We present a generic solution to a problem which lies at the heart of the unpredictable worst-case performance characteristics of a wide class of multi-dimensional index designs: those which employ a recursive partitioning of the data space. We then show how this solution can produce modified designs with fully predictable and controllable worst-case characteristics. In particular, we show how the recursive partitioning of an n-dimensional dataspace can be represented in such a way that the characteristics of the one-dimensional B-tree are preserved in n dimensions, as far as is topologically possible i.e. a representation guaranteeing logarithmic access and update time, while also guaranteeing a one-third minimum occupancy of both data and index nodes.

J. Nievergelt, M. Freeston; Special Issue Editorial Spatial data: applications, concepts, techniques, The Computer Journal, Volume 37, Issue 1, 1 January