Extended Operators of Relational Algebra

Extended Operators of Relational Algebra

Section outline

• Introduction to the need for extended relational operators beyond the classical set-based algebra, particularly for bag-based relations and modern query languages like SQL.
• Overview of the new operators to be introduced: duplicate-elimination (\(\delta\)), aggregation (SUM, AVG, etc.), grouping (\(\gamma\)), extended projection (\(\pi\)), sorting (\(\tau\)), and outerjoin.

#1

read: Section 5.2: Extended Operators of Relational Algebra (section)

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Present the following.  Use definition formal definition when it applies.

- Introduction to the need for extended relational operators beyond the classical set-based algebra, particularly for bag-based relations and modern query languages like SQL.
- Overview of the new operators to be introduced: duplicate-elimination ($\delta$), aggregation (SUM, AVG, etc.), grouping ($\gamma$), extended projection ($\pi$), sorting ($\tau$), and outerjoin.

Classical relational algebra provides a foundational framework for querying data. However, its set-based semantics and limited set of operators are insufficient for the needs of modern query languages like SQL, which operate on bags (multisets) and require more powerful data manipulation capabilities. To bridge this gap, we introduce several extended operators to the algebra.

• The Duplicate-Elimination Operator (\(\delta\))

  This operator transforms a bag into a set. Given a relation \(R\) that may contain duplicate tuples, \(\delta(R)\) produces a new relation containing exactly one instance of each unique tuple from \(R\).

• Aggregation Operators

  These operators are used to summarize the values within a column of a relation. They are not standalone relational algebra operators but are integral components of the grouping operator. Common aggregators include SUM, AVG, MIN, MAX, and COUNT.

• The Grouping Operator (\(\gamma\))

  The grouping operator, denoted \(\gamma_L(R)\), partitions the tuples of a relation \(R\) into groups. All tuples in a group share the same values for the grouping attributes specified in the list \(L\). For each group, the operator computes one or more aggregate values, also specified in \(L\), producing a single summary tuple as a result.

• Extended Projection (\(\pi\))

  This operator generalizes the standard projection. In addition to selecting attributes, the extended projection \(\pi_L(R)\) allows the list \(L\) to contain expressions that compute new attribute values. These expressions can involve arithmetic operations, string manipulations, and constants applied to the attributes of each tuple in \(R\).

• The Sorting Operator (\(\tau\))

  The sorting operator, \(\tau_L(R)\), orders the tuples of a relation \(R\) based on a list of attributes \(L\). The result is a list of tuples, not a set or a bag, as the order of tuples is now significant.

• The Outerjoin Operator (\(\bowtie\))

  The outerjoin is a form of join that preserves tuples that do not have matching tuples in the other relation. These “dangling tuples” are included in the result, padded with NULL values for the attributes of the relation in which they had no match. Variants include the left outerjoin (\(\bowtie^L\)), right outerjoin (\(\bowtie^R\)), and full outerjoin (\(\bowtie\)).

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Duplicate Elimination

Section outline

• Defines the duplicate-elimination operator \(\delta(R)\) which converts a multiset (bag) R into a set by removing all but one copy of each unique tuple.

#2

read: Section 5.2.1: Duplicate Elimination (section)

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Present the following.  Use definition formal definition when it applies.

- Defines the duplicate-elimination operator $\delta(R)$ which converts a multiset (bag) `R` into a set by removing all but one copy of each unique tuple.

• The duplicate-elimination operator, denoted by \(\delta\), transforms a bag of tuples into a set.

  Its function is to remove all duplicate tuples from a relation, thereby ensuring that each tuple in the output appears only once.

Definition: Duplicate-Elimination Operator (\(\delta\))

Let \(R\) be a relation, which is a bag of tuples. The operation \(\delta(R)\) results in a set containing every tuple that appears one or more times in \(R\). Formally, \[ \delta(R) = \{ t \mid t \in R \} \] The resulting relation \(\delta(R)\) contains exactly one instance of each distinct tuple from the bag \(R\).

• The \(\delta\) operator is a special case of the grouping operator, \(\gamma\).

