1. Lecture #4: Relational Algebra
Faloutsos - Pavlo
CMU SCS /615
Carnegie Mellon Univ. School of Computer Science /615 - DB Applications
C. Faloutsos & A. Pavlo
Lecture #4: Relational Algebra

2. Overview history concepts Formal query languages relational algebra
Faloutsos - Pavlo
CMU SCS /615
Overview
history
concepts
Formal query languages
relational algebra
rel. tuple calculus
rel. domain calculus
Faloutsos - Pavlo
CMU SCS /615

3. History before: records, pointers, sets etc
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CMU SCS /615
History
before: records, pointers, sets etc
introduced by E.F. Codd in 1970
revolutionary!
first systems: (System R; Ingres)
Turing award in 1981
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CMU SCS /615

4. Concepts - reminder Database: a set of relations (= tables)
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CMU SCS /615
Concepts - reminder
Database: a set of relations (= tables)
rows: tuples
columns: attributes (or keys)
superkey, candidate key, primary key
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CMU SCS /615

5. Example Database: Faloutsos - Pavlo CMU SCS 15-415/615

6. Example: cont’d Database: k-th attribute (Dk domain)
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CMU SCS /615
Example: cont’d
k-th attribute
(Dk domain)
Database:
rel. schema (attr+domains)
tuple
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CMU SCS /615

7. Example: cont’d rel. schema (attr+domains) instance Faloutsos - Pavlo
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Example: cont’d
rel. schema (attr+domains)
instance
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8. Example: cont’d Di: the domain of the i-th attribute (eg., char(10)
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Example: cont’d
Di: the domain of the i-th attribute (eg., char(10)
rel. schema (attr+domains)
instance
Faloutsos - Pavlo
CMU SCS /615

9. Overview history concepts Formal query languages relational algebra
Faloutsos - Pavlo
CMU SCS /615
Overview
history
concepts
Formal query languages
relational algebra
rel. tuple calculus
rel. domain calculus
Faloutsos - Pavlo
CMU SCS /615

10. Formal query languages
Faloutsos - Pavlo
CMU SCS /615
Formal query languages
How do we collect information?
Eg., find ssn’s of people in 415
(recall: everything is a set!)
One solution: Rel. algebra, ie., set operators
Q1: Which ones??
Q2: what is a minimal set of operators?
Faloutsos - Pavlo
CMU SCS /615

11. Relational operators . set union U set difference ‘-’
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CMU SCS /615
Relational operators .
set union U
set difference ‘-’
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CMU SCS /615

12. Example: Q: find all students (part or full time)
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Example:
Q: find all students (part or full time)
A: PT-STUDENT union FT-STUDENT
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CMU SCS /615

13. Faloutsos - Pavlo
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Observations:
two tables are ‘union compatible’ if they have the same attributes (‘domains’)
Q: how about intersection U
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14. Observations: A: redundant: STUDENT intersection STAFF = STAFF STUDENT
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CMU SCS /615
Observations:
A: redundant:
STUDENT intersection STAFF =
STAFF
STUDENT
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CMU SCS /615

15. Observations: A: redundant: STUDENT intersection STAFF = STAFF STUDENT
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CMU SCS /615
Observations:
A: redundant:
STUDENT intersection STAFF =
STAFF
STUDENT
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CMU SCS /615

16. Observations: A: redundant: STUDENT intersection STAFF =
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CMU SCS /615
Observations:
A: redundant:
STUDENT intersection STAFF =
STUDENT - (STUDENT - STAFF)
STAFF
STUDENT
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CMU SCS /615

17. Observations: A: redundant: STUDENT intersection STAFF =
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CMU SCS /615
Observations:
A: redundant:
STUDENT intersection STAFF =
STUDENT - (STUDENT - STAFF)
Double negation:
We’ll see it again, later…
Faloutsos - Pavlo
CMU SCS /615

18. Relational operators . set union set difference ‘-’ U
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CMU SCS /615
Relational operators .
set union
set difference ‘-’ U
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CMU SCS /615

19. Other operators? eg, find all students on ‘Main street’ A: ‘selection’
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CMU SCS /615
Other operators?
eg, find all students on ‘Main street’
A: ‘selection’
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CMU SCS /615

20. Faloutsos - Pavlo
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Other operators?
Notice: selection (and rest of operators) expect tables, and produce tables (-> can be cascaded!!)
For selection, in general:
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CMU SCS /615

