uni/year2/semester1/logseq-stuff/pages/Query Processing%3A Relational Algebra.md

• #CT230 - Database Systems I
• Previous Topic: Normalisation
• Next Topic: Query Processing & Optimisation
• Relevant Slides:

• Query Processing

  • What is Query Processing? #card card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T16:16:17.961Z card-last-score:: 1
    • Query Processing transforms SQL (a high-level language) into a correct & efficient low-level language representation of relational algebra.
    • Each relational algebra operator has code associated with it, which, when run, performs the operation on the data specified, allowing the specified data to be output as the result.

  • Steps Involved in Processing an SQL Query #card

    card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T15:52:53.439Z card-last-score:: 1
    • 1. Process (Parse & Translate) the query and create an internal representation of the query.
      • This may be an Operator Tree, Query Tree, or Query Graph (for more complicated queries).
    • 2. Optimise.
      3. Execute / Evaluate returning results.
  • What do you need to translate SQL to Relational Algebra? #card card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T16:21:00.003Z card-last-score:: 1
    • To translate SQL to Relational Algebra, you must have a meaningful set of relational algebra operators, and a mapping (translation) between SQL code & relational algebra expressions.

• Relational Algebra

  • Two formal languages exist for the relational model:
    • Relational Algebra (procedural).
    • Relational Calculus (non-procedural).
  • Both are logically equivalent.

  • Relational Algebra Operations

    • A basic set of operations exist for the relational model.
      • These allow for the specification of basic retrieval requests.
    • A sequence of Relational Algebra (RA) operations forms a relational algebra expression.
    • RA operations are divided into two groups:
      • Operations based on mathematical set theory (e.g., union, product, etc.).
      • Specific relational database operations.

  • Relational Algebra vs SQL #card

    card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T15:53:39.774Z card-last-score:: 1
    • The core operations & functions (i.e., programs) in the internal modules of most relational database systems are based on relational algebra.
    • SQL is a declarative language - It allows you to specify the results that you require, not the order of the operations to retrieve these results.
    • Relational Algebra is procedural - We must specify exactly how to retrieve results when using relational algebra.

  • Relational Algebra Expressions

    • A valid relational algebra expression is built by connecting tables or expressions with defined unary & binary operators & their arguments (if applicable).
    • Temporary relations resulting from a relational algebra expression can be used as input to a new relational algebra expression.
    • Expressions in brackets are evaluated first.
    • Relational Algebra operators are either unary or binary.

  • Working With the RelaX Calculator

    • There is no standard language for relational algebra like there is for SQL.
    • One University group have developed a calculator that supports a fairly command standard.
    • Note that it is Case Sensitive.
    • The RelaX calculator provides a number of datasets with the option of also using your own dataset.

    • Loading a Dataset

      • 1. Go to the "Group Editor" tab.
        2. Copy text from the file on Blackboard and add.
        3. Then choose "Preview".
        4. Then choose "Use group in Editor".
      • Note: Only stored temporarily.

    • Note on Degrees

      • The degree of the relation resulting from a selection of a table R is the same as the degree of R, i.e., they have the same number of attributes (columns).
      • The operation is commutative - i.e., a sequence of selects can be applied in any order.
      • E.g.:
        • \sigma_{\text{hours < 20 and pno = 10}}\text{works\_on}
        • \sigma_{\text{pno = 10 and hours < 20}}\text{works\_on}

• Relational Algebra: Unary Operators

  • Each operation takes one relation or expression as input and gives a new relation as a result.

  • Selection Operator (\sigma) #card

    card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T16:16:42.201Z card-last-score:: 1
    • Used to select certain tuples (rows) from a relation R.
    • Notation: \sigma_pR, where p is the selection predicate (i.e., a condidtion) and R is a relation / table name.
    • Note: The Selection (\sigma) operator in relational algebra is not the same as the SELECT clause in an SQL query.
      • An SQL SELECT query could be equivalent to a combination of relational algebra operators, (\sigma, \pi, or JOIN).

    • Example (Using Company Schema)

      • Find the projects with pno = 10 and hours worked < 20. background-color:: green
        • \sigma_{\text{hours < 20 AND pno = 10}}\text{works\_on}
        • Returns the set:
          • {(333445555, 10, 10.0 ), (999887777, 10, 10.0)}

  • Projection Operator (\pi) #card

    card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T16:20:37.894Z card-last-score:: 1
    • Used to return certain attributes / columns.
    • Notation:
      • \pi_{A_1, A_2, \cdots, A_k}(R)
      • Where A_1, \cdots, A_k are attribute names, R is a relation name.
    • The result is a relation with the k attributes listed in the same order as they appear in the list.
      • Duplicate tuples are removed from the result.
    • Note: Commutativity does not hold.

    • Example (Using Company Schema)

      • List all the department numbers where employees work. background-color:: green
        • \pi_\text{dno}\text{employee}
        • Returns: {5,4,1}.

  • Rename Operators (\rho & \leftarrow) #card

    card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T16:23:44.470Z card-last-score:: 1
    • Rename Operation (\rho).
    • Notation: \rho_x(E), where the result of the expression E is saved with the name x.

  • Order Operator (\tau) #card

    card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-22T00:00:00.000Z card-last-reviewed:: 2022-11-21T13:09:00.157Z card-last-score:: 1
    • Used to order by certain columns from a relation R.
    • Notation: \tau_{A_1, A_2, \cdots, A_k}R where A_1, A_2, \cdots, A_k are attributes with either ASC or DESC.

  • Group By Operator (\gamma)

    • Used to group by certain columns from a relation R.

  • Aggregate Functions Supported by RelaX

    • (Not part of Relation Algebra),
    • COUNT(*).
    • COUNT(column).
    • MIN(column).
    • MAX(column).
    • SUM(column).
    • AVG(column).

• Binary Operators

  • General syntax: (child_expression) function argument (child_argument).

  • Union Operator (\cup) #card

    card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T15:54:46.077Z card-last-score:: 1
    • Notation: (R) \cup (S), where R & S are relations.
    • Returns all tuples from R and all tuples from S.
    • Note: No duplicates will be returned.

  • Intersection Operator (\cap) #card

    card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T20:25:05.543Z card-last-score:: 1
    • Notation: (R) \cap (S), where R & S are relations.
    • Returns all tuples from R that are also in S.

  • Set Difference (-) #card

    card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T20:26:25.449Z card-last-score:: 1
    • Notation: (R) - (S) where R & S are relations.
    • Returns tuples that are in relation R but not in S.
    • Note: (R) - (S) and (S) - (R) are not the same.

  • Union Compatibility

    • What is Union Compatibility? #card card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T15:50:22.644Z card-last-score:: 1
      • For union, intersection, & minus, relations must be union compatible.
        • That is, schemas of relations must match, i.e., have the same number of attributes and each corresponding attributes have the same domain.

  • Cartesian Product Operator (\times) (Cross-Join) #card

    id:: 636a860b-5170-4227-befa-68876a53c856 card-last-interval:: 0.98 card-repeats:: 2 card-ease-factor:: 2.36 card-next-schedule:: 2022-11-22T12:05:57.590Z card-last-reviewed:: 2022-11-21T13:05:57.590Z card-last-score:: 3
    • Notation: (R) \times (S) where R & S are relations / tables.
    • Returns: Tuples comprising the concatenation (combination) of every tuple in R with every tuple in S.
    • Note: No condition specified.

    • Cartesian Product vs Join #card

      card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T16:20:31.096Z card-last-score:: 1
      • The main difference between a Cartesian product operator and a Join operator is that, with a Join, only tuples satisfying a condition appear in the result, while in a Cartesian product operator, all combinations of tuples are included in the result.

  • Join Operator (\Join) #card

    card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T16:23:36.847Z card-last-score:: 1
    • The Join Operator is a hybrid operator - it is a combination of the Cartesian Product operator (\times) & a Select operator (\sigma).
    • Tables are joined together based on the condition specified.

  • Equi & Theta Joins #card

    card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-19T00:00:00.000Z card-last-reviewed:: 2022-11-18T18:37:52.403Z card-last-score:: 1
    • Notation: (R_1) \Join p (R_2) where p is the join condition and R_1 & R_2 are relations.
    • Result: The JOIN operation returns all combinations of tuples from relation R_1 & relation R_2 satisfying the join condition p.
    • Note: EQUI JOINS use only equality comparisons (=) in the join condition p.