        int         nmatched;
        int         nexprs;
        int         k;
        MVDependencies *deps;
 
        /* skip statistics that are not of the correct type */
        if (stat->kind != STATS_EXT_DEPENDENCIES)
            continue;
 
        /* skip statistics with mismatching stxdinherit value */
        if (stat->inherit != rte->inh)
            continue;
 
        /*
         * Count matching attributes - we have to undo the attnum offsets. The
         * input attribute numbers are not offset (expressions are not
         * included in stat->keys, so it's not necessary). But we need to
         * offset it before checking against clauses_attnums.
         */
        nmatched = 0;
        k = -1;
        while ((k = bms_next_member(stat->keys, k)) >= 0)
        {
            AttrNumber  attnum = (AttrNumber) k;
 
            /* skip expressions */
            if (!AttrNumberIsForUserDefinedAttr(attnum))
                continue;
 
            /* apply the same offset as above */
            attnum += attnum_offset;
 
            if (bms_is_member(attnum, clauses_attnums))
                nmatched++;
        }
 
        /* count matching expressions */
        nexprs = 0;
        for (i = 0; i < unique_exprs_cnt; i++)
        {
            ListCell   *lc;
 
            foreach(lc, stat->exprs)
            {
                Node       *stat_expr = (Node *) lfirst(lc);
 
                /* try to match it */
                if (equal(stat_expr, unique_exprs[i]))
                    nexprs++;
            }
        }
 
        /*
         * Skip objects matching fewer than two attributes/expressions from
         * clauses.
         */
        if (nmatched + nexprs < 2)
            continue;
 
        deps = statext_dependencies_load(stat->statOid, rte->inh);
 
        /*
         * The expressions may be represented by different attnums in the
         * stats, we need to remap them to be consistent with the clauses.
         * That will make the later steps (e.g. picking the strongest item and
         * so on) much simpler and cheaper, because it won't need to care
         * about the offset at all.
         *
         * When we're at it, we can ignore dependencies that are not fully
         * matched by clauses (i.e. referencing attributes or expressions that
         * are not in the clauses).
         *
         * We have to do this for all statistics, as long as there are any
         * expressions - we need to shift the attnums in all dependencies.
         *
         * XXX Maybe we should do this always, because it also eliminates some
         * of the dependencies early. It might be cheaper than having to walk
         * the longer list in find_strongest_dependency later, especially as
         * we need to do that repeatedly?
         *
         * XXX We have to do this even when there are no expressions in
         * clauses, otherwise find_strongest_dependency may fail for stats
         * with expressions (due to lookup of negative value in bitmap). So we
         * need to at least filter out those dependencies. Maybe we could do
         * it in a cheaper way (if there are no expr clauses, we can just
         * discard all negative attnums without any lookups).
         */
        if (unique_exprs_cnt > 0 || stat->exprs != NIL)
        {
            int         ndeps = 0;
 
            for (i = 0; i < deps->ndeps; i++)
            {
                bool        skip = false;
                MVDependency *dep = deps->deps[i];
                int         j;
 
                for (j = 0; j < dep->nattributes; j++)
                {
                    int         idx;
                    Node       *expr;
                    AttrNumber  unique_attnum = InvalidAttrNumber;
                    AttrNumber  attnum;
 
                    /* undo the per-statistics offset */
                    attnum = dep->attributes[j];
 
                    /*
                     * For regular attributes we can simply check if it
                     * matches any clause. If there's no matching clause, we
                     * can just ignore it. We need to offset the attnum
                     * though.
                     */
                    if (AttrNumberIsForUserDefinedAttr(attnum))
                    {
                        dep->attributes[j] = attnum + attnum_offset;
 
                        if (!bms_is_member(dep->attributes[j], clauses_attnums))
                        {
                            skip = true;
                            break;
                        }
 
                        continue;
                    }
 
                    /*
                     * the attnum should be a valid system attnum (-1, -2,
                     * ...)
                     */
                    Assert(AttributeNumberIsValid(attnum));
 
                    /*
                     * For expressions, we need to do two translations. First
                     * we have to translate the negative attnum to index in
                     * the list of expressions (in the statistics object).
                     * Then we need to see if there's a matching clause. The
                     * index of the unique expression determines the attnum
                     * (and we offset it).
                     */
                    idx = -(1 + attnum);
 
                    /* Is the expression index is valid? */
                    Assert((idx >= 0) && (idx < list_length(stat->exprs)));
 
                    expr = (Node *) list_nth(stat->exprs, idx);
 
                    /* try to find the expression in the unique list */
                    for (int m = 0; m < unique_exprs_cnt; m++)
                    {
                        /*
                         * found a matching unique expression, use the attnum
                         * (derived from index of the unique expression)
                         */
                        if (equal(unique_exprs[m], expr))
                        {
                            unique_attnum = -(m + 1) + attnum_offset;
                            break;
                        }
                    }
 
                    /*
                     * Found no matching expression, so we can simply skip
                     * this dependency, because there's no chance it will be
                     * fully covered.
                     */
                    if (unique_attnum == InvalidAttrNumber)
                    {
                        skip = true;
                        break;
                    }
 
                    /* otherwise remap it to the new attnum */
                    dep->attributes[j] = unique_attnum;
                }
 
                /* if found a matching dependency, keep it */
                if (!skip)
                {
                    /* maybe we've skipped something earlier, so move it */
                    if (ndeps != i)
                        deps->deps[ndeps] = deps->deps[i];
 
                    ndeps++;
                }
            }
 
            deps->ndeps = ndeps;
        }
 
        /*
         * It's possible we've removed all dependencies, in which case we
         * don't bother adding it to the list.
         */
        if (deps->ndeps > 0)
        {
            func_dependencies[nfunc_dependencies] = deps;
            total_ndeps += deps->ndeps;
            nfunc_dependencies++;
        }
    }
 
    /* if no matching stats could be found then we've nothing to do */
    if (nfunc_dependencies == 0)
    {
        pfree(func_dependencies);
        bms_free(clauses_attnums);
        pfree(list_attnums);
        pfree(unique_exprs);
        return 1.0;
    }
 
    /*
     * Work out which dependencies we can apply, starting with the
     * widest/strongest ones, and proceeding to smaller/weaker ones.
     */
    dependencies = (MVDependency **) palloc(sizeof(MVDependency *) *
                                            total_ndeps);
    ndependencies = 0;
 
    while (true)
    {
        MVDependency *dependency;
        AttrNumber  attnum;
 
        /* the widest/strongest dependency, fully matched by clauses */
        dependency = find_strongest_dependency(func_dependencies,
                                               nfunc_dependencies,
                                               clauses_attnums);
        if (!dependency)
            break;
 
        dependencies[ndependencies++] = dependency;
 
        /* Ignore dependencies using this implied attribute in later loops */
        attnum = dependency->attributes[dependency->nattributes - 1];
        clauses_attnums = bms_del_member(clauses_attnums, attnum);
    }
 
    /*
     * If we found applicable dependencies, use them to estimate all
     * compatible clauses on attributes that they refer to.
     */
    if (ndependencies != 0)
        s1 = clauselist_apply_dependencies(root, clauses, varRelid, jointype,
                                           sjinfo, dependencies, ndependencies,
                                           list_attnums, estimatedclauses);
 
    /* free deserialized functional dependencies (and then the array) */
    for (i = 0; i < nfunc_dependencies; i++)
        pfree(func_dependencies[i]);
 
    pfree(dependencies);
    pfree(func_dependencies);
    bms_free(clauses_attnums);
    pfree(list_attnums);
    pfree(unique_exprs);
 
    return s1;
}