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Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe  CHAPTER 14 Basics of Functional Dependencies and Normalization for Relational Databases Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 2 Chapter Outline 1 Informal...

Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe  CHAPTER 14 Basics of Functional Dependencies and Normalization for Relational Databases Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 2 Chapter Outline 1 Informal Design Guidelines for Relational Databases 1.1 Semantics of the Relation Attributes 1.2 Redundant Information in Tuples and Update Anomalies 1.3 Null Values in Tuples 1.4 Spurious Tuples 2 Functional Dependencies (FDs) 2.1 Definition of Functional Dependency Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 3 Chapter Outline 3 Normal Forms Based on Primary Keys 3.1 Normalization of Relations 3.2 Practical Use of Normal Forms 3.3 Definitions of Keys and Attributes Participating in Keys 3.4 First Normal Form 3.5 Second Normal Form 3.6 Third Normal Form 4 General Normal Form Definitions for 2NF and 3NF (For Multiple Candidate Keys) 5 BCNF (Boyce-Codd Normal Form) Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 4 Chapter Outline 6 Multivalued Dependency and Fourth Normal Form 7 Join Dependencies and Fifth Normal Form Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 5 1. Informal Design Guidelines for Relational Databases (1) What is relational database design? The grouping of attributes to form "good" relation schemas Two levels of relation schemas The logical "user view" level The storage "base relation" level Design is concerned mainly with base relations What are the criteria for "good" base relations? Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 6 Informal Design Guidelines for Relational Databases (2) We first discuss informal guidelines for good relational design Then we discuss formal concepts of functional dependencies and normal forms - 1NF (First Normal Form) - 2NF (Second Normal Form) - 3NF (Third Noferferferfewrmal Form) - BCNF (Boyce-Codd Normal Form) Additional types of dependencies, further normal forms, relational design algorithms by synthesis are discussed in Chapter 15 Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 7 1.1 Semantics of the Relational Attributes must be clear GUIDELINE 1: Informally, each tuple in a relation should represent one entity or relationship instance. (Applies to individual relations and their attributes). Attributes of different entities (EMPLOYEEs, DEPARTMENTs, PROJECTs) should not be mixed in the same relation Only foreign keys should be used to refer to other entities Entity and relationship attributes should be kept apart as much as possible. Bottom Line: Design a schema that can be explained easily relation by relation. The semantics of attributes should be easy to interpret. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 8 Figure 14.1 A simplified COMPANY relational database schema Figure 14.1 A simplified COMPANY relational database schema. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 9 1.2 Redundant Information in Tuples and Update Anomalies Information is stored redundantly Wastes storage Causes problems with update anomalies Insertion anomalies Deletion anomalies Modification anomalies Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 10 EXAMPLE OF AN UPDATE ANOMALY Consider the relation: EMP_PROJ(Emp#, Proj#, Ename, Pname, No_hours) Update Anomaly: Changing the name of project number P1 from “Billing” to “Customer-Accounting” may cause this update to be made for all 100 employees working on project P1. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 11 EXAMPLE OF AN INSERT ANOMALY Consider the relation: EMP_PROJ(Emp#, Proj#, Ename, Pname, No_hours) Insert Anomaly: Cannot insert a project unless an employee is assigned to it. Conversely Cannot insert an employee unless an he/she is assigned to a project. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 12 EXAMPLE OF A DELETE ANOMALY Consider the relation: EMP_PROJ(Emp#, Proj#, Ename, Pname, No_hours) Delete Anomaly: When a project is deleted, it will result in deleting all the employees who work on that project. Alternately, if an employee is the sole employee on a project, deleting that employee would result in deleting the corresponding project. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 13 Figure 14.3 Two relation schemas suffering from update anomalies Figure 14.3 Two relation schemas suffering from update anomalies. (a) EMP_DEPT and (b) EMP_PROJ. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 14 Figure 14.4 Sample states for EMP_DEPT and EMP_PROJ Figure 14.4 Sample states for EMP_DEPT and EMP_PROJ resulting from applying NATURAL JOIN to the relations in Figure 14.2. These may be stored as base relations for performance reasons. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 15 Guideline for Redundant Information in Tuples and Update Anomalies GUIDELINE 2: Design a schema that does not suffer from the insertion, deletion and update anomalies. If there are any anomalies present, then note them so that applications can be made to take them into account. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 16 1.3 Null Values in Tuples GUIDELINE 3: Relations should be designed such that their tuples will have as few NULL values as possible Attributes that are NULL frequently could be placed in separate relations (with the primary key) Reasons for nulls: Attribute not applicable or invalid Attribute value unknown (may exist) Value known to exist, but unavailable Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 17 1.4 Generation of Spurious Tuples – avoid at any cost Bad designs for a relational database may result in erroneous results for certain JOIN operations The "lossless join" property is used to guarantee meaningful results for join operations GUIDELINE 4: The relations should be designed to satisfy the lossless join condition. No spurious tuples should be generated by doing a natural-join of any relations. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 18 Spurious Tuples (2) There are two important properties of decompositions: a) Non-additive or losslessness of the corresponding join b) Preservation of the functional dependencies. Note that: Property (a) is extremely important and cannot be sacrificed. Property (b) is less stringent and may be sacrificed. (See Chapter 15). Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 19 2. Functional Dependencies Functional dependencies (FDs) Are used to specify formal measures of the "goodness" of relational designs And keys are used to define normal forms for relations Are constraints that are derived from the meaning and interrelationships of the data attributes A set of attributes X functionally determines a set of attributes Y if the value of X determines a unique value for Y Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 20 2.1 Defining Functional Dependencies X  Y holds if whenever two tuples have the same value for X, they must have the same value for Y For any two tuples t1 and t2 in any relation instance r(R): If t1[X]=t2[X], then t1[Y]=t2[Y] X  Y in R specifies a constraint on all relation instances r(R) Written as X  Y; can be displayed graphically on a relation schema as in Figures. ( denoted by the arrow: ). FDs are derived from the real-world constraints on the attributes Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 21 Examples of FD constraints (1) Social security number determines employee name SSN  ENAME Project number determines project name and location PNUMBER  {PNAME, PLOCATION} Employee ssn and project number determines the hours per week that the employee works on the project {SSN, PNUMBER}  HOURS Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 22 Examples of FD constraints (2) An FD is a property of the attributes in the schema R The constraint must hold on every relation instance r(R) If K is a key of R, then K functionally determines all attributes in R (since we never have two distinct tuples with t1[K]=t2[K]) Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 23 Defining FDs from instances Note that in order to define the FDs, we need to understand the meaning of the attributes involved and the relationship between them. An FD is a property of the attributes in the schema R Given the instance (population) of a relation, all we can conclude is that an FD may exist between certain attributes. What we can definitely conclude is – that certain FDs do not exist because there are tuples that show a violation of those dependencies. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 24 Figure 14.7 Ruling Out FDs Note that given the state of the TEACH relation, we can say that the FD: Text → Course may exist. However, the FDs Teacher → Course, Teacher → Text and Couse → Text are ruled out. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 25 Figure 14.8 What FDs may exist? A relation R(A, B, C, D) with its extension. Which FDs may exist in this relation? Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 26 Inference Rules for FDs (2) Armstrong's inference rules: IR1. (Reflexive) If Y subset-of X, then X → Y IR2. (Augmentation) If X → Y, then XZ → YZ (Notation: XZ stands for X U Z) IR3. (Transitive) If X → Y and Y → Z, then X → Z IR1, IR2, IR3 form a sound and complete set of inference rules These are rules hold and all other rules that hold can be deduced from these Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 27 Inference Rules for FDs (3) Some additional inference rules that are useful: Decomposition: If X → YZ, then X → Y and X → Z Union: If X → Y and X → Z, then X → YZ Psuedotransitivity: If X → Y and WY → Z, then WX → Z The last three inference rules, as well as any other inference rules, can be deduced from IR1, IR2, and IR3 (completeness property) Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15 - 28 3 Normal Forms Based on Primary Keys 3.1 Normalization of Relations 3.2 Practical Use of Normal Forms 3.3 Definitions of Keys and Attributes Participating in Keys 3.4 First Normal Form 3.5 Second Normal Form 3.6 Third Normal Form Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 29 3.1 Normalization of Relations (1) Normalization: The process of decomposing unsatisfactory "bad" relations by breaking up their attributes into smaller relations Normal form: Condition using keys and FDs of a relation to certify whether a relation schema is in a particular normal form Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 30 Normalization of Relations (2) 2NF, 3NF, BCNF based on keys and FDs of a relation schema 4NF based on keys, multi-valued dependencies : MVDs; 5NF based on keys, join dependencies : JDs Additional properties may be needed to ensure a good relational design (lossless join, dependency preservation; see Chapter 15) Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 31 3.2 Practical Use of Normal Forms Normalization is carried out in practice so that the resulting designs are of high quality and meet the desirable properties The practical utility of these normal forms becomes questionable when the constraints on which they are based are hard to understand or to detect The database designers need not normalize to the highest possible normal form (usually up to 3NF and BCNF. 4NF rarely used in practice.) Denormalization: The process of storing the join of higher normal form relations as a base relation—which is in a lower normal form Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 32 3.3 Definitions of Keys and Attributes Participating in Keys (1) A superkey of a relation schema R = {A1, A2,...., An} is a set of attributes S subset-of R with the property that no two tuples t1 and t2 in any legal relation state r of R will have t1[S] = t2[S] A key K is a superkey with the additional property that removal of any attribute from K will cause K not to be a superkey any more. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 33 Definitions of Keys and Attributes Participating in Keys (2) If a relation schema has more than one key, each is called a candidate key. One of the candidate keys is arbitrarily designated to be the primary key, and the others are called secondary keys. A Prime attribute must be a member of some candidate key A Nonprime attribute is not a prime attribute— that is, it is not a member of any candidate key. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 34 3.4 First Normal Form Disallows composite attributes multivalued attributes nested relations; attributes whose values for an individual tuple are non-atomic Considered to be part of the definition of a relation Most RDBMSs allow only those relations to be defined that are in First Normal Form Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 35 Figure 14.9 Normalization into 1NF Figure 14.9 Normalization into 1NF. (a) A relation schema that is not in 1NF. (b) Sample state of relation DEPARTMENT. (c) 1NF version of the same relation with redundancy. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 36 Figure 14.10 Normalizing nested relations into 1NF Figure 14.10 Normalizing nested relations into 1NF. (a) Schema of the EMP_PROJ relation with a nested relation attribute PROJS. (b) Sample extension of the EMP_PROJ relation showing nested relations within each tuple. (c) Decomposition of EMP_PROJ into relations EMP_PROJ1 and EMP_PROJ2 by propagating the primary key. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 37 3.5 Second Normal Form (1) Uses the concepts of FDs, primary key Definitions Prime attribute: An attribute that is member of the primary key K Full functional dependency: a FD Y -> Z where removal of any attribute from Y means the FD does not hold any more Examples: {SSN, PNUMBER} -> HOURS is a full FD since neither SSN -> HOURS nor PNUMBER -> HOURS hold {SSN, PNUMBER} -> ENAME is not a full FD (it is called a partial dependency ) since SSN -> ENAME also holds Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 38 Second Normal Form (2) A relation schema R is in second normal form (2NF) if every non-prime attribute A in R is fully functionally dependent on the primary key R can be decomposed into 2NF relations via the process of 2NF normalization or “second normalization” Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 39 Figure 14.11 Normalizing into 2NF and 3NF Figure 14.11 Normalizing into 2NF and 3NF. (a) Normalizing EMP_PROJ into 2NF relations. (b) Normalizing EMP_DEPT into 3NF relations. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 40 Figure 14.12 Normalization into 2NF and 3NF Figure 14.12 Normalization into 2NF and 3NF. (a) The LOTS relation with its functional dependencies FD1 through FD4. (b) Decomposing into the 2NF relations LOTS1 and LOTS2. (c) Decomposing LOTS1 into the 3NF relations LOTS1A and LOTS1B. (d) Progressive normalization of LOTS into a 3NF design. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 41 3.6 Third Normal Form (1) Definition: Transitive functional dependency: a FD X -> Z that can be derived from two FDs X -> Y and Y -> Z Examples: SSN -> DMGRSSN is a transitive FD Since SSN -> DNUMBER and DNUMBER -> DMGRSSN hold SSN -> ENAME is non-transitive Since there is no set of attributes X where SSN -> X and X -> ENAME Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 42 Third Normal Form (2) A relation schema R is in third normal form (3NF) if it is in 2NF and no non-prime attribute A in R is transitively dependent on the primary key R can be decomposed into 3NF relations via the process of 3NF normalization NOTE: In X -> Y and Y -> Z, with X as the primary key, we consider this a problem only if Y is not a candidate key. When Y is a candidate key, there is no problem with the transitive dependency. E.g., Consider EMP (SSN, Emp#, Salary ). Here, SSN -> Emp# -> Salary and Emp# is a candidate key. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 43 Normal Forms Defined Informally 1st normal form All attributes depend on the key 2nd normal form All attributes depend on the whole key 3rd normal form All attributes depend on nothing but the key Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 44 4. General Normal Form Definitions (For Multiple Keys) (1) The above definitions consider the primary key only The following more general definitions take into account relations with multiple candidate keys Any attribute involved in a candidate key is a prime attribute All other attributes are called non-prime attributes. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 45 4.1 General Definition of 2NF (For Multiple Candidate Keys) A relation schema R is in second normal form (2NF) if every non-prime attribute A in R is fully functionally dependent on every key of R In Figure 14.12 the FD County_name → Tax_rate violates 2NF. So second normalization converts LOTS into LOTS1 (Property_id#, County_name, Lot#, Area, Price) LOTS2 ( County_name, Tax_rate) Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 46 4.2 General Definition of Third Normal Form Definition: Superkey of relation schema R - a set of attributes S of R that contains a key of R A relation schema R is in third normal form (3NF) if whenever a FD X → A holds in R, then either: (a) X is a superkey of R, or (b) A is a prime attribute of R LOTS1 relation violates 3NF because Area → Price ; and Area is not a superkey in LOTS1. (see Figure 14.12). Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 47 4.3 Interpreting the General Definition of Third Normal Form Consider the 2 conditions in the Definition of 3NF: A relation schema R is in third normal form (3NF) if whenever a FD X → A holds in R, then either: (a) X is a superkey of R, or (b) A is a prime attribute of R Condition (a) catches two types of violations : - one where a prime attribute functionally determines a non-prime attribute. This catches 2NF violations due to non-full functional dependencies. -second, where a non-prime attribute functionally determines a non-prime attribute. This catches 3NF violations due to a transitive dependency. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 48 4.3 Interpreting the General Definition of Third Normal Form (2) ALTERNATIVE DEFINITION of 3NF: We can restate the definition as: A relation schema R is in third normal form (3NF) if every non-prime attribute in R meets both of these conditions: It is fully functionally dependent on every key of R It is non-transitively dependent on every key of R Note that stated this way, a relation in 3NF also meets the requirements for 2NF. The condition (b) from the last slide takes care of the dependencies that “slip through” (are allowable to) 3NF but are “caught by” BCNF which we discuss next. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 49 5. BCNF (Boyce-Codd Normal Form) A relation schema R is in Boyce-Codd Normal Form (BCNF) if whenever an FD X → A holds in R, then X is a superkey of R Each normal form is strictly stronger than the previous one Every 2NF relation is in 1NF Every 3NF relation is in 2NF Every BCNF relation is in 3NF There exist relations that are in 3NF but not in BCNF Hence BCNF is considered a stronger form of 3NF The goal is to have each relation in BCNF (or 3NF) Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 50 Figure 14.13 Boyce-Codd normal form Figure 14.13 Boyce-Codd normal form. (a) BCNF normalization of LOTS1A with the functional dependency FD2 being lost in the decomposition. (b) A schematic relation with FDs; it is in 3NF, but not in BCNF due to the f.d. C → B. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 51 Figure 14.14 A relation TEACH that is in 3NF but not in BCNF Figure 14.14 A relation TEACH that is in 3NF but not BCNF. