Definitions

Database normalization

Database normalization

Database normalization, sometimes referred to as canonical synthesis, is a technique for designing relational database tables to minimize duplication of information and, in so doing, to safeguard the database against certain types of logical or structural problems, namely data anomalies. For example, when multiple instances of a given piece of information occur in a table, the possibility exists that these instances will not be kept consistent when the data within the table is updated, leading to a loss of data integrity. A table that is sufficiently normalized is less vulnerable to problems of this kind, because its structure reflects the basic assumptions for when multiple instances of the same information should be represented by a single instance only.

Higher degrees of normalization typically involve more tables and create the need for a larger number of joins, which can reduce performance. Accordingly, more highly normalized tables are typically used in database applications involving many isolated transactions (e.g. an automated teller machine), while less normalized tables tend to be used in database applications that need to map complex relationships between data entities and data attributes (e.g. a reporting application, or a full-text search application).

Database theory describes a table's degree of normalization in terms of normal forms of successively higher degrees of strictness. A table in third normal form (3NF), for example, is consequently in second normal form (2NF) as well; but the reverse is not necessarily the case.

Although the normal forms are often defined informally in terms of the characteristics of tables, rigorous definitions of the normal forms are concerned with the characteristics of mathematical constructs known as relations. Whenever information is represented relationally, it is meaningful to consider the extent to which the representation is normalized.

Problems addressed by normalization

A table that is not sufficiently normalized can suffer from logical inconsistencies of various types, and from anomalies involving data operations. In such a table:

  • The same information can be expressed on multiple records; therefore updates to the table may result in logical inconsistencies. For example, each record in an "Employees' Skills" table might contain an Employee ID, Employee Address, and Skill; thus a change of address for a particular employee will potentially need to be applied to multiple records (one for each of his skills). If the update is not carried through successfully—if, that is, the employee's address is updated on some records but not others—then the table is left in an inconsistent state. Specifically, the table provides conflicting answers to the question of what this particular employee's address is. This phenomenon is known as an update anomaly.
  • There are circumstances in which certain facts cannot be recorded at all. For example, each record in a "Faculty and Their Courses" table might contain a Faculty ID, Faculty Name, Faculty Hire Date, and Course Code—thus we can record the details of any faculty member who teaches at least one course, but we cannot record the details of a newly-hired faculty member who has not yet been assigned to teach any courses. This phenomenon is known as an insertion anomaly.
  • There are circumstances in which the deletion of data representing certain facts necessitates the deletion of data representing completely different facts. The "Faculty and Their Courses" table described in the previous example suffers from this type of anomaly, for if a faculty member temporarily ceases to be assigned to any courses, we must delete the last of the records on which that faculty member appears. This phenomenon is known as a deletion anomaly.

Ideally, a relational database table should be designed in such a way as to exclude the possibility of update, insertion, and deletion anomalies. The normal forms of relational database theory provide guidelines for deciding whether a particular design will be vulnerable to such anomalies. It is possible to correct an unnormalized design so as to make it adhere to the demands of the normal forms: this is called normalization. Removal of redundancies of the tables will lead to several tables, with referential integrity restrictions between them.

Normalization typically involves decomposing an unnormalized table into two or more tables that, were they to be combined (joined), would convey exactly the same information as the original table.

Background to normalization: definitions

  • Functional dependency: Attribute B has a functional dependency on attribute A i.e. A → B if, for each value of attribute A, there is exactly one value of attribute B. If value of A is repeating in tuples then value of B will also repeat. In our example, Employee Address has a functional dependency on Employee ID, because a particular Employee ID value corresponds to one and only one Employee Address value. (Note that the reverse need not be true: several employees could live at the same address and therefore one Employee Address value could correspond to more than one Employee ID. Employee ID is therefore not functionally dependent on Employee Address.) An attribute may be functionally dependent either on a single attribute or on a combination of attributes. It is not possible to determine the extent to which a design is normalized without understanding what functional dependencies apply to the attributes within its tables; understanding this, in turn, requires knowledge of the problem domain. For example, an Employer may require certain employees to split their time between two locations, such as New York City and London, and therefore want to allow Employees to have more than one Employee Address. In this case, Employee Address would no longer be functionally dependent on Employee ID.