21. Selection - examples Find all ‘Smiths’ on ‘Forbes Ave’
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Selection - examples
Find all ‘Smiths’ on ‘Forbes Ave’
‘condition’ can be any boolean combination of ‘=‘, ‘>’, ‘>=‘, ...
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CMU SCS /615

22. Relational operators selection . set union set difference R - S R U S
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CMU SCS /615
Relational operators
selection .
set union
set difference R - S
R U S
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CMU SCS /615

23. Relational operators selection picks rows - how about columns?
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CMU SCS /615
Relational operators
selection picks rows - how about columns?
A: ‘projection’ - eg.:
finds all the ‘ssn’ - removing duplicates
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CMU SCS /615

24. Relational operators Cascading: ‘find ssn of students on ‘forbes ave’
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CMU SCS /615
Relational operators
Cascading: ‘find ssn of students on ‘forbes ave’
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CMU SCS /615

25. Relational operators selection projection . set union
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CMU SCS /615
Relational operators
selection
projection .
set union
set difference R - S
R U S
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CMU SCS /615

26. Relational operators Are we done yet?
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Relational operators
Are we done yet?
Q: Give a query we can not answer yet!
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27. Relational operators A: any query across two or more tables,
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Relational operators
A: any query across two or more tables,
eg., ‘find names of students in ’
Q: what extra operator do we need??
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CMU SCS /615

28. Relational operators A: any query across two or more tables,
Faloutsos - Pavlo
CMU SCS /615
Relational operators
A: any query across two or more tables,
eg., ‘find names of students in ’
Q: what extra operator do we need??
A: surprisingly, cartesian product is enough!
Faloutsos - Pavlo
CMU SCS /615

29. Cartesian product eg., dog-breeding: MALE x FEMALE
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Cartesian product
eg., dog-breeding: MALE x FEMALE
gives all possible couples = x
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CMU SCS /615

30. so what? Eg., how do we find names of students taking 415?
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CMU SCS /615
so what?
Eg., how do we find names of students taking 415?
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CMU SCS /615

31. Cartesian product A: Faloutsos - Pavlo CMU SCS 15-415/615

32. Cartesian product Faloutsos - Pavlo CMU SCS 15-415/615

33. Faloutsos - Pavlo
CMU SCS /615
Faloutsos - Pavlo
CMU SCS /615

34. FUNDAMENTAL Relational operators
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FUNDAMENTAL Relational operators
selection
projection
cartesian product MALE x FEMALE
set union
set difference R - S
R U S
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CMU SCS /615

35. Faloutsos - Pavlo
CMU SCS /615
Relational ops
Surprisingly, they are enough, to help us answer almost any query we want!!
derived/convenience operators:
set intersection
join (theta join, equi-join, natural join)
‘rename’ operator
division
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CMU SCS /615

36. Joins Equijoin: Faloutsos - Pavlo CMU SCS 15-415/615 Faloutsos - Pavlo

37. Cartesian product A: Faloutsos - Pavlo CMU SCS 15-415/615

38. Joins Equijoin: theta-joins:
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CMU SCS /615
Joins
Equijoin:
theta-joins:
generalization of equi-join - any condition
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CMU SCS /615

39. Joins very popular: natural join: R S
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Joins
very popular: natural join: R S
like equi-join, but it drops duplicate columns:
STUDENT (ssn, name, address)
TAKES (ssn, cid, grade)
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CMU SCS /615

40. Joins nat. join has 5 attributes equi-join: 6 Faloutsos - Pavlo
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Joins
nat. join has 5 attributes
equi-join: 6
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CMU SCS /615

41. Natural Joins - nit-picking
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CMU SCS /615
Natural Joins - nit-picking
if no attributes in common between R, S:
nat. join -> cartesian product
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CMU SCS /615

42. Overview - rel. algebra fundamental operators derived operators
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CMU SCS /615
Overview - rel. algebra
fundamental operators
derived operators
joins etc
rename
division
examples
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CMU SCS /615

43. Rename op. Q: why? A: shorthand; self-joins; …
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CMU SCS /615
Rename op.
Q: why?
A: shorthand; self-joins; …
for example, find the grand-parents of ‘Tom’, given PC (parent-id, child-id)
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CMU SCS /615

44. Rename op. PC (parent-id, child-id) Faloutsos - Pavlo
CMU SCS /615
Rename op.
PC (parent-id, child-id)
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CMU SCS /615

45. Rename op. first, WRONG attempt: (why? how many columns?)
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CMU SCS /615
Rename op.
first, WRONG attempt:
(why? how many columns?)
Second WRONG attempt:
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CMU SCS /615

