Relational Algebra Database Systems

What is an “Algebra” Mathematical system consisting of: Operands --- variables or values from which new values can be constructed. Operators --- symbols denoting procedures that construct new values from given values.

What is Relational Algebra? An algebra whose operands are relations or variables that represent relations. Operators are designed to do the most common things that we need to do with relations in a database. The result is an algebra that can be used as a query language for relations.

Roadmap There is a core relational algebra that has traditionally been thought of as the relational algebra. But there are several other operators we shall add to the core in order to model better the language SQL --- the principal language used in relational database systems.

Core Relational Algebra Union, intersection, and difference. Usual set operations, but require both operands have the same relation schema. Selection: picking certain rows. Projection: picking certain columns. Products and joins: compositions of relations. Renaming of relations and attributes.

The SELECT Operation SELECT extracts tuples from a relation result has same relation schema as operand SELECT requires a selection condition selection condition is a boolean expression to filter tuple values Syntax: <selection condition> (R) Selection condition may contain AND, OR, NOT, =, <, , >, , 

Selection R1 := SELECTC (R2) C is a condition (as in “if” statements) that refers to attributes of R2. R1 is all those tuples of R2 that satisfy C.

Example Relation Sells: bar beer price Joe’s Bud 2.50 Joe’s Miller 2.75 Sue’s Bud 2.50 Sue’s Miller 3.00 JoeMenu := SELECTbar=“Joe’s”(Sells): bar beer price Joe’s Bud 2.50 Joe’s Miller 2.75

SELECT Examples StoreId = "S047" (STORESTOCK) < "S002", "I065", 120 >, < "S333", "I954", 198 >, < "S047", "I099", 267 >, < "S047", "I954", 300 > r2(STORESTOCK) = StoreId = "S047" (STORESTOCK) quantity < 200 (STORESTOCK) < "S333", "I954", 198 >

The PROJECT Operation PROJECT extracts attributes from a relation result schema attributes are a subset of the operand schema PROJECT requires a attribute list Syntax: <attribute list> (R) Duplicates are not kept in result result is a relation, which is a set

Projection R1 := PROJL (R2) L is a list of attributes from the schema of R2. R1 is constructed by looking at each tuple of R2, extracting the attributes on list L, in the order specified, and creating from those components a tuple for R1. Eliminate duplicate tuples, if any.

Example Relation Sells: bar beer price Joe’s Bud 2.50 Joe’s Miller 2.75 Sue’s Bud 2.50 Sue’s Miller 3.00 Prices := PROJbeer,price(Sells): beer price Bud 2.50 Miller 2.75 Miller 3.00

PROJECT Examples StoreId, Item (STORESTOCK) < "S002", "I065", 120 >, < "S333", "I954", 198 >, < "S047", "I099", 267 >, < "S047", "I954", 300 > r2(STORESTOCK) = StoreId, Item (STORESTOCK) < "S002", "I065" >, < "S333", "I954" >, < "S047", "I099" >, < "S047", "I954" >

PROJECT Examples Description, Price (STOREITEM) r(STOCKITEM) = < "I075", "Ice Cream", $1.49, false >, < "I345", "Cupcakes", $1.99, false >, < "I333", "Twinkies", $1.98, false > < "Ice Cream", $1.49 >, < "Cupcakes", $1.99 >, < "Twinkies", $1.98 >

Composing Operations Price ( Description="Ice Cream" (STOREITEM)) r(STOCKITEM) = < "I075", "Ice Cream", $1.49, false >, < "I345", "Cupcakes", $1.99, false >, < "I333", "Twinkies", $1.98, false > < "I075", "Ice Cream", $1.49, false > < $1.49 > SELECT result: PROJECT result:

Product R3 := R1 * R2 Pair each tuple t1 of R1 with each tuple t2 of R2. Concatenation t1t2 is a tuple of R3. Schema of R3 is the attributes of R1 and R2, in order. But beware attribute A of the same name in R1 and R2: use R1.A and R2.A.

