join_funcs.go

// Copyright 2020 The Cockroach Authors.
//
// Use of this software is governed by the Business Source License
// included in the file licenses/BSL.txt.
//
// As of the Change Date specified in that file, in accordance with
// the Business Source License, use of this software will be governed
// by the Apache License, Version 2.0, included in the file
// licenses/APL.txt.

package xform

import (
	"fmt"

	"github.com/cockroachdb/cockroach/pkg/sql/opt"
	"github.com/cockroachdb/cockroach/pkg/sql/opt/cat"
	"github.com/cockroachdb/cockroach/pkg/sql/opt/invertedidx"
	"github.com/cockroachdb/cockroach/pkg/sql/opt/memo"
	"github.com/cockroachdb/cockroach/pkg/sql/opt/props/physical"
	"github.com/cockroachdb/cockroach/pkg/sql/sem/tree"
	"github.com/cockroachdb/cockroach/pkg/sql/types"
	"github.com/cockroachdb/cockroach/pkg/util"
	"github.com/cockroachdb/errors"
)

// GenerateMergeJoins spawns MergeJoinOps, based on any interesting orderings.
func (c *CustomFuncs) GenerateMergeJoins(
	grp memo.RelExpr,
	originalOp opt.Operator,
	left, right memo.RelExpr,
	on memo.FiltersExpr,
	joinPrivate *memo.JoinPrivate,
) {
	if joinPrivate.Flags.Has(memo.DisallowMergeJoin) {
		return
	}

	leftProps := left.Relational()
	rightProps := right.Relational()

	leftEq, rightEq := memo.ExtractJoinEqualityColumns(
		leftProps.OutputCols, rightProps.OutputCols, on,
	)
	n := len(leftEq)
	if n == 0 {
		return
	}

	// We generate MergeJoin expressions based on interesting orderings from the
	// left side. The CommuteJoin rule will ensure that we actually try both
	// sides.
	orders := DeriveInterestingOrderings(left).Copy()
	orders.RestrictToCols(leftEq.ToSet(), nil /* equivCols */)

	if !c.NoJoinHints(joinPrivate) || c.e.evalCtx.SessionData.ReorderJoinsLimit == 0 {
		// If we are using a hint, or the join limit is set to zero, the join won't
		// be commuted. Add the orderings from the right side.
		rightOrders := DeriveInterestingOrderings(right).Copy()
		rightOrders.RestrictToCols(leftEq.ToSet(), nil /* equivCols */)
		orders = append(orders, rightOrders...)

		// If we don't allow hash join, we must do our best to generate a merge
		// join, even if it means sorting both sides. We append an arbitrary
		// ordering, in case the interesting orderings don't result in any merge
		// joins.
		o := make(opt.Ordering, len(leftEq))
		for i := range o {
			o[i] = opt.MakeOrderingColumn(leftEq[i], false /* descending */)
		}
		orders.Add(o)
	}

	if len(orders) == 0 {
		return
	}

	var colToEq util.FastIntMap
	for i := range leftEq {
		colToEq.Set(int(leftEq[i]), i)
		colToEq.Set(int(rightEq[i]), i)
	}

	var remainingFilters memo.FiltersExpr

	for _, o := range orders {
		if len(o) < n {
			// TODO(radu): we have a partial ordering on the equality columns. We
			// should augment it with the other columns (in arbitrary order) in the
			// hope that we can get the full ordering cheaply using a "streaming"
			// sort. This would not useful now since we don't support streaming sorts.
			continue
		}

		if remainingFilters == nil {
			remainingFilters = memo.ExtractRemainingJoinFilters(on, leftEq, rightEq)
		}

		merge := memo.MergeJoinExpr{Left: left, Right: right, On: remainingFilters}
		merge.JoinPrivate = *joinPrivate
		merge.JoinType = originalOp
		merge.LeftEq = make(opt.Ordering, n)
		merge.RightEq = make(opt.Ordering, n)
		merge.LeftOrdering.Columns = make([]physical.OrderingColumnChoice, 0, n)
		merge.RightOrdering.Columns = make([]physical.OrderingColumnChoice, 0, n)
		for i := 0; i < n; i++ {
			eqIdx, _ := colToEq.Get(int(o[i].ID()))
			l, r, descending := leftEq[eqIdx], rightEq[eqIdx], o[i].Descending()
			merge.LeftEq[i] = opt.MakeOrderingColumn(l, descending)
			merge.RightEq[i] = opt.MakeOrderingColumn(r, descending)
			merge.LeftOrdering.AppendCol(l, descending)
			merge.RightOrdering.AppendCol(r, descending)
		}

		// Simplify the orderings with the corresponding FD sets.
		merge.LeftOrdering.Simplify(&leftProps.FuncDeps)
		merge.RightOrdering.Simplify(&rightProps.FuncDeps)

		c.e.mem.AddMergeJoinToGroup(&merge, grp)
	}
}

// GenerateLookupJoins looks at the possible indexes and creates lookup join
// expressions in the current group. A lookup join can be created when the ON
// condition has equality constraints on a prefix of the index columns.
//
// There are two cases:
//
//  1. The index has all the columns we need; this is the simple case, where we
//     generate a LookupJoin expression in the current group:
//
//         Join                       LookupJoin(t@idx)
//         /   \                           |
//        /     \            ->            |
//      Input  Scan(t)                   Input
//
//
//  2. The index is not covering. We have to generate an index join above the
//     lookup join. Note that this index join is also implemented as a
//     LookupJoin, because an IndexJoin can only output columns from one table,
//     whereas we also need to output columns from Input.
//
//         Join                       LookupJoin(t@primary)
//         /   \                           |
//        /     \            ->            |
//      Input  Scan(t)                LookupJoin(t@idx)
//                                         |
//                                         |
//                                       Input
//
//     For example:
//      CREATE TABLE abc (a INT PRIMARY KEY, b INT, c INT)
//      CREATE TABLE xyz (x INT PRIMARY KEY, y INT, z INT, INDEX (y))
//      SELECT * FROM abc JOIN xyz ON a=y
//
//     We want to first join abc with the index on y (which provides columns y, x)
//     and then use a lookup join to retrieve column z. The "index join" (top
//     LookupJoin) will produce columns a,b,c,x,y; the lookup columns are just z
//     (the original index join produced x,y,z).
//
//     Note that the top LookupJoin "sees" column IDs from the table on both
//     "sides" (in this example x,y on the left and z on the right) but there is
//     no overlap.
//
// A lookup join can be created when the ON condition or implicit filters from
// CHECK constraints and computed columns constrain a prefix of the index
// columns to non-ranging constant values. To support this, the constant values
// are cross-joined with the input and used as key columns for the parent lookup
// join.
//
// For example, consider the tables and query below.
//
//   CREATE TABLE abc (a INT PRIMARY KEY, b INT, c INT)
//   CREATE TABLE xyz (
//     x INT PRIMARY KEY,
//     y INT,
//     z INT NOT NULL,
//     CHECK z IN (1, 2, 3),
//     INDEX (z, y)
//   )
//   SELECT a, x FROM abc JOIN xyz ON a=y
//
// GenerateLookupJoins will perform the following transformation.
//
//         Join                       LookupJoin(t@idx)
//         /   \                           |
//        /     \            ->            |
//      Input  Scan(t)                   Join
//                                       /   \
//                                      /     \
//                                    Input  Values(1, 2, 3)
//
// If a column is constrained to a single constant value, inlining normalization
// rules will reduce the cross join into a project.
//
//         Join                       LookupJoin(t@idx)
//         /   \                           |
//        /     \            ->            |
//      Input  Scan(t)                  Project
//                                         |
//                                         |
//                                       Input
//
func (c *CustomFuncs) GenerateLookupJoins(
	grp memo.RelExpr,
	joinType opt.Operator,
	input memo.RelExpr,
	scanPrivate *memo.ScanPrivate,
	on memo.FiltersExpr,
	joinPrivate *memo.JoinPrivate,
) {
	if joinPrivate.Flags.Has(memo.DisallowLookupJoinIntoRight) {
		return
	}
	md := c.e.mem.Metadata()
	inputProps := input.Relational()