Another way to look at the above is by reviewing basic mathematical functions:

Let F(x) be a mathematical function of one independent variable. The independent variable is analogous to the attribute A. The dependent variable (or the dependent attribute using the lingo above), and hence the term functional dependency, is the value of F(A); A is an independent attribute. As we know, mathematical functions can have only one output. Notationally speaking, it is common to express this relationship in mathematics as F(A) = B; or, B → F(A).

There are also functions of more than one independent variable--commonly, this is referred to as multivariable functions. This idea represents an attribute being functionally dependent on a combination of attributes. Hence, F(x,y,z) contains three independent variables, or independent attributes, and one dependent attribute, namely, F(x,y,z). In multivariable functions, there can only be one output, or one dependent variable, or attribute.

  • Trivial functional dependency: A trivial functional dependency is a functional dependency of an attribute on a superset of itself. {Employee ID, Employee Address} → {Employee Address} is trivial, as is {Employee Address} → {Employee Address}.
  • Full functional dependency: An attribute is fully functionally dependent on a set of attributes X if it is
    • functionally dependent on X, and
    • not functionally dependent on any proper subset of X. {Employee Address} has a functional dependency on {Employee ID, Skill}, but not a full functional dependency, because is also dependent on {Employee ID}.
  • Transitive dependency: A transitive dependency is an indirect functional dependency, one in which XZ only by virtue of XY and YZ.
  • Multivalued dependency: A multivalued dependency is a constraint according to which the presence of certain rows in a table implies the presence of certain other rows: see the Multivalued Dependency article for a rigorous definition.
  • Join dependency: A table T is subject to a join dependency if T can always be recreated by joining multiple tables each having a subset of the attributes of T.
  • Superkey: A superkey is an attribute or set of attributes that uniquely identifies rows within a table; in other words, two distinct rows are always guaranteed to have distinct superkeys. {Employee ID, Employee Address, Skill} would be a superkey for the "Employees' Skills" table; {Employee ID, Skill} would also be a superkey.
  • Candidate key: A candidate key is a minimal superkey, that is, a superkey for which we can say that no proper subset of it is also a superkey. {Employee Id, Skill} would be a candidate key for the "Employees' Skills" table.
  • Non-prime attribute: A non-prime attribute is an attribute that does not occur in any candidate key. Employee Address would be a non-prime attribute in the "Employees' Skills" table.
  • Primary key: Most DBMSs require a table to be defined as having a single unique key, rather than a number of possible unique keys. A primary key is a key which the database designer has designated for this purpose.

History

Edgar F. Codd first proposed the process of normalization and what came to be known as the 1st normal form:

In his paper, Edgar F. Codd used the term "non-simple" domains to describe a heterogeneous data structure, but later researchers would refer to such a structure as an abstract data type.

Normal forms

The normal forms (abbrev. NF) of relational database theory provide criteria for determining a table's degree of vulnerability to logical inconsistencies and anomalies. The higher the normal form applicable to a table, the less vulnerable it is to inconsistencies and anomalies. Each table has a "highest normal form" (HNF): by definition, a table always meets the requirements of its HNF and of all normal forms lower than its HNF; also by definition, a table fails to meet the requirements of any normal form higher than its HNF.

The normal forms are applicable to individual tables; to say that an entire database is in normal form n is to say that all of its tables are in normal form n.

Newcomers to database design sometimes suppose that normalization proceeds in an iterative fashion, i.e. a 1NF design is first normalized to 2NF, then to 3NF, and so on. This is not an accurate description of how normalization typically works. A sensibly designed table is likely to be in 3NF on the first attempt; furthermore, if it is 3NF, it is overwhelmingly likely to have an HNF of 5NF. Achieving the "higher" normal forms (above 3NF) does not usually require an extra expenditure of effort on the part of the designer, because 3NF tables usually need no modification to meet the requirements of these higher normal forms.