  If a relation \(R\) has attributes \(A_1, A_2, \ldots, A_n\), then the elimination of duplicates is equivalent to grouping by all attributes of the relation without performing any aggregation. \[ \delta(R) \equiv \gamma_{A_1, A_2, \ldots, A_n}(R) \]

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Aggregation Operators

Section outline

• Introduces aggregation operators which apply to columns of a relation to produce a single summary value.
• Lists and describes common aggregation operators: SUM, AVG, MIN, MAX, and COUNT.

#3

read: Section 5.2.2: Aggregation Operators (section)

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Present the following.  Use definition formal definition when it applies.

- Introduces aggregation operators which apply to columns of a relation to produce a single summary value.
- Lists and describes common aggregation operators: SUM, AVG, MIN, MAX, and COUNT.

Aggregation operators are functions that apply to the set or bag of values in a single column of a relation. They are used to produce a single summary value from these values. These operators are not part of the classical relational algebra but are essential components for modern query languages, particularly in conjunction with the grouping operator.

Definition: Aggregation Operator

An aggregation operator, denoted as AGG, is a function that maps a multiset (bag) of values, corresponding to a column in a relation, to a single summary value.

The standard aggregation operators are as follows:

• SUM

  Produces the arithmetic sum of a column with numerical values.

• AVG

  Produces the average of a column with numerical values.

• MIN and MAX

  For a column with numerical values, these operators produce the smallest and largest values, respectively. When applied to character-string values, they yield the lexicographically first and last values.

• COUNT

  Produces the number of values in a column, including duplicates. This is equivalent to the number of tuples in the relation or group to which it is applied.

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Grouping

Section outline

• Explains the concept of grouping, which partitions the tuples of a relation into groups based on the values of one or more attributes.
• Describes how aggregation can be applied to each group independently, enabling more complex queries than possible with classical relational algebra.

#4

read: Section 5.2.3: Grouping (section)

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Present the following.  Use definition formal definition when it applies.

- Explains the concept of grouping, which partitions the tuples of a relation into groups based on the values of one or more attributes.
- Describes how aggregation can be applied to each group independently, enabling more complex queries than possible with classical relational algebra.

• Grouping

  Grouping is an operation that partitions the tuples of a relation into groups. Tuples are placed into the same group if they share the same value for one or more specified “grouping attributes.” This process does not, by itself, alter the tuples but organizes them for subsequent operations.

• Aggregation on Groups

  After partitioning a relation into groups, an aggregation operator (such as SUM, AVG, COUNT) can be applied independently to each group. This allows for the computation of summary data for each distinct value of the grouping attributes. For instance, given a Movies relation, one could group tuples by the studioName attribute and then apply SUM(length) to each group to find the total running time of movies for each studio.

• The Grouping Operator (\(\gamma\))

  The grouping and aggregation process is formally expressed by the grouping operator, \(\gamma\). The operator takes a list of grouping attributes and a list of aggregation functions to be applied. The expression \(\gamma_L(R)\) partitions the relation \(R\) and computes the specified aggregations.

Definition: Grouping

Let \(R\) be a relation. Grouping partitions the multiset of tuples in \(R\) into disjoint groups \(\{G_1, G_2, \dots, G_k\}\). All tuples within a single group \(G_i\) must have identical values for a specified set of grouping attributes. For any two tuples \(t_1, t_2\) in the same group \(G_i\), and for any grouping attribute \(A\), it holds that \(t_1[A] = t_2[A]\). An aggregation function can then be applied to each group \(G_i\) to produce a single summary tuple.

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The Grouping Operator

Section outline

• Formally defines the grouping operator \(\gamma_L(R)\), where L is a list of grouping attributes and/or aggregation functions.
• Describes the construction of the result: tuples in R are partitioned into groups based on the grouping attributes, and then one result tuple is generated for each group by applying the specified aggregations.

#5

read: Section 5.2.4: The Grouping Operator (section)

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Present the following.  Use definition formal definition when it applies.

- Formally defines the grouping operator $\gamma_L(R)$, where `L` is a list of grouping attributes and/or aggregation functions.
- Describes the construction of the result: tuples in `R` are partitioned into groups based on the grouping attributes, and then one result tuple is generated for each group by applying the specified aggregations.

The grouping operator, denoted by \(\gamma\), allows for the grouping of tuples in a relation and the application of aggregation functions to these groups.

