Schema Refinement, Functional Dependencies, Normalization
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1 Schema Refinement, Functional Dependencies, Normalization MSCI 346: Database Systems Güneş Aluç, University of Waterloo Spring 2015 MSCI 346: Database Systems Chapter 19 1 / 42
2 Outline 1 Introduction Design Principles Problems due to Poor Designs 2 Functional Dependencies Logical Implication of FDs Attribute Closure 3 Schema Decomposition Lossless-Join Decompositions Dependency Preservation 4 Normal Forms based on FDs Boyce-Codd Normal Form Third Normal Form MSCI 346: Database Systems Chapter 19 2 / 42
3 Design Process Where are we? Conceptual Design Conceptual Schema (ER Model) Step 1 ER-to-relational mapping Logical Design Step 2 Normalization: Improving the design Logical Schema (Relational Model) MSCI 346: Database Systems Chapter 19 3 / 42
4 Relational Design Principles Relations should have semantic unity Information repetition should be avoided Anomalies: insertion, deletion, modification Avoid null values as much as possible Certainly avoid excessive null values Avoid spurious joins MSCI 346: Database Systems Chapter 19 4 / 42
5 A Parts/Suppliers Database Example Description of a parts/suppliers database: Each type of part has a name and an identifying number, and may be supplied by zero or more suppliers. Each supplier may offer the part at a different price. Each supplier has an identifying number, a name, and a contact location for ordering parts. City Pno Sno Supplier Supplies Part Sname Price Pname MSCI 346: Database Systems Chapter 19 5 / 42
6 Parts/Suppliers Example (cont.) Suppliers Sno Sname City S1 Magna Ajax S2 Budd Hull Parts Pno Pname P1 P2 P3 Bolt Nut Screw Supplies Sno Pno Price S1 P S1 P S1 P S2 P An instance of the parts/suppliers database. MSCI 346: Database Systems Chapter 19 6 / 42
7 Alternative Parts/Suppliers Database Sno Sname Pno Supplied_Items City Pname Price An alternative E-R model for the parts/suppliers database. MSCI 346: Database Systems Chapter 19 7 / 42
8 Alternative Example (cont.) Supplied Items Sno Sname City Pno Pname Price S1 Magna Ajax P1 Bolt 0.50 S1 Magna Ajax P2 Nut 0.25 S1 Magna Ajax P3 Screw 0.30 S2 Budd Hull P3 Screw 0.40 A database instance corresponding to the alternative E-R model. MSCI 346: Database Systems Chapter 19 8 / 42
9 Change Anomalies Consider Is one schema better than the other? What does it mean for a schema to be good? The single-table schema suffers from several kinds of problems: Update problems (e.g. changing name of supplier) Insert problems (e.g. add a new item) Delete problems (e.g. Budd no longer supplies screws) Likely increase in space requirements The multi-table schema does not have these problems. MSCI 346: Database Systems Chapter 19 9 / 42
10 Another Alternative Parts/Supplier Database Is more tables always better? Snos Sno S1 S2 Inums Inum I1 I2 I3 Snames Sname Magna Budd Inames Iname Bolt Nut Screw Cities City Ajax Hull Prices Price Information about relationships is lost! MSCI 346: Database Systems Chapter / 42
11 Designing Good Databases Goals A methodology for evaluating schemas (detecting anomalies). A methodology for transforming bad schemas into good schemas (repairing anomalies). How do we know an anomaly exists? Certain types of integrity constraints reveal regularities in database instances that lead to anomalies. What should we do if an anomaly exists? Certain schema decompositions can avoid anomalies while retaining all information in the instances MSCI 346: Database Systems Chapter / 42
12 Functional Dependencies (FDs) Idea: Express the fact that in a relation schema (values of) a set of attributes uniquely determine (values of) another set of attributes. Definition (Functional Dependency) Let R be a relation schema, and X, Y R sets of attributes. The functional dependency X Y holds on R if whenever an instance of R contains two tuples t and u such that t[x ] = u[x ] then it is also true that t[y ] = u[y ]. We say that X functionally determines Y (in R). Notation: t[a 1,..., A k ] means projection of tuple t onto the attributes A 1,..., A k. In other words, (t.a 1,..., t.a k ). MSCI 346: Database Systems Chapter / 42
![The functional dependency X Y holds on R if whenever an instance of R contains two tuples t and u such that t[x ] = u[x ] then it is also true that t[y ] = u[y ].](/docs-images/44/13177686/images/page_12.jpg "We say that X functionally determines Y (in R). Notation: t[a 1,..., A k ] means projection of tuple t onto the attributes A 1,..., A k. In other words, (t.a 1,..., t.")
