# Algebra

Relational algebra is at the heart of Calcite. Every query is represented as a tree of relational operators. You can translate from SQL to relational algebra, or you can build the tree directly.

Planner rules transform expression trees using mathematical identities that preserve semantics. For example, it is valid to push a filter into an input of an inner join if the filter does not reference columns from the other input.

Calcite optimizes queries by repeatedly applying planner rules to a relational expression. A cost model guides the process, and the planner engine generates an alternative expression that has the same semantics as the original but a lower cost.

The planning process is extensible. You can add your own relational operators, planner rules, cost model, and statistics.

## Algebra builder

The simplest way to build a relational expression is to use the algebra builder, RelBuilder. Here is an example:

### TableScan

(You can find the full code for this and other examples in RelBuilderExample.java.)

The code prints

It has created a scan of the `EMP` table; equivalent to the SQL

### Adding a Project

Now, let’s add a Project, the equivalent of

We just add a call to the `project` method before calling `build`:

and the output is

The two calls to `builder.field` create simple expressions that return the fields from the input relational expression, namely the TableScan created by the `scan` call.

Calcite has converted them to field references by ordinal, `\$7` and `\$1`.

### Adding a Filter and Aggregate

A query with an Aggregate, and a Filter:

is equivalent to SQL

and produces

### Push and pop

The builder uses a stack to store the relational expression produced by one step and pass it as an input to the next step. This allows the methods that produce relational expressions to produce a builder.

Most of the time, the only stack method you will use is `build()`, to get the last relational expression, namely the root of the tree.

Sometimes the stack becomes so deeply nested it gets confusing. To keep things straight, you can remove expressions from the stack. For example, here we are building a bushy join:

We build it in three stages. Store the intermediate results in variables `left` and `right`, and use `push()` to put them back on the stack when it is time to create the final `Join`:

### Field names and ordinals

You can reference a field by name or ordinal.

Ordinals are zero-based. Each operator guarantees the order in which its output fields occur. For example, `Project` returns the fields in the generated by each of the scalar expressions.

The field names of an operator are guaranteed to be unique, but sometimes that means that the names are not exactly what you expect. For example, when you join EMP to DEPT, one of the output fields will be called DEPTNO and another will be called something like DEPTNO_1.

Some relational expression methods give you more control over field names:

• `project` lets you wrap expressions using `alias(expr, fieldName)`. It removes the wrapper but keeps the suggested name (as long as it is unique).
• `values(String[] fieldNames, Object... values)` accepts an array of field names. If any element of the array is null, the builder will generate a unique name.

If an expression projects an input field, or a cast of an input field, it will use the name of that input field.

Once the unique field names have been assigned, the names are immutable. If you have a particular `RelNode` instance, you can rely on the field names not changing. In fact, the whole relational expression is immutable.

But if a relational expression has passed through several rewrite rules (see RelOptRule), the field names of the resulting expression might not look much like the originals. At that point it is better to reference fields by ordinal.

When you are building a relational expression that accepts multiple inputs, you need to build field references that take that into account. This occurs most often when building join conditions.

Suppose you are building a join on EMP, which has 8 fields [EMPNO, ENAME, JOB, MGR, HIREDATE, SAL, COMM, DEPTNO] and DEPT, which has 3 fields [DEPTNO, DNAME, LOC]. Internally, Calcite represents those fields as offsets into a combined input row with 11 fields: the first field of the left input is field #0 (0-based, remember), and the first field of the right input is field #8.

But through the builder API, you specify which field of which input. To reference “SAL”, internal field #5, write `builder.field(2, 0, "SAL")`, `builder.field(2, "EMP", "SAL")`, or `builder.field(2, 0, 5)`. This means “the field #5 of input #0 of two inputs”. (Why does it need to know that there are two inputs? Because they are stored on the stack; input #1 is at the top of the stack, and input #0 is below it. If we did not tell the builder that were two inputs, it would not know how deep to go for input #0.)

Similarly, to reference “DNAME”, internal field #9 (8 + 1), write `builder.field(2, 1, "DNAME")`, `builder.field(2, "DEPT", "DNAME")`, or `builder.field(2, 1, 1)`.

### Recursive Queries

Warning: The current API is experimental and subject to change without notice. A SQL recursive query, e.g. this one that generates the sequence 1, 2, 3, …10:

can be generated using a scan on a TransientTable and a RepeatUnion:

which produces:

### API summary

#### Relational operators

The following methods create a relational expression (RelNode), push it onto the stack, and return the `RelBuilder`.

