dependencies.c

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/*-------------------------------------------------------------------------
 *
 * dependencies.c
 *    POSTGRES functional dependencies
 *
 * Portions Copyright (c) 1996-2020, 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 "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/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(int numrows, HeapTuple *rows, int k,
                                AttrNumber *dependency, VacAttrStats **stats, Bitmapset *attrs);
static bool dependency_is_fully_matched(MVDependency *dependency,
                                        Bitmapset *attnums);
static bool dependency_is_compatible_clause(Node *clause, Index relid,
                                            AttrNumber *attnum);
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(int numrows, HeapTuple *rows, int k, AttrNumber *dependency,
                  VacAttrStats **stats, Bitmapset *attrs)
{
    int         i,
                nitems;
    MultiSortSupport mss;
    SortItem   *items;
    AttrNumber *attnums;
    AttrNumber *attnums_dep;
    int         numattrs;

    /* 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);

    /*
     * Transform the attrs from bitmap to an array to make accessing the i-th
     * member easier, and then construct a filtered version with only attnums
     * referenced by the dependency we validate.
     */
    attnums = build_attnums_array(attrs, &numattrs);

    attnums_dep = (AttrNumber *) palloc(k * sizeof(AttrNumber));
    for (i = 0; i < k; i++)
        attnums_dep[i] = 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 = 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(numrows, &nitems, rows, stats[0]->tupDesc,
                               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++;
    }

    if (items)
        pfree(items);

    pfree(mss);
    pfree(attnums);
    pfree(attnums_dep);

    /* Compute the 'degree of validity' as (supporting/total). */
    return (n_supporting_rows * 1.0 / 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(int numrows, HeapTuple *rows, Bitmapset *attrs,
                           VacAttrStats **stats)
{
    int         i,
                k;
    int         numattrs;
    AttrNumber *attnums;

    /* result */
    MVDependencies *dependencies = NULL;

    /*
     * Transform the bms into an array, to make accessing i-th member easier.
     */
    attnums = build_attnums_array(attrs, &numattrs);

    Assert(numattrs >= 2);

    /*
     * 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 <= numattrs; k++)
    {
        AttrNumber *dependency; /* array with k elements */

        /* prepare a DependencyGenerator of variation */
        DependencyGenerator DependencyGenerator = DependencyGenerator_init(numattrs, k);

        /* generate all possible variations of k values (out of n) */
        while ((dependency = DependencyGenerator_next(DependencyGenerator)))
        {
            double      degree;
            MVDependency *d;

            /* compute how valid the dependency seems */
            degree = dependency_degree(numrows, rows, k, dependency, stats, attrs);

            /*
             * 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] = 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);
    }

    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 %zd (expected at least %zd)",
             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 %zd (expected at least %zd)",
             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)
{
    MVDependencies *result;
    bool        isnull;
    Datum       deps;
    HeapTuple   htup;

    htup = SearchSysCache1(STATEXTDATASTXOID, ObjectIdGetDatum(mvoid));
    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 statistic 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;

    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)))
            var = linitial(expr->args);
        else if (is_pseudo_constant_clause(linitial(expr->args)))
            var = 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;

        var = 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   *expr = (BoolExpr *) clause;
        ListCell   *lc;

        /* start with no attribute number */
        *attnum = InvalidAttrNumber;

        foreach(lc, 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;

            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.
         */
        var = (Var *) get_notclausearg(clause);
    }
    else
    {
        /*
         * A boolean expression "x" can be interpreted as "x = true", so
         * proceed with seeing if it's a suitable Var.
         */
        var = (Var *) 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(var, RelabelType))
        var = (Var *) ((RelabelType *) var)->arg;

    /* We only support plain Vars for now */
    if (!IsA(var, Var))
        return false;

    /* 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 selecvitity 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_simple(root, attr_clauses, varRelid,
                                                   jointype, sjinfo, NULL);
        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;
}

/*
 * 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;

    /* 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));

    /*
     * 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).
     */
    listidx = 0;
    foreach(l, clauses)
    {
        Node       *clause = (Node *) lfirst(l);
        AttrNumber  attnum;

        if (!bms_is_member(listidx, *estimatedclauses) &&
            dependency_is_compatible_clause(clause, rel->relid, &attnum))
        {
            list_attnums[listidx] = attnum;
            clauses_attnums = bms_add_member(clauses_attnums, attnum);
        }
        else
            list_attnums[listidx] = InvalidAttrNumber;

        listidx++;
    }

    /*
     * If there's not at least two distinct attnums then reject the whole list
     * of clauses. We must return 1.0 so the calling function's selectivity is
     * unaffected.
     */
    if (bms_num_members(clauses_attnums) < 2)
    {
        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);
        Bitmapset  *matched;
        int         num_matched;

        /* skip statistics that are not of the correct type */
        if (stat->kind != STATS_EXT_DEPENDENCIES)
            continue;

        matched = bms_intersect(clauses_attnums, stat->keys);
        num_matched = bms_num_members(matched);
        bms_free(matched);

        /* skip objects matching fewer than two attributes from clauses */
        if (num_matched < 2)
            continue;

        func_dependencies[nfunc_dependencies]
            = statext_dependencies_load(stat->statOid);

        total_ndeps += func_dependencies[nfunc_dependencies]->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);
        return 1.0;
    }

    /*
     * Work out which dependencies we can apply, starting with the
     * widest/stongest 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);

    return s1;
}