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					                         Chapter 4

     Relational Algebra
                                                               p
   By relieving the brain of all unnecessary work, a
   good notation sets it free to concentrate on more
   advanced problems, and, in effect, increases the
                            mental power of the race.

-- Alfred North Whitehead (1861 - 1947)




        Ch4-1: Relational Algebra              S. Pisitkasem       1
Relational Query Languages
   • Query languages: Allow manipulation and retrieval of
     data from a database.
   • Relational model supports simple, powerful QLs:
      – Strong formal foundation based on logic.
      – Allows for much optimization.
   • Query Languages != programming languages!
      – QLs not expected to be “Turing complete”.
      – QLs not intended to be used for complex calculations.
      – QLs support easy, efficient access to large data sets.

                                                            2
Ch4-1: Relational Algebra   S. Pisitkasem
Formal Relational Query Languages
 Two mathematical Query Languages form the basis for
   “real” languages (e.g. SQL), and for implementation:
 Relational Algebra: More operational, very useful for
   representing execution plans.

 Relational Calculus: Lets users describe what they want,
   rather than how to compute it. (Non-procedural,
   declarative.)

 Understanding Algebra & Calculus is key to
  understanding SQL, query processing!
                                                          3
Ch4-1: Relational Algebra   S. Pisitkasem
    Preliminaries
  • A query is applied to relation instances, and the result
    of a query is also a relation instance.
     – Schemas of input relations for a query are fixed (but
       query will run over any legal instance)
     – The schema for the result of a given query is also
       fixed. It is determined by the definitions of the query
       language constructs.
  • Positional vs. named-field notation:
     – Positional notation easier for formal definitions,
       named-field notation more readable.
     – Both used in SQL
         • Though positional notation is not encouraged
                                                             4
Ch4-1: Relational Algebra   S. Pisitkasem
     Relational Algebra: 5 Basic Operations
    • Selection (  ) Selects a subset of rows from relations.
      (horizontal)
    • Projection ( p ) Retains only wanted columns from
      (vertical)        relation.
    • Cross-product (  ) Allows us to combine two relations.

    • Set-difference ( — ) Tuples in r1, but not in r2.
    • Union (  ) Tuples in r1 and/or in r2.

    Since each operation returns a relation, operations can be
       composed! (Algebra is “closed”.)

                                                                 5
Ch4-1: Relational Algebra   S. Pisitkasem
      Example Instances                                R1   sid      bid            day
                                                            22       101        10/10/96
                                                            58       103        11/12/96

                                               S1     sid   sname          rating         age
                                                      22    dustin             7          45.0
bid     bname                 color
                                                      31    lubber             8          55.5
101     Interlake             blue
102     Interlake             red                     58    rusty              10         35.0
103     Clipper               green
104     Marine                red
                                               S2     sid   sname          rating     age
Boats
                                                      28    yuppy              9      35.0
                                                      31    lubber             8      55.5
                                                      44    guppy              5      35.0
                                                      58    rusty              10     35.0
                                                                                           6
  Ch4-1: Relational Algebra           S. Pisitkasem
     Projection

        •Examples:
                   p age(S2)                p sname,rating(S2)

         • Retains only attributes that are in the “projection list”.
         • Schema of result:
            – exactly the fields in the projection list, with the same
              names that they had in the input relation.
         • Projection operator has to eliminate duplicates (How
           do they arise? Why remove them?)
            – Note: real systems typically don’t do duplicate
              elimination unless the user explicitly asks for it.
              (Why not?)

                                                                   7
Ch4-1: Relational Algebra   S. Pisitkasem
                                                     sname    rating
Projection                                           yuppy    9
                                                     lubber   8
                                                     guppy    5
                                                     rusty    10
sid       sname             rating    age            p sname,rating (S 2)
28        yuppy                 9     35.0
31        lubber                8     55.5
44        guppy                 5     35.0                 age
58        rusty                 10    35.0
                                                           35.0
                       S2
                                                           55.5

                                                          p age(S2)    8
Ch4-1: Relational Algebra            S. Pisitkasem
    Selection ()
         • Selects rows that satisfy selection condition (=,<,>,NOT,AND, OR)
         • Result is a relation.
            Schema of result is same as that of the input relation.
         • Do we need to do duplicate elimination?


      sid        sname rating age                sname       rating
      28         yuppy   9    35.0
                                                 yuppy       9
      31         lubber  8    55.5
      44         guppy   5    35.0               rusty       10
      58         rusty   10 35.0
     rating 8(S2)                       p sname,rating( rating 8(S2))
                                                                        9
Ch4-1: Relational Algebra     S. Pisitkasem
       Union and Set-Difference

     • All of these operations take two input relations,
       which must be union-compatible:
       – Same number of fields.
       – `Corresponding’ fields have the same type.