Example: R3 := R1 * R2 R3( A, R1.B, R2.B, C ) R1( A, B ) 1 2 5 6 1 2 1 2 3 4 R2( B, C ) 5 6 7 8 9 10 R3( A, R1.B, R2.B, C ) 1 2 5 6 1 2 7 8 1 2 9 10 3 4 5 6 3 4 7 8 3 4 9 10

CROSS PRODUCT: example STOCKITEM ItemId Description Taxable Price I075 Ice Cream FALSE $1.49 I333 Twinkies $1.98 I345 Cup Cakes $1.99 StoreId Item Quantity STORESTOCK S002 I075 120 S047 I333 267 STOCKITEM  STORESTOCK ItemId Description Taxable Price I075 Ice Cream FALSE $1.49 I333 Twinkies $1.98 I345 Cup Cakes $1.99 StoreId Item Quantity S047 267 S002 120

JOIN = XPROD and SELECT ItemId=Item (STOCKITEM  STORESTOCK) STORESTOCK ⋈ <Item = ItemId> STOCKITEM ItemId Description Taxable Price I075 Ice Cream FALSE $1.49 StoreId Item Quantity S002 120 S047 I333 267 Twinkies $1.98 ItemId=Item (STOCKITEM  STORESTOCK) ItemId Description Taxable Price I075 Ice Cream FALSE $1.49 I333 Twinkies $1.98 I345 Cup Cakes $1.99 StoreId Item Quantity S047 267 S002 120

The JOIN Operation JOIN combines tuples from two tables based on values of related attributes (usually a FK) JOIN requires a join condition boolean expression comparing attributes from each operand Syntax: R⋈ <join condition> S The join condition may contain AND, =, <, , >, , 

Theta-Join R3 := R1 JOINC R2 Take the product R1 * R2. Then apply SELECTC to the result. As for SELECT, C can be any boolean-valued condition. Historic versions of this operator allowed only A theta B, where theta was =, <, etc.; hence the name “theta-join.”

Example Sells( bar, beer, price ) Bars( name, addr ) Joe’s Bud 2.50 Joe’s Maple St. Joe’s Miller 2.75 Sue’s River Rd. Sue’s Bud 2.50 Sue’s Coors 3.00 BarInfo := Sells JOIN Sells.bar = Bars.name Bars BarInfo( bar, beer, price, name, addr ) Joe’s Bud 2.50 Joe’s Maple St. Joe’s Miller 2.75 Joe’s Maple St. Sue’s Bud 2.50 Sue’s River Rd. Sue’s Coors 3.00 Sue’s River Rd.

JOIN Examples STORESTOCK ⋈<Item = ItemId> STOCKITEM STORESTOCK( StoreId, Item, Quantity ) STOCKITEM( ItemId, Description, Price, Taxable ) < "S002", "I075", 120 >, < "S047", "I333", 267 > r(STORESTOCK) = < "I075", "Ice Cream", $1.49, false >, < "I345", "Cupcakes", $1.99, false >, < "I333", "Twinkies", $1.98, false > r(STOCKITEM) = STORESTOCK ⋈<Item = ItemId> STOCKITEM

JOIN Examples STORESTOCK ⋈<Item = ItemId> STOCKITEM r(RESULT) = RESULT( StoreId, Item, Quantity, ItemId, Description, Price, Taxable ) < "S002", "I075", 120, "I075", "Ice Cream", $1.49, false >, < "S047", "I333", 267, "I333", "Twinkies", $1.98, false > r(RESULT) = STORESTOCK ⋈<Item = ItemId> STOCKITEM

Natural Join A frequent type of join connects two relations by: Equating attributes of the same name, and Projecting out one copy of each pair of equated attributes. Called natural join. Denoted R3 := R1 JOIN R2.

Example Sells( bar, beer, price ) Bars( bar, addr ) Joe’s Bud 2.50 Joe’s Maple St. Joe’s Miller 2.75 Sue’s River Rd. Sue’s Bud 2.50 Sue’s Coors 3.00 BarInfo := Sells JOIN Bars Note Bars.name has become Bars.bar to make the natural join “work.” BarInfo( bar, beer, price, addr ) Joe’s Bud 2.50 Maple St. Joe’s Milller 2.75 Maple St. Sue’s Bud 2.50 River Rd. Sue’s Coors 3.00 River Rd.

Renaming The RENAME operator gives a new schema to a relation. R1 := RENAMER1(A1,…,An)(R2) makes R1 be a relation with attributes A1,…,An and the same tuples as R2. Simplified notation: R1(A1,…,An) := R2.

Example Bars( name, addr ) Joe’s Maple St. Sue’s River Rd. R(bar, addr) := Bars R( bar, addr ) Joe’s Maple St. Sue’s River Rd.

RENAME Rename the attributes of a relation or change the relation name The general RENAME operation : S (B1, B2, …, Bn )(R) changes both: the relation name to S, and the column (attribute) names to B1, B1, …..Bn S(R) changes: the relation name only to S (B1, B2, …, Bn )(R) changes: the column (attribute) names only to B1, B1, …..Bn

RENAME (shorthand) shorthand for renaming attributes : R1   FNAME, LNAME, SALARY (DEP5_EMPS) R1 has same attribute names as DEP5_EMPS R2   (F,L,S) ( FNAME, LNAME, SALARY (DEP5_EMPS)) R2 has attributes renamed to F, L and S R3 (F,L,S)   FNAME, LNAME, SALARY (DEP5_EMPS) R3 also has attributes renamed to F, L and S

Building Complex Expressions Algebras allow us to express sequences of operations in a natural way. Example: in arithmetic --- (x + 4)*(y - 3). Relational algebra allows the same. Three notations, just as in arithmetic: Sequences of assignment statements. Expressions with several operators. Expression trees.