	leftEq, rightEq := memo.ExtractJoinEqualityColumns(inputProps.OutputCols, scanPrivate.Cols, on)
	n := len(leftEq)
	if n == 0 {
		return
	}

	// Generate implicit filters from CHECK constraints and computed columns as
	// optional filters to help generate lookup join keys.
	optionalFilters := c.checkConstraintFilters(scanPrivate.Table)
	computedColFilters := c.computedColFilters(scanPrivate, on, optionalFilters)
	optionalFilters = append(optionalFilters, computedColFilters...)

	var pkCols opt.ColList
	var iter scanIndexIter
	iter.Init(c.e.evalCtx, c.e.f, c.e.mem, &c.im, scanPrivate, on, rejectInvertedIndexes)
	iter.ForEach(func(index cat.Index, onFilters memo.FiltersExpr, indexCols opt.ColSet, isCovering bool, constProj memo.ProjectionsExpr) {
		// Find the longest prefix of index key columns that are constrained by
		// an equality with another column or a constant.
		numIndexKeyCols := index.LaxKeyColumnCount()

		var constFilters memo.FiltersExpr
		allFilters := append(onFilters, optionalFilters...)

		// Check if the first column in the index has an equality constraint, or if
		// it is constrained to a constant value. This check doesn't guarantee that
		// we will find lookup join key columns, but it avoids the unnecessary work
		// in most cases.
		firstIdxCol := scanPrivate.Table.IndexColumnID(index, 0)
		if _, ok := rightEq.Find(firstIdxCol); !ok {
			if _, _, ok := c.findJoinFilterConstants(allFilters, firstIdxCol); !ok {
				return
			}
		}

		lookupJoin := memo.LookupJoinExpr{Input: input}
		lookupJoin.JoinPrivate = *joinPrivate
		lookupJoin.JoinType = joinType
		lookupJoin.Table = scanPrivate.Table
		lookupJoin.Index = index.Ordinal()

		lookupJoin.KeyCols = make(opt.ColList, 0, numIndexKeyCols)
		rightSideCols := make(opt.ColList, 0, numIndexKeyCols)

		shouldBuildMultiSpanLookupJoin := false

		// All the lookup conditions must apply to the prefix of the index and so
		// the projected columns created must be created in order.
		for j := 0; j < numIndexKeyCols; j++ {
			idxCol := scanPrivate.Table.IndexColumnID(index, j)
			if eqIdx, ok := rightEq.Find(idxCol); ok {
				lookupJoin.KeyCols = append(lookupJoin.KeyCols, leftEq[eqIdx])
				rightSideCols = append(rightSideCols, idxCol)
				continue
			}

			// Try to find a filter that constrains this column to non-NULL
			// constant values. We cannot use a NULL value because the lookup
			// join implements logic equivalent to simple equality between
			// columns (where NULL never equals anything).
			foundVals, allIdx, ok := c.findJoinFilterConstants(allFilters, idxCol)
			if !ok {
				break
			}

			if len(foundVals) > 1 && (joinType == opt.LeftJoinOp || joinType == opt.AntiJoinOp) {
				// We cannot use the method constructJoinWithConstants to create a cross
				// join for left or anti joins, because constructing a cross join with
				// foundVals will increase the size of the input. As a result,
				// non-matching input rows will show up more than once in the output,
				// which is incorrect (see #59615).
				shouldBuildMultiSpanLookupJoin = true
				break
			}

			// We will join these constant values with the input to make
			// equality columns for the lookup join.
			if constFilters == nil {
				constFilters = make(memo.FiltersExpr, 0, numIndexKeyCols-j)
			}

			idxColType := c.e.f.Metadata().ColumnMeta(idxCol).Type
			constColAlias := fmt.Sprintf("lookup_join_const_col_@%d", idxCol)
			join, constColID := c.constructJoinWithConstants(
				lookupJoin.Input,
				foundVals,
				idxColType,
				constColAlias,
			)

			lookupJoin.Input = join
			lookupJoin.KeyCols = append(lookupJoin.KeyCols, constColID)
			rightSideCols = append(rightSideCols, idxCol)
			constFilters = append(constFilters, allFilters[allIdx])
		}

		if shouldBuildMultiSpanLookupJoin {
			// Some of the index columns were constrained to multiple constant values,
			// and this is a left or anti join. As described above, we cannot use the
			// method constructJoinWithConstants to create a cross join as the input
			// for left or anti joins, since it would produce incorrect results.
			//
			// As an alternative, we store all the filters needed for the lookup in
			// LookupExpr, which will be used to construct spans at execution time.
			// The result is that each input row will generate multiple spans to
			// lookup in the index. For example, if the index cols are (region, id)
			// and the LookupExpr is `region in ('east', 'west') AND id = input.id`,
			// each input row will generate two spans to be scanned in the lookup:
			//   [/'east'/<id> - /'east'/<id>] [/'west'/<id> - /'west'/<id>]
			// where <id> is the value of input.id for the current input row.
			var eqFilters memo.FiltersExpr
			extractEqualityFilter := func(leftCol, rightCol opt.ColumnID) memo.FiltersItem {
				return memo.ExtractJoinEqualityFilter(
					leftCol, rightCol, inputProps.OutputCols, scanPrivate.Cols, on,
				)
			}
			eqFilters, constFilters, rightSideCols = c.findFiltersForIndexLookup(
				allFilters, scanPrivate.Table, index, leftEq, rightEq, extractEqualityFilter,
			)
			lookupJoin.LookupExpr = append(eqFilters, constFilters...)