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 52 Achieving the BCNF by Decomposition (1) Two FDs exist in the relation TEACH: fd1: { student, course} -> instructor fd2: instructor -> course {student, course} is a candidate key for this relation and that the dependencies shown follow the pattern in Figure 14.13 (b). So this relation is in 3NF but not in BCNF A relation NOT in BCNF should be decomposed so as to meet this property, while possibly forgoing the preservation of all functional dependencies in the decomposed relations. (See Algorithm 15.3) Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 53 Achieving the BCNF by Decomposition (2) Three possible decompositions for relation TEACH D1: {student, instructor} and {student, course} D2: {course, instructor } and {course, student} D3: {instructor, course } and {instructor, student}  All three decompositions will lose fd1. We have to settle for sacrificing the functional dependency preservation. But we cannot sacrifice the non-additivity property after decomposition. Out of the above three, only the 3rd decomposition will not generate spurious tuples after join.(and hence has the non-additivity property). A test to determine whether a binary decomposition (decomposition into two relations) is non-additive (lossless) is discussed under Property NJB on the next slide. We then show how the third decomposition above meets the property. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 54 Test for checking non-additivity of Binary Relational Decompositions Testing Binary Decompositions for Lossless Join (Non-additive Join) Property Binary Decomposition: Decomposition of a relation R into two relations. PROPERTY NJB (non-additive join test for binary decompositions): A decomposition D = {R1, R2} of R has the lossless join property with respect to a set of functional dependencies F on R if and only if either The f.d. ((R1 ∩ R2)  (R1- R2)) is in F+, or The f.d. ((R1 ∩ R2)  (R2 - R1)) is in F+. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 55 Test for checking non-additivity of Binary Relational Decompositions If you apply the NJB test to the 3 decompositions of the TEACH relation: D1 gives Student  Instructor or Student  Course, none of which is true. D2 gives Course  Instructor or Course  Student, none of which is true. However, in D3 we get Instructor  Course or Instructor  Student. Since Instructor  Course is indeed true, the NJB property is satisfied and D3 is determined as a non- additive (good) decomposition. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 56 General Procedure for achieving BCNF when a relation fails BCNF Here we make use the algorithm from Chapter 15 (Algorithm 15.5): Let R be the relation not in BCNF, let X be a subset-of R, and let X A be the FD that causes a violation of BCNF. Then R may be decomposed into two relations: (i) R –A and (ii) X υ A. If either R –A or X υ A. is not in BCNF, repeat the process. Note that the f.d. that violated BCNF in TEACH was Instructor Course. Hence its BCNF decomposition would be : (TEACH – COURSE) and (Instructor υ Course), which gives the relations: (Instructor, Student) and (Instructor, Course) that we obtained before in decomposition D3. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14-55 5. Multivalued Dependencies and Fourth Normal Form (1) Definition: A multivalued dependency (MVD) X —>> Y specified on relation schema R, where X and Y are both subsets of R, specifies the following constraint on any relation state r of R: If two tuples t1 and t2 exist in r such that t1[X] = t2[X], then two tuples t3 and t4 should also exist in r with the following properties, where we use Z to denote (R 2 (X υ Y)): t3[X] = t4[X] = t1[X] = t2[X]. t3[Y] = t1[Y] and t4[Y] = t2[Y]. t3[Z] = t2[Z] and t4[Z] = t1[Z]. An MVD X —>> Y in R is called a trivial MVD if (a) Y is a subset of X, or (b) X υ Y = R. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14-56 Multivalued Dependencies and Fourth Normal Form (3) Definition: A relation schema R is in 4NF with respect to a set of dependencies F (that includes functional dependencies and multivalued dependencies) if, for every nontrivial multivalued dependency X —>> Y in F+, X is a superkey for R. Note: F+ is the (complete) set of all dependencies (functional or multivalued) that will hold in every relation state r of R that satisfies F. It is also called the closure of F. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14-57 Figure 14.15 Fourth and fifth normal forms. Figure 14.15 Fourth and fifth normal forms. (a) The EMP relation with two MVDs: Ename –>> Pname and Ename –>> Dname. (b) Decomposing the EMP relation into two 4NF relations EMP_PROJECTS and EMP_DEPENDENTS. (c) The relation SUPPLY with no MVDs is in 4NF but not in 5NF if it has the JD(R1, R2, R3). (d) Decomposing the relation SUPPLY into the 5NF relations R1, R2, R3. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 60 6. Join Dependencies and Fifth Normal Form (1) Definition: A join dependency (JD), denoted by JD(R1, R2,..., Rn), specified on relation schema R, specifies a constraint on the states r of R. The constraint states that every legal state r of R should have a non-additive join decomposition into R1, R2,..., Rn; that is, for every such r we have * (R1(r), R2(r),..., Rn(r)) = r Note: an MVD is a special case of a JD where n = 2. A join dependency JD(R1, R2,..., Rn), specified on relation schema R, is a trivial JD if one of the relation schemas Ri in JD(R1, R2,..., Rn) is equal to R. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 61 Join Dependencies and Fifth Normal Form (2) Definition: A relation schema R is in fifth normal form (5NF) (or Project-Join Normal Form (PJNF)) with respect to a set F of functional, multivalued, and join dependencies if, for every nontrivial join dependency JD(R1, R2,..., Rn) in F+ (that is, implied by F), every Ri is a superkey of R. Discovering join dependencies in practical databases with hundreds of relations is next to impossible. Therefore, 5NF is rarely used in practice. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 62 Chapter Summary Informal Design Guidelines for Relational Databases Functional Dependencies (FDs) Normal Forms (1NF, 2NF, 3NF)Based on Primary Keys General Normal Form Definitions of 2NF and 3NF (For Multiple Keys) BCNF (Boyce-Codd Normal Form) Fourth and Fifth Normal Forms Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 63 Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe  CHAPTER 15 Relational Database Design Algorithms and Further Dependencies Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 2 Chapter Outline 1. Further topics in Functional Dependencies 1.1 Inference Rules for FDs 1.2 Equivalence of Sets of FDs 1.3 Minimal Sets of FDs 2. Properties of Relational Decompositions 3. Algorithms for Relational Database Schema Design 4. Nulls, Dangling Tuples, Alternative Relational Designs Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 3 Chapter Outline 5. Multivalued Dependencies and Fourth Normal Form – further discussion 6. Other Dependencies and Normal Forms 6.1 Join Dependencies 6.2 Inclusion Dependencies 6.3 Dependencies based on Arithmetic Functions and Procedures 6.2 Domain-Key Normal Form Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 4 1. Functional Dependencies : Inference Rules, Equivalence and Minimal Cover We discussed functional dependencies in the last chapter. To recollect: A set of attributes X functionally determines a set of attributes Y if the value of X determines a unique value for Y. Our goal here is to determine the properties of functional dependencies and to find out the ways of manipulating them. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 5 Defining Functional Dependencies X → Y holds if whenever two tuples have the same value for X, they must have the same value for Y For any two tuples t1 and t2 in any relation instance r(R): If t1[X]=t2[X], then t1[Y]=t2[Y] X → Y in R specifies a constraint on all relation instances r(R) Written as X → Y; can be displayed graphically on a relation schema as in Figures in Chapter 14. ( denoted by the arrow: ). FDs are derived from the real-world constraints on the attributes Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 6 1.1 Inference Rules for FDs (1) Definition: An FD X Y is inferred from or implied by a set of dependencies F specified on R if X Y holds in every legal relation state r of R; that is, whenever r satisfies all the dependencies in F, X Y also holds in r. Given a set of FDs F, we can infer additional FDs that hold whenever the FDs in F hold Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 7 Inference Rules for FDs (2) Armstrong's inference rules: IR1. (Reflexive) If Y subset-of X, then X → Y IR2. (Augmentation) If X → Y, then XZ → YZ (Notation: XZ stands for X U Z) IR3. (Transitive) If X → Y and Y → Z, then X → Z IR1, IR2, IR3 form a sound and complete set of inference rules These are rules hold and all other rules that hold can be deduced from these Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 8 Inference Rules for FDs (3) Some additional inference rules that are useful: Decomposition: If X → YZ, then X → Y and X → Z Union: If X → Y and X → Z, then X → YZ Psuedotransitivity: If X → Y and WY → Z, then WX → Z The last three inference rules, as well as any other inference rules, can be deduced from IR1, IR2, and IR3 (completeness property) Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15 - 9 Closure Closure of a set F of FDs is the set F+ of all FDs that can be inferred from F Closure of a set of attributes X with respect to F is the set X+ of all attributes that are functionally determined by X X+ can be calculated by repeatedly applying IR1, IR2, IR3 using the FDs in F Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15 - 10 Algorithm to determine Closure Algorithm 15.1. Determining X+, the Closure of X under F Input: A set F of FDs on a relation schema R, and a set of attributes X, which is a subset of R. X+ := X; repeat oldX+ := X+; for each functional dependency Y Z in F do if X+  Y then X+ := X+  Z; until (X+ = oldX+); Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15 - 11 Example of Closure (1) For example, consider the following relation schema about classes held at a university in a given academic year. CLASS ( Classid, Course#, Instr_name, Credit_hrs, Text, Publisher, Classroom, Capacity). Let F, the set of functional dependencies for the above relation include the following f.d.s: FD1: Classid Course#, Instr_name, Credit_hrs, Text, Publisher, Classroom, Capacity; FD2: Course# Credit_hrs; FD3: {Course#, Instr_name} Text, Classroom; FD4: Text Publisher FD5: Classroom Capacity These f.d.s above represent the meaning of the individual attributes and the relationship among them and defines certain rules about the classes. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15 - 12 Example of Closure (2) The closures of attributes or sets of attributes for some example sets: { Classid } + = { Classid , Course#, Instr_name, Credit_hrs, Text, Publisher, Classroom, Capacity } = CLASS { Course#} + = { Course#, Credit_hrs} { Course#, Instr_name } + = { Course#, Credit_hrs, Text, Publisher, Classroom, Capacity } Note that each closure above has an interpretation that is revealing about the attribute(s) on the left-hand-side. The closure of { Classid } + is the entire relation CLASS indicating that all attributes of the relation can be determined from Classid and hence it is a key. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15 - 13 1.2 Equivalence of Sets of FDs Two sets of FDs F and G are equivalent if: Every FD in F can be inferred from G, and Every FD in G can be inferred from F Hence, F and G are equivalent if F+ =G+ Definition (Covers): F covers G if every FD in G can be inferred from F (i.e., if G+ subset-of F+) F and G are equivalent if F covers G and G covers F There is an algorithm for checking equivalence of sets of FDs Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15 - 14 1.3 Finding Minimal Cover of F.D.s (1) Just as we applied inference rules to expand on a set F of FDs to arrive at F+, its closure, it is possible to think in the opposite direction to see if we could shrink or reduce the set F to its minimal form so that the minimal set is still equivalent to the original set F. Definition: An attribute in a functional dependency is considered extraneous attribute if we can remove it without changing the closure of the set of dependencies. Formally, given F, the set of functional dependencies and a functional dependency X A in F , attribute Y is extraneous in X if Y is a subset of X, and F logically implies (F- (X A)  { (X – Y) A } ) Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15 - 15 Minimal Sets of FDs (2) A set of FDs is minimal if it satisfies the following conditions: 1. Every dependency in F has a single attribute for its RHS. 2. We cannot remove any dependency from F and have a set of dependencies that is equivalent to F. 3. We cannot replace any dependency X A in F with a dependency Y A, where Y is a proper- subset-of X and still have a set of dependencies that is equivalent to F. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15 - 16 Minimal Sets of FDs (3) Algorithm 15.2. Finding a Minimal Cover F for a Set of Functional Dependencies E Input: A set of functional dependencies E. 1. Se tF:=E. 2. Replace each functional dependency X → {A1, A2,..., An} in F by the n functional dependencies X →A1, X →A2,..., X → An. 3. For each functional dependency X → A in F for each attribute B that is an element of X if { {F – {X → A} } ∪ { (X – {B} ) → A} } is equivalent to F then replace X → A with (X – {B} ) → A in F. (* The above constitutes a removal of the extraneous attribute B from X *) 4. For each remaining functional dependency X → A in F if {F – {X → A} } is equivalent to F, then remove X → A from F. (* The above constitutes a removal of the redundant dependency X A from F *) Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 17 Computing the Minimal Sets of FDs (4) We illustrate algorithm 15.2 with the following: Let the given set of FDs be E : {B → A, D → A, AB → D}.We have to find the minimum cover of E. All above dependencies are in canonical form; so we have completed step 1 of Algorithm 10.2 and can proceed to step 2. In step 2 we need to determine if AB → D has any redundant attribute on the left-hand side; that is, can it be replaced by B → D or A → D? Since B → A, by augmenting with B on both sides (IR2), we have BB → AB, or B → AB (i). However, AB → D as given (ii). Hence by the transitive rule (IR3), we get from (i) and (ii), B → D. Hence AB → D may be replaced by B → D. We now have a set equivalent to original E , say E′ : {B → A, D → A, B → D}. No further reduction is possible in step 2 since all FDs have a single attribute on the left-hand side. In step 3 we look for a redundant FD in E′. By using the transitive rule on B → D and D → A, we derive B → A. Hence B → A is redundant in E’ and can be eliminated. Hence the minimum cover of E is {B → D, D → A}. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 18 Minimal Sets of FDs (5) Every set of FDs has an equivalent minimal set There can be several equivalent minimal sets There is no simple algorithm for computing a minimal set of FDs that is equivalent to a set F of FDs. The process of Algorithm 15.2 is used until no further reduction is possible. To synthesize a set of relations, we assume that we start with a set of dependencies that is a minimal set E.g., see algorithm 15.4 Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 19 DESIGNING A SET OF RELATIONS (1) The Approach of Relational Synthesis (Bottom-up Design): Assumes that all possible functional dependencies are known. First constructs a minimal set of FDs Then applies algorithms that construct a target set of 3NF or BCNF relations. Additional criteria may be needed to ensure the the set of relations in a relational database are satisfactory (see Algorithm 15.3). Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 20 DESIGNING A SET OF RELATIONS (2) Goals: Lossless join property (a must) Algorithm 15.3 tests for general losslessness. Dependency preservation property Observe as much as possible Algorithm 15.5 decomposes a relation into BCNF components by sacrificing the dependency preservation. Additional normal forms 4NF (based on multi-valued dependencies) 5NF (based on join dependencies) Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 21 Algorithm to determine the key of a relation Algorithm 15.2a Finding a Key K for R, given a set F of Functional Dependencies Input: A universal relation R and a set of functional dependencies F on the attributes of R. 1. Set K := R; 2. For each attribute A in K { Compute (K - A)+ with respect to F; If (K - A)+ contains all the attributes in R, then set K := K - {A}; } Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 22 2. Properties of Relational Decompositions (1) Relation Decomposition and Insufficiency of Normal Forms: Universal Relation Schema: A relation schema R = {A1, A2, …, An} that includes all the attributes of the database. Universal relation assumption: Every attribute name is unique. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 23 Properties of Relational Decompositions (2) 2.1 Relation Decomposition and Insufficiency of Normal Forms (cont.): Decomposition: The process of decomposing the universal relation schema R into a set of relation schemas D = {R1,R2, …, Rm} that will become the relational database schema by using the functional dependencies. Attribute preservation condition: Each attribute in R will appear in at least one relation schema Ri in the decomposition so that no attributes are “lost”. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 24 Properties of Relational Decompositions (3) Another goal of decomposition is to have each individual relation Ri in the decomposition D be in BCNF or 3NF. Additional properties of decomposition are needed to prevent from generating spurious tuples Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 25 Properties of Relational Decompositions (4) 2.2 Dependency Preservation Property of a Decomposition: Definition: Given a set of dependencies F on R, the projection of F on Ri, denoted by pRi(F) where Ri is a subset of R, is the set of dependencies X  Y in F+ such that the attributes in X υ Y are all contained in Ri. Hence, the projection of F on each relation schema Ri in the decomposition D is the set of functional dependencies in F+, the closure of F, such that all their left- and right-hand-side attributes are in Ri. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 26 Properties of Relational Decompositions (5) Dependency Preservation Property of a Decomposition (cont.): Dependency Preservation Property: A decomposition D = {R1, R2,..., Rm} of R is dependency-preserving with respect to F if the union of the projections of F on each Ri in D is equivalent to F; that is ((R1(F)) υ... υ (Rm(F)))+ = F+ (See examples in Fig 14.13a and Fig 14.12) Claim 1: It is always possible to find a dependency- preserving decomposition D with respect to F such that each relation Ri in D is in 3nf. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 27 Properties of Relational Decompositions (6) 2.3 Non-additive (Lossless) Join Property of a Decomposition: Definition: Lossless join property: a decomposition D = {R1, R2,..., Rm} of R has the lossless (nonadditive) join property with respect to the set of dependencies F on R if, for every relation state r of R that satisfies F, the following holds, where * is the natural join of all the relations in D: * ( R1(r),..., Rm(r)) = r Note: The word loss in lossless refers to loss of information, not to loss of tuples. In fact, for “loss of information” a better term is “addition of spurious information” Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 28 Properties of Relational Decompositions (7) Lossless (Non-additive) Join Property of a Decomposition : Algorithm 15.3: Testing for Lossless Join Property Input: A universal relation R, a decomposition D = {R1, R2,..., Rm} of R, and a set F of functional dependencies. 