13 Examples of Functional Dependencies Consider the following relation schema: EmpProj SIN PNum Hours EName PName PLoc Allowance SIN determines employee name SIN EName project number determines project name and location PNum PName, PLoc allowances are always the same for the same number of hours at the same location PLoc, Hours Allowance MSCI 346: Database Systems Chapter / 42
14 Functional Dependencies and Keys Keys (as defined previously): A superkey is a set of attributes such that no two tuples (in an instance) agree on their values for those attributes. A candidate key is a minimal superkey. A primary key is a candidate key chosen by the DBA Relating keys and FDs: If K R is a superkey for relation schema R, then dependency K R holds on R. If dependency K R holds on R and we assume that R does not contain duplicate tuples (i.e. relational model) then K R is a superkey for relation schema R MSCI 346: Database Systems Chapter / 42
15 Closure of FD Sets How do we know what additional FDs hold in a schema? The closure of the set of functional dependencies F (denoted F + ) is the set of all functional dependencies that are satisfied by every relational instance that satisfies F. Informally, F + includes all of the dependencies in F, plus any dependencies they imply. MSCI 346: Database Systems Chapter / 42
16 Reasoning About FDs Logical implications can be derived by using inference rules called Armstrong s axioms (reflexivity) Y X X Y (augmentation) X Y XZ YZ (transitivity) X Y, Y Z X Z The axioms are sound (anything derived from F is in F + ) complete (anything in F + can be derived) Additional rules can be derived (union) X Y, X Z X YZ (decomposition) X YZ X Y MSCI 346: Database Systems Chapter / 42
17 Reasoning About FDs (example) Example: F = { SIN, PNum Hours SIN EName PNum PName, PLoc PLoc, Hours Allowance } A derivation of SIN, PNum Allowance: SIN, PNum Hours ( F ) PNum PName, PLoc ( F ) PLoc, Hours Allowance ( F ) SIN, PNum PNum (reflexivity) SIN, PNum PName, PLoc (transitivity, 4 and 2) SIN, PNum PLoc (decomposition, 5) SIN, PNum PLoc, Hours (union, 6, 1) SIN, PNum Allowance (transitivity, 7 and 3) MSCI 346: Database Systems Chapter / 42
18 Computing Attribute Closures There is a more efficient way of using Armstrong s axioms, if we only want to derive the maximal set of attributes functionally determined by some X (called the attribute closure of X ). function ComputeX + (X, F ) begin X + := X ; while true do if there exists (Y Z) F such that (1) Y X +, and (2) Z X + then X + := X + Z else exit; return X + ; end MSCI 346: Database Systems Chapter / 42
19 Computing Attribute Closures (cont d) Let R be a relational schema and F a set of functional dependencies on R. Then Theorem: X is a superkey of R if and only if ComputeX + (X, F ) = R Theorem: X Y F + if and only if Y ComputeX + (X, F ) MSCI 346: Database Systems Chapter / 42
20 Attribute Closure Example Example: F = { SIN EName PNum PName, PLoc PLoc, Hours Allowance } ComputeX + ({Pnum,Hours},F ): FD X + initial Pnum,Hours Pnum Pname,Ploc Pnum,Hours,Pname,Ploc PLoc,Hours Allowance Pnum,Hours,Pname,Ploc,Allowance MSCI 346: Database Systems Chapter / 42
21 Schema Decomposition Definition (Schema Decomposition) Let R be a relation schema (= set of attributes). The collection {R 1,..., R n } of relation schemas is a decomposition of R if R = R 1 R 2 R n A good decomposition does not lose information complicate checking of constraints contain anomalies (or at least contains fewer anomalies) MSCI 346: Database Systems Chapter / 42
22 Lossless-Join Decompositions We should be able to construct the instance of the original table from the instances of the tables in the decomposition Example: Consider replacing Marks Student Assignment Group Mark Ann A1 G1 80 Ann A2 G3 60 Bob A1 G2 60 by decomposing (i.e. projecting) into two tables SGM Student Group Mark Ann G1 80 Ann G3 60 Bob G2 60 AM Assignment Mark A1 80 A2 60 A1 60 MSCI 346: Database Systems Chapter / 42
23 Lossless-Join Decompositions (cont.) But computing the natural join of SGM and AM produces Student Assignment Group Mark Ann A1 G1 80 Ann A2 G3 60 Ann A1 G3 60 Bob A2 G2 60 Bob A1 G and we get extra data (spurious tuples). We would therefore lose information if we were to replace Marks by SGM and AM. If re-joining SGM and AM would always produce exactly the tuples in Marks, then we call SGM and AM a lossless-join decomposition. MSCI 346: Database Systems Chapter / 42