			// Reset KeyCols since we're not using it anymore.
			lookupJoin.KeyCols = opt.ColList{}
		}

		if len(lookupJoin.KeyCols) == 0 && len(lookupJoin.LookupExpr) == 0 {
			// We couldn't find equality columns which we can lookup.
			return
		}

		tableFDs := memo.MakeTableFuncDep(md, scanPrivate.Table)
		// A lookup join will drop any input row which contains NULLs, so a lax key
		// is sufficient.
		lookupJoin.LookupColsAreTableKey = tableFDs.ColsAreLaxKey(rightSideCols.ToSet())

		// Remove redundant filters from the ON condition if columns were
		// constrained by equality filters or constant filters.
		lookupJoin.On = onFilters
		if len(lookupJoin.KeyCols) > 0 {
			lookupJoin.On = memo.ExtractRemainingJoinFilters(lookupJoin.On, lookupJoin.KeyCols, rightSideCols)
		}
		lookupJoin.On = lookupJoin.On.Difference(lookupJoin.LookupExpr)
		lookupJoin.On = lookupJoin.On.Difference(constFilters)
		lookupJoin.ConstFilters = constFilters

		// Add input columns and lookup expression columns, since these will be
		// needed for all join types and cases.
		lookupJoin.Cols = lookupJoin.LookupExpr.OuterCols()
		lookupJoin.Cols.UnionWith(inputProps.OutputCols)

		// TODO(mgartner): The right side of the join can "produce" columns held
		// constant by a partial index predicate, but the lookup joiner does not
		// currently support this. For now, if constProj is non-empty we
		// consider the index non-covering.
		if isCovering && len(constProj) == 0 {
			// Case 1 (see function comment).
			lookupJoin.Cols.UnionWith(scanPrivate.Cols)

			// If some optional filters were used to build the lookup expression, we may
			// need to wrap the final expression with a project. We only need to do this
			// for left joins, since anti joins have an implicit projection that removes
			// all right-side columns.
			needsProject := joinType == opt.LeftJoinOp &&
				!lookupJoin.Cols.SubsetOf(grp.Relational().OutputCols)
			if !needsProject {
				c.e.mem.AddLookupJoinToGroup(&lookupJoin, grp)
				return
			}
			var project memo.ProjectExpr
			project.Input = c.e.f.ConstructLookupJoin(
				lookupJoin.Input,
				lookupJoin.On,
				&lookupJoin.LookupJoinPrivate,
			)
			project.Passthrough = grp.Relational().OutputCols
			c.e.mem.AddProjectToGroup(&project, grp)
			return
		}

		_, isPartial := index.Predicate()
		if isPartial && (joinType == opt.SemiJoinOp || joinType == opt.AntiJoinOp) {
			// Typically, the index must cover all columns in the scanPrivate in
			// order to generate a lookup join without an additional index join
			// (case 1, see function comment). However, if the index is a
			// partial index, the filters remaining after proving
			// filter-predicate implication may no longer reference some
			// columns. A lookup semi- or anti-join can be generated if the
			// columns in the new filters from the right side of the join are
			// covered by the index. Consider the example:
			//
			//   CREATE TABLE a (a INT)
			//   CREATE TABLE xy (x INT, y INT, INDEX (x) WHERE y > 0)
			//
			//   SELECT a FROM a WHERE EXISTS (SELECT 1 FROM xyz WHERE a = x AND y > 0)
			//
			// The original ON filters of the semi-join are (a = x AND y > 0).
			// The (y > 0) expression in the filter is an exact match to the
			// partial index predicate, so the remaining ON filters are (a = x).
			// Column y is no longer referenced, so a lookup semi-join can be
			// created despite the partial index not covering y.
			//
			// Note that this is a special case that only works for semi- and
			// anti-joins because they never include columns from the right side
			// in their output columns. Other joins include columns from the
			// right side in their output columns, so even if the ON filters no
			// longer reference an un-covered column, they must be fetched (case
			// 2, see function comment).
			filterColsFromRight := scanPrivate.Cols.Intersection(onFilters.OuterCols())
			if filterColsFromRight.SubsetOf(indexCols) {
				lookupJoin.Cols.UnionWith(filterColsFromRight)
				c.e.mem.AddLookupJoinToGroup(&lookupJoin, grp)
				return
			}
		}

		// All code that follows is for case 2 (see function comment).

		if scanPrivate.Flags.NoIndexJoin {
			return
		}
		if joinType == opt.SemiJoinOp || joinType == opt.AntiJoinOp {
			// We cannot use a non-covering index for semi and anti join. Note that
			// since the semi/anti join doesn't pass through any columns, "non
			// covering" here means that not all columns in the ON condition are
			// available.
			//
			// TODO(radu): We could create a semi/anti join on top of an inner join if
			// the lookup columns form a key (to guarantee that input rows are not
			// duplicated by the inner join).
			return
		}

		if pkCols == nil {
			pkCols = c.getPkCols(scanPrivate.Table)
		}

		// The lower LookupJoin must return all PK columns (they are needed as key
		// columns for the index join).
		lookupJoin.Cols.UnionWith(scanPrivate.Cols.Intersection(indexCols))
		for i := range pkCols {
			lookupJoin.Cols.Add(pkCols[i])
		}

		var indexJoin memo.LookupJoinExpr

		// onCols are the columns that the ON condition in the (lower) lookup join
		// can refer to: input columns, or columns available in the index.
		onCols := indexCols.Union(inputProps.OutputCols)
		if c.FiltersBoundBy(lookupJoin.On, onCols) {
			// The ON condition refers only to the columns available in the index.
			//
			// For LeftJoin, both LookupJoins perform a LeftJoin. A null-extended row
			// from the lower LookupJoin will not have any matches in the top
			// LookupJoin (it has NULLs on key columns) and will get null-extended
			// there as well.
			indexJoin.On = memo.TrueFilter
		} else {
			// ON has some conditions that are bound by the columns in the index (at
			// the very least, the equality conditions we used for KeyCols), and some
			// conditions that refer to other columns. We can put the former in the
			// lower LookupJoin and the latter in the index join.
			//
			// This works for InnerJoin but not for LeftJoin because of a
			// technicality: if an input (left) row has matches in the lower
			// LookupJoin but has no matches in the index join, only the columns
			// looked up by the top index join get NULL-extended.
			if joinType == opt.LeftJoinOp {
				// TODO(radu): support LeftJoin, perhaps by looking up all columns and
				// discarding columns that are already available from the lower
				// LookupJoin. This requires a projection to avoid having the same
				// ColumnIDs on both sides of the index join.
				return
			}
			conditions := lookupJoin.On
			lookupJoin.On = c.ExtractBoundConditions(conditions, onCols)
			indexJoin.On = c.ExtractUnboundConditions(conditions, onCols)
		}

		indexJoin.Input = c.e.f.ConstructLookupJoin(
			lookupJoin.Input,
			lookupJoin.On,
			&lookupJoin.LookupJoinPrivate,
		)
		indexJoin.JoinType = joinType
		indexJoin.Table = scanPrivate.Table
		indexJoin.Index = cat.PrimaryIndex
		indexJoin.KeyCols = pkCols
		indexJoin.Cols = scanPrivate.Cols.Union(inputProps.OutputCols)
		indexJoin.LookupColsAreTableKey = true

