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Relational Algebra

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Relational Algebra IS0: Relational Languages Relational Languages - Relational Algebra - SQL (Structured Query Language) I (’86) en II (’92) (wij kijken slechts naar enkele ‘extra’ aspecten van SQL ’92) Literatuur: Boek Halpin hfdst 11 Dictaat: meerdere stukken over SQL ( I en II) Relational Algebra Studying this algebra first clarifies the basic query operations without getting distracted by the specific syntax of commercial query languages. In the relational model of data, all facts are stored in tables (or relations). New tables may be formed from existing tables by applying operations in the relational algebra. The original relational algebra defined by Codd contained eight relational operators: four based on traditional set operations (union, intersection, difference, and Cartesian product) and four special operations (selection, projection, join, and division). Each of these eight relational operators is a table-forming operator on tables. Relational algebra includes six comparison operators (=, <>, <, >, <=, >=). These are proposition-forming operators on terms. For example, x <> 0 asserts that x is not equal to zero. It also includes three logical operators (and, or, not). These are proposition-forming operators on propositions (e.g., x > 0 and x < 8). Since the algebra does not include arithmetic operators (e.g., +) or functions (e.g., count), it is less expressive than SQL. Union (“A B” or “A union B”) Two tables are union-compatible if and only if they have the same number of columns, and their corresponding columns are based on the same domain (the columns may have different names). The union of tables A and B is the set of all rows belonging to A or B (or both). Intersection (“A B” or “A intersect B”) and Difference (“A - B” or “A minus B” or “A except B”) The intersection of tables A and B is the set of rows common to A or B. The difference between tables A and B is the set of rows belonging to A but not B. Cartesian Product (Unrestricted Join) (ªA Bº or ªA times Bº) If A and B are tables, A B is formed by pairing each row of A with each row of B. (Here “pairing” means “prepending”. The number of rows in A B is the product of the number of rows in A and B. The number of columns in A B is the sum of the number of columns in A and B. The operations of union, intersection and Cartesian product are associative (e.g., A (B C) = (A B) C ). Difference is not associative (e.g., A – ( B – C ) is not equal to ( A – B ) – C ) What is wrong with this Cartesian Product? Table aliases are needed to multiply a table by itself: Codds table operations: a) Relational Selection The selection operation chooses just those rows that satisfy a specified condition. The selection operation may be specified using an expression of the form T where c. T denotes a table expression (i.e., an expression whose value is a table) and c denotes a condition. The ªwhere c” part is called a where clause. 1 The alternative notation c(T) is often used in academic journals. The ª1º is sigma, the Greek s (which is the first letter of ªselectionº). Examples Unless parentheses are used, and is evaluated before or. Two equivalent queries; if in doubt, include parentheses. Three equivalent queries. De Morgan’s laws: not ( p and q ) å not p or not q not ( p or q ) å not p and not q Codds table operations: b) Relational Projection Relational projection involves choosing one or more columns from a table, and then eliminating any duplicate rows that might result. We may represent the projection operation as T [ a, b, ...] where T is a table expression and a, b, ... are the names of the required columns. The alternative notation a,b..(T) is common in academic journals. The ªº is pi, the Greek p (the first letter of ªselectionº). Projection involves picking the columns and removing any duplicates Codds table operations: c) Relational Joins We now consider the relational join operator between two tables, which compares attribute values from the tables, using the comparison operators (=, <, >, <>, <=, >=). There are several kinds of join operations, and we discuss only some of these here. Let , (theta) denote any comparison operator (=, <, etc.). Then the ,-join of tables A and B on attributes a of A and b of B equals the Cartesian product A B , restricted to those rows where A.a , B.b We write this as shown below, an academic notation is shown in braces. A B where c { or A L c B } The condition c used to express this comparison of attributes between tables is called the join condition. The join condition may be composite (e.g., a1 < b1 and a2 < b2 ). With most joins, the comparison operator used is = . The ,-join is then called an equijoin. Examples: A B where A.a = B.b or: A B where a = b { if a, b occur in only one of A, B} Base for the following examples The ORM schema (a) maps to the relational schema (b). An example population for the schema. Example: an equijoin If the matching columns actually refer to the same thing in the UoD (and they typically do), then one of these columns is redundant. In this case we lose no information if we delete one of these matching columns (by performing a projection on all but the column to delete). Example: from equijoin to natural join (a) If the columns used for joining have the same name in both tables, then the unqualified name is used in the join result. The resulting table is then said to be the natural inner join of the original tables. Since ‘inner’ is assumed by default, the natural inner join may be expressed simply as “natural join”. This is by far the most common join operation in practice. It may be written as: A L B , or in words: “A natural join B” To compute A L B : Form A B For each column name c that occurs in both A and B Apply the restriction A.c = B.c Remove B.c Rename A.c to c Note that “L” looks like a cross “” with two vertical lines added, suggesting that a natural join is a Cartesian product plus two other operations (selection of rows with equal values for the common columns, followed by projection to delete redundant columns). Example: from equijoin to natural join (b) The duplicate column is removed by projection on the equijoin. A natural join. Example: with natural joins Tables being joined may have zero, one, or more common columns. Examples: Account has a composite identification scheme. Example: with natural joins (cont.) Sample population for the bank schema: Two queries using natural joins: Other join types In rare cases, comparison operators other than equality are used in joins. As a simple example: Suppose we want a list of drinker-smoker pairs, where the drinker and smoker are distinct persons. Other kinds of joins can be defined. For example, left, right, and full outer joins are used to include various cases with null values. An outer join is basically an inner join, with extra rows padded with nulls when the join condition is not satisfied. For example, Client left outer join AcUser includes a row to indicate that client 8005 has the name ªShankara, TAº but uses no account (branchNr a nd accountNr are assigned null values on this row). Codds table operations: d) Relational Division A table A is divisible by another table B only if A has more columns. Let B have n columns. The operation A ÷ B ( A divide-by B) is defined if and only if the domains of the last n columns of A match the domains of the columns in B (in order). Examples: Query strategies (in relational algebra . ) • Phrase the query in natural language, and understand what it means. • Which tables hold the information? • If you have table data, answer the query yourself, then retrace your mental steps. • Divide the query up into steps or subproblems. • If the columns to be listed are in different tables, declare joins between the tables. • If you need to relate two or more rows of the same table, use an alias to perform a self-join. Example: how can the next query be formulated in relational algebra? Solution: query in relational algebra (Account where balance > 700 ) [ branchNr, accountNr ] L AcUser L Client [ clientNr, clientName ] This is not the only way we could express the query. For instance, using a top-down approach, both joins could have been done before selecting or projecting, thus: ( Account L AcUser L Client ) where balance > 700 [ clientNr, clientName ] Although these two queries are logically equivalent, if executed in the evaluation order shown, the second query is less efficient because it involves a larger join. Relational algebra can be used to specify transformation rules between equivalent queries to obtain an optimally efficient, or at least more efficient, formulation. SQL database systems include a query optimizer to translate queries into an optimized form before executing them. Relational Database Systems / SQL We may now define a relational DBMS as a DBMS that has the relational table as its only essential data structure, and supports the selection, projection and join operations without needing specification of physical access paths. A relational system that supports all eight table operations of the relational algebra is said to be relationally complete. This doesn’t entail eight distinct operators for these tasks; ra ther, the eight operations must be expressible in terms of the table operations provided by the system.
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