B-Trees for Disk Storage: Balanced Search Trees Explained | Chapter 18 of introduction to algorithms

B-trees are a cornerstone of external memory data structures, providing efficient, balanced search trees optimized for disk-based storage systems. Chapter 18 of introduction to algorithms details how B-trees manage massive datasets with minimal disk access, making them the backbone of databases and file systems.

Watch the full chapter summary below for an in-depth explanation of B-trees, their operations, and practical use in storage systems. Subscribe to Last Minute Lecture for more introduction to algorithms tutorials!

Structure and Advantages of B-Trees

B-trees are rooted, balanced trees where all leaves are at the same depth. Unlike binary search trees, B-tree nodes can contain multiple keys and have a high branching factor (often between 50 and 2000 children), minimizing tree height and reducing the number of slow disk accesses per operation. Each node stores:

• x.n: Number of keys
• x.keys[1 to n]: Sorted keys
• x.leaf: Indicates if the node is a leaf
• For internal nodes: n + 1 child pointers

The minimum degree t defines the bounds for the number of keys and children per node, ensuring the tree remains balanced. B-trees are designed so each node fits in a single disk block, significantly improving performance for large, disk-resident datasets.

Operations and Performance

• Search (B-TREE-SEARCH): Generalizes binary search tree search, requiring O(log_t n) disk accesses.
• Create (B-TREE-CREATE): Initializes an empty B-tree in O(1) disk operations.
• Insert (B-TREE-INSERT): Descends to the appropriate leaf, splitting full nodes on the way down, and may increase tree height. Insertion and splitting require O(t log_t n) CPU time and O(log_t n) disk accesses.
• Delete (B-TREE-DELETE): Removes keys in a single downward pass, handling underfull nodes by merging or redistributing. Complexity is O(t log_t n).

B-trees maintain balance automatically, so their height remains O(log_t n) regardless of insertion or deletion order, making them highly efficient for massive datasets.

Variants and Practical Considerations

• B+-Tree: Stores all actual data in the leaves, while internal nodes serve only as navigation.
• B-tree*: Requires internal nodes to be at least two-thirds full, improving space utilization and further reducing tree height.
• Disk block size determines node capacity, balancing CPU and disk access costs.

By grouping many keys per node and keeping all leaves at the same depth, B-trees enable databases and file systems to efficiently manage billions of records with minimal disk reads and writes.

Glossary of Key Terms

• B-Tree: Balanced multi-way search tree optimized for external memory
• Minimum Degree (t): Parameter controlling minimum and maximum keys/children per node
• Branching Factor: Maximum number of children per node (up to 2t)
• Disk Access: Primary performance cost for external data structures
• Block: Unit of disk I/O; B-tree nodes are sized to match
• Full Node: Contains the maximum number of keys (2t – 1)
• Underfull Node: Fewer than t – 1 keys (invalid except root)
• Split: Restructuring full nodes during insertion
• Predecessor/Successor: Nearest keys in in-order traversal
• Latency: Mechanical delay in disk access
• B+-Tree: Variant with all data in leaf nodes
• B-Tree*: Tighter packing and higher minimum utilization

Conclusion: The Foundation of Efficient Storage

B-trees are essential for high-performance storage, enabling databases and filesystems to scale effortlessly. Their self-balancing, multi-key nodes keep disk operations minimal while ensuring reliable, efficient management of vast data collections. For more in-depth algorithm tutorials, subscribe to Last Minute Lecture and explore the entire introduction to algorithms series!