Sequences of Assignments Create temporary relation names. Renaming can be implied by giving relations a list of attributes. Example: R3 := R1 JOINC R2 can be written: R4 := R1 * R2 R3 := SELECTC (R4)

Writing Queries STORESTOCK Query: Get the Description and Price for all Items stocked by Store S002 StoreId Item Quantity S002 I075 120 S047 I333 267 S002 I333 1200 STOCKITEM ItemId Description Taxable Price I075 Ice Cream FALSE $1.49 I333 Twinkies $1.98 I345 Cup Cakes $1.99 result only includes tuples with certain ItemIds result has these attributes

Writing Queries STORESTOCK Query: Get the Description and Price for all Items stocked by Store S002 StoreId Item Quantity S002 I075 120 S047 I333 267 We need a join to merge data across relations S002 I333 1200 STOCKITEM ItemId Description Taxable Price I075 Ice Cream FALSE $1.49 I333 Twinkies $1.98 I345 Cup Cakes $1.99

Writing Queries STORESTOCK ⋈<Item = ItemId> STOCKITEM StoreId Quantity S002 I075 120 S047 I333 267 ItemId Description Taxable Price Ice Cream FALSE $1.49 Twinkies $1.98 1200 Query: Get the Description and Price for all Items stocked by Store S002 Now we can select and project to extract the information we want

Writing Queries R1 = STORESTOCK ⋈<Item = ItemId> STOCKITEM StoreId Item Quantity S002 I075 120 S047 I333 267 ItemId Description Taxable Price Ice Cream FALSE $1.49 Twinkies $1.98 1200 R2 = StoreId = "S002" (R1) StoreId Item Quantity S002 I075 120 ItemId Description Taxable Price Ice Cream FALSE $1.49 I333 Twinkies $1.98 1200

Writing Queries R2 = StoreId = "S002" (R1) Item Quantity S002 I075 120 ItemId Description Taxable Price Ice Cream FALSE $1.49 I333 Twinkies $1.98 1200 R3 = <Description, Price> (R2) Description Price Ice Cream $1.49 Twinkies $1.98

Composing Queries R1 = STORESTOCK ⋈<Item = ItemId> STOCKITEM Since the operands and results of every query are relations, we can compose or chain queries. R2 = StoreId = "S002" (R1) R3 = <Description, Price> (R2) R1 = STORESTOCK ⋈<Item = ItemId> STOCKITEM <Description, Price>(StoreId = "S002" (STORESTOCK ⋈<Item = ItemId> STOCKITEM))

UNION: example StoreId Item Quantity STORESTOCK S002 I075 120 S047 267 StoreId Item Quantity WAREHOUSESTOCK W998 I075 1200 W087 I001 5000 W222 I188 11500 W023 300 STORESTOCK ⋃ WAREHOUSESTOCK Id Item Quantity S002 I075 120 S047 I333 267 W998 1200 W087 I001 5000 W222 I188 11500 W023 300

INTERSECTION: example StoreId Item Quantity WAREHOUSESTOCK W998 I075 1200 W087 I001 5000 W222 I188 11500 W023 300 StoreId Item Quantity STORESTOCK S002 I075 120 S047 I333 267 ItemID (STORESTOCK) ⋂ Item (WAREHOUSESTOCK) Item I075 I001 I188 Item I075 I333 Item I075 ⋂ =

DIFFERENCE: example ItemID (STOCKITEM) - Item (STORESTOCK) - = Description Taxable Price I075 Ice Cream FALSE $1.49 I333 Twinkies $1.98 I345 Cup Cakes $1.99 StoreId Item Quantity STORESTOCK S002 I075 120 S047 I333 267 ItemID (STOCKITEM) - Item (STORESTOCK) ItemId I075 I333 I345 Item - =

Expressions in a Single Assignment Example: the theta-join R3 := R1 JOINC R2 can be written: R3 := SELECTC (R1 * R2) Precedence of relational operators: Unary operators --- select, project, rename --- have highest precedence, bind first. Then come products and joins. Then intersection. Finally, union and set difference bind last. But you can always insert parentheses to force the order you desire.