		// Create the LookupJoin for the index join in the same group.
		c.e.mem.AddLookupJoinToGroup(&indexJoin, grp)
	})
}

// findFiltersForIndexLookup finds the equality and constant filters in
// filters that can be used to constrain the given index.
func (c *CustomFuncs) findFiltersForIndexLookup(
	filters memo.FiltersExpr,
	tabID opt.TableID,
	index cat.Index,
	leftEq, rightEq opt.ColList,
	extractEqualityFilter func(opt.ColumnID, opt.ColumnID) memo.FiltersItem,
) (eqFilters, constFilters memo.FiltersExpr, rightSideCols opt.ColList) {
	numIndexKeyCols := index.LaxKeyColumnCount()

	eqFilters = make(memo.FiltersExpr, 0, len(filters))
	rightSideCols = make(opt.ColList, 0, len(filters))

	// All the lookup conditions must apply to the prefix of the index.
	for j := 0; j < numIndexKeyCols; j++ {
		idxCol := tabID.IndexColumnID(index, j)
		if eqIdx, ok := rightEq.Find(idxCol); ok {
			eqFilter := extractEqualityFilter(leftEq[eqIdx], rightEq[eqIdx])
			eqFilters = append(eqFilters, eqFilter)
			rightSideCols = append(rightSideCols, idxCol)
			continue
		}

		// Try to find a filter that constrains this column to non-NULL
		// constant values. We cannot use a NULL value because the lookup
		// join implements logic equivalent to simple equality between
		// columns (where NULL never equals anything).
		values, allIdx, ok := c.findJoinFilterConstants(filters, idxCol)
		if !ok {
			break
		}

		if constFilters == nil {
			constFilters = make(memo.FiltersExpr, 0, numIndexKeyCols-j)
		}

		// Ensure that the constant filter is either an equality or an IN expression.
		// These are the only two types of expressions currently supported by the
		// lookupJoiner for building lookup spans.
		constFilter := filters[allIdx]
		if !c.isCanonicalConstFilter(constFilter) {
			constFilter = c.makeConstFilter(idxCol, values)
		}
		constFilters = append(constFilters, constFilter)
	}

	if len(eqFilters) == 0 {
		// We couldn't find equality columns which we can lookup.
		return nil, nil, nil
	}

	return eqFilters, constFilters, rightSideCols
}

// isCanonicalConstFilter checks that the given filter is a constant filter in
// one of two possible canonical formats:
//  1. It is an equality between a variable and a constant.
//  2. It is an IN expression between a variable and a tuple of constants.
// Returns true if the filter matches one of these two formats. Otherwise
// returns false.
func (c *CustomFuncs) isCanonicalConstFilter(filter memo.FiltersItem) bool {
	switch t := filter.Condition.(type) {
	case *memo.EqExpr:
		if t.Left.Op() == opt.VariableOp && opt.IsConstValueOp(t.Right) {
			return true
		}
	case *memo.InExpr:
		if t.Left.Op() == opt.VariableOp && memo.CanExtractConstTuple(t.Right) {
			return true
		}
	}
	return false
}

// makeConstFilter builds a filter that constrains the given column to the given
// set of constant values. This is performed by either constructing an equality
// expression or an IN expression.
func (c *CustomFuncs) makeConstFilter(col opt.ColumnID, values tree.Datums) memo.FiltersItem {
	if len(values) == 1 {
		return c.e.f.ConstructFiltersItem(c.e.f.ConstructEq(
			c.e.f.ConstructVariable(col),
			c.e.f.ConstructConstVal(values[0], values[0].ResolvedType()),
		))
	}
	elems := make(memo.ScalarListExpr, len(values))
	elemTypes := make([]*types.T, len(values))
	for i := range values {
		typ := values[i].ResolvedType()
		elems[i] = c.e.f.ConstructConstVal(values[i], typ)
		elemTypes[i] = typ
	}
	return c.e.f.ConstructFiltersItem(c.e.f.ConstructIn(
		c.e.f.ConstructVariable(col),
		c.e.f.ConstructTuple(elems, types.MakeTuple(elemTypes)),
	))
}

// constructContinuationColumnForPairedJoin constructs a continuation column
// ID for the paired-joiners used for left outer/semi/anti joins when the
// first join generates false positives (due to an inverted index or
// non-covering index). The first join will be either a left outer join or
// an inner join.
func (c *CustomFuncs) constructContinuationColumnForPairedJoin() opt.ColumnID {
	return c.e.f.Metadata().AddColumn("continuation", c.BoolType())
}

// GenerateInvertedJoins is similar to GenerateLookupJoins, but instead
// of generating lookup joins with regular indexes, it generates lookup joins
// with inverted indexes. Similar to GenerateLookupJoins, there are two cases
// depending on whether or not the index is covering. See the comment above
// GenerateLookupJoins for details.
func (c *CustomFuncs) GenerateInvertedJoins(
	grp memo.RelExpr,
	joinType opt.Operator,
	input memo.RelExpr,
	scanPrivate *memo.ScanPrivate,
	on memo.FiltersExpr,
	joinPrivate *memo.JoinPrivate,
) {
	if joinPrivate.Flags.Has(memo.DisallowInvertedJoinIntoRight) {
		return
	}

	inputCols := input.Relational().OutputCols
	var pkCols opt.ColList
	var newScanPrivate *memo.ScanPrivate

	eqColsAndOptionalFiltersCalculated := false
	var leftEqCols opt.ColList
	var rightEqCols opt.ColList
	var optionalFilters memo.FiltersExpr

	var iter scanIndexIter
	iter.Init(c.e.evalCtx, c.e.f, c.e.mem, &c.im, scanPrivate, on, rejectNonInvertedIndexes)
	iter.ForEach(func(index cat.Index, onFilters memo.FiltersExpr, indexCols opt.ColSet, _ bool, _ memo.ProjectionsExpr) {
		invertedJoin := memo.InvertedJoinExpr{Input: input}
		numPrefixCols := index.NonInvertedPrefixColumnCount()

		var allFilters memo.FiltersExpr
		if numPrefixCols > 0 {
			// Only calculate the left and right equality columns and optional
			// filters if there is a multi-column inverted index.
			if !eqColsAndOptionalFiltersCalculated {
				inputProps := input.Relational()
				leftEqCols, rightEqCols = memo.ExtractJoinEqualityColumns(inputProps.OutputCols, scanPrivate.Cols, onFilters)

				// Generate implicit filters from CHECK constraints and computed
				// columns as optional filters. We build the computed column
				// optional filters from the original on filters, not the
				// filters within the context of the iter.ForEach callback. The
				// latter may be reduced during partial index implication and
				// using them here would result in a reduced set of optional
				// filters.
				optionalFilters = c.checkConstraintFilters(scanPrivate.Table)
				computedColFilters := c.computedColFilters(scanPrivate, on, optionalFilters)
				optionalFilters = append(optionalFilters, computedColFilters...)