46. Faloutsos - Pavlo
CMU SCS /615
Rename op.
we clearly need two different names for the same table - hence, the ‘rename’ op.
Faloutsos - Pavlo
CMU SCS /615

47. Overview - rel. algebra fundamental operators derived operators
Faloutsos - Pavlo
CMU SCS /615
Overview - rel. algebra
fundamental operators
derived operators
joins etc
rename
division
examples
Faloutsos - Pavlo
CMU SCS /615

48. Division Rarely used, but powerful.
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CMU SCS /615
Division
Rarely used, but powerful.
Example: find suspicious suppliers, ie., suppliers that supplied all the parts in A_BOMB
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CMU SCS /615

49. Division Faloutsos - Pavlo CMU SCS 15-415/615 Faloutsos - Pavlo

50. Division Observations: ~reverse of cartesian product
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Division
Observations: ~reverse of cartesian product
It can be derived from the 5 fundamental operators (!!)
How?
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CMU SCS /615

51. Faloutsos - Pavlo
CMU SCS /615
Division
Answer:
Observation: find ‘good’ suppliers, and subtract! (double negation)
Faloutsos - Pavlo
CMU SCS /615

52. Faloutsos - Pavlo
CMU SCS /615
Division
Answer:
Observation: find ‘good’ suppliers, and subtract! (double negation)
Faloutsos - Pavlo
CMU SCS /615

53. Division Answer: All suppliers All bad parts Faloutsos - Pavlo
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Division
Answer:
All suppliers
All bad parts
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CMU SCS /615

54. Division Answer: all possible suspicious shipments Faloutsos - Pavlo
CMU SCS /615
Division
Answer:
all possible
suspicious shipments
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CMU SCS /615

55. Division Answer: all possible suspicious shipments that didn’t happen
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CMU SCS /615
Division
Answer:
all possible
suspicious shipments
that didn’t happen
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CMU SCS /615

56. all suppliers who missed at least one suspicious shipment,
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CMU SCS /615
Division
Answer:
all suppliers who missed
at least one suspicious shipment,
i.e.: ‘good’ suppliers
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CMU SCS /615

57. Overview - rel. algebra fundamental operators derived operators
Faloutsos - Pavlo
CMU SCS /615
Overview - rel. algebra
fundamental operators
derived operators
joins etc
rename
division
examples
Faloutsos - Pavlo
CMU SCS /615

58. Sample schema find names of students that take 15-415
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CMU SCS /615
Sample schema
find names of students that take
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CMU SCS /615

59. Examples find names of students that take 15-415 Faloutsos - Pavlo
CMU SCS /615
Examples
find names of students that take
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CMU SCS /615

60. Examples find names of students that take 15-415 Faloutsos - Pavlo
CMU SCS /615
Examples
find names of students that take
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CMU SCS /615

61. Sample schema find course names of ‘smith’ Faloutsos - Pavlo
CMU SCS /615
Sample schema
find course names of ‘smith’
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CMU SCS /615

62. Examples find course names of ‘smith’ Faloutsos - Pavlo
CMU SCS /615
Examples
find course names of ‘smith’
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CMU SCS /615

63. Faloutsos - Pavlo
CMU SCS /615
Examples
find ssn of ‘overworked’ students, ie., that take 412, 413, 415
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CMU SCS /615

64. Faloutsos - Pavlo
CMU SCS /615
Examples
find ssn of ‘overworked’ students, ie., that take 412, 413, 415: almost correct answer:
Faloutsos - Pavlo
CMU SCS /615

65. Faloutsos - Pavlo
CMU SCS /615
Examples
find ssn of ‘overworked’ students, ie., that take 412, 413, Correct answer:
c-name=413
c-name=415
Faloutsos - Pavlo
CMU SCS /615

66. Faloutsos - Pavlo
CMU SCS /615
Examples
find ssn of students that work at least as hard as ssn=123, ie., they take all the courses of ssn=123, and maybe more
Faloutsos - Pavlo
CMU SCS /615

67. Sample schema Faloutsos - Pavlo CMU SCS 15-415/615 Faloutsos - Pavlo

68. Faloutsos - Pavlo
CMU SCS /615
Examples
find ssn of students that work at least as hard as ssn=123 (ie., they take all the courses of ssn=123, and maybe more
Faloutsos - Pavlo
CMU SCS /615

69. Conclusions Relational model: only tables (‘relations’)
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CMU SCS /615
Conclusions
Relational model: only tables (‘relations’)
relational algebra: powerful, minimal: 5 operators can handle almost any query!
Faloutsos - Pavlo
CMU SCS /615