     • For which, if any, is duplicate elimination
       required?


                                                       10
Ch4-1: Relational Algebra   S. Pisitkasem
           Union
sid       sname               rating   age               sid sname rating age
22        dustin                  7    45.0              22   dustin   7    45.0
                                                         31   lubber   8    55.5
31        lubber                  8    55.5              58   rusty    10   35.0
58        rusty                   10   35.0              44   guppy    5    35.0
                     S1                                  28   yuppy    9    35.0
sid        sname              rating   age                      S1 S2
28         yuppy                  9    35.0
31         lubber                 8    55.5
44         guppy                  5    35.0
58         rusty                  10   35.0
                        S2                                                   11
      Ch4-1: Relational Algebra          S. Pisitkasem
           Set Difference
sid         sname                 rating   age              sid sname rating age
22          dustin                    7    45.0             22    dustin     7        45.0
31          lubber                    8    55.5                           S1 S2
58          rusty                     10   35.0
                     S1
sid          sname                rating   age              sid   sname      rating   age
28           yuppy                    9    35.0             28    yuppy         9     35.0
31           lubber                   8    55.5             44    guppy         5     35.0
44           guppy                    5    35.0
58           rusty                    10   35.0                           S2 – S1
                        S2
                                                                                       12
      Ch4-1: Relational Algebra             S. Pisitkasem
     Cross-Product
            • S1  R1: Each row of S1 paired with each
              row of R1.
            • Q: How many rows in the result?
            • Result schema has one field per field of S1
              and R1, with field names `inherited’ if
              possible.
               – May have a naming conflict: Both S1 and
                 R1 have a field with the same name.
               – In this case, can use the renaming
                 operator:
             (C(1 sid1, 5  sid2), S1 R1)
                                                       13
Ch4-1: Relational Algebra   S. Pisitkasem
          Cross Product Example
sid          bid                  day                            sid      sname      rating   age
22           101          10/10/96                               22       dustin         7    45.0
58           103          11/12/96                               31       lubber         8    55.5
                                                                 58       rusty          10   35.0
                    R1                                                             S1
                                   (sid) sname rating age                  (sid) bid day
                                        22   dustin         7      45.0      22    101 10/10/96
                                        22   dustin         7      45.0      58    103 11/12/96
R1 X S1 =                               31   lubber         8      55.5      22    101 10/10/96
                                        31   lubber         8      55.5      58    103 11/12/96
                                        58   rusty          10     35.0      22    101 10/10/96
                                        58   rusty          10     35.0      58    103 11/12/96
                                                                                               14
      Ch4-1: Relational Algebra                 S. Pisitkasem
       Compound Operator: Intersection
• In addition to the 5 basic operators, there are several
  additional “Compound Operators”
   – These add no computational power to the language,
     but are useful shorthands.
   – Can be expressed solely with the basic ops.

• Intersection takes two input relations, which must be
  union-compatible.
• Q: How to express it using basic operators?
                            R  S = R  (R  S)
                                                   15
Ch4-1: Relational Algebra        S. Pisitkasem
           Intersection
sid       sname              rating    age
22        dustin                 7     45.0
31        lubber                 8     55.5
58        rusty                  10    35.0              sid sname rating age
                                                         31 lubber 8      55.5
                     S1
                                                         58 rusty  10     35.0
sid       sname              rating    age
28
31
          yuppy
          lubber
                                 9
                                 8
                                       35.0
                                       55.5
                                                           S1 S2
44        guppy                  5     35.0
58        rusty                  10    35.0
                                  S2                                       16
      Ch4-1: Relational Algebra          S. Pisitkasem
     Compound Operator: Join
       • Joins are compound operators involving cross
         product, selection, and (sometimes) projection.
       • Most common type of join is a “natural join” (often
         just called “join”). R       S conceptually is:
          – Compute R  S
          – Select rows where attributes that appear in both
             relations have equal values
          – Project all unique atttributes and one copy of each
             of the common ones.
       • Note: Usually done much more efficiently than this.
       • Useful for putting “normalized” relations back
         together.

