B-Trees

Introduction

A B-tree is a specialized multiway tree designed especially for use on disk. In a B-tree each node may contain a large number of keys. The number of subtrees of each node, then, may also be large. A B-tree is designed to branch out in this large number of directions and to contain a lot of keys in each node so that the height of the tree is relatively small. This means that only a small number of nodes must be read from disk to retrieve an item. The goal is to get fast access to the data, and with disk drives this means reading a very small number of records. Note that a large node size (with lots of keys in the node) also fits with the fact that with a disk drive one can usually read a fair amount of data at once.

Definitions

A multiway tree of order m is an ordered tree where each node has at most m children. For each node, if k is the actual number of children in the node, then k - 1 is the number of keys in the node. If the keys and subtrees are arranged in the fashion of a search tree, then this is called a multiway search tree of order m. For example, the following is a multiway search tree of order 4. 

A formal definition is given below:-

A B-tree of order m is a m-way tree in which

1. All leaves are on the same level.

2. All internal nodes except the root have at most m (nonempty) children, and at least ceil (m/2) (nonempty) children.

3. The number of keys in each internal node is one less than the number of its children , and these keys partition the keys in the children in the fashion of a search tree.

4. The root has at most m children, but may have as few as 2 if it is not a leaf, or none if the tree consists of the root alone.

 Note that the first row in each node shows the keys, while the second row shows the pointers to the child nodes. Of course, in any useful application there would be a record of data associated with each key, so that the first row in each node might be an array of records where each record contains a key and its associated data. Another approach would be to have the first row of each node contain an array of records where each record contains a key and a record number for the associated data record, which is found in another file. This last method is often used when the data records are large.

Example B-Tree

The following is an example of a B-tree of order 5. This means that (other that the root node) all internal nodes have at least ceil(5 / 2) = ceil(2.5) = 3 children (and hence at least 2 keys). Of course, the maximum number of children that a node can have is 5 (so that 4 is the maximum number of keys). According to condition 4, each leaf node must contain at least 2 keys. In practice B-trees usually have orders a lot bigger than 5.

Operations on a B-Tree

Question: How would you search in the above tree to look up S? How about J? How would you do a sort-of "in-order" traversal, that is, a traversal that would produce the letters in ascending order? (One would only do such a traversal on rare occasion as it would require a large amount of disk activity and thus be very slow!)

Inserting a New Item

According to Kruse (see reference at the end of this file) the insertion algorithm proceeds as follows: When inserting an item, first do a search for it in the B-tree. If the item is not already in the B-tree, this unsuccessful search will end at a leaf. If there is room in this leaf, just insert the new item here. Note that this may require that some existing keys be moved one to the right to make room for the new item. If instead this leaf node is full so that there is no room to add the new item, then the node must be "split" with about half of the keys going into a new node to the right of this one. The median (middle) key is moved up into the parent node. (Of course, if that node has no room, then it may have to be split as well.) Note that when adding to an internal node, not only might we have to move some keys one position to the right, but the associated pointers have to be moved right as well. If the root node is ever split, the median key moves up into a new root node, thus causing the tree to increase in height by one.

Let's work our way through an example similar to that given by Kruse. Insert the following letters into what is originally an empty B-tree of order 5: C N G A H E K Q M F W L T Z D P R X Y S Order 5 means that a node can have a maximum of 5 children and 4 keys. All nodes other than the root must have a minimum of 2 keys. The first 4 letters get inserted into the same node, resulting in this picture:

When we try to insert the H, we find no room in this node, so we split it into 2 nodes, moving the median item G up into a new root node. Note that in practice we just leave the A and C in the current node and place the H and N into a new node to the right of the old one.

Inserting E, K, and Q proceeds without requiring any splits:

Inserting M requires a split. Note that M happens to be the median key and so is moved up into the parent node.

The letters F, W, L, and T are then added without needing any split.

When Z is added, the rightmost leaf must be split. The median item T is moved up into the parent node. Note that by moving up the median key, the tree is kept fairly balanced, with 2 keys in each of the resulting nodes.

The insertion of D causes the leftmost leaf to be split. D happens to be the median key and so is the one moved up into the parent node. The letters P, R, X, and Y are then added without any need of splitting:

Finally, when S is added, the node with N, P, Q, and R splits, sending the median Q up to the parent. However, the parent node is full, so it splits, sending the median M up to form a new root node. Note how the 3 pointers from the old parent node stay in the revised node that contains D and G.

