Partitioning

Partitioning is the act of dividing a set, space, or system into parts that are disjoint and collectively cover the whole. In mathematics, a partition of a set S is a collection of nonempty subsets called blocks, such that every element of S belongs to exactly one block and the blocks are pairwise disjoint. Each two elements in the same block are related by an equivalence relation; conversely, the equivalence classes of any relation form a partition.

Partitions can be compared by refinement; a partition P1 is finer than P2 if every block of P1 is contained in a block of P2. The set of partitions of a finite n-element set forms a lattice under refinement. The Bell number Bn counts partitions of an n-element set, while the Stirling numbers of the second kind count partitions into k blocks.

In computer storage, partitioning refers to dividing a physical disk into independent regions that can host separate file systems or operating systems. The common schemes include the older MBR (Master Boot Record) and the modern GPT (GUID Partition Table). Primary partitions can host operating systems; extended partitions can contain logical drives. Partitioning supports isolation, boot management, and easier backup, but resizing and data risk must be managed.

In databases and distributed systems, partitioning splits data across multiple locations or storage units. Horizontal partitioning (sharding) distributes rows by a key, while vertical partitioning splits by columns. Partitioning improves scalability and performance, but introduces complexity for queries spanning partitions and for maintaining transactional consistency; rebalancing and schema design are important considerations.

Partitioning also appears in data processing to enable parallel computation and sampling. In statistics, the sample space can be partitioned into disjoint events.