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In previous sections of this chapter, you have studied the terminology and integrity rules of relational databases with the goal of understanding existing relational databases In particu lar, understanding connections among tables was emphasized as a prerequisite to retrieving useful information This section describes some fundamental operators that can be used to retrieve useful information from a relational database You can think of relational algebra similarly to the algebra of numbers except that the objects are different: algebra applies to numbers and relational algebra applies to tables In algebra, each operator transforms one or more numbers into another number Similarly, each operator of relational algebra transforms a table (or two tables) into a new table This section emphasizes the study of each relational algebra operator in isolation For each operator, you should understand its purpose and inputs While it is possible to com bine operators to make complicated formulas, this level of understanding is not important for developing query formulation skills Using relational algebra by itself to write queries can be awkward because of details such as ordering of operations and parentheses There fore, you should seek only to understand the meaning of each operator, not how to combine operators to write expressions The coverage of relational algebra groups the operators into three categories The most widely used operators (restrict, project, and join) are presented first The extended cross product operator is also presented to provide background for the join operator Knowledge of these operators will help you to formulate a large percentage of queries More special ized operators are covered in latter parts of the section The more specialized operators include the traditional set operators (union, intersection, and difference) and advanced operators (summarize and divide) Knowledge of these operators will help you formulate more difficult queries

To understand how to index spatial data consisting of two or more dimensions, we consider rst the indexing of points in one-dimensional data Tree structures, such as binary trees and B-trees, operate by successively dividing space into smaller parts For instance, each internal node of a binary tree partitions a one-dimensional interval

The McGraw Hill Companies, 2001

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Our analysis determined that there were eight major transaction types to test After a capture is complete, the last packet in the captured file is a summary log that Network Monitor places there for analysis You can see how many packets were sent and received, as well as their cumulative size The bytes total was divided by the time it would normally take for a user to issue the command to get a rate The analysis shows that the highest rate is only 165 Kbps Therefore, no new links are going to be installed to run this application To see how this new traffic will influence the rest of the network requires modeling (see the later section "Simulations and Capacity Planning") TipThe captured packets are in bytes and must be converted to bits, by multiplying them by eight

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an operator that re trieves a subset of the rows of the input table that satisfy a given condition

in two Points that lie in the left partition go into the left subtree; points that lie in the right partition go into the right subtree In a balanced binary tree, the partition is chosen so that approximately one-half of the points stored in the subtree fall in each partition Similarly, each level of a B-tree splits a one-dimensional interval into multiple parts We can use that intuition to create tree structures for two-dimensional space, as well as in higher-dimensional spaces A tree structure called a k-d tree was one of the early structures used for indexing in multiple dimensions Each level of a k-d tree partitions the space into two The partitioning is done along one dimension at the node at the top level of the tree, along another dimension in nodes at the next level, and so on, cycling through the dimensions The partitioning proceeds in such a way that, at each node, approximately one-half of the points stored in the subtree fall on one side, and one-half fall on the other Partitioning stops when a node has less than a given maximum number of points Figure 234 shows a set of points in two-dimensional space, and a k-d tree representation of the set of points Each line corresponds to a node in the tree, and the maximum number of points in a leaf node has been set at 1 Each line in the gure (other than the outside box) corresponds to a node in the k-d tree The numbering of the lines in the gure indicates the level of the tree at which the corresponding node appears The k-d-B tree extends the k-d tree to allow multiple child nodes for each internal node, just as a B-tree extends a binary tree, to reduce the height of the tree k-d-B trees are better suited for secondary storage than k-d trees.

The McGraw Hill Companies, 2001

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