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Because we did not list functional dependencies that rule out the preceding case, loannumber is not a candidate key However, the functional dependency loan-number amount is nontrivial Therefore, Loan-info-schema does not satisfy the de nition of BCNF We claim that Loan-info-schema is not in a desirable form, since it suffers from the problem of repetition of information that we described in Section 72 We observe that, if there are several customer names associated with a loan, in a relation on Loan-infoschema, then we are forced to repeat the branch name and the amount once for each customer We can eliminate this redundancy by redesigning our database such that all schemas are in BCNF One approach to this problem is to take the existing nonBCNF design as a starting point, and to decompose those schemas that are not in BCNF Consider the decomposition of Loan-info-schema into two schemas: Loan-schema = (loan-number, branch-name, amount) Borrower-schema = (customer-name, loan-number) This decomposition is a lossless-join decomposition To determine whether these schemas are in BCNF, we need to determine what functional dependencies apply to them In this example, it is easy to see that loan-number amount branch-name applies to the Loan-schema, and that only trivial functional dependencies apply to Borrower-schema Although loan-number is not a superkey for Loan-info-schema, it is a candidate key for Loan-schema Thus, both schemas of our decomposition are in BCNF It is now possible to avoid redundancy in the case where there are several customers associated with a loan There is exactly one tuple for each loan in the relation on Loan-schema, and one tuple for each customer of each loan in the relation on Borrower-schema Thus, we do not have to repeat the branch name and the amount once for each customer associated with a loan Often testing of a relation to see if it satis es BCNF can be simpli ed: To check if a nontrivial dependency causes a violation of BCNF, compute + (the attribute closure of ), and verify that it includes all attributes of R; that is, it is a superkey of R To check if a relation schema R is in BCNF, it suf ces to check only the dependencies in the given set F for violation of BCNF, rather than check all dependencies in F + We can show that if none of the dependencies in F causes a violation of BCNF, then none of the dependencies in F + will cause a violation of BCNF either Unfortunately, the latter procedure does not work when a relation is decomposed That is, it does not suf ce to use F when we test a relation Ri , in a decomposition of R, for violation of BCNF For example, consider relation schema R (A, B, C, D, E), with functional dependencies F containing A B and BC D Suppose this were.

The McGraw Hill Companies, 2001

Server41" "[132189304]" "[1321435783]"

.

The extended cross product operator is also known as the Cartesian product after French mathematician Rene Descartes

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A storage structure called an R-tree is useful for indexing of rectangles and other polygons An R-tree is a balanced tree structure with the indexed polygons stored in leaf nodes, much like a B+ -tree However, instead of a range of values, a rectangular bounding box is associated with each tree node The bounding box of a leaf node is the smallest rectangle parallel to the axes that contains all objects stored in the leaf node The bounding box of internal nodes is, similarly, the smallest rectangle parallel to the axes that contains the bounding boxes of its child nodes The bounding box of a polygon is de ned, similarly, as the smallest rectangle parallel to the axes that contains the polygon Each internal node stores the bounding boxes of the child nodes along with the pointers to the child nodes Each leaf node stores the indexed polygons, and may optionally store the bounding boxes of the polygons; the bounding boxes help speed up checks for overlaps of the rectangle with the indexed polygons if a query rectangle does not overlap with the bounding box of a polygon, it cannot overlap with the polygon either (If the indexed polygons are rectangles, there is of course no need to store bounding boxes since they are identical to the rectangles) Figure 236 shows an example of a set of rectangles (drawn with a solid line) and the bounding boxes (drawn with a dashed line) of the nodes of an R-tree for the set of rectangles Note that the bounding boxes are shown with extra space inside them, to make them stand out pictorially In reality, the boxes would be smaller and t tightly on the objects that they contain; that is, each side of a bounding box B would touch at least one of the objects or bounding boxes that are contained in B The R-tree itself is at the right side of Figure 236 The gure refers to the coordinates of bounding box i as BBi in the gure.

#Pkts (A->B)

The McGraw Hill Companies, 2001

23

Understanding Relational Databases number of rows in the result The number of rows in the resulting table is the product of the number of rows of the two input tables In contrast, the number of result columns is the sum of the columns of the two input tables In Figure 34, the result table has nine rows and two columns As another example, consider the product of the sample Student Enrollment (Table 310) and (Table 311) tables The resulting table (Table 312) has 9 rows (3 X 3) and 7

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#Bytes(A->B)

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