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Chapter 16: Relational Database Design and Further Dependencies Ref: Elmasri, Navathe, Fundamentals of Database Systems, 6th, Addison Wesley, ISBN-10: TA: Zhe Jiang

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Outline Big picture & motivation Simple case algorithm (part of ) Formal algorithm – Basic concepts (16.1): – General case algorithm (16.3.3)

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Big Picture: Database Design Phases ER-Diagram Relational Tables Which choice is good? How to guarantee it? Formal Norm Theory

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Motivation We have: Universal relational schema U(A1,A2, … An). A set of functional dependencies (FDs) from domain knowledge. Question: How do we decompose U into sub-relations, so as to satisfy 3NF?

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Simple Case Decomposition Algorithm Motivation: – Decompose universal relational schema into sub relations which satisfy 3NF Properties: – Preserve dependencies (nonlossy design) – Non-additive join property (no spurious tuples) – Resulting relational schemas are in 3NF Problem Definition: – Input: Universal Relation R and a set of functional dependencies F on the attributes of R – Output: Sub-relations, FDs. – Constraint: the three properties above

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Simple Case Decomposition Algorithm Suppose the FD set given is already “good” minimal cover (defined later) Approach: 1.For each LHS X in F, create a relation schema in D {X U {A1} U {A2} … U {Ak} }.where X Ai only dependency with X as LHS. 2.If none of the relation schemas in D contains a key of R, create one relation with key. (How? Introduce later) 3.Eliminate redundant relations.

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Simple Case Decomposition Algorithm Exercise: – Universal relation – FD: {P LC, LC AP, A C} – Q: Does it satisfy 1NF, 2NF, 3NF? – Q: How to decompose the relation to satisfy 3NF? Solution: 1.R1(P,L,C); R2(L,C,A,P); R3(A,C) 2.Already contains key. 3.Remove redundant relations R1 and R3, final answer is R2(L,C,A,P).

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General Case Decomposition Algorithm New info: Transform the given FD set into minimal cover New info: If no key exists, find key of U, then create a relation contain key We will introduce some basic concepts, then formal algorithm

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Basic Concept Inference rules: One FD could infer another – trivial: IR1: IR1 (reflexive rule) If X Y, then X Y. – non-trivial: IR2-IR4 {X Y} |= XZ YZ {X Y, Y Z} |= X Z {X YZ} |=X Y Closure of set of dependencies – Closure of F: F +, set of all FDs could be inferred. – Use IR1 to IR3;

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Basic Concepts Closure of left-hand-side under dependency set Algorithm Start: X+={X} 2.Grow X+ with new attributes determined by elements in X+ 3.Repeat 2 until can’t grow any more. Exercise: Given: F={X YZ, X W, W U, Y V}, U(X,Y,Z,W,U,V) Find: X+ ?

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Basic Concepts Equivalence of functional dependencies sets – Definition Cover: F covers E if F+ contains E. Equivalent FD sets: – Algorithm Check if all left-hand-sides’ closures are same Minimal Cover of dependency set F – definition: Can’t find subset that is equivalent to F

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Basic Concept Minimal Cover of dependency set F 1.break down right-hand-side, X {A1,A2,…An} to X A1, X A2, …X An 2.Try reduce size of LHS X in F, e.g. changing X into {X-B} still equivalent to F? 3.Try reduce unnecessary FD in F, e.g. remove X A in F, if result still equivalent to F. Example: – F={P LCA, LC AP, A C} – What is “minimal cover” of F?

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Basic Concepts Algorithm to find key of R&F – Start with K=R. – Find A in R such that (K-A)+ contain all attributes. – Repeat until size of K is as small as possible Example: – U(Emp_ssn, Pno, Esal, Ephone, Dno, Pname, Plocation) – F={Emp_ssn Esal, Ephone, Dno; Pno Pname, plocation}; – What is the key?

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Decomposition Algorithm: Exercise FD3 FD1: Property_id Lot#, County, Area FD2: Lot#, County Area, Property_id FD3: Area County Simpler Version: F={P LCA, LC AP, A C} 1.What is the minimal cover G? 2.Decompose G

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Decomposition Algorithm Example Simpler Version: F={P LCA, LC AP, A C} First Case: Minimal cover GX: 1.F: {P L, P C, P A, LC A, LC P,A C} 2.Minimal cover GX: {P LC, LC AP, A C} Design X: 3. R1(P,L,C), R2(L,C,A,P), and R3(A,C) 4. R2(L,C,A,P)

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Decomposition Algorithm Example Simpler Version: F={P LCA, LC AP, A C} Second Case: Minimal cover GX: 1.F: {P L, P C, P A, LC A, LC P,A C} 2.Minimal cover GX: {P LA, LC P, A C} Design Y: 3. S1(P,A,L), S2(L,C,P), and S3(A,C) 4. No redundant relations.

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Exercise Given: – Universal relation U(A,B,C,D,E,F,G,H,I,J) – Functional dependencies F={ {A,B} {C}, {B,D} {E,F}, {A,D} {G,H}, {A} {I}, {H} {J} }. – Decompose it into 3NF?

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