1. Create an initial matrix S with one row i for each relation Ri in D, and one column j for each attribute Aj in R. 2. Set S(i,j):=bij for all matrix entries. (* each bij is a distinct symbol associated with indices (i,j) *). 3. For each row i representing relation schema Ri {for each column j representing attribute Aj {if (relation Ri includes attribute Aj) then set S(i,j):= aj;};}; (* each aj is a distinct symbol associated with index (j) *) CONTINUED on NEXT SLIDE Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 29 Properties of Relational Decompositions (8) Lossless (Non-additive) Join Property of a Decomposition (cont.): Algorithm 15.3: Testing for Lossless Join Property (continued) 4. Repeat the following loop until a complete loop execution results in no changes to S {for each functional dependency X Y in F {for all rows in S which have the same symbols in the columns corresponding to attributes in X {make the symbols in each column that correspond to an attribute in Y be the same in all these rows as follows: If any of the rows has an “a” symbol for the column, set the other rows to that same “a” symbol in the column. If no “a” symbol exists for the attribute in any of the rows, choose one of the “b” symbols that appear in one of the rows for the attribute and set the other rows to that same “b” symbol in the column ;}; }; }; 5. If a row is made up entirely of “a” symbols, then the decomposition has the lossless join property; otherwise it does not. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 30 Properties of Relational Decompositions (9) Figure 15.1 Nonadditive join test for n-ary decompositions. (a) Case 1: Decomposition of EMP_PROJ into EMP_PROJ1 and EMP_LOCS fails test. (b) A decomposition of EMP_PROJ that has the lossless join property. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 31 Properties of Relational Decompositions (10) Nonadditive join test for n- ary decompositions. (Figure 15.1) (c) Case 2: Decomposition of EMP_PROJ into EMP, PROJECT, and WORKS_ON satisfies test. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 32 Test for checking non-additivity of Binary Relational Decompositions (11) 2.4 Testing Binary Decompositions for Non- additive Join (Lossless Join) Property Binary Decomposition: Decomposition of a relation R into two relations. PROPERTY NJB (non-additive join test for binary decompositions): A decomposition D = {R1, R2} of R has the lossless join property with respect to a set of functional dependencies F on R if and only if either The f.d. ((R1 ∩ R2)  (R1- R2)) is in F+, or The f.d. ((R1 ∩ R2)  (R2 - R1)) is in F+. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 33 Properties of Relational Decompositions (12) 2.5 Successive Non-additive Join Decomposition: Claim 2 (Preservation of non-additivity in successive decompositions): If a decomposition D = {R1, R2,..., Rm} of R has the lossless (non-additive) join property with respect to a set of functional dependencies F on R, and if a decomposition Di = {Q1, Q2,..., Qk} of Ri has the lossless (non-additive) join property with respect to the projection of F on Ri, then the decomposition D2 = {R1, R2,..., Ri-1, Q1, Q2,..., Qk, Ri+1,..., Rm} of R has the non-additive join property with respect to F. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 34 3. Algorithms for Relational Database Schema Design (1) Design of 3NF Schemas: Algorithm 15.4 Relational Synthesis into 3NF with Dependency Preservation and Non-Additive (Lossless) Join Property Input: A universal relation R and a set of functional dependencies F on the attributes of R. 1. Find a minimal cover G for F (use Algorithm 15.0). 2. For each left-hand-side X of a functional dependency that appears in G, create a relation schema in D with attributes {X υ {A1} υ {A2}... υ {Ak}}, where X  A1, X  A2,..., X –>Ak are the only dependencies in G with X as left-hand-side (X is the key of this relation). 3. If none of the relation schemas in D contains a key of R, then create one more relation schema in D that contains attributes that form a key of R. (Use Algorithm 15.4a to find the key of R) Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 35 Algorithms for Relational Database Schema Design (2) Design of BCNF Schemas Algorithm 15.5: Relational Decomposition into BCNF with Lossless (non-additive) join property Input: A universal relation R and a set of functional dependencies F on the attributes of R. 1. Set D := {R}; 2. While there is a relation schema Q in D that is not in BCNF do { choose a relation schema Q in D that is not in BCNF; find a functional dependency X  Y in Q that violates BCNF; replace Q in D by two relation schemas (Q - Y) and (X υ Y); }; Assumption: No null values are allowed for the join attributes. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 36 4. Problems with Null Values and Dangling Tuples (1) 4.1 Problems with NULL values when some tuples have NULL values for attributes that will be used to join individual relations in the decomposition that may lead to incomplete results. E.g., see Figure 15.2(a), where two relations EMPLOYEE and DEPARTMENT are shown. The last two employee tuples—‘Berger’ and ‘Benitez’—represent newly hired employees who have not yet been assigned to a department (assume that this does not violate any integrity constraints). If we want to retrieve a list of (Ename, Dname) values for all the employees. If we apply the NATURAL JOIN operation on EMPLOYEE and DEPARTMENT (Figure 15.2(b)), the two aforementioned tuples will not appear in the result. In such cases, LEFT OUTER JOIN may be used. The result is shown in Figure 15.2 (c).. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 37 Problems with Null Values and Dangling Tuples (2) Figure 15.2 Issues with NULL-value joins. (a) Some EMPLOYEE tuples have NULL for the join attribute Dnum. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 38 Problems with Null Values and Dangling Tuples (3) Figure 15.2 Issues with NULL- value joins. (b) Result of applying NATURAL JOIN to the EMPLOYEE and DEPARTMENT relations. (c) Result of applying LEFT OUTER JOIN to EMPLOYEE and DEPARTMENT Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 39 Problems with Null Values and Dangling Tuples (4) Problems with Dangling Tuples Consider the decomposition of EMPLOYEE into EMPLOYEE_1 and EMPLOYEE_2 as shown in Figure 15.3 (a) and !5.3 (b). Their NATURAL JOIN yields the original relation EMPLOYEE in Figure 15.2(a). We may use the alternative representation, shown in Figure 15.3(c), where we do not include a tuple in EMPLOYEE_3 if the employee has not been assigned a department (instead of including a tuple with NULL for Dnum as in EMPLOYEE_2). If we use EMPLOYEE_3 instead of EMPLOYEE_2 and apply a NATURAL JOIN on EMPLOYEE_1 and EMPLOYEE_3, the tuples for Berger and Benitez will not appear in the result; these are called dangling tuples in EMPLOYEE. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 40 Problems with Null Values and Dangling Tuples (5) Figure 15.3 The dangling tuple problem. (a) The relation EMPLOYEE_1 (includes all attributes of EMPLOYEE from Figure 15.2(a) except Dnum). (b) The relation EMPLOYEE_2 (includes Dnum attribute with NULL values). (c) The relation EMPLOYEE_3 (includes Dnum attribute but does not include tuples for which Dnum has NULL values). Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 41 About Normalization Algorithms 4.2 Discussion of Normalization Algorithms: Problems: The database designer must first specify all the relevant functional dependencies among the database attributes. These algorithms are not deterministic in general. It is not always possible to find a decomposition into relation schemas that preserves dependencies and allows each relation schema in the decomposition to be in BCNF (instead of 3NF as in Algorithm 15.5). Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 42 Summary of Algorithms for Relational Database Schema Design (1) Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 43 Summary of Algorithms for Relational Database Schema Design (2) Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 44 5. Multivalued Dependencies and Fourth Normal Form – Further Discussion (1) Definition: A multivalued dependency (MVD) X —>> Y specified on relation schema R, where X and Y are both subsets of R, specifies the following constraint on any relation state r of R: If two tuples t1 and t2 exist in r such that t1[X] = t2[X], then two tuples t3 and t4 should also exist in r with the following properties, where we use Z to denote (R 2 (X υ Y)): t3[X] = t4[X] = t1[X] = t2[X]. t3[Y] = t1[Y] and t4[Y] = t2[Y]. t3[Z] = t2[Z] and t4[Z] = t1[Z]. An MVD X —>> Y in R is called a trivial MVD if (a) Y is a subset of X, or (b) X υ Y = R. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 45 Multivalued Dependencies and Fourth Normal Form (2) Inference Rules for Functional and Multivalued Dependencies: IR1 (reflexive rule for FDs): If X  Y, then X → Y. IR2 (augmentation rule for FDs): {X → Y}  XZ → YZ. IR3 (transitive rule for FDs): {X → Y, Y → Z}  X → Z. IR4 (complementation rule for MVDs): {X —>> Y}  X —>> (R – (X  Y))}. IR5 (augmentation rule for MVDs): If X —>> Y and W  Z then WX —>> YZ. IR6 (transitive rule for MVDs): {X —>> Y, Y —>> Z}  X —>> (Z - Y). IR7 (replication rule for FD to MVD): {X –> Y}  X —>> Y. IR8 (coalescence rule for FDs and MVDs): If X —>> Y and there exists W with the properties that (a) W  Y is empty, (b) W –> Z, and (c) Y  Z, then X –> Z. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 46 Multivalued Dependencies and Fourth Normal Form (3) Definition: A relation schema R is in 4NF with respect to a set of dependencies F (that includes functional dependencies and multivalued dependencies) if, for every nontrivial multivalued dependency X —>> Y in F+, X is a superkey for R. Note: F+ is the (complete) set of all dependencies (functional or multivalued) that will hold in every relation state r of R that satisfies F. It is also called the closure of F. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 47 Multivalued Dependencies and Fourth Normal Form (4) Fig. 15.4 Decomposing a relation state of EMP that is not in 4NF. (a) EMP relation with additional tuples. (b) Two corresponding 4NF relations EMP_PROJECTS and EMP_DEPENDENTS. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 48 Multivalued Dependencies and Fourth Normal Form (5) 5.3 Non-additive( Lossless) Join Decomposition into 4NF Relations: PROPERTY NJB’ The relation schemas R1 and R2 form a lossless (non-additive) join decomposition of R with respect to a set F of functional and multivalued dependencies if and only if (R1 ∩ R2) —>> (R1 - R2) or by symmetry, if and only if (R1 ∩ R2) —>> (R2 - R1)). Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 49 Multivalued Dependencies and Fourth Normal Form (6) Algorithm 15.7: Relational decomposition into 4NF relations with non-additive join property Input: A universal relation R and a set of functional and multivalued dependencies F. 1. Set D := { R }; 2. While there is a relation schema Q in D that is not in 4NF do { choose a relation schema Q in D that is not in 4NF; find a nontrivial MVD X —>> Y in Q that violates 4NF; replace Q in D by two relation schemas (Q - Y) and (X  Y); }; Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 50 6. Other Dependencies and Normal Forms Join Dependency was defined in Chapter 14: Definition: A join dependency (JD), denoted by JD(R1, R2,..., Rn), specified on relation schema R, specifies a constraint on the states r of R. The constraint states that every legal state r of R should have a non-additive join decomposition into R1, R2,..., Rn; that is, for every such r we have * (R1(r), R2(r),..., Rn(r)) = r Note: an MVD is a special case of a JD where n = 2. A join dependency JD(R1, R2,..., Rn), specified on relation schema R, is a trivial JD if one of the relation schemas Ri in JD(R1, R2,..., Rn) is equal to R. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 51 Join Dependencies and Fifth Normal Form Definition of 5NF: A relation schema R is in fifth normal form (5NF) (or Project-Join Normal Form (PJNF)) with respect to a set F of functional, multivalued, and join dependencies if, for every nontrivial join dependency JD(R1, R2,..., Rn) in F+ (that is, implied by F), every Ri is a superkey of R. Discovering join dependencies in practical databases with hundreds of relations is next to impossible. Therefore, 5NF is rarely used in practice. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 52 Inclusion Dependencies (1) Definition: An inclusion dependency R.X < S.Y between two sets of attributes—X of relation schema R, and Y of relation schema S—specifies the constraint that, at any specific time when r is a relation state of R and s a relation state of S, we must have X(r(R))  Y(s(S)) Note: The  (subset) relationship does not necessarily have to be a proper subset. The sets of attributes on which the inclusion dependency is specified—X of R and Y of S—must have the same number of attributes. In addition, the domains for each pair of corresponding attributes should be compatible. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 53 Inclusion Dependencies (2) Objective of Inclusion Dependencies: To formalize two types of interrelational constraints which cannot be expressed using F.D.s or MVDs: Referential integrity constraints Class/subclass relationships Inclusion dependency inference rules IDIR1 (reflexivity): R.X < R.X. IDIR2 (attribute correspondence): If R.X < S.Y where X = {A1, A2 ,..., An} and Y = {B1, B2,..., Bn} and Ai Corresponds-to Bi, then R.Ai < S.Bi for 1 ≤ i ≤ n. IDIR3 (transitivity): If R.X < S.Y and S.Y < T.Z, then R.X < T.Z. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 54  Functional Dependencies based on Arithmetic functions and procedures (1) Arithmetic Functions: As long as a unique value of Y is associated with every X, we can still consider that the FD X Y exists. For example,consider the relation: ORDER_LINE (Order#, Item#, Quantity, Unit_price, Extended_price, Discounted_price) each tuple represents an item from an order with a particular quantity, and the price per unit for that item. In this relation, (Quantity, Unit_price ) Extended_price by the formula Extended_price = Quantity * Unit_price. Hence, there is a unique value for Extended_price for every pair (Quantity, Unit_price ), and thus it conforms to the definition of functional dependency. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 55 Functional Dependencies based on Arithmetic functions and procedures (2) Procedures: There may be a procedure that takes into account the quantity discounts, the type of item, and so on and computes a discounted price for the total quantity ordered for that item. Therefore, we can say (Item#, Quantity, Unit_price ) Discounted_price, or (Item#, Quantity, Extended_price) Discounted_price. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 56 Other Dependencies and Normal Forms (3) 6.4 Domain-Key Normal Form (DKNF): Definition: A relation schema is said to be in DKNF if all constraints and dependencies that should hold on the valid relation states can be enforced simply by enforcing the domain constraints and key constraints on the relation. The idea is to specify (theoretically, at least) the “ultimate normal form” that takes into account all possible types of dependencies and constraints.. For a relation in DKNF, it becomes very straightforward to enforce all database constraints by simply checking that each attribute value in a tuple is of the appropriate domain and that every key constraint is enforced. The practical utility of DKNF is limited Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 57 Recap Functional Dependencies Revisited Designing a Set of Relations by Synthesis Properties of Relational Decompositions Algorithms for Relational Database Schema Design in 3Nf and BCNF Multivalued Dependencies and Fourth Normal Form Other Dependencies and Normal Forms Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 15- 58 Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe  CHAPTER 16 Disk Storage, Basic File Structures, Hashing, and Modern Storage Architectures Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe 16.1 Introduction Databases typically stored on magnetic disks Accessed using physical database file structures Storage hierarchy Primary storage CPU main memory, cache memory Secondary storage Magnetic disks, flash memory, solid-state drives Tertiary storage Removable media Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 16- 3 Memory Hierarchies and Storage Devices Cache memory Static RAM DRAM Mass storage Magnetic disks CD-ROM, DVD, tape drives Flash memory Nonvolatile Tertiary Storage Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 16- 4 Storage Types and Characteristics Table 16.1 Types of Storage with Capacity, Access Time, Max Bandwidth (Transfer Speed), and Commodity Cost Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 16-5 Storage Organization of Databases Persistent data Most databases Transient data Exists only during program execution File organization Determines how records are physically placed on the disk Determines how records are accessed Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 16- 6 16.2 Secondary Storage Devices Hard disk drive Bits (ones and zeros) Grouped into bytes or characters Disk capacity measures storage size Disks may be single or double-sided Concentric circles called tracks Tracks divided into blocks or sectors Disk packs Cylinder Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 16- 7 Single-Sided Disk and Disk Pack Figure 16.1 (a) A single-sided disk with read/write hardware (b) A disk pack with read/write hardware Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 16-8 Sectors on a Disk Figure 16.2 Different sector organizations on disk (a) Sectors subtending a fixed angle (b) Sectors maintaining a uniform recording density Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 16-9 Secondary Storage Devices (cont’d.) Formatting Divides tracks into equal-sized disk blocks Blocks separated by interblock gaps Data transfer in units of disk blocks Hardware address supplied to disk I/O hardware Buffer Used in read and write operations Read/write head Hardware mechanism for read and write operations Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 16- 10 Secondary Storage Devices (cont’d.) Disk controller Interfaces disk drive to computer system Standard interfaces SCSI SATA SAS Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 16- 11 Secondary Storage Devices (cont’d.) Techniques for efficient data access Data buffering Proper organization of data on disk Reading data ahead of request Proper scheduling of I/O requests Use of log disks to temporarily hold writes Use of SSDs or flash memory for recovery purposes Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 16- 12 Solid State Device Storage Sometimes called flash storage Main component: controller Set of interconnected flash memory cards No moving parts Data less likely to be fragmented More costly than HDDs DRAM-based SSDs available Faster access times compared with flash Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 16- 13 Magnetic Tape Storage Devices Sequential access Must scan preceding blocks Tape is mounted and scanned until required block is under read/write head Important functions Backup Archive Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 16- 14 16.3 Buffering