Definition: The Grouping Operator (\(\gamma\))

The expression \(\gamma_L(R)\) applies the grouping and aggregation operations specified in the list \(L\) to the relation \(R\). The subscript \(L\) is a list of elements, where each element is either a grouping attribute or an aggregation function applied to an attribute.

• Elements of the Subscript List \(L\)

  The list \(L\) consists of one or both of the following types of elements:

  1. Grouping Attribute: An attribute of the relation \(R\) that is used to partition the tuples.
  2. Aggregated Attribute: An aggregation operator (e.g., SUM, AVG, MIN, MAX, COUNT) applied to an attribute of \(R\). To provide a name for the resulting column, a new name is appended using an arrow, such as AGG(A) -> newName.

• Construction of the Result Relation

  The relation returned by the expression \(\gamma_L(R)\) is constructed through a two-step process:

  1. Partition: The tuples of relation \(R\) are partitioned into groups. Each group consists of all tuples that have the same values for the specified grouping attributes in list \(L\). If no grouping attributes are provided, the entire relation is treated as a single group.
  2. Generation: For each group, one tuple is generated in the result. This tuple contains:
     1. The values of the grouping attributes for that group.
     2. The results of the specified aggregations, computed over all tuples within that group.

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Extending the Projection Operator

Section outline

• Introduces an extension to the projection operator \(\pi_L(R)\), allowing the list L to include more than just attribute names.
• Describes the three types of elements allowed in the extended projection list L: a single attribute, an attribute renaming (\(x \rightarrow y\)), and an expression involving attributes and constants (\(E \rightarrow z\)).

#6

read: Section 5.2.5: Extending the Projection Operator (section)

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Present the following.  Use definition formal definition when it applies.

- Introduces an extension to the projection operator $\pi_L(R)$, allowing the list `L` to include more than just attribute names.
- Describes the three types of elements allowed in the extended projection list `L`: a single attribute, an attribute renaming ($x \rightarrow y$), and an expression involving attributes and constants ($E \rightarrow z$).

The classical projection operator, \(\pi_L(R)\), is extended to provide additional computational power. In this generalized form, the projection list \(L\) is not limited to attributes of the relation \(R\). It can also include expressions that perform calculations on the components of tuples to produce new columns in the resulting relation.

Definition: Extended Projection

Let \(R\) be a relation. The extended projection of \(R\), denoted \(\pi_L(R)\), is a relation whose tuples are derived by applying the specifications in the list \(L\) to each tuple in \(R\). The schema of the resulting relation is determined by the names specified in \(L\).

The list \(L\) can contain elements of the following types:

• A single attribute

  This is the classical behavior where an attribute of \(R\) is selected for inclusion in the result.

• Attribute renaming: \(x \rightarrow y\)

  The attribute \(x\) from relation \(R\) is selected, but it is renamed to \(y\) in the resulting relation’s schema.

• Computed expression: \(E \rightarrow z\)

  An expression \(E\) is evaluated for each tuple of \(R\). The expression may consist of attributes of \(R\), constants, arithmetic operators, or string operators. The result of the expression \(E\) for a given tuple becomes the value for a new attribute named \(z\). For example, the element unit_price * quantity -> total_cost would compute a new attribute total_cost.

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The Sorting Operator

Section outline

• Defines the sorting operator \(\tau_L(R)\), which returns the tuples of relation R as a list, sorted according to the attributes in list L.
• Discusses how sorting is used in query results and for query optimization, and how subsequent operations might affect the sorted order.

#7

read: Section 5.2.6: The Sorting Operator (section)

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Present the following.  Use definition formal definition when it applies.

- Defines the sorting operator $\tau_L(R)$, which returns the tuples of relation `R` as a list, sorted according to the attributes in list `L`.
- Discusses how sorting is used in query results and for query optimization, and how subsequent operations might affect the sorted order.

The sorting operator, \(\tau\), arranges the tuples of a relation into a specific sequence, thereby transforming the relation from an unordered set or bag into an ordered list.

Definition: The Sorting Operator (\(\tau\))

Let \(R\) be a relation and \(L\) be an ordered list of attributes of \(R\), say \(L = (A_1, A_2, \dots, A_n)\). The expression \(\tau_L(R)\) denotes a list of all tuples in \(R\), sorted according to the attributes in \(L\).

The sorting is performed lexicographically. Tuples are ordered primarily based on their values for attribute \(A_1\). Ties are resolved by comparing values for attribute \(A_2\), and so on, for all attributes in \(L\). Any tuples that are identical in all attributes listed in \(L\) may appear in an arbitrary order relative to one another.

• Purpose of Sorting

  The sorting operator is primarily used for two purposes. First, it is employed to present the final result of a query in an ordered fashion, which is often more useful for human interpretation. For example, one might wish to see a list of employees sorted by last name. Second, sorting is a critical internal operation for the efficient execution of other relational operators, such as joins (e.g., the sort-merge join algorithm).

• Interaction with Other Operators

  The result of \(\tau_L(R)\) is a list, not a relation in the classical sense (which is an unordered set or bag). Consequently, the sorted order is typically not preserved when subsequent relational algebra operators, such as a join, are applied. The result of such an operation on a list is usually treated as a bag. However, certain operations, like selection and projection, can be implemented to maintain the sorted order of the tuples that remain in the result.

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Outerjoins

Section outline

• Introduces the problem of “dangling tuples” in standard joins, where tuples from one relation fail to match any tuple in the other.
• Defines the outerjoin operator as a variant of the join that preserves dangling tuples by padding them with NULL values (\(\perp\)) for attributes of the other relation.
• Describes variations: natural outerjoin (\(\bowtie\)), left outerjoin (\(\bowtie_L\)), right outerjoin (\(\bowtie_R\)), and theta-outerjoin (\(\bowtie_C\)).

#8

read: Section 5.2.7: Outerjoins (section)

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Present the following.  Use definition formal definition when it applies.

- Introduces the problem of "dangling tuples" in standard joins, where tuples from one relation fail to match any tuple in the other.
- Defines the outerjoin operator as a variant of the join that preserves dangling tuples by padding them with NULL values ($\perp$) for attributes of the other relation.
- Describes variations: natural outerjoin ($\bowtie$), left outerjoin ($\bowtie_L$), right outerjoin ($\bowtie_R$), and theta-outerjoin ($\bowtie_C$).

• The standard join operator may lose information. Tuples from one relation that do not match any tuple in the other relation, based on the join attributes, are absent from the result. These are known as “dangling tuples.”

• The outerjoin is a variant of the join designed to prevent the loss of information caused by dangling tuples. It includes the results of the standard join and also preserves the dangling tuples by padding them with a special NULL value, denoted as \(\perp\).

Definition: Natural Outerjoin

Let \(R\) and \(S\) be two relations. The natural outerjoin of \(R\) and \(S\), denoted \(R \bowtie S\), is formed by taking the natural join \(R \bowtie S\) and augmenting it with the dangling tuples from both \(R\) and \(S\).

A tuple \(t \in R\) is dangling if there is no tuple in \(S\) that agrees with \(t\) on all common attributes. A dangling tuple from \(R\) is included in the result, padded with \(\perp\) for all attributes unique to \(S\).

Symmetrically, a tuple \(t' \in S\) is dangling if it does not join with any tuple in \(R\). It is included in the result, padded with \(\perp\) for all attributes unique to \(R\).

• Several variations of the outerjoin exist to control which dangling tuples are preserved.

  Left Outerjoin (\(R \bowtie_L S\)): This operation preserves only the dangling tuples from the left argument, \(R\). The result consists of the natural join \(R \bowtie S\) plus the dangling tuples of \(R\), padded with \(\perp\).

  Right Outerjoin (\(R \bowtie_R S\)): This operation preserves only the dangling tuples from the right argument, \(S\). The result consists of the natural join \(R \bowtie S\) plus the dangling tuples of \(S\), padded with \(\perp\).

  Theta-Outerjoin (\(R \bowtie_C S\)): This operation is an analog to the theta-join. It first computes the theta-join based on condition \(C\). Then, any tuple from \(R\) or \(S\) that failed to satisfy the condition \(C\) with any tuple of the other relation is padded with \(\perp\) and added to the result. This operator can also be modified with \(L\) or \(R\) to indicate left- or right-outerjoin variants.

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