24 Lossless-Join Decompositions (cont.) A decomposition {R 1, R 2 } of R is lossless if and only if the common attributes of R 1 and R 2 form a superkey for either schema, that is R 1 R 2 R 1 or R 1 R 2 R 2 Example: In the previous example we had R = {Student, Assignment, Group, Mark}, F = {(Student, Assignment Group, Mark)}, R 1 = {Student, Group, Mark}, R 2 = {Assignment, Mark} Decomposition {R 1, R 2 } is lossy because R 1 R 2 (= {Mark}) is not a superkey of either {Student, Group, Mark} or {Assignment, Mark} MSCI 346: Database Systems Chapter / 42
25 Dependency Preservation How do we test/enforce constraints on the decomposed schema? Example: A table for a company database could be R Proj Dept Div FD1: Proj Dept, FD2: Dept Div, and FD3: Proj Div and two decompositions D 1 = {R1[Proj, Dept], R2[Dept, Div]} D 2 = {R1[Proj, Dept], R3[Proj, Div]} Both are lossless. (Why?) MSCI 346: Database Systems Chapter / 42
26 Dependency Preservation (cont.) Which decomposition is better? Decomposition D 1 lets us test FD1 on table R1 and FD2 on table R2; if they are both satisfied, FD3 is automatically satisfied. In decomposition D 2 we can test FD1 on table R1 and FD3 on table R3. Dependency FD2 is an interrelational constraint: testing it requires joining tables R1 and R3. D 1 is better! Given a schema R and a set of functional dependencies F, decomposition D = {R 1,..., R n } of R is dependency preserving if there is an equivalent set of functional dependencies F, none of which is interrelational in D. MSCI 346: Database Systems Chapter / 42
27 Normal Forms What is a good relational database schema? Rule of thumb: Independent facts in separate tables: Each relation schema should consist of a primary key and a set of mutually independent attributes This is achieved by transforming a schema into a normal form. Goals: Intuitive and straightforward transformation Anomaly-free/Nonredundant representation of data Normal Forms based on Functional Dependencies: Boyce-Codd Normal Form (BCNF) Third Normal Form (3NF) MSCI 346: Database Systems Chapter / 42
28 Normal Forms Based on FDs 1NF eliminates relations within relations or relations as attributes of tuples Lossless First Normal Form (1NF) eliminate the par-al func-onal dependencies of non- prime a5ributes to key a5ributes Lossless & Second Normal Form (2NF) Dependency preserving eliminate the transi-ve func-onal dependencies of non- prime a5ributes to key a5ributes Third Normal Form (3NF) eliminate the par-al and transi-ve func-onal dependencies of prime (ke a5ributes to key. Boyce- Codd Normal Form (BCNF) MSCI 346: Database Systems Chapter / 42
29 Boyce-Codd Normal Form (BCNF) - Informal BCNF formalizes the goal that in a good database schema, independent relationships are stored in separate tables. Given a database schema and a set of functional dependencies for the attributes in the schema, we can determine whether the schema is in BCNF. A database schema is in BCNF if each of its relation schemas is in BCNF. Informally, a relation schema is in BCNF if and only if any group of its attributes that functionally determines any others of its attributes functionally determines all others, i.e., that group of attributes is a superkey of the relation. MSCI 346: Database Systems Chapter / 42
30 Formal Definition of BCNF Let R be a relation schema and F a set of functional dependencies. Schema R is in BCNF (w.r.t. F ) if and only if whenever (X Y ) F + and XY R, then either (X Y ) is trivial (i.e., Y X ), or X is a superkey of R A database schema {R 1,..., R n } is in BCNF if each relation schema R i is in BCNF. MSCI 346: Database Systems Chapter / 42
31 BCNF and Redundancy Why does BCNF avoid redundancy? Consider: Supplied Items Sno Sname City Pno Pname Price The following functional dependency holds: Sno Sname, City Therefore, supplier name Magna and city Ajax must be repeated for each item supplied by supplier S1. Assume the above FD holds over a schema R that is in BCNF. This implies that: Sno is a superkey for R each Sno value appears on one row only no need to repeat Sname and City values MSCI 346: Database Systems Chapter / 42