Expression Trees Leaves are operands --- either variables standing for relations or particular, constant relations. Interior nodes are operators, applied to their child or children.

Example Using the relations Bars(name, addr) and Sells(bar, beer, price), find the names of all the bars that are either on Maple St. or sell Bud for less than $3.

As a Tree: UNION RENAMER(name) PROJECTname PROJECTbar SELECTaddr = “Maple St.” SELECT price<3 AND beer=“Bud” Bars Sells

Example Using Sells(bar, beer, price), find the bars that sell two different beers at the same price. Strategy: by renaming, define a copy of Sells, called S(bar, beer1, price). The natural join of Sells and S consists of quadruples (bar, beer, beer1, price) such that the bar sells both beers at this price.

The Tree PROJECTbar SELECTbeer != beer1 JOIN RENAMES(bar, beer1, price) Sells Sells

Query Trees Query trees are representations of queries that can be manipulated by query optimizers, according to mathematical properties of the operators.

Schemas for Interior Nodes An expression tree defines a schema for the relation associated with each interior node. Similarly, a sequence of assignments defines a schema for each relation on the left of the := sign.

Schema-Defining Rules 1 For union, intersection, and difference, the schemas of the two operands must be the same, so use that schema for the result. Selection: schema of the result is the same as the schema of the operand. Projection: list of attributes tells us the schema.

Schema-Defining Rules 2 Product: the schema is the attributes of both relations. Use R.A, etc., to distinguish two attributes named A. Theta-join: same as product. Natural join: use attributes of both relations. Shared attribute names are merged. Renaming: the operator tells the schema.

Relational Algebra on Bags A bag is like a set, but an element may appear more than once. Multiset is another name for “bag.” Example: {1,2,1,3} is a bag. {1,2,3} is also a bag that happens to be a set. Bags also resemble lists, but order in a bag is unimportant. Example: {1,2,1} = {1,1,2} as bags, but [1,2,1] != [1,1,2] as lists.

Example: Bag Selection R( A, B ) S( B, C ) 1 2 3 4 5 6 7 8 1 2 SELECTA+B<5 (R) = A B 1 2

Example: Bag Projection R( A, B ) S( B, C ) 1 2 3 4 5 6 7 8 1 2 PROJECTA (R) = A 1 5

Example: Bag Product R( A, B ) S( B, C ) 1 2 3 4 5 6 7 8 1 2 1 2 3 4 5 6 7 8 1 2 R * S = A R.B S.B C 1 2 3 4 1 2 7 8 5 6 3 4 5 6 7 8

Example: Bag Theta-Join R( A, B ) S( B, C ) 1 2 3 4 5 6 7 8 1 2 R JOIN R.B<S.B S = A R.B S.B C 1 2 3 4 1 2 7 8 5 6 7 8

Bag Union Union, intersection, and difference need new definitions for bags. An element appears in the union of two bags the sum of the number of times it appears in each bag. Example: {1,2,1} UNION {1,1,2,3,1} = {1,1,1,1,1,2,2,3}

Bag Intersection An element appears in the intersection of two bags the minimum of the number of times it appears in either. Example: {1,2,1} INTER {1,2,3} = {1,2}.

Bag Difference An element appears in the difference A – B of bags as many times as it appears in A, minus the number of times it appears in B. But never less than 0 times. Example: {1,2,1} – {1,2,3} = {1}.

Beware: Bag Laws != Set Laws Not all algebraic laws that hold for sets also hold for bags. For one example, the commutative law for union (R UNION S = S UNION R ) does hold for bags. Since addition is commutative, adding the number of times x appears in R and S doesn’t depend on the order of R and S.

An Example of Inequivalence Set union is idempotent, meaning that S UNION S = S. However, for bags, if x appears n times in S, then it appears 2n times in S UNION S. Thus S UNION S != S in general.

The Extended Algebra DELTA = eliminate duplicates from bags. TAU = sort tuples. Extended projection : arithmetic, duplication of columns. GAMMA = grouping and aggregation. OUTERJOIN: avoids “dangling tuples” = tuples that do not join with anything.

Duplicate Elimination R1 := DELTA(R2). R1 consists of one copy of each tuple that appears in R2 one or more times.

Example: Duplicate Elimination R = A B 1 2 3 4 DELTA(R) = A B 1 2 3 4

Sorting R1 := TAUL (R2). L is a list of some of the attributes of R2. R1 is the list of tuples of R2 sorted first on the value of the first attribute on L, then on the second attribute of L, and so on. Break ties arbitrarily. TAU is the only operator whose result is neither a set nor a bag.