Edgar F. Codd originally defined the first three normal forms (1NF, 2NF, and 3NF). These normal forms have been summarized as requiring that all non-key attributes be dependent on "the key, the whole key and nothing but the key". The fourth and fifth normal forms (4NF and 5NF) deal specifically with the representation of many-to-many and one-to-many relationships among attributes. Sixth normal form (6NF) incorporates considerations relevant to temporal databases.

First normal form

A table is in first normal form (1NF) if and only if it represents a relation. Given that database tables embody a relation-like form, the defining characteristic of one in first normal form is that it does not allow duplicate rows or nulls. Simply put, a table with a unique key (which, by definition, prevents duplicate rows) and without any nullable columns is in 1NF.

Note that the restriction on nullable columns as a 1NF requirement, as espoused by Chris Date, et. al., is controversial. This particular requirement for 1NF is a direct contradiction to Dr. Codd's vision of the relational database, in which he stated that "null values" must be supported in a fully relational DBMS in order to represent "missing information and inapplicable information in a systematic way, independent of data type." By redefining 1NF to exclude nullable columns in 1NF, no level of normalization can ever be achieved unless all nullable columns are completely eliminated from the entire database. This is in line with Date's and Darwen's vision of the perfect relational database, but can introduce additional complexities in SQL databases to the point of impracticality.

One requirement of a relation is that every table contains exactly one value for each attribute. This is sometimes expressed as "no repeating groups". While that statement itself is axiomatic, experts disagree about what qualifies as a "repeating group", in particular whether a value may be a relation value; thus the precise definition of 1NF is the subject of some controversy. Notwithstanding, this theoretical uncertainty applies to relations, not tables. Table manifestations are intrinsically free of variable repeating groups because they are structurally constrained to the same number of columns in all rows.

Put at its simplest; when applying 1NF to a database, every record must be the same length. This means that each record has the same number of fields, and none of them contain a null value.

Second normal form



The criteria for second normal form (2NF) are:

  • The table must be in 1NF.
  • None of the non-prime attributes of the table are functionally dependent on a part (proper subset) of a candidate key; in other words, all functional dependencies of non-prime attributes on candidate keys are full functional dependencies. For example, consider an "Employees' Skills" table whose attributes are Employee ID, Employee Name, and Skill; and suppose that the combination of Employee ID and Skill uniquely identifies records within the table. Given that Employee Name depends on only one of those attributes – namely, Employee ID – the table is not in 2NF.
  • In simple terms, a table is 2NF if it is in 1NF and all fields are dependent on the whole of the primary key, or a relation is in 2NF if it is in 1NF and every non-key attribute is fully dependent on each candidate key of the relation.
  • Note that if none of a 1NF table's candidate keys are composite – i.e. every candidate key consists of just one attribute – then we can say immediately that the table is in 2NF.
  • All columns must be a fact about the entire key, and not a subset of the key.

Third normal form

The criteria for third normal form (3NF) are:

  • The table must be in 2NF.
  • Transitive dependencies must be eliminated. All attributes must rely only on the primary key. So, if a database has a table with columns Student ID, Student, Company, and Company Phone Number, it is not in 3NF. This is because the Phone number relies on the Company. So, for it to be in 3NF, there must be a second table with Company and Company Phone Number columns; the Phone Number column in the first table would be removed.

Boyce-Codd normal form

A table is in Boyce-Codd normal form (BCNF) if and only if, for every one of its non-trivial functional dependencies X → Y, X is a superkey—that is, X is either a candidate key or a superset thereof.

Fourth normal form

A table is in fourth normal form (4NF) if and only if, for every one of its non-trivial multivalued dependencies X Y, X is a superkey—that is, X is either a candidate key or a superset thereof.

  • For example, if you can have two phone numbers values and two email address values, then you should not have them in the same table.

Fifth normal form

The criteria for fifth normal form (5NF and also PJ/NF) are:

  • The table must be in 4NF.
  • There must be no non-trivial join dependencies that do not follow from the key constraints. A 4NF table is said to be in the 5NF if and only if every join dependency in to is implied by the candidate keys.

