In one embodiment, the present invention is directed to a data structure for representing a spatial region. The data structure comprises a hierarchical arrangement of nodes associated with a plurality of refinement levels, wherein each node of the hierarchical arrangement of nodes is a regular spatial subdivision of the spatial region or another node that is associated with a preceding refinement level. The hierarchical arrangement of nodes forms a directed acyclic graph. The hierarchical arrangement of nodes comprises at least two nodes that have respective edges that are traversed to a common child node such that the hierarchical arrangement of nodes does not comprise a repeated pattern from any two nodes of a common refinement level of the data structure.