B-Tree 文件组织：成功的原因与变种分析

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"这篇文档是关于B-树的广泛应用及其成功原因的深入探讨。作者Douglas Comer来自普渡大学计算机科学系，文中详细介绍了B-树在文件组织、数据库系统以及通用访问方法中的重要角色。B-树因其高效性能而被广泛采用，包括B+树在内的各种变体也在文中进行了比较分析。此外，文中还提供了一个利用B-树实现通用访问方法的实例。关键词包括B-树、B*树、B+树、文件组织和索引，属于计算机科学领域的数据存储与检索类别。" 本文主要围绕B-树这一数据结构展开，B-树是一种自平衡的树，特别适合于磁盘等外部存储设备的访问，因为它能够减少磁盘I/O操作，提高数据检索效率。B-树的节点通常包含多个键和指向子节点的指针，使得搜索、插入和删除操作可以在对数时间内完成，这对于大规模数据的管理至关重要。 B+树是B-树的一个变种，它优化了数据存储和检索的性能，特别是对于范围查询和顺序遍历。B+树的所有数据都存储在叶子节点，且叶子节点之间通过链表连接，这样可以保证所有数据在同一层上，减少了磁盘跳转次数，提高了查询效率。而B*树则是B+树的进一步改进，它引入了父节点到叶子节点的指针，降低了查找最近邻节点的时间复杂度。文件组织是计算机系统中不可或缺的一部分，有效的文件组织策略能大大提高数据处理的效率。B-树因其高效性和适应性，被广泛应用在用户文件索引、专用数据库系统以及通用访问方法中。文章通过对比B-树的不同变体，如B-树、B+树和B*树，讨论了它们各自的优点和实现成本，帮助读者理解如何根据实际需求选择最适合的数据结构。作者还介绍了一个使用B-树作为基础的通用访问方法，展示了如何在实际应用中利用B-树的优势来处理数据访问问题。这表明，无论是在传统的文件系统还是现代的数据库管理系统中，B-树都是一个不可或缺的工具。 B-树是计算机科学领域中数据存储和检索的基石之一，它的成功在于其对磁盘I/O操作的有效管理和高效的查询性能。通过深入理解和灵活运用B-树及其变体，开发者可以构建更加高效和可靠的系统，满足大数据时代的需求。

124 • D. Comer

Recall that in a binary search tree the

branch taken at a node depends on the

outcome of a comparison of the query key

and the key stored at the node. If the query

is less than the stored key, the left branch

is taken; if it is greater, the right branch is

followed. Figure 2 shows part of such a tree

used to store employee numbers, and the

path taken for the query "15."

Now consider Figure 3 which shows a

modified search tree with two keys stored

in each node. Searching proceeds by choos-

ing one of three paths at each node. In the

figure, the query, 15, is less than 42 so the

leftmost would be taken at the root. For

those queries between 42 and 81 the center

path would be selected, while the rightmost

path would be followed for queries greater

than 81. The decision procedure is repeated

at each node until an exact match occurs

(success) or a leaf is encountered (failure).

In general, each node in a B-tree of order

d contains at most 2d keys and 2d + 1

pointers, as shown in Figure 4. Actually,

the number of keys may vary from node to

node, but each must have at least d keys

and d + 1 pointers. As a result, each node

is at least 1/~ full. In the usual implementa-

tion a node forms one record of the index

f'fle, has a fixed length capable of accom-

modating 2d keys and 2d pointers, and con-

tains additional information telling how

many keys correctly reside in the node.

/ \ / \

FIGURE 2. Part of a binary search tree for employee

numbers The path taken for query "15" is darkened.

I N\

!~2 I,I ~3, I

1 1

FIGURE 3. A search tree with 2 keys and 3 branches

per node. The path taken for query "15" is darkened.

Ugually, large, multikey nodes cannot be

kept in main memory and require an access

to secondary storage each time they are to

be inspected. Later, we will see how, under

our cost criterion, maintaining more than

one key per node lowers the cost of find,

insert, and delete operations.

Balancing

The beauty of B-trees lies in the methods

for inserting and deleting records that al-

ways leave the tree balanced. As in the case

of binary search trees, random insertions of

records into a file can leave a tree unbal-

anced. While an unbalanced tree, like the

one shown in Figure 5a has some long paths

and some short ones, a balanced tree, like

the one shown in Figure 5b, has all leaves

at the same depth. Intuitively, B-trees have

a shape as shown in Figure 6. The longest

path in a B-tree of n keys contains at most

about logdn nodes, d being the order of the

B-tree. A find operation may visit n nodes

FIGURE 4. A node in a B-tree of order d with 2d

keys and 2d + 1 pointers.

I,, , -, \1

I/t',

--1

I/i,, \1

I/ \ -..I

--....

(b)

FIGURE 5. (a) An unbalanced tree with many long

paths, and (b) a balanced tree with all paths to

leaves exactly the same length.

Computing Surveys, Vol 11, No 2, June 1979

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