32 Lossless-Join BCNF Decomposition function DecomposeBCNF (R, F ) begin Result := {R}; while some R i Result and (X Y ) F + violate the BCNF condition do begin Replace R i by R i (Y X ); Add {X, Y } to Result; end; return Result; end MSCI 346: Database Systems Chapter / 42
33 Lossless-Join BCNF Decomposition No efficient procedure to do this exists. Results depend on sequence of FDs used to decompose the relations. It is possible that no lossless join dependency preserving BCNF decomposition exists Consider R = {A, B, C} and F = {AB C, C B}. MSCI 346: Database Systems Chapter / 42
34 BCNF Decomposition - An Example R = {Sno,Sname,City,Pno,Pname,Price} functional dependencies: Sno Sname,City Pno Pname Sno,Pno Price This schema is not in BCNF because, for example, Sno determines Sname and City, but is not a superkey of R. MSCI 346: Database Systems Chapter / 42
35 BCNF Decomposition - An Example (cont.) Decomposition Diagram: {Sno,Sname,City,Pno,Pname,Price} {Sno,Pno,Price} {Sno,Pno,Pname,Price} Pno > Pname {Pno,Pname} Sno > Sname,City {Sno,Sname,City} The complete schema is now R 1 = {Sno,Sname,City} R 2 = {Sno,Pno,Price} R 3 = {Pno,Pname} This schema is a lossless-join, BCNF decomposition of the original schema R. MSCI 346: Database Systems Chapter / 42
36 Third Normal Form (3NF) Schema R is in 3NF (w.r.t. F ) if and only if whenever (X Y ) F + and XY R, then either (X Y ) is trivial, or X is a superkey of R, or each attribute in Y X is contained in a candidate key of R A database schema {R 1,..., R n } is in 3NF if each relation schema R i is in 3NF. 3NF is looser than BCNF allows more redundancy e.g. R = {A, B, C} and F = {AB C, C B}. lossless-join, dependency-preserving decomposition into 3NF relation schemas always exists. MSCI 346: Database Systems Chapter / 42
37 Minimal Cover Definition: Two sets of dependencies F and G are equivalent iff F + = G +. There are different sets of functional dependencies that have the same logical implications. Simple sets are desirable. Definition: A set of dependencies G is minimal if every right-hand side of an dependency in F is a single attribute. for no X A is the set F {X A} equivalent to F. for no X A and Z a proper subset of X is the set F {X A} {Z A} equivalent to F. Theorem: For every set of dependencies F there is an equivalent minimal set of dependencies (minimal cover). MSCI 346: Database Systems Chapter / 42
38 Finding Minimal Covers A minimal cover for F can be computed in three steps. Note that each step must be repeated until it no longer succeeds in updating F. Step 1. Replace X YZ with the pair X Y and X Z. Step 2. Remove A from the left-hand-side of X B in F if B is in ComputeX + (X {A}, F ). Step 3. Remove X A from F if A ComputeX + (X, F {X A}). MSCI 346: Database Systems Chapter / 42
39 Dependency-Preserving 3NF Decomposition Idea: Decompose into 3NF relations and then repair function Decompose3NF (R, F ) begin Result := {R}; while some R i Result and (X Y ) F + violate the 3NF condition do begin Replace R i by R i (Y X ); Add {X, Y } to Result; end; N := (a minimal cover for F ) ( i F i) + for each (X Y ) N do Add {X, Y } to Result; end; return Result; end MSCI 346: Database Systems Chapter / 42
40 Dep-Preserving 3NF Decomposition - An Example R = {Sno,Sname,City,Pno,Pname,Price} Functional dependencies: Sno Sname,City Sno,Pno Price Pno Pname Sno, Pname Price Following same decomposition tree as BCNF example: Minimal cover: Sno Sname Sno City R 1 = {Sno,Sname,City} R 2 = {Sno,Pno,Price} R 3 = {Pno,Pname} Pno Pname Sno, Pname Price Add relation to preserve missing dependency R 4 = {Sno, Pname, Price} MSCI 346: Database Systems Chapter / 42
41 3NF Synthesis - An Example R = {Sno,Sname,City,Pno,Pname,Price} Functional dependencies: Sno Sname,City Pno Pname Sno,Pno Price Sno, Pname Price Minimal cover: Sno Sname R 1 = {Sno, Sname} Sno City R 2 = {Sno, City} Pno Pname R 3 = {Pno, Pname} Sno, Pname Price R 4 = {Sno, Pname, Price} Add relation for candidate key R 5 = {Sno, Pno} Optimization: combine relations R 1 and R 2 (same key) MSCI 346: Database Systems Chapter / 42
42 Summary Functional dependencies provide clues towards elimination of (some) redundancies in a relational schema. Goals: to decompose relational schemas in such a way that the decomposition is (1) lossless-join (2) dependency preserving (3) BCNF (and if we fail here, at least 3NF) MSCI 346: Database Systems Chapter / 42
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