B-tree indexing is a fundamental technique used in databases to enhance the efficiency of data retrieval. Here’s a detailed explanation of how B-tree indexes work:
Structure of B-trees
A B-tree is a self-balancing tree data structure that maintains sorted data and allows searches, sequential access, insertions, and deletions in logarithmic time. Unlike binary search trees, B-trees can have multiple children per node, which reduces the tree's height and thus the number of disk accesses required to find an element[1][2].
Key Properties
Nodes and Keys: Each node in a B-tree contains a certain number of keys and child pointers. The keys within a node are sorted in ascending order. Each key separates the range of values in the child nodes, ensuring that all keys in the left child are less than the parent key and all keys in the right child are greater[1][2][9].
Balanced Structure: B-trees are balanced, meaning all leaf nodes are at the same depth. This balance ensures that the time complexity for search, insert, and delete operations remains $$O(\log n)$$[2][7].
Order of the Tree: The order of a B-tree (denoted as $$m$$) determines the maximum number of children a node can have. A B-tree of order $$m$$ can have at most $$m-1$$ keys and $$m$$ children[2].
Operations
Search: Searching in a B-tree involves traversing from the root to the leaf nodes. At each node, the search algorithm compares the target key with the keys in the node to decide which child pointer to follow. This process continues until the key is found or a leaf node is reached[1][2][9].
Insertion: When inserting a new key, the algorithm first finds the appropriate leaf node where the key should be added. If the node has space, the key is inserted directly. If the node is full, it is split into two nodes, and the middle key is promoted to the parent node. This splitting and promotion may propagate up to the root, potentially increasing the tree's height[2][9].
Deletion: Deleting a key from a B-tree involves finding the key and removing it. If the key is in an internal node, it is replaced with its predecessor or successor key from a leaf node. If this causes the leaf node to have fewer keys than the minimum required, nodes are merged or keys are redistributed to maintain the tree's properties[2][9].
