The A/D bits exist entirely for the operating system's benefit. The Accessed bit tells the OS which pages have been recently used, enabling page replacement algorithms to choose pages to evict when memory runs low. The Dirty bit tells the OS which pages have been modified and must be written back to disk before eviction; clean pages can simply be discarded and re-read from disk later.
Quadtrees are everywhere spatial data exists. Mapping services use quadtree-like tile pyramids to serve map tiles at different zoom levels (Bing's quadkey system, for example, addresses tiles as base-4 paths). Game engines use them for collision detection and visibility culling. Geographic information systems use spatial indexes to store and query spatial datasets. PostGIS uses GiST indexes (R-tree-style) for spatial queries on geometries, while PostgreSQL's core supports quadtree-like SP-GiST indexes for certain data types like points.
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