Dictionaries for very large files typically reside on secondary storage such as a disk. The dictionary is implemented as an index to the actual file and contains the key and record address of data. To implement a dictionary we could use red-black trees replacing pointers with offsets from the beginning of the index file and use random access to reference nodes of the tree. However every transition on a link would imply a disk access and would be prohibitively expensive. Recall that low-level disk I/O accesses disk by sectors (typically 512 bytes). We could equate node size to sector size and group several keys together in each node to minimize the number of I/O operations. This is the principle behind B-trees. Good references for B-trees include Knuth [1998] and Cormen [2001]. For B^{+}-trees consult Aho [1983].
Figure 4-3 illustrates a B-tree with 3 keys/node. Keys in internal nodes are surrounded by pointers or record offsets to keys that are less than or greater than the key value. For example all keys less than 22 are to the left and all keys greater than 22 are to the right. For simplicity I have not shown the record address associated with each key.
We can locate any key in this 2-level tree with three disk accesses. If we were to group 100 keys/node we could search over 1 000 000 keys in only three reads. To ensure this property holds we must maintain a balanced tree during insertion and deletion. During insertion we examine the child node to verify that it is able to hold an additional node. If not then a new sibling node is added to the tree and the child’s keys are redistributed to make room for the new node. When descending for insertion and the root is full then the root is spilled to new children and the level of the tree increases. A similar action is taken on deletion where child nodes may be absorbed by the root. This technique for altering the height of the tree maintains a balanced tree.
B-Tree | B*-Tree | B^{+}-Tree | B^{++}-Tree | |
---|---|---|---|---|
data stored in | any node | any node | leaf only | leaf only |
on insert split | 1 x 1 –>2 x 1/2 | 2 x 1 –>3 x 2/3 | 1 x 1 –>2 x 1/2 | 3 x 1 –>4 x 3/4 |
on delete join | 2 x 1/2 –>1 x 1 | 3 x 2/3 –>2 x 1 | 2 x 1/2 –>1 x 1 | 3 x 1/2 –>2 x 3/4 |
Several variants on the B-tree are listed in Table 4-1. The standard B-tree stores keys and data in both internal and leaf nodes. When descending the tree during insertion a full child node is first redistributed to adjacent nodes. If the adjacent nodes are also full then a new node is created and half the keys in the child are moved to the newly created node. During deletion children that are 1/2 full first attempt to obtain keys from adjacent nodes. If the adjacent nodes are also 1/2 full then two nodes are joined to form one full node. B*-trees are similar only the nodes are kept 2/3 full. This results in better utilization of space in the tree and slightly better performance.
Figure 4-4 illustrates a B^{+}-tree. All keys are stored at the leaf level with their associated data values. Duplicates of the keys appear in internal parent nodes to guide the search. Pointers have a slightly different meaning than in conventional B-trees. The left pointer designates all keys less than the value while the right pointer designates all keys greater than or equal to (GE) the value. For example all keys less than 22 are on the left pointer and all keys greater than or equal to 22 are on the right. Notice that key 22 is duplicated in the leaf where the associated data may be found. During insertion and deletion care must be taken to properly update parent nodes. When modifying the first key in a leaf the tree is walked from leaf to root. The last GE pointer found while descending the tree will require modification to reflect the new key value. Since all keys are in the leaf nodes we may link them for sequential access.
The last method B^{++}-trees is something of my own invention. The organization is similar to B^{+}-trees except for the split/join strategy. Assume each node can hold k keys and the root node holds 3k keys. Before we descend to a child node during insertion we check to see if it is full. If it is the keys in the child node and two nodes adjacent to the child are all merged and redistributed. If the two adjacent nodes are also full then another node is added resulting in four nodes each 3/4 full. Before we descend to a child node during deletion we check to see if it is 1/2 full. If it is the keys in the child node and two nodes adjacent to the child are all merged and redistributed. If the two adjacent nodes are also 1/2 full then they are merged into two nodes each 3/4 full. This is halfway between 1/2 full and completely full allowing for an equal number of insertions or deletions in the future.
Recall that the root node holds 3k keys. If the root is full during insertion we distribute the keys to four new nodes each 3/4 full. This increases the height of the tree. During deletion we inspect the child nodes. If there are only three child nodes and they are all 1/2 full they are gathered into the root and the height of the tree decreases.
Another way of expressing the operation is to say we are gathering three nodes and then scattering them. In the case of insertion where we need an extra node we scatter to four nodes. For deletion where a node must be deleted we scatter to two nodes. The symmetry of the operation allows the gather/scatter routines to be shared by insertion and deletion in the implementation.
An ANSI-C implementation of a B^{++}-tree is included.
In the implementation-dependent section
you’ll need to define bAdrType
and
eAdrType
the types associated with B-tree file offsets and data file offsets
respectively. You’ll also need to provide a callback function which is used by the B^{++}-tree
algorithm to compare keys. Functions are provided to insert/delete keys
find keys
and access
keys sequentially. Function main
at the bottom of the file
provides a simple
illustration for insertion.
The code provided allows for multiple indices to the same data. This was implemented by
returning a handle when the index is opened. Subsequent accesses are done using the
supplied handle. Duplicate keys are allowed. Within one index
all keys must be the same length.
A binary search was implemented to search each node. A flexible buffering scheme allows nodes
to be retained in memory until the space is needed. If you expect access to be somewhat ordered
increasing the bufCt
will reduce paging.
Not implemented.