Introduction to Data Management. Lecture #12 (Relational Algebra II)

Transcription

1 Introduction to Data Management Lecture #12 (Relational Algebra II) Instructor: Mike Carey Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 1 Announcements v HW and exams: HW #4 info Came out yesterday a little before 6PM As always, due one week later (to be on time) Midterm Exam 1 Over! (And hopefully easier than you expected) Grading is commencing goal is 2-week turnaround It doesn t matter now, in a material sense! (Reset!) v Today s plan: Relational query languages, Episode 2 Relational calculus, Episode 1 (if time) Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 2

2 (Cache Reload J) Relational Query Languages v Two mathematical Query Languages form the basis for real languages (e.g., SQL), and for their 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-operational, declarative.) Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 3 (Cache Reload J) Relational Algebra v Basic operations: σ Selection ( ) Selects a subset of rows from relation. π Projection ( ) Omits unwanted columns from relation. Cross-product ( ) Allows us to combine two relations. Set-difference ( ) Tuples in reln. 1, but not in reln. 2. Union ( ) Tuples in reln. 1 and in reln. 2. v Additional operations: Intersection, join, division, renaming: Not essential, but (very!) useful. (I.e., don t add expressive power, but ) v Since each operation returns a relation, operations can be composed! (Algebra is closed.) Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 4

3 Ex: Wisconsin Sailing Club Database Sailors Reserves Boats sid sname rating age sid date bname color 22 Dustin Brutus Lubber Andy Rusty Horatio Zorba Horatio Art Bob /10/ /10/ /8/ /7/ /10/ /6/ /12/ /5/ /8/ /8/ Interlake blue 102 Interlake red 103 Clipper green 104 Marine red Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 5 Find names of sailors who ve reserved boat #103 v Solution 1: π sname ((σ =103 Reserves) Sailors) v Solution 2: ρ ( Temp1, σ Re serves) =103 ρ ( Tem, Temp1 Sailors) π sname ( Tem) v Solution 3: π sname (σ =103 (Reserves Sailors)) Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 6

4 Find names of sailors who ve reserved boat #103 v Solution 1: π sname ((σ =103 Reserves) Sailors) v Solution 2: Temp1= σ =103 Reserves Tem= Temp1 Sailors π sname ( Tem) v Solution 3: π sname (σ =103 (Reserves Sailors)) Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 7 Ex: Wisconsin Sailing Club Database σ =103 Reserves sid date /8/ /6/ /8/93 (σ =103 Reserves) Sailors π sname ((σ =103 Reserves) Sailors) sname Dustin Lubber Horatio sid date sname rating age /8/ /6/ /8/93 Dustin Lubber Horatio (Solution 1) Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 8

5 Find names of sailors who ve reserved a red boat v Information about boat color only available in Boats; so need to do another join: π sname (( σ Boats) Re serves Sailors) color = ' red' v A more efficient solution: π sname ( π π σ sid (( Boats) Re s) Sailors) color = ' red' A query optimizer will find the latter, given the 1 st query! Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 9 Find sailors who ve reserved a red or a green boat v Can identify all red or green boats, then find sailors who ve reserved one of these boats: ρ ( Tempboats, ( σ )) color = ' red ' color = ' green' Boats π sname ( Tempboats Re serves Sailors) v Can also define Tempboats using union! (Q: How?) v What happens if is replaced by in this query? Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 10

6 Find sailors who ve reserved a red and a green boat v Previous approach won t work! Must identify sailors who reserved red boats and sailors who ve reserved green boats, then find their intersection (notice that sid is a key for Sailors!): ρ ( Tempred, π (( σ Boats) Re serves)) sid color = ' red ' ρ ( Tempgreen, π (( σ Boats) Re serves)) sid color = ' green' π sname (( Tempred Tempgreen) Sailors) Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 11 Find the names of sailors who ve reserved all boats v Uses division; schemas of the input relations feeding the / operator must be carefully chosen: ρ ( Tempsids, ( π Re serves) / ( π )) sid, Boats π sname ( Tempsids Sailors) v To find sailors who ve reserved all Interlake boats:... / π ( σ ) bname= ' Interlake' Boats Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 12