				eqColsAndOptionalFiltersCalculated = true
			}

			// Combine the ON filters and optional filters together. This set of
			// filters will be used to attempt to constrain non-inverted prefix
			// columns of the multi-column inverted index.
			allFilters = append(onFilters, optionalFilters...)
		}

		// The non-inverted prefix columns of a multi-column inverted index must
		// be constrained in order to perform an inverted join. We attempt to
		// constrain each prefix column to non-ranging constant values. These
		// values are joined with the input to create key columns for the
		// InvertedJoin, similar to GenerateLookupJoins.
		var constFilters memo.FiltersExpr
		var rightSideCols opt.ColList
		for i := 0; i < numPrefixCols; i++ {
			prefixCol := scanPrivate.Table.IndexColumnID(index, i)

			// Check if prefixCol is constrained by an equality constraint.
			if eqIdx, ok := rightEqCols.Find(prefixCol); ok {
				invertedJoin.PrefixKeyCols = append(invertedJoin.PrefixKeyCols, leftEqCols[eqIdx])
				rightSideCols = append(rightSideCols, prefixCol)
				continue
			}

			// Try to constrain prefixCol to constant, non-ranging values.
			foundVals, allIdx, ok := c.findJoinFilterConstants(allFilters, prefixCol)
			if !ok {
				// Cannot constrain prefix column and therefore cannot generate
				// an inverted join.
				return
			}

			if len(foundVals) > 1 && (joinType == opt.LeftJoinOp || joinType == opt.AntiJoinOp) {
				// We cannot create an inverted join in this case, because constructing
				// a cross join with foundVals will increase the size of the input. As a
				// result, non-matching input rows will show up more than once in the
				// output, which is incorrect (see #59615).
				// TODO(rytaft,mgartner): find a way to create an inverted join for this
				// case.
				return
			}

			// We will join these constant values with the input to make
			// equality columns for the inverted join.
			if constFilters == nil {
				constFilters = make(memo.FiltersExpr, 0, numPrefixCols)
			}

			prefixColType := c.e.f.Metadata().ColumnMeta(prefixCol).Type
			constColAlias := fmt.Sprintf("inverted_join_const_col_@%d", prefixCol)
			join, constColID := c.constructJoinWithConstants(
				invertedJoin.Input,
				foundVals,
				prefixColType,
				constColAlias,
			)

			invertedJoin.Input = join
			invertedJoin.PrefixKeyCols = append(invertedJoin.PrefixKeyCols, constColID)
			constFilters = append(constFilters, allFilters[allIdx])
			rightSideCols = append(rightSideCols, prefixCol)
		}

		// Remove redundant filters from the ON condition if non-inverted prefix
		// columns were constrained by equality filters or constant filters.
		onFilters = memo.ExtractRemainingJoinFilters(onFilters, invertedJoin.PrefixKeyCols, rightSideCols)
		onFilters = onFilters.Difference(constFilters)
		invertedJoin.ConstFilters = constFilters

		// Check whether the filter can constrain the inverted column.
		invertedExpr := invertedidx.TryJoinInvertedIndex(
			c.e.evalCtx.Context, c.e.f, onFilters, scanPrivate.Table, index, inputCols,
		)
		if invertedExpr == nil {
			return
		}

		// All geospatial and JSON inverted joins that are currently supported
		// are not covering, so we must wrap them in an index join.
		// TODO(rytaft): Avoid adding an index join if possible for Array
		// inverted joins.
		if scanPrivate.Flags.NoIndexJoin {
			return
		}

		if pkCols == nil {
			pkCols = c.getPkCols(scanPrivate.Table)
		}

		// Though the index is marked as containing the column being indexed, it
		// doesn't actually, and it is only valid to extract the primary key
		// columns and non-inverted prefix columns from it.
		indexCols = pkCols.ToSet()
		for i, n := 0, index.NonInvertedPrefixColumnCount(); i < n; i++ {
			prefixCol := scanPrivate.Table.IndexColumnID(index, i)
			indexCols.Add(prefixCol)
		}

		// Create a new ScanPrivate, which will be used below for the inverted join.
		// Note: this must happen before the continuation column is created to ensure
		// that the continuation column will have the highest column ID.
		//
		// See the comment where this newScanPrivate is used below in mapInvertedJoin
		// for details about why it's needed.
		if newScanPrivate == nil {
			newScanPrivate = c.DuplicateScanPrivate(scanPrivate)
		}

		continuationCol := opt.ColumnID(0)
		invertedJoinType := joinType
		// Anti joins are converted to a pair consisting of a left inverted join
		// and anti lookup join.
		if joinType == opt.LeftJoinOp || joinType == opt.AntiJoinOp {
			continuationCol = c.constructContinuationColumnForPairedJoin()
			invertedJoinType = opt.LeftJoinOp
		} else if joinType == opt.SemiJoinOp {
			// Semi joins are converted to a pair consisting of an inner inverted
			// join and semi lookup join.
			continuationCol = c.constructContinuationColumnForPairedJoin()
			invertedJoinType = opt.InnerJoinOp
		}
		invertedJoin.JoinPrivate = *joinPrivate
		invertedJoin.JoinType = invertedJoinType
		invertedJoin.Table = scanPrivate.Table
		invertedJoin.Index = index.Ordinal()
		invertedJoin.InvertedExpr = invertedExpr
		invertedJoin.Cols = indexCols.Union(inputCols)
		if continuationCol != 0 {
			invertedJoin.Cols.Add(continuationCol)
			invertedJoin.IsFirstJoinInPairedJoiner = true
			invertedJoin.ContinuationCol = continuationCol
		}

		var indexJoin memo.LookupJoinExpr

		// ON may have some conditions that are bound by the columns in the index
		// and some conditions that refer to other columns. We can put the former
		// in the InvertedJoin and the latter in the index join.
		invertedJoin.On = c.ExtractBoundConditions(onFilters, invertedJoin.Cols)
		indexJoin.On = c.ExtractUnboundConditions(onFilters, invertedJoin.Cols)

		// Map the inverted join to use the new table and column IDs from the
		// newScanPrivate created above. We want to make sure that the column IDs
		// returned by the inverted join are different from the IDs that will be
		// returned by the top level index join.
		//
		// In addition to avoiding subtle bugs in the optimizer when the same
		// column ID is reused, this mapping is also essential for correct behavior
		// at execution time in the case of a left paired join. This is because a
		// row that matches in the first left join (the inverted join) might be a
		// false positive and fail to match in the second left join (the lookup
		// join). If an original left row has no matches after the second left join,
		// it must appear as a null-extended row with all right-hand columns null.
		// If one of the right-hand columns comes from the inverted join, however,
		// it might incorrectly show up as non-null (see #58892).
		c.mapInvertedJoin(&invertedJoin, indexCols, newScanPrivate)

		indexJoin.Input = c.e.f.ConstructInvertedJoin(
			invertedJoin.Input,
			invertedJoin.On,
			&invertedJoin.InvertedJoinPrivate,
		)
		indexJoin.JoinType = joinType
		indexJoin.Table = scanPrivate.Table
		indexJoin.Index = cat.PrimaryIndex
		indexJoin.KeyCols = c.getPkCols(invertedJoin.Table)
		indexJoin.Cols = scanPrivate.Cols.Union(inputCols)
		indexJoin.LookupColsAreTableKey = true
		if continuationCol != 0 {
			indexJoin.IsSecondJoinInPairedJoiner = true
		}

		// If this is a semi- or anti-join, ensure the columns do not include any
		// unneeded right-side columns.
		if joinType == opt.SemiJoinOp || joinType == opt.AntiJoinOp {
			indexJoin.Cols = inputCols.Union(indexJoin.On.OuterCols())
		}

		// Create the LookupJoin for the index join in the same group.
		c.e.mem.AddLookupJoinToGroup(&indexJoin, grp)
	})
}

// getPkCols gets the primary key columns for the given table as a ColList.
func (c *CustomFuncs) getPkCols(tabID opt.TableID) opt.ColList {
	tab := c.e.mem.Metadata().Table(tabID)
	pkIndex := tab.Index(cat.PrimaryIndex)
	pkCols := make(opt.ColList, pkIndex.KeyColumnCount())
	for i := range pkCols {
		pkCols[i] = tabID.IndexColumnID(pkIndex, i)
	}
	return pkCols
}

// mapInvertedJoin maps the given inverted join to use the table and columns
// provided in newScanPrivate. The inverted join is modified in place. indexCols
// contains the pre-calculated index columns used by the given invertedJoin.
//
// Note that columns from the input are not mapped. For example, PrefixKeyCols
// does not need to be mapped below since it only contains input columns.
func (c *CustomFuncs) mapInvertedJoin(
	invertedJoin *memo.InvertedJoinExpr, indexCols opt.ColSet, newScanPrivate *memo.ScanPrivate,
) {
	tabID := invertedJoin.Table
	newTabID := newScanPrivate.Table

	// Get the catalog index (same for both new and old tables).
	index := c.e.mem.Metadata().TableMeta(tabID).Table.Index(invertedJoin.Index)

	// Though the index is marked as containing the column being indexed, it
	// doesn't actually, and it is only valid to extract the primary key
	// columns and non-inverted prefix columns from it.
	newPkCols := c.getPkCols(newTabID)
	newIndexCols := newPkCols.ToSet()
	for i, n := 0, index.NonInvertedPrefixColumnCount(); i < n; i++ {
		prefixCol := newTabID.IndexColumnID(index, i)
		newIndexCols.Add(prefixCol)
	}

	// Get the source and destination ColSets, including the inverted source
	// columns, which will be used in the invertedExpr.
	srcCols := indexCols.Copy()
	dstCols := newIndexCols.Copy()
	ord := index.VirtualInvertedColumn().InvertedSourceColumnOrdinal()
	invertedSourceCol := tabID.ColumnID(ord)
	newInvertedSourceCol := newTabID.ColumnID(ord)
	srcCols.Add(invertedSourceCol)
	dstCols.Add(newInvertedSourceCol)

	invertedJoin.Table = newTabID
	invertedJoin.InvertedExpr = c.mapScalarExprCols(invertedJoin.InvertedExpr, srcCols, dstCols)
	invertedJoin.Cols = invertedJoin.Cols.Difference(indexCols).Union(newIndexCols)
	invertedJoin.ConstFilters = c.MapFilterCols(invertedJoin.ConstFilters, srcCols, dstCols)
	invertedJoin.On = c.MapFilterCols(invertedJoin.On, srcCols, dstCols)
}

// findJoinFilterConstants tries to find a filter that is exactly equivalent to
// constraining the given column to a constant value or a set of constant
// values. If successful, the constant values and the index of the constraining
// FiltersItem are returned. Note that the returned constant values do not
// contain NULL.
func (c *CustomFuncs) findJoinFilterConstants(
	filters memo.FiltersExpr, col opt.ColumnID,
) (values tree.Datums, filterIdx int, ok bool) {
	for filterIdx := range filters {
		props := filters[filterIdx].ScalarProps()
		if props.TightConstraints {
			constCol, constVals, ok := props.Constraints.HasSingleColumnConstValues(c.e.evalCtx)
			if !ok || constCol != col {
				continue
			}
			hasNull := false
			for i := range constVals {
				if constVals[i] == tree.DNull {
					hasNull = true
					break
				}
			}
			if !hasNull {
				return constVals, filterIdx, true
			}
		}
	}
	return nil, -1, false
}

// constructJoinWithConstants constructs a cross join that joins every row in
// the input with every value in vals. The cross join will be converted into a
// projection by inlining normalization rules if vals contains only a single
// value. The constructed expression and the column ID of the constant value
// column are returned.
func (c *CustomFuncs) constructJoinWithConstants(
	input memo.RelExpr, vals tree.Datums, typ *types.T, columnAlias string,
) (join memo.RelExpr, constColID opt.ColumnID) {
	constColID = c.e.f.Metadata().AddColumn(columnAlias, typ)
	tupleType := types.MakeTuple([]*types.T{typ})

	constRows := make(memo.ScalarListExpr, len(vals))
	for i := range constRows {
		constRows[i] = c.e.f.ConstructTuple(
			memo.ScalarListExpr{c.e.f.ConstructConst(vals[i], typ)},
			tupleType,
		)
	}

	values := c.e.f.ConstructValues(
		constRows,
		&memo.ValuesPrivate{
			Cols: opt.ColList{constColID},
			ID:   c.e.mem.Metadata().NextUniqueID(),
		},
	)

	// We purposefully do not propagate any join flags into this JoinPrivate. If
	// a LOOKUP join hint was propagated to this cross join, the cost of the
	// cross join would be artificially inflated and the lookup join would not
	// be selected as the optimal plan.
	join = c.e.f.ConstructInnerJoin(input, values, nil /* on */, &memo.JoinPrivate{})
	return join, constColID
}

// ShouldReorderJoins returns whether the optimizer should attempt to find
// a better ordering of inner joins. This is the case if the given expression is
// the first expression of its group, and the join tree rooted at the expression
// has not previously been reordered. This is to avoid duplicate work. In
// addition, a join cannot be reordered if it has join hints.
func (c *CustomFuncs) ShouldReorderJoins(root memo.RelExpr) bool {
	// Only match the first expression of a group to avoid duplicate work.
	if root != root.FirstExpr() {
		return false
	}

	private, ok := root.Private().(*memo.JoinPrivate)
	if !ok {
		panic(errors.AssertionFailedf("operator does not have a join private: %v", root.Op()))
	}

	// Ensure that this join expression was not added to the memo by a previous
	// reordering, as well as that the join does not have hints.
	return !private.SkipReorderJoins && c.NoJoinHints(private)
}

// ReorderJoins adds alternate orderings of the given join tree to the memo. The
// first expression of the memo group is used for construction of the join
// graph. For more information, see the comment in join_order_builder.go.
func (c *CustomFuncs) ReorderJoins(grp memo.RelExpr) memo.RelExpr {
	c.e.o.JoinOrderBuilder().Init(c.e.f, c.e.evalCtx)
	c.e.o.JoinOrderBuilder().Reorder(grp.FirstExpr())
	return grp
}

// IsSimpleEquality returns true if all of the filter conditions are equalities
// between simple data types (constants, variables, tuples and NULL).
func (c *CustomFuncs) IsSimpleEquality(filters memo.FiltersExpr) bool {
	for i := range filters {
		eqFilter, ok := filters[i].Condition.(*memo.EqExpr)
		if !ok {
			return false
		}

		left, right := eqFilter.Left, eqFilter.Right
		switch left.Op() {
		case opt.VariableOp, opt.ConstOp, opt.NullOp, opt.TupleOp:
		default:
			return false
		}

		switch right.Op() {
		case opt.VariableOp, opt.ConstOp, opt.NullOp, opt.TupleOp:
		default:
			return false
		}
	}

	return true
}

// ConvertIndexToLookupJoinPrivate constructs a new LookupJoinPrivate using the
// given IndexJoinPrivate with the given output columns.
func (c *CustomFuncs) ConvertIndexToLookupJoinPrivate(
	indexPrivate *memo.IndexJoinPrivate, outCols opt.ColSet,
) *memo.LookupJoinPrivate {
	// Retrieve an ordered list of primary key columns from the lookup table;
	// these will form the lookup key.
	md := c.e.mem.Metadata()
	primaryIndex := md.Table(indexPrivate.Table).Index(cat.PrimaryIndex)
	lookupCols := make(opt.ColList, primaryIndex.KeyColumnCount())
	for i := 0; i < primaryIndex.KeyColumnCount(); i++ {
		lookupCols[i] = indexPrivate.Table.IndexColumnID(primaryIndex, i)
	}

	return &memo.LookupJoinPrivate{
		JoinType:              opt.InnerJoinOp,
		Table:                 indexPrivate.Table,
		Index:                 cat.PrimaryIndex,
		KeyCols:               lookupCols,
		Cols:                  outCols,
		LookupColsAreTableKey: true,
		ConstFilters:          nil,
		JoinPrivate:           memo.JoinPrivate{},
	}
}

// HasVolatileProjection returns true if any of the projection items of the
// ProjectionsExpr contains a volatile expression.
func (c *CustomFuncs) HasVolatileProjection(projections memo.ProjectionsExpr) bool {
	for i := range projections {
		if projections[i].ScalarProps().VolatilitySet.HasVolatile() {
			return true
		}
	}
	return false
}

// FindLeftJoinCanaryColumn tries to find a "canary" column from the right input
// of a left join. This is a column that is NULL in the join output iff the row
// is an "outer left" row that had no match in the join.
//
// Returns 0 if we couldn't find such a column.
func (c *CustomFuncs) FindLeftJoinCanaryColumn(
	right memo.RelExpr, on memo.FiltersExpr,
) opt.ColumnID {
	canaryCol, ok := right.Relational().NotNullCols.Next(0)
	if ok {
		// The right expression has a non-null column; we can use it as a canary
		// column.
		return canaryCol
	}

	// Find any column from the right which is null-rejected by the ON condition.
	// right rows where such a column is NULL will never contribute to the join
	// result.
	nullRejectedCols := memo.NullColsRejectedByFilter(c.e.evalCtx, on)
	nullRejectedCols.IntersectionWith(right.Relational().OutputCols)

	canaryCol, ok = nullRejectedCols.Next(0)
	if ok {
		return canaryCol
	}

	return 0
}

// FoundCanaryColumn returns true if the given column returned by
// FindLeftJoinCanaryColum indicates that we found a canary column.
func (c *CustomFuncs) FoundCanaryColumn(canaryCol opt.ColumnID) bool {
	return canaryCol != 0
}

// MakeProjectionsForOuterJoin takes a set of projections and wraps them in a
// conditional which overrides them to NULL whenever the canary column is NULL.
// TODO(radu): detect projections that evaluate to NULL on NULL inputs anyway,
// and leave those alone.
func (c *CustomFuncs) MakeProjectionsForOuterJoin(
	canaryCol opt.ColumnID, proj memo.ProjectionsExpr,
) memo.ProjectionsExpr {
	result := make(memo.ProjectionsExpr, len(proj))

	for i := range proj {
		// Construct "IF(canaryCol IS NULL, NULL, <projection>)".
		ifExpr := c.e.f.ConstructCase(
			c.e.f.ConstructIs(
				c.e.f.ConstructVariable(
					canaryCol,
				),
				memo.NullSingleton,
			),
			memo.ScalarListExpr{
				c.e.f.ConstructWhen(memo.TrueSingleton, c.e.f.ConstructNull(proj[i].Typ)),
			},
			proj[i].Element,
		)
		result[i] = c.e.f.ConstructProjectionsItem(ifExpr, proj[i].Col)
	}
	return result
}

// LocalAndRemoteLookupExprs is used by the GenerateLocalityOptimizedAntiJoin
// rule to hold two sets of filters: one targeting local partitions and one
// targeting remote partitions.
type LocalAndRemoteLookupExprs struct {
	Local  memo.FiltersExpr
	Remote memo.FiltersExpr
}

// LocalAndRemoteLookupExprsSucceeded returns true if the
// LocalAndRemoteLookupExprs is not empty.
func (c *CustomFuncs) LocalAndRemoteLookupExprsSucceeded(le LocalAndRemoteLookupExprs) bool {
	return len(le.Local) != 0 && len(le.Remote) != 0
}

// CreateLocalityOptimizedAntiLookupJoinPrivate creates a new lookup join
// private from the given private and replaces the LookupExpr with the given
// filters. It also marks the private as locality optimized.
func (c *CustomFuncs) CreateLocalityOptimizedAntiLookupJoinPrivate(
	lookupExpr memo.FiltersExpr, private *memo.LookupJoinPrivate,
) *memo.LookupJoinPrivate {
	newPrivate := *private
	newPrivate.LookupExpr = lookupExpr
	newPrivate.LocalityOptimized = true
	return &newPrivate
}

// LocalLookupExpr extracts the Local filters expr from the given
// LocalAndRemoteLookupExprs.
func (c *CustomFuncs) LocalLookupExpr(le LocalAndRemoteLookupExprs) memo.FiltersExpr {
	return le.Local
}

// RemoteLookupExpr extracts the Remote filters expr from the given
// LocalAndRemoteLookupExprs.
func (c *CustomFuncs) RemoteLookupExpr(le LocalAndRemoteLookupExprs) memo.FiltersExpr {
	return le.Remote
}

// GetLocalityOptimizedAntiJoinLookupExprs gets the lookup expressions needed to
// build a locality optimized anti join if possible from the given lookup join
// private. See the comment above the GenerateLocalityOptimizedAntiJoin rule for
// more details.
func (c *CustomFuncs) GetLocalityOptimizedAntiJoinLookupExprs(
	input memo.RelExpr, private *memo.LookupJoinPrivate,
) LocalAndRemoteLookupExprs {
	// Respect the session setting LocalityOptimizedSearch.
	if !c.e.evalCtx.SessionData.LocalityOptimizedSearch {
		return LocalAndRemoteLookupExprs{}
	}

	// Check whether this lookup join has already been locality optimized.
	if private.LocalityOptimized {
		return LocalAndRemoteLookupExprs{}
	}

	// We can only apply this optimization to anti-joins.
	if private.JoinType != opt.AntiJoinOp {
		return LocalAndRemoteLookupExprs{}
	}

	// This lookup join cannot not be part of a paired join.
	if private.IsSecondJoinInPairedJoiner {
		return LocalAndRemoteLookupExprs{}
	}

	// This lookup join should have the LookupExpr filled in, indicating that one
	// or more of the join filters constrain an index column to multiple constant
	// values.
	if private.LookupExpr == nil {
		return LocalAndRemoteLookupExprs{}
	}

	// The local region must be set, or we won't be able to determine which
	// partitions are local.
	localRegion, found := c.e.evalCtx.Locality.Find(regionKey)
	if !found {
		return LocalAndRemoteLookupExprs{}
	}

	// There should be at least two partitions, or we won't be able to
	// differentiate between local and remote partitions.
	tabMeta := c.e.mem.Metadata().TableMeta(private.Table)
	index := tabMeta.Table.Index(private.Index)
	if index.PartitionCount() < 2 {
		return LocalAndRemoteLookupExprs{}
	}

	// Determine whether the index has both local and remote partitions.
	var localPartitions util.FastIntSet
	for i, n := 0, index.PartitionCount(); i < n; i++ {
		part := index.Partition(i)
		if isZoneLocal(part.Zone(), localRegion) {
			localPartitions.Add(i)
		}
	}
	if localPartitions.Len() == 0 || localPartitions.Len() == index.PartitionCount() {
		// The partitions are either all local or all remote.
		return LocalAndRemoteLookupExprs{}
	}

	// Find a filter that constrains the first column of the index.
	filterIdx, ok := c.getConstPrefixFilter(index, private.Table, private.LookupExpr)
	if !ok {
		return LocalAndRemoteLookupExprs{}
	}
	filter := private.LookupExpr[filterIdx]

	// Check whether the filter constrains the first column of the index
	// to at least two constant values. We need at least two values so that one
	// can target a local partition and one can target a remote partition.
	col, vals, ok := filter.ScalarProps().Constraints.HasSingleColumnConstValues(c.e.evalCtx)
	if !ok || len(vals) < 2 {
		return LocalAndRemoteLookupExprs{}
	}

	// Determine whether the values target both local and remote partitions.
	localValOrds := c.getLocalValues(index, localPartitions, vals)
	if localValOrds.Len() == 0 || localValOrds.Len() == len(vals) {
		// The values target all local or all remote partitions.
		return LocalAndRemoteLookupExprs{}
	}

	// Split the values into local and remote sets.
	localValues, remoteValues := c.splitValues(vals, localValOrds)

	// Copy all of the filters from the LookupExpr, and replace the filter that
	// constrains the first index column with a filter targeting only local
	// partitions or only remote partitions.
	localExpr := make(memo.FiltersExpr, len(private.LookupExpr))
	copy(localExpr, private.LookupExpr)
	localExpr[filterIdx] = c.makeConstFilter(col, localValues)

	remoteExpr := make(memo.FiltersExpr, len(private.LookupExpr))
	copy(remoteExpr, private.LookupExpr)
	remoteExpr[filterIdx] = c.makeConstFilter(col, remoteValues)

	// Return the two sets of lookup expressions. They will be used to construct
	// two nested anti joins.
	return LocalAndRemoteLookupExprs{
		Local:  localExpr,
		Remote: remoteExpr,
	}
}

// getConstPrefixFilter finds the position of the filter in the given slice of
// filters that constrains the first index column to one or more constant
// values. If such a filter is found, getConstPrefixFilter returns the position
// of the filter and ok=true. Otherwise, returns ok=false.
func (c CustomFuncs) getConstPrefixFilter(
	index cat.Index, table opt.TableID, filters memo.FiltersExpr,
) (pos int, ok bool) {
	idxCol := table.IndexColumnID(index, 0)
	for i := range filters {
		props := filters[i].ScalarProps()
		if !props.TightConstraints {
			continue
		}
		if props.OuterCols.Len() != 1 {
			continue
		}
		col, _ := props.OuterCols.Next(0)
		if col == idxCol {
			return i, true
		}
	}
	return 0, false
}

// getLocalValues returns the indexes of the values in the given Datums slice
// that target local partitions.
func (c *CustomFuncs) getLocalValues(
	index cat.Index, localPartitions util.FastIntSet, values tree.Datums,
) util.FastIntSet {
	// Collect all the prefixes from all the different partitions (remembering
	// which ones came from local partitions), and sort them so that longer
	// prefixes come before shorter prefixes. For each value in the given Datums,
	// we will iterate through the list of prefixes until we find a match, so
	// ordering them with longer prefixes first ensures that the correct match is
	// found.
	allPrefixes := getSortedPrefixes(index, localPartitions)

	// TODO(rytaft): Sort the prefixes by key in addition to length, and use
	// binary search here.
	var localVals util.FastIntSet
	for i, val := range values {
		for j := range allPrefixes {
			prefix := allPrefixes[j].prefix
			isLocal := allPrefixes[j].isLocal
			if len(prefix) > 1 {
				continue
			}
			if val.Compare(c.e.evalCtx, prefix[0]) == 0 {
				if isLocal {
					localVals.Add(i)
				}
				break
			}
		}
	}
	return localVals
}

// splitValues splits the given slice of Datums into local and remote slices
// by putting the Datums at positions identified by localValOrds into the local
// slice, and the remaining Datums into the remote slice.
func (c *CustomFuncs) splitValues(
	values tree.Datums, localValOrds util.FastIntSet,
) (localVals, remoteVals tree.Datums) {
	localVals = make(tree.Datums, 0, localValOrds.Len())
	remoteVals = make(tree.Datums, 0, len(values)-len(localVals))
	for i, val := range values {
		if localValOrds.Contains(i) {
			localVals = append(localVals, val)
		} else {
			remoteVals = append(remoteVals, val)
		}
	}
	return localVals, remoteVals
}