of Blocks Buffering most useful when processes can run concurrently in parallel Figure 16.3 Interleaved concurrency versus parallel execution Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 16- 15 Buffering of Blocks (cont’d.) Double buffering can be used to read continuous stream of blocks Figure 16.4 Use of two buffers, A and B, for reading from disk Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 16- 16 Buffer Management and Replacement Strategies Buffer management information Pin count Dirty bit Buffer replacement strategies Least recently used (LRU) Clock policy First-in-first-out (FIFO) Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 16- 17 16.4 Placing File Records on Disk Record: collection of related data values or items Values correspond to record field Data types Numeric String Boolean Date/time Binary large objects (BLOBs) Unstructured objects Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 16- 18 Placing File Records on Disk (cont’d.) Reasons for variable-length records One or more fields have variable length One or more fields are repeating One or more fields are optional File contains records of different types Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 16- 19 Record Blocking and Spanned Versus Unspanned Records File records allocated to disk blocks Spanned records Larger than a single block Pointer at end of first block points to block containing remainder of record Unspanned Records not allowed to cross block boundaries Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 16- 20 Record Blocking and Spanned Versus Unspanned Records (cont’d.) Blocking factor Average number of records per block for the file Figure 16.6 Types of record organization (a) Unspanned (b) Spanned Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 16-21 Record Blocking and Spanned Versus Unspanned Records (cont’d.) Allocating file blocks on disk Contiguous allocation Linked allocation Indexed allocation File header (file descriptor) Contains file information needed by system programs Disk addresses Format descriptions Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 16- 22 16.5 Operations on Files Retrieval operations No change to file data Update operations File change by insertion, deletion, or modification Records selected based on selection condition Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 16- 23 Operations on Files (cont’d.) Examples of operations for accessing file records Open Find Read FindNext Delete Insert Close Scan Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 16- 24 16.6 Files of Unordered Records (Heap Files) Heap (or pile) file Records placed in file in order of insertion Inserting a new record is very efficient Searching for a record requires linear search Deletion techniques Rewrite the block Use deletion marker Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 16- 25 16.7 Files of Ordered Records (Sorted Files) Ordered (sequential) file Records sorted by ordering field Called ordering key if ordering field is a key field Advantages Reading records in order of ordering key value is extremely efficient Finding next record Binary search technique Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 16- 26 Access Times for Various File Organizations Table 16.3 Average access times for a file of b blocks under basic file organizations Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 16-27 16.8 Hashing Techniques Hash function (randomizing function) Applied to hash field value of a record Yields address of the disk block of stored record Organization called hash file Search condition is equality condition on the hash field Hash field typically key field Hashing also internal search structure Used when group of records accessed exclusively by one field value Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 16- 28 Hashing Techniques (cont’d.) Internal hashing Hash table Collision Hash field value for inserted record hashes to address already containing a different record Collision resolution Open addressing Chaining Multiple hashing Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 16- 29 Hashing Techniques (cont’d.) External hashing for disk files Target address space made of buckets Bucket: one disk block or contiguous blocks Hashing function maps a key into relative bucket Table in file header converts bucket number to disk block address Collision problem less severe with buckets Static hashing Fixed number of buckets allocated Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 16- 30 Hashing Techniques (cont’d.) Hashing techniques that allow dynamic file expansion Extendible hashing File performance does not degrade as file grows Dynamic hashing Maintains tree-structured directory Linear hashing Allows hash file to expand and shrink buckets without needing a directory Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 16- 31 16.9 Other Primary File Organizations Files of mixed records Relationships implemented by logical field references Physical clustering B-tree data structure Column-based data storage Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 16- 32 Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe  CHAPTER 17 Indexing Structures for Files and Physical Database Design Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Introduction Indexes used to speed up record retrieval in response to certain search conditions Index structures provide secondary access paths Any field can be used to create an index Multiple indexes can be constructed Most indexes based on ordered files Tree data structures organize the index Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 17- 3 17.1 Types of Single-Level Ordered Indexes Ordered index similar to index in a textbook Indexing field (attribute) Index stores each value of the index field with list of pointers to all disk blocks that contain records with that field value Values in index are ordered Primary index Specified on the ordering key field of ordered file of records Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 17- 4 Types of Single-Level Ordered Indexes (cont’d.) Clustering index Used if numerous records can have the same value for the ordering field Secondary index Can be specified on any nonordering field Data file can have several secondary indexes Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 17- 5 Primary Indexes Ordered file with two fields Primary key, K(i) Pointer to a disk block, P(i) One index entry in the index file for each block in the data file Indexes may be dense or sparse Dense index has an index entry for every search key value in the data file Sparse index has entries for only some search values Refer Example 1 in book Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 17- 6 Primary Indexes (cont’d.) Figure 17.1 Primary index on the ordering key field of the file shown in Figure 16.7 Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 17-7 Primary Indexes (cont’d.) Major problem: insertion and deletion of records Move records around and change index values Solutions Use unordered overflow file Use linked list of overflow records Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 17- 8 Clustering Indexes Clustering field File records are physically ordered on a nonkey field without a distinct value for each record Ordered file with two fields Same type as clustering field Disk block pointer Refer Example 2 in book Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 17- 9 Clustering Indexes (cont’d.) Figure 17.2 A clustering index on the Dept_number ordering nonkey field of an EMPLOYEE file Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 17-10 Secondary Indexes Provide secondary means of accessing a data file Some primary access exists Ordered file with two fields Indexing field, K(i) Block pointer or record pointer, P(i) Usually need more storage space and longer search time than primary index Improved search time for arbitrary record Refer Example 3 in book Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 17- 11 Secondary Indexes (cont’d.) Figure 17.4 Dense secondary index (with block pointers) on a nonordering key field of a file. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 17-12 Types of Single-Level Ordered Indexes (cont’d.) Table 17.1 Types of indexes based on the properties of the indexing field Table 17.2 Properties of index types Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 17-13 17.2 Multilevel Indexes Designed to greatly reduce remaining search space as search is conducted Index file Considered first (or base level) of a multilevel index Second level Primary index to the first level Third level Primary index to the second level Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 17- 14 Figure 17.6 A two-level primary index resembling ISAM (indexed sequential access method) organization Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 17-15 17.3 Dynamic Multilevel Indexes Using B-Trees and B+ -Trees Tree data structure terminology Tree is formed of nodes Each node (except root) has one parent and zero or more child nodes Leaf node has no child nodes Unbalanced if leaf nodes occur at different levels Nonleaf node called internal node Subtree of node consists of node and all descendant nodes Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 17- 16 Tree Data Structure Figure 17.7 A tree data structure that shows an unbalanced tree Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 17-17 Search Trees and B-Trees Search tree used to guide search for a record Given value of one of record’s fields Figure 17.8 A node in a search tree with pointers to subtrees below it Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 17- 18 Search Trees and B-Trees (cont’d.) Algorithms necessary for inserting and deleting search values into and from the tree Figure 17.9 A search tree of order p = 3 Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 17- 