vb.net code to generate barcode Figure 12.2 Inserting 66 into a five-way search tree in Java

Print EAN-13 Supplement 5 in Java Figure 12.2 Inserting 66 into a five-way search tree

Figure 12.2 Inserting 66 into a five-way search tree
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Insert the new key 66 in that last parent node between the bracketing keys 65 and 67 as shown in Figure 12.3 on page 232. Now that node contains five keys, which violates the four-key limit for a five-way tree. So the node gets split, shifting its middle key 65 up to its parent node as shown in Figure 12.4 on page 232.
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Node splitting occurs relatively infrequently, especially if m is large. For example, if m = 50, then on average only 2 percent of the nodes would be full, so a bottom-level split would be
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Figure 12.3 Inserting 66 into a five-way search tree
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Figure 12.4 Inserting 66 into a five-way search tree
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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
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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.
Figure 12.5 A B-tree of order 5
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.
Note how similar this process is to looking up a topic in the index of a book. Each page of the index is labeled with a word or letter that represents the topics listed on that page. The page labels are analogous to the keys in the internal nodes of the search tree. The actual page number listed next to the topic in the book s index is analogous to the disk address of file name that leads you to the actual data. The last step of the search process is searching through that page in the book, or through that file on the disk. This analogy is closer if the book s index itself had an index. Each internal level of the multiway tree corresponds to another index level. Algorithm 12.2 Inserting into a B-Tree To insert a record with key k using a B-tree index of order m: 1. If the tree is empty, create a root node with two dummy leaves, insert k there, and return true (indicating that the insertion was successful). 2. Let x be the root. 3. Repeat steps 4 6 until x is a leaf node. 4. Apply the binary search to node x for the key k i , where k i 1 < k k i (regarding k 0 = and k m = ).
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