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required for only about 2 percent of the insertions. Furthermore, a second-from-bottom-level split (i.e., a double split) would be required for only about 2 percent of 2 percent of the insertions, that is, with probability 0.0004. And the probability of a triple split would be 0.000008. So the chances of the root being split are very small. And since that is the only way that the tree can grow vertically, it tends to remain a very shallow, very broad tree, providing very fast search time. B-TREES A B-tree of order m is an m-way search tree that satisfies the following extra conditions: 1. The root has at least two children. 2. All other internal nodes have at least m/2 children. 3. All leaf nodes are at the same level. These conditions make the tree more balanced (and thus more efficient), and they simplify the insertion and deletion algorithms. B-trees are used as indexes for large data sets stored on disk. In a relational database, data are organized in separate sequences of records called tables. Each table could be stored as a sequential data file in which the records are numbered like the elements of an array. Or the database system might access the records directly by their disk addresses. Either way, each record is directly accessible on disk via some addressing scheme. So once we have the record s disk address, we can access it immediately (i.e., with a single disk read). So the key that is stored in

CHAP. 12]

Intellectual property can be protected as a trade secret provided the owner of the secret takes care to protect the information The owner must restrict access to the information, so that only those with authorization can get to it The source code could be kept in a secure archive, for instance The owner must also limit access to the information Even though the code is in a secure archive, the owner also must be vigilant in providing access only to authorized individuals The owner must also require those who have access to the information to sign an agreement not to disclose to others anything about the code Such an agreement is called a nondisclosure agreement, and it may be required of employees as well as any outsiders who might have a need to see the source.

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the B-tree is actually a key/address pair containing the record s actual key value (e.g., a U.S. Social Security number for personnel records, or an ISBN for books) together with its disk address. In the outline that follows, only the key value is shown, the accompanying disk address being understood to accompany it. EXAMPLE 12.3 A B-Tree

Figure 12.5 shows a B-tree of order 5. Each of its internal nodes has 3, 4, or 5 children, and all the leaves are at level 3.

To protect their trade secret rights, the owner must also mark any material related to the secret as proprietary The source code itself, training manuals, and other documentation should include a statement that the software is proprietary A trade secret can last indefinitely, and it costs nothing except being serious about protecting the secret The courts will enforce the owner s rights to the trade secret, as long as the owner remains diligent in efforts to protect the secret However, if the owner becomes sloppy about security surrounding the secret, the owner can lose the trade secret rights Suppose, for instance, that (perhaps mistakenly) the owner posts the source code on a web page for a while Such disclosure of the secret to the public could invalidate any future claim of the owner to legal protection of the secret.

Algorithm 12.1 Searching in a B-Tree To find a record with key k using a B-tree index of order m: 1. If the tree is empty, return null. 2. Let x be the root. 3. Repeat steps 4 6 until x is a leaf node. 4. Apply the binary search (page 31) to node x for the key k i , where k i 1 < k (regarding k 0 = and k m = ). 5. If k i = k, retrieve the record from disk and return it. 6. Let x be the root of subtree S i . Return null.

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