7 (Cache Reload) Examples of Division A/B sno s2 s2 s3 R pno p1 p3 p4 p1 p4 pno B1 sno s2 s3 pno p4 B2 sno pno p1 p4 B3 sno A/B1 A/B2 A/B3 Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 13 Examples of Division R/B sid s2 s2 s3 b1 b3 b4 b1 b4 R B1 b4 B2 b1 b4 B3 sid s2 sid s3 sid R/B1 R/B2 R/B3 Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 14

8 Find the names of sailors who ve reserved all boats v Uses division; schemas of the input relations feeding the / operator must be carefully chosen: ρ ( Tempsids, ( π Re serves) / ( π )) sid, Boats π sname ( Tempsids Sailors) v To find sailors who ve reserved all Interlake boats:... / π ( σ ) bname= ' Interlake' Boats Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 15 Find the names of sailors who ve reserved all boats v Uses division; schemas of the input relations feeding the / operator must be carefully chosen: ρ ( Tempsids, ( π Re serves) / ( π )) sid, Boats π sname ( Tempsids Sailors) v To find sailors who ve reserved all Interlake boats:... / π ( σ ) bname= ' Interlake' Boats Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 16

9 Relational Algebra Summary v The relational model has (several) rigorously defined query languages that are both simple and powerful in nature. v Relational algebra is more operational; very useful as an internal representation for query evaluation plans. v Several ways of expressing a given query; a query optimizer should choose the most efficient version. (Take CS122C...! J) v We ll add a few more operators later on v Next up for now: Relational Calculus Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 17 Relational Calculus v Comes in two flavors: Tuple relational calculus (TRC) and Domain relational calculus (DRC). v Calculus has variables, constants, comparison ops, logical connectives and quantifiers. TRC: Variables range over (i.e., get bound to) tuples. DRC: Variables range over domain elements (= field values). Both TRC and DRC are simple subsets of first-order logic. v Expressions in the calculus are called formulas. An answer tuple is essentially an assignment of constants to variables that make the formula evaluate to true. v TRC is the basis for various query languages (Quel, SQL, OQL, XQuery, ), while DRC is the basis for examplebased relational query UIs. We ll study TRC! Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 18

10 Tuple Relational Calculus v Query in TRC has the form: { t(attrlist) P(t) } v Answer includes all tuples t with (optionally) specified schema (attrlist) that cause formula P(t) to be true. v Formula is recursively defined, starting with simple atomic formulas (getting tuples from relations or making comparisons of values), and building up bigger and better Boolean formulas using logical connectives. Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 19 TRC Formulas v Atomic formula: r R, or r R, or r.a op s.b, or r.a op constant op is one of <, >,,,,= v Formula: an atomic formula, or P, P Q, P Q, where P and Q are formulas, or r R (P(r)), where variable r is free in P( ), or r R (P(r)), where variable r is free in P( ), or P Q (pronounced implies, equivalent to ( P) Q)) Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 20

11 Free and Bound Variables v The use of a quantifier such as t T or t T in a formula is said to bind t. A variable that is not bound is free. v Now let us revisit the definition of a TRC query: { t(a1, a2, ) P(t) } v There is an important restriction: the variable t that appears to the left of the ( such that ) symbol must be the only free variable in the formula P(...). v Let s look at some examples! Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 21 Find sailors with a rating above 7 { s s Sailors s.rating > 7 } v This is equivalent to the more general form: { t(id, nm, rtg, age) s Sailors ( t.id = s.sid t.nm = s.sname t.rtg=s.rating t.age = s.age s.rating > 7 ) } (Q: See how each one specifies the answer s schema and values...? Note that the second one s schema is different, as we ve specified it.) Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 22

12 Find ids of sailors who are older than 18 or who have a rating under 9 and are named Joe { t(sid) s Sailors ( (s.age > 18 (s.rating < 9 s.sname = Joe )) v Things to notice: t.sid = s.sid ) } Again, how result schema and values are specified Use of Boolean formula to specify the query constraints Highly declarative nature of this form of query language! Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 23 To Be Continued! v A few more TRC query examples next time v Then we ll be on to SQL! Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 24