                                                              17
Ch4-1: Relational Algebra   S. Pisitkasem
          Natural Join Example
sid          bid                  day                   sid   sname         rating   age
22           101          10/10/96                      22    dustin            7    45.0
58           103          11/12/96                      31    lubber            8    55.5
                                                        58    rusty             10   35.0
                    R1
                                                                       S1

                             R1    S1 =
         sid             sname rating age bid day
         22              dustin 7       45.0 101 10/10/96
         58              rusty  10      35.0 103 11/12/96

                                                                                      18
      Ch4-1: Relational Algebra         S. Pisitkasem
     Other Types of Joins
• Condition Join (or “theta-join”):
                            R    c S =  c ( R  S)
   (sid) sname rating age (sid) bid day
   22    dustin 7     45.0 58   103 11/12/96
   31    lubber 8     55.5 58   103 11/12/96
                            S1          < R1.sid R1
                                 S1.sid
• Result schema same as that of cross-product.
• May have fewer tuples than cross-product.
• Equi-Join: Special case: condition c contains only
  conjunction of equalities.

                                                       19
Ch4-1: Relational Algebra          S. Pisitkasem
     Semijoin
    Applications in distributed database:
    • Product(pid,cid,pname,…) at site 1
    • Company(cid,cname,…) at site 2
    •Query: Find product which price > 1000 which
             company producer ay site 2.
           price>1000(Product) < Company
    • Compute as follows:
                  T1 = price>1000(Product)        site 1
                  T2 = pcid(T1)                    site 1
                  send T2 to site 2                (T2 smaller than T1)
                  T3 = T2  < Company              site 2 (semijoin)
                  send T3 to site 1                (T3 smaller than Company)
                  Answer = T1  < T3               site 1 (semijoin)

                                                                         20
Ch4-1: Relational Algebra          S. Pisitkasem
   Complex Queries
        Product (pid, name, price, category, maker-cid)
        Purchase(ssn, storeID ,pid)
        Company (cid, name, stock_price,country)
        Person(ssn, name, phone number, city)
        Store(storeID, name, phone, city)
        Note
        • in purchase : ssn, storeID and pid are foreign key in Person,
                       Store and Product
        • in Product make-cid is a foreign key in Company
        Query
        •Find phone numbers of people who bought TV
               from ‘Bestbuy’
        •Find IT products that somebody bought
                                                                          21
Ch4-1: Relational Algebra       S. Pisitkasem
Exercises
     Product (pid, name, price, category, maker-cid)
     Purchase(ssn, storeID ,pid)
     Company (cid, name, stock_price,country)
     Person(ssn, name, phone number, city)
     Store(storeID, name, phone, city)

     Ex#1: Find people who bought IT product
     Ex#2: Find names of people who bought Thai products
     Ex#3: Find names of people who bought Thai products and did not buy
          American products
     Ex#4: Find name of people who bought American products from Depo
     Ex#5: Which store in his living city which “smith” bought stuff from

                                                                       22
Ch4-1: Relational Algebra    S. Pisitkasem
       Compound Operator: Division

• Useful for expressing “for all” queries like:
  Find sids of sailors who have reserved all boats.
• For A/B attributes of B are subset of attrs of A.
   – May need to “project” to make this happen.
• E.g., let A have 2 fields, x and y; B have only field y:

                    AB =     x  y                          
                                                B( x, y  A)

          A/B contains all tuples (x) such that for every y tuple in
          B, there is an xy tuple in A.


                                                                 23
 Ch4-1: Relational Algebra     S. Pisitkasem
Examples of Division A/B
  sno          pno          pno              pno    pno
  s1           p1           p2               p2     p1
  s1           p2                            p4
                             B1                     p2
  s1           p3
                                              B2    p4
  s1           p4
  s2           p1           sno
                                                    B3
  s2           p2           s1
  s3           p2           s2               sno
  s4           p2           s3               s1     sno
  s4           p4           s4               s4     s1
           A                A/B1             A/B2   A/B3
                                                      24
Ch4-1: Relational Algebra    S. Pisitkasem
     Expressing A/B Using Basic Operators
• Division is not essential op; just a useful shorthand.
   – (Also true of joins, but joins are so common that systems
     implement joins specially.)
• Idea: For A/B, compute all x values that are not `disqualified’
  by some y value in B.
   – x value is disqualified if by attaching y value from B, we obtain
     an xy tuple that is not in A.
            Disqualified x values:                   p x ((p x ( A)  B)  A)
              A/B:              p x ( A)            Disqualified x values

                                                                                25
    Ch4-1: Relational Algebra        S. Pisitkasem
                                                      sid      bid            day
                                       Reserves
Examples                                              22       101        10/10/96
                                                      58       103        11/12/96
                                          sid         sname          rating         age
                                  Sailors 22          dustin             7          45.0
                                          31          lubber             8          55.5
                                          58          rusty              10         35.0
                            bid    bname          color
        Boats
                            101   Interlake       Blue
                            102   Interlake       Red
                            103   Clipper         Green
                            104   Marine          Red
                                                                                    26
Ch4-1: Relational Algebra         S. Pisitkasem
Find names of sailors who’ve reserved boat #103
       Solution 1:
             p sname ((                            
                                        Re serves)  Sailors)
                            bid =103

         Solution 2:
            p sname (                                      
                                                (Re serves  Sailors))
                            bid =103
          Which solution is more efficient ?




                                                                         27
Ch4-1: Relational Algebra       S. Pisitkasem
         Find names of sailors who’ve reserved a red boat

•Information about boat color only available in Boats;
 so need an extra join;

p sname(( color = ‘red’Boats)                Reserves     Sailors)
                                                 p sname



                                                           Sailors
             color = ‘red’         Reserves

               Boats
                                                                      28
Ch4-1: Relational Algebra     S. Pisitkasem
     Find names of sailors who’ve reserved a red boat

•A more efficient solution

p sname(p sid((p bid  color = ‘red’ Boats) Res)    Sailors)
                                        p sname

                                    p sid
                                                 Sailors
             p bid
                                      Reserves
        color = ‘red’

            Boats
     A query optimizer can find this given the first solution!
                                                               29
Ch4-1: Relational Algebra   S. Pisitkasem
 Find names of sailors who’ve reserved a red or a
 green boat

•Can identify all red or green boats, then find sailors
 who’ve reserved one of these boats:

       (Tempboats,( color = ‘red’ V       color = ‘green’ Boats))

       p sname (Tempboats        Reserves          Sailors)


Can also define Tempboats using union! (How?)




                                                                      30
Ch4-1: Relational Algebra   S. Pisitkasem
      Find names of sailors who’ve reserved a red and a
      green boat
• Previous approach won’t work! Must identify sailors who’ve
  reserved red boats, sailors who’ve reserved green boats, then
  find the intersection (note that sid is a key for Sailors):



  (Tempred , p                     ( (                    Boats)     Re serves))
                              sid          color =' red '
  (Tempgreen, p                     ((                      Boats)      Reserves))
               sid                          color =' green'
p sname ((Tempred                           Tempgreen )          Sailors )

                                                                              31
  Ch4-1: Relational Algebra            S. Pisitkasem
           Find the names of sailors who’ve
           reserved all boats
       • Use division; schemas of the input relations to /
         must be carefully chosen:

                      (Tempsids, (p sid, bid Reserves) / (p bid Boats))

                      p sname (Tempsids              Sailors)


       To      find sailors who’ve reserved all ‘red’ boats:
                     …. / p bid ( color = ‘red’ Boats)




                                                                           32
Ch4-1: Relational Algebra            S. Pisitkasem
Relational Algebra

• Why bother ? Can write any RA expression directly
  in C++/Java , seem easy.
• Two reasons:
  - Each Operator admits sophisticated implementations
    (think of  , C)
  - Expressions in relational algebra can be rewritten :
     optimized



                                                   33
Ch4-1: Relational Algebra   S. Pisitkasem
Efficient Implementations of Operators
      • (age>=30 AND age<=35)(Employees)
        - Method 1 : scan the file,test each employee
        - Method 2 : use an index on age
        - Which one is better ? Well, depends….
      • Employees  Relatives
        - Iterate over Employees, then over Relatives
        - Iterate over Relatives, then over Employees
        - Sort Employees, Relatives, do “merge-join”
        - “hash-join”
        - etc

                                                        34
Ch4-1: Relational Algebra   S. Pisitkasem
Optimizations
Product (pid, name, price, category, maker-cid)
Purchase(ssn, storeID ,pid)
Person(ssn, name, phone number, city)

• Which is better :
  price>100(Product)  (Purchase   city=sea People)
                                  
  (price>100(Product)  Purchase)   city=sea People
                                   
• Depends! This is the optimizer’s job…


                                                         35
Ch4-1: Relational Algebra   S. Pisitkasem
Finally : RA has Limitations!
• Cannot compute “transitive closure”
                        Name1       Name2            Name3
                            Fred    Mary             Father
                            Mary     Joe             Cousin
                            Mary     Bill            Spouse
                            Nancy    Lou             Sister




• Find all direct and indirect relatives of Fred
• Cannot express in RA!!! Need to write C program

                                                              36
Ch4-1: Relational Algebra            S. Pisitkasem
Summary

• Relational Algebra: a small set of operators
  mapping relations to relations
   – Operational, in the sense that you specify the
     explicit order of operations
   – A closed set of operators! Can mix and match.
• Basic ops include: s, p, , , —
• Important compound ops: , , /


                                                  37
Ch4-1: Relational Algebra   S. Pisitkasem

				
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