Method Description
`scan(tableName)` Creates a TableScan.
`functionScan(operator, n, expr...)`
`functionScan(operator, n, exprList)`
Creates a TableFunctionScan of the `n` most recent relational expressions.
`transientScan(tableName [, rowType])` Creates a TableScan on a [TransientTable]](/apidocs/org/apache/calcite/schema/TransientTable.html) with the given type (if not specified, the most recent relational expression’s type will be used).
`values(fieldNames, value...)`
`values(rowType, tupleList)`
Creates a Values.
`filter([variablesSet, ] exprList)`
`filter([variablesSet, ] expr...)`
Creates a Filter over the AND of the given predicates; if `variablesSet` is specified, the predicates may reference those variables.
`project(expr...)`
`project(exprList [, fieldNames])`
Creates a Project. To override the default name, wrap expressions using `alias`, or specify the `fieldNames` argument.
`projectPlus(expr...)`
`projectPlus(exprList)`
Variant of `project` that keeps original fields and appends the given expressions.
`permute(mapping)` Creates a Project that permutes the fields using `mapping`.
`convert(rowType [, rename])` Creates a Project that converts the fields to the given types, optionally also renaming them.
`aggregate(groupKey, aggCall...)`
`aggregate(groupKey, aggCallList)`
Creates an Aggregate.
`distinct()` Creates an Aggregate that eliminates duplicate records.
`sort(fieldOrdinal...)`
`sort(expr...)`
`sort(exprList)`
Creates a Sort.

In the first form, field ordinals are 0-based, and a negative ordinal indicates descending; for example, -2 means field 1 descending.

In the other forms, you can wrap expressions in `as`, `nullsFirst` or `nullsLast`.
`sortLimit(offset, fetch, expr...)`
`sortLimit(offset, fetch, exprList)`
Creates a Sort with offset and limit.
`limit(offset, fetch)` Creates a Sort that does not sort, only applies with offset and limit.
`exchange(distribution)` Creates an Exchange.
`sortExchange(distribution, collation)` Creates a SortExchange.
`correlate(joinType, correlationId, requiredField...)`
`correlate(joinType, correlationId, requiredFieldList)`
Creates a Correlate of the two most recent relational expressions, with a variable name and required field expressions for the left relation.
`join(joinType, expr...)`
`join(joinType, exprList)`
`join(joinType, fieldName...)`
Creates a Join of the two most recent relational expressions.

The first form joins on a boolean expression (multiple conditions are combined using AND).

The last form joins on named fields; each side must have a field of each name.
`semiJoin(expr)` Creates a Join with SEMI join type of the two most recent relational expressions.
`antiJoin(expr)` Creates a Join with ANTI join type of the two most recent relational expressions.
`union(all [, n])` Creates a Union of the `n` (default two) most recent relational expressions.
`intersect(all [, n])` Creates an Intersect of the `n` (default two) most recent relational expressions.
`minus(all)` Creates a Minus of the two most recent relational expressions.
`repeatUnion(tableName, all [, n])` Creates a RepeatUnion associated to a [TransientTable]](/apidocs/org/apache/calcite/schema/TransientTable.html) of the two most recent relational expressions, with `n` maximum number of iterations (default -1, i.e. no limit).
`snapshot(period)` Creates a Snapshot of the given snapshot period.
`match(pattern, strictStart,` `strictEnd, patterns, measures,` `after, subsets, allRows,` `partitionKeys, orderKeys,` `interval)` Creates a Match.

Argument types:

• `expr`, `interval` RexNode
• `expr...`, `requiredField...` Array of RexNode
• `exprList`, `measureList`, `partitionKeys`, `orderKeys`, `requiredFieldList` Iterable of RexNode
• `fieldOrdinal` Ordinal of a field within its row (starting from 0)
• `fieldName` Name of a field, unique within its row
• `fieldName...` Array of String
• `fieldNames` Iterable of String
• `rowType` RelDataType
• `groupKey` RelBuilder.GroupKey
• `aggCall...` Array of RelBuilder.AggCall
• `aggCallList` Iterable of RelBuilder.AggCall
• `value...` Array of Object
• `value` Object
• `tupleList` Iterable of List of RexLiteral
• `all`, `distinct`, `strictStart`, `strictEnd`, `allRows` boolean
• `alias` String
• `correlationId` CorrelationId
• `variablesSet` Iterable of CorrelationId
• `varHolder` Holder of RexCorrelVariable
• `patterns` Map whose key is String, value is RexNode
• `subsets` Map whose key is String, value is a sorted set of String
• `distribution` RelDistribution
• `collation` RelCollation
• `operator` SqlOperator
• `joinType` JoinRelType

The builder methods perform various optimizations, including:

• `project` returns its input if asked to project all columns in order
• `filter` flattens the condition (so an `AND` and `OR` may have more than 2 children), simplifies (converting say `x = 1 AND TRUE` to `x = 1`)
• If you apply `sort` then `limit`, the effect is as if you had called `sortLimit`

There are annotation methods that add information to the top relational expression on the stack:

Method Description
`as(alias)` Assigns a table alias to the top relational expression on the stack
`variable(varHolder)` Creates a correlation variable referencing the top relational expression

#### Stack methods

Method Description
`build()` Pops the most recently created relational expression off the stack
`push(rel)` Pushes a relational expression onto the stack. Relational methods such as `scan`, above, call this method, but user code generally does not
`pushAll(collection)` Pushes a collection of relational expressions onto the stack
`peek()` Returns the relational expression most recently put onto the stack, but does not remove it

#### Scalar expression methods

The following methods return a scalar expression (RexNode).

