hierpart

Hierpart, short for hierarchical partition, is a concept used in mathematics and computer science to describe a multi-level subdivision of a domain into smaller, simpler parts. In its standard form, a hierpart is a sequence of partitions P0, P1, P2, ... of a bounded domain Ω in R^d where P0 consists of a single cell Ω and each successive partition Pi+1 refines Pi by subdividing some cells into a fixed number of subcells. The subdivision is typically represented as a tree: each node corresponds to a cell, and its children are the subcells produced by refinement.

Common concrete instantiations include dyadic partitions in one dimension (splitting intervals in half), quadtrees in two dimensions (splitting squares into four), and octrees in three dimensions (splitting cubes into eight). Adaptive hierparts refine only where needed, guided by error estimators, data density, or feature complexity, producing nonuniform partitions suited to the problem’s local structure.

Key properties include disjoint interiors, full coverage of Ω, and refinement consistency (a refinement replaces a cell with smaller ones without overlapping). Performance considerations depend on the maximum depth and branching; operations such as range search, nearest-neighbor queries, and updates can be implemented efficiently via the tree representation.

Hierparts are used in various applications: spatial indexing and databases, computer graphics and collision detection, finite element mesh generation and numerical integration, image processing, and geographic information systems. The term hierpart is not universally standardized; many authors prefer the broader phrasing hierarchical partitioning or hierarchical spatial partitioning, and related structures include kd-trees, quadtrees, and adaptive mesh refinement.