Two comments regarding the growth of B-trees are in order. First, when a node splits, it produces two nodes that are now only half full. Later insertions, therefore, can more likely be made without the need to split the nodes again. Hence, one splitting leads the way for several simple insertions. Second, it is always a median key that is sent upward, not necessarily the key being inserted. Hence, repeated insertions tend to improve the balance of the tree, no matter in what order the keys happen to arrive.

Deleting an Item

In the B-tree as we left it at the end of the last section, delete H. Of course, we first do a lookup to find H. Since H is in a leaf and the leaf has more than the minimum number of keys, this is easy. We move the K over where the H had been and the L over where the K had been. This gives:

Next, delete the T. Since T is not in a leaf, we find its successor (the next item in ascending order), which happens to be W, and move W up to replace the T. That way, what we really have to do is to delete W from the leaf, which we already know how to do, since this leaf has extra keys. In ALL cases we reduce deletion to a deletion in a leaf, by using this method.

Next, delete R. Although R is in a leaf, this leaf does not have an extra key; the deletion results in a node with only one key, which is not acceptable for a B-tree of order 5. If the sibling node to the immediate left or right has an extra key, we can then borrow a key from the parent and move a key up from this sibling. In our specific case, the sibling to the right has an extra key. So, the successor W of S (the last key in the node where the deletion occurred), is moved down from the parent, and the X is moved up. (Of course, the S is moved over so that the W can be inserted in its proper place.)

Finally, let's delete E. This one causes lots of problems. Although E is in a leaf, the leaf has no extra keys, nor do the siblings to the immediate right or left. In such a case the leaf has to be combined with one of these two siblings. This includes moving down the parent's key that was between those of these two leaves. In our example, let's combine the leaf containing F with the leaf containing A C. We also move down the D.

Of course, you immediately see that the parent node now contains only one key, G. This is not acceptable. If this problem node had a sibling to its immediate left or right that had a spare key, then we would again "borrow" a key. Suppose for the moment that the right sibling (the node with Q X) had one more key in it somewhere to the right of Q. We would then move M down to the node with too few keys and move the Q up where the M had been. However, the old left subtree of Q would then have to become the right subtree of M. In other words, the N P node would be attached via the pointer field to the right of M's new location. Since in our example we have no way to borrow a key from a sibling, we must again combine with the sibling, and move down the M from the parent. In this case, the tree shrinks in height by one.

Databases

A database is a collection of data organized in a fashion that facilitates updating, retrieving, and managing the data. The data can consist of anything, including, but not limited to names, addresses, pictures, and numbers. Databases are commonplace and are used everyday. For example, an airline reservation system might maintain a database of available flights, customers, and tickets issued. A teacher might maintain a database of student names and grades.

Because computers excel at quickly and accurately manipulating, storing, and retrieving data, databases are often maintained electronically using a database management system. Database management systems are essential components of many everyday business operations. Database products like Microsoft SQL Server, Sybase Adaptive Server, IBM DB2, and Oracle serve as a foundation for accounting systems, inventory systems, medical recordkeeping sytems, airline reservation systems, and countless other important aspects of modern businesses.

It is not uncommon for a database to contain millions of records requiring many gigabytes of storage. For examples, TELSTRA, an Australian telecommunications company, maintains a customer billing database with 51 billion rows (yes, billion) and 4.2 terabytes of data. In order for a database to be useful and usable, it must support the desired operations, such as retrieval and storage, quickly. Because databases cannot typically be maintained entirely in memory, b-trees are often used to index the data and to provide fast access. For example, searching an unindexed and unsorted database containing n key values will have a worst case running time of O(n); if the same data is indexed with a b-tree, the same search operation will run in O(log n). To perform a search for a single key on a set of one million keys (1,000,000), a linear search will require at most 1,000,000 comparisons. If the same data is indexed with a b-tree of minimum degree 10, 114 comparisons will be required in the worst case. Clearly, indexing large amounts of data can significantly improve search performance. Although other balanced tree structures can be used, a b-tree also optimizes costly disk accesses that are of concern when dealing with large data sets.