Domain/key normal form

Domain/key normal form (or DKNF) requires that a table not be subject to any constraints other than domain constraints and key constraints.

Sixth normal form

According to the definition by Christopher J. Date and others, who extended database theory to take account of temporal and other interval data, a table is in sixth normal form (6NF) if and only if it satisfies no non-trivial (in the formal sense) join dependencies at all,, meaning that the fifth normal form is also satisfied. When referring to "join" in this context it should be noted that Date et al. additionally use generalized definitions of relational operators that also take account of interval data (e.g. from-date to-date) by conceptually breaking them down ("unpacking" them) into atomic units (e.g. individual days), with defined rules for joining interval data, for instance.

Sixth normal form is intended to decompose relation variables to irreducible components. Though this may be relatively unimportant for non-temporal relation variables, it can be important when dealing with temporal variables or other interval data. For instance, if a relation comprises a supplier's name, status, and city, we may also want to add temporal data, such as the time during which these values are, or were, valid (e.g. for historical data) but the three values may vary independently of each other and at different rates. We may, for instance, wish to trace the history of changes to Status.

For further discussion on Temporal Aggregation in SQL, see also Zimyani, For a non-relational approach, see TSQL2.

In a different meaning, sixth normal form may also be used by some to refer to Domain/key normal form (DKNF).

Denormalization

Databases intended for Online Transaction Processing (OLTP) are typically more normalized than databases intended for Online Analytical Processing (OLAP). OLTP Applications are characterized by a high volume of small transactions such as updating a sales record at a super market checkout counter. The expectation is that each transaction will leave the database in a consistent state. By contrast, databases intended for OLAP operations are primarily "read mostly" databases. OLAP applications tend to extract historical data that has accumulated over a long period of time. For such databases, redundant or "denormalized" data may facilitate business intelligence applications. Specifically, dimensional tables in a star schema often contain denormalized data. The denormalized or redundant data must be carefully controlled during ETL processing, and users should not be permitted to see the data until it is in a consistent state. The normalized alternative to the star schema is the snowflake schema. It has never been proven that this denormalization itself provides any increase in performance, or if the concurrent removal of data constraints is what increases the performance. In many cases, the need for denormalization has waned as computers and RDBMS software have become more powerful, but since data volumes have generally increased along with hardware and software performance, OLAP databases often still use denormalized schemas.

Denormalization is also used to improve performance on smaller computers as in computerized cash-registers and mobile devices, since these may use the data for look-up only (e.g. price lookups). Denormalization may also be used when no RDBMS exists for a platform (such as Palm), or no changes are to be made to the data and a swift response is crucial.

Non-first normal form (NF² or N1NF)

In recognition that denormalization can be deliberate and useful, the non-first normal form is a definition of database designs which do not conform to the first normal form, by allowing "sets and sets of sets to be attribute domains" (Schek 1982). This extension is a (non-optimal) way of implementing hierarchies in relations. Some academics have dubbed this practitioner developed method, "First Ab-normal Form", Codd defined a relational database as using relations, so any table not in 1NF could not be considered to be relational.

Consider the following table:

Non-First Normal Form
Person Favorite Colors
Bob blue, red
Jane green, yellow, red

Assume a person has several favorite colors. Obviously, favorite colors consist of a set of colors modeled by the given table.

To transform this NF² table into a 1NF an "unnest" operator is required which extends the relational algebra of the higher normal forms. The reverse operator is called "nest" which is not always the mathematical inverse of "unnest", although "unnest" is the mathematical inverse to "nest". Another constraint required is for the operators to be bijective, which is covered by the Partitioned Normal Form (PNF).

Further reading

Notes and References

  • Paper: "Non First Normal Form Relations" by G. Jaeschke, H. -J Schek ; IBM Heidelberg Scientific Center. -> Paper studying normalization and denormalization operators nest and unnest as mildly described at the end of this wiki page. The paper contains the Formalization through Set Theory of 1NF and NF^2 relations.

See also

External links

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