19 B-Trees Provide multi-level access structure Tree is always balanced Space wasted by deletion never becomes excessive Each node is at least half-full Each node in a B-tree of order p can have at most p-1 search values Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 17- 20 B-Tree Structures Figure 17.10 B-tree structures (a) A node in a B-tree with q−1 search values (b) A B-tree of order p=3. The values were inserted in the order 8, 5, 1, 7, 3, 12, 9, 6 Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 17-21 B+ -Trees Data pointers stored only at the leaf nodes Leaf nodes have an entry for every value of the search field, and a data pointer to the record if search field is a key field For a nonkey search field, the pointer points to a block containing pointers to the data file records Internal nodes Some search field values from the leaf nodes repeated to guide search Refer Example 6 and 7 in book Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 17- 22 B+ -Trees Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 16- 23 B+ -Trees (cont’d.) Figure 17.11 The nodes of a B+-tree (a) Internal node of a B+-tree with q−1 search values (b) Leaf node of a B+-tree with q−1 search values and q−1 data pointers Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 17-24 Searching for a Record With Search Key Field Value K, Using a B+ -Tree Algorithm 17.2 Searching for a record with search key field value K, using a B+ -Tree Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 17- 25 17.4 Indexes on Multiple Keys Multiple attributes involved in many retrieval and update requests Composite keys Access structure using key value that combines attributes Partitioned hashing Suitable for equality comparisons Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 17- 26 Indexes on Multiple Keys (cont’d.) Grid files Array with one dimension for each search attribute Figure 17.14 Example of a grid array on Dno and Age attributes Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 17- 27 17.5 Other Types of Indexes Hash indexes Secondary structure for file access Uses hashing on a search key other than the one used for the primary data file organization Index entries of form (K, Pr) or (K, P) Pr: pointer to the record containing the key P: pointer to the block containing the record for that key Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 17- 28 Hash Indexes (cont’d.) Figure 17.15 Hash-based indexing Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 17-29 Bitmap Indexes Used with a large number of rows Creates an index for one or more columns Each value or value range in the column is indexed Built on one particular value of a particular field Array of bits Existence bitmap Bitmaps for B+ -tree leaf nodes Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 17- 30 Function-Based Indexing Value resulting from applying some function on a field (or fields) becomes the index key Introduced in Oracle relational DBMS Example Function UPPER(Lname) returns uppercase representation Query Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 17- 31 Index Creation General form of the command to create an index Unique and cluster keywords optional Order can be ASC or DESC Secondary indexes can be created for any primary record organization Complements other primary access methods Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 17- 32 Indexing of Strings Strings can be variable length Strings may be too long, limiting the fan-out Prefix compression Stores only the prefix of the search key adequate to distinguish the keys that are being separated and directed to the subtree Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 17- 33 17.8 Summary Indexes are access structures that improve efficiency of record retrieval from a data file Ordered single-level index types Primary, clustering, and secondary Multilevel indexes can be implemented as B-trees and B+ -trees Dynamic structures Multiple key access methods Logical and physical indexes Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 17- 34 Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe  CHAPTER 8 The Relational Algebra and The Relational Calculus (plus QBE- Appendix C) Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 8- 2 Chapter Outline Relational Algebra Unary Relational Operations Relational Algebra Operations From Set Theory Binary Relational Operations Additional Relational Operations Examples of Queries in Relational Algebra Relational Calculus Tuple Relational Calculus Domain Relational Calculus Example Database Application (COMPANY) Overview of the QBE language (appendix D) Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 8- 3 Relational Algebra Overview Relational algebra is the basic set of operations for the relational model These operations enable a user to specify basic retrieval requests (or queries) The result of an operation is a new relation, which may have been formed from one or more input relations This property makes the algebra “closed” (all objects in relational algebra are relations) Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 8- 4 Relational Algebra Overview (continued) The algebra operations thus produce new relations These can be further manipulated using operations of the same algebra A sequence of relational algebra operations forms a relational algebra expression The result of a relational algebra expression is also a relation that represents the result of a database query (or retrieval request) Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 8- 5 Relational Algebra Overview Relational Algebra consists of several groups of operations Unary Relational Operations SELECT (symbol:  (sigma)) PROJECT (symbol:  (pi)) RENAME (symbol:  (rho)) Relational Algebra Operations From Set Theory UNION (  ), INTERSECTION (  ), DIFFERENCE (or MINUS, – ) CARTESIAN PRODUCT ( x ) Binary Relational Operations JOIN (several variations of JOIN exist) DIVISION Additional Relational Operations OUTER JOINS, OUTER UNION AGGREGATE FUNCTIONS (These compute summary of information: for example, SUM, COUNT, AVG, MIN, MAX) Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 8- 7 Database State for COMPANY All examples discussed below refer to the COMPANY database shown here. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 8- 8 Unary Relational Operations: SELECT The SELECT operation (denoted by  (sigma)) is used to select a subset of the tuples from a relation based on a selection condition. The selection condition acts as a filter Keeps only those tuples that satisfy the qualifying condition Tuples satisfying the condition are selected whereas the other tuples are discarded (filtered out) Examples: Select the EMPLOYEE tuples whose department number is 4:  DNO = 4 (EMPLOYEE) Select the employee tuples whose salary is greater than $30,000:  SALARY > 30,000 (EMPLOYEE) Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 8- 9 Unary Relational Operations: SELECT In general, the select operation is denoted by  (R) where the symbol  (sigma) is used to denote the select operator the selection condition is a Boolean (conditional) expression specified on the attributes of relation R tuples that make the condition true are selected appear in the result of the operation tuples that make the condition false are filtered out discarded from the result of the operation Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 8- 10 Unary Relational Operations: SELECT (continued) SELECT Operation Properties The SELECT operation  (R) produces a relation S that has the same schema (same attributes) as R SELECT  is commutative:  ( < condition2> (R)) =  ( < condition1> (R)) Because of commutativity property, a cascade (sequence) of SELECT operations may be applied in any order: ( ( (R)) =  ( ( ( R))) A cascade of SELECT operations may be replaced by a single selection with a conjunction of all the conditions: (< cond2> ((R)) =  AND < cond2> AND < cond3>(R))) The number of tuples in the result of a SELECT is less than (or equal to) the number of tuples in the input relation R Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 8- 11 The following query results refer to this database state Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 8- 12 Unary Relational Operations: PROJECT PROJECT Operation is denoted by  (pi) This operation keeps certain columns (attributes) from a relation and discards the other columns. PROJECT creates a vertical partitioning The list of specified columns (attributes) is kept in each tuple The other attributes in each tuple are discarded Example: To list each employee’s first and last name and salary, the following is used: LNAME, FNAME,SALARY(EMPLOYEE) Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 8- 13 Unary Relational Operations: PROJECT (cont.) The general form of the project operation is: (R)  (pi) is the symbol used to represent the project operation is the desired list of attributes from relation R. The project operation removes any duplicate tuples This is because the result of the project operation must be a set of tuples Mathematical sets do not allow duplicate elements. Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 8- 14 Unary Relational Operations: PROJECT (contd.) PROJECT Operation Properties The number of tuples in the result of projection (R) is always less or equal to the number of tuples in R If the list of attributes includes a key of R, then the number of tuples in the result of PROJECT is equal to the number of tuples in R PROJECT is not commutative  ( (R) ) =  (R) as long as contains the attributes in Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 8- 15 Examples of applying SELECT and PROJECT operations Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 8- 16 Relational Algebra Expressions We may want to apply several relational algebra operations one after the other Either we can write the operations as a single relational al