Many of them use the contents of the stack. For example, `field("DEPTNO")` returns a reference to the “DEPTNO” field of the relational expression just added to the stack.

Method Description
`literal(value)` Constant
`field(fieldName)` Reference, by name, to a field of the top-most relational expression
`field(fieldOrdinal)` Reference, by ordinal, to a field of the top-most relational expression
`field(inputCount, inputOrdinal, fieldName)` Reference, by name, to a field of the (`inputCount` - `inputOrdinal`)th relational expression
`field(inputCount, inputOrdinal, fieldOrdinal)` Reference, by ordinal, to a field of the (`inputCount` - `inputOrdinal`)th relational expression
`field(inputCount, alias, fieldName)` Reference, by table alias and field name, to a field at most `inputCount - 1` elements from the top of the stack
`field(alias, fieldName)` Reference, by table alias and field name, to a field of the top-most relational expressions
`field(expr, fieldName)` Reference, by name, to a field of a record-valued expression
`field(expr, fieldOrdinal)` Reference, by ordinal, to a field of a record-valued expression
`fields(fieldOrdinalList)` List of expressions referencing input fields by ordinal
`fields(mapping)` List of expressions referencing input fields by a given mapping
`fields(collation)` List of expressions, `exprList`, such that `sort(exprList)` would replicate collation
`call(op, expr...)`
`call(op, exprList)`
Call to a function or operator
`and(expr...)`
`and(exprList)`
Logical AND. Flattens nested ANDs, and optimizes cases involving TRUE and FALSE.
`or(expr...)`
`or(exprList)`
Logical OR. Flattens nested ORs, and optimizes cases involving TRUE and FALSE.
`not(expr)` Logical NOT
`equals(expr, expr)` Equals
`isNull(expr)` Checks whether an expression is null
`isNotNull(expr)` Checks whether an expression is not null
`alias(expr, fieldName)` Renames an expression (only valid as an argument to `project`)
`cast(expr, typeName)`
`cast(expr, typeName, precision)`
`cast(expr, typeName, precision, scale)`
Converts an expression to a given type
`desc(expr)` Changes sort direction to descending (only valid as an argument to `sort` or `sortLimit`)
`nullsFirst(expr)` Changes sort order to nulls first (only valid as an argument to `sort` or `sortLimit`)
`nullsLast(expr)` Changes sort order to nulls last (only valid as an argument to `sort` or `sortLimit`)
`cursor(n, input)` Reference to `input`th (0-based) relational input of a `TableFunctionScan` with `n` inputs (see `functionScan`)

#### Pattern methods

The following methods return patterns for use in `match`.

Method Description
`patternConcat(pattern...)` Concatenates patterns
`patternAlter(pattern...)` Alternates patterns
`patternQuantify(pattern, min, max)` Quantifies a pattern
`patternPermute(pattern...)` Permutes a pattern
`patternExclude(pattern)` Excludes a pattern

#### Group key methods

The following methods return a RelBuilder.GroupKey.

Method Description
`groupKey(fieldName...)`
`groupKey(fieldOrdinal...)`
`groupKey(expr...)`
`groupKey(exprList)`
Creates a group key of the given expressions
`groupKey(exprList, exprListList)` Creates a group key of the given expressions with grouping sets
`groupKey(bitSet [, bitSets])` Creates a group key of the given input columns, with multiple grouping sets if `bitSets` is specified

#### Aggregate call methods

The following methods return an RelBuilder.AggCall.

Method Description
`aggregateCall(op, expr...)`
`aggregateCall(op, exprList)`
Creates a call to a given aggregate function
`count([ distinct, alias, ] expr...)`
`count([ distinct, alias, ] exprList)`
Creates a call to the `COUNT` aggregate function
`countStar(alias)` Creates a call to the `COUNT(*)` aggregate function
`sum([ distinct, alias, ] expr)` Creates a call to the `SUM` aggregate function
`min([ alias, ] expr)` Creates a call to the `MIN` aggregate function
`max([ alias, ] expr)` Creates a call to the `MAX` aggregate function

To further modify the `AggCall`, call its methods:

Method Description
`approximate(approximate)` Allows approximate value for the aggregate of `approximate`
`as(alias)` Assigns a column alias to this expression (see SQL `AS`)
`distinct()` Eliminates duplicate values before aggregating (see SQL `DISTINCT`)
`distinct(distinct)` Eliminates duplicate values before aggregating if `distinct`
`filter(expr)` Filters rows before aggregating (see SQL `FILTER (WHERE ...)`)
`sort(expr...)`
`sort(exprList)`
Sorts rows before aggregating (see SQL `WITHIN GROUP`)