Quadtrees

Quadtrees are a tree data structure that partitions a two-dimensional space by recursively subdividing it into four quadrants. In a quadtree, each internal node has four children, and every node represents a square region of space. Leaves store data about the region, such as a point, a summary, or an indication that the region is homogeneous. The root covers the entire area of interest, and subdivision follows a predefined criterion.

Variants include point quadtrees, region quadtrees, and PR-quadtrees. Point quadtrees index individual points by subdividing until each leaf contains at most one point. Region quadtrees store information about homogeneous regions and stop subdividing when a region is uniform. PR-quadtrees combine these approaches and are commonly used in image processing and spatial indexing.

Construction and operations: Build from a root covering the full area; subdivide a region into four equal subregions when the criterion is not met. Typical operations include insertion, deletion, range queries, and collision detection. Quadtrees support hierarchical rendering, view culling, and compression by representing large uniform areas with a single leaf. Time complexity depends on the tree height; average cases are near logarithmic in n, with worst-case linear for degenerate distributions.

Applications: Quadtrees are used in computer graphics, geographic information systems, image processing, and game development to accelerate queries and manage large two-dimensional datasets. Related structures include k-d trees and octrees for higher dimensions; quadtrees are often extended or combined with other methods for dynamic datasets.