Example: Sorting R = A B 1 2 3 4 5 2 TAUB (R) = [(5,2), (1,2), (3,4)]

Extended Projection Using the same PROJL operator, we allow the list L to contain arbitrary expressions involving attributes, for example: Arithmetic on attributes, e.g., A+B. Duplicate occurrences of the same attribute.

Example: Extended Projection R = A B 1 2 3 4 PROJA+B,A,A (R) = A+B A1 A2 3 1 1 7 3 3

Aggregation Operators Aggregation operators are not operators of relational algebra. Rather, they apply to entire columns of a table and produce a single result. The most important examples: SUM, AVG, COUNT, MIN, and MAX.

Example: Aggregation R = A B 1 3 3 4 3 2 SUM(A) = 7 COUNT(A) = 3 1 3 3 4 3 2 SUM(A) = 7 COUNT(A) = 3 MAX(B) = 4 AVG(B) = 3

Grouping Operator R1 := GAMMAL (R2). L is a list of elements that are either: Individual (grouping ) attributes. AGG(A ), where AGG is one of the aggregation operators and A is an attribute.

Applying GAMMAL(R) Group R according to all the grouping attributes on list L. That is, form one group for each distinct list of values for those attributes in R. Within each group, compute AGG(A ) for each aggregation on list L. Result has grouping attributes and aggregations as attributes. One tuple for each list of values for the grouping attributes and their group’s aggregations.

Example: Grouping/Aggregation R = A B C 1 2 3 4 5 6 1 2 5 GAMMAA,B,AVG(C) (R) = ?? Then, average C within groups: A B AVG(C) 1 2 4 4 5 6 First, group R : A B C 1 2 3 1 2 5 4 5 6

Outerjoin Suppose we join R JOINC S. A tuple of R that has no tuple of S with which it joins is said to be dangling. Similarly for a tuple of S. Outerjoin preserves dangling tuples by padding them with a special NULL symbol in the result.

Example: Outerjoin R = A B S = B C 1 2 2 3 4 5 6 7 1 2 2 3 4 5 6 7 (1,2) joins with (2,3), but the other two tuples are dangling. R OUTERJOIN S = A B C 1 2 3 4 5 NULL NULL 6 7

Division Operation Notation: r  s Notation: Suited to queries that include the phrase “for all”. Let r and s be relations on schemas R and S respectively where R = (A1, …, Am , B1, …, Bn ) S = (B1, …, Bn) The result of r  s is a relation on schema R – S = (A1, …, Am) r  s = { t | t   R-S (r)   u  s ( tu  r ) } Where tu means the concatenation of tuples t and u to produce a single tuple

Division Operation – Example Relations r, s: A B B      1 2 3 4 6 1 2 s r  s: A r  

Another Division Example Relations r, s: A B C D E D E    a    a b 1 3 a b 1 s r r  s: A B C   a 

Division Operation (Cont.) Property Let q = r  s Then q is the largest relation satisfying q x s  r Definition in terms of the basic algebra operation Let r(R) and s(S) be relations, and let S  R r  s = R-S (r ) – R-S ( ( R-S (r ) x s ) – R-S,S(r )) To see why R-S,S (r) simply reorders attributes of r R-S (R-S (r ) x s ) – R-S,S(r) ) gives those tuples t in R-S (r ) such that for some tuple u  s, tu  r.

EXERCISE 1: Queries Express the following queries in the algebra: First and last name of employees who have no supervisor. First and last name of employees supervised by Franklin Wong. Last name of employees who have dependents. Last name of employees who have daughters. Last name of employees in department 5 who work more than 10 hours/week on ProductX. Last name of supervisors of employees in department 5 who work more than 10 hours/week on ProductX.

EXERCISE 1: Schema

EXERCISE 2: Schema

EXERCISE 2: Queries First and last names of all department managers. Salaries of all employees who have worked on the Reorganization project. SSN of all employees who have worked on a project that is controlled by a department different than the department that they are assigned to. Last name of all employees who are not married.

Exercise 3: Schema

EXERCISE 3: Queries List all airplane types that can land at any airport in San Francisco. List the ids and number of seats for all airplanes that can land at any airport in Chicago. List the name and phone number of all customers with a seat reserved on a flight that leaves Chicago O’Hara airport (ORD) on October 31, 2008. List all airlines that have seats available for flights leaving Los Angeles (LAX) on September 25, 2008. List all airlines that operate at San Jose International Airport (SJC).

EXERCISE 4: Schema

EXERCISE 4: Queries Count the number of overdue books. How many books by author Harry Crews are in the database? Determine the number of library cards assigned to each borrower phone number. Find names of all borrowers who do not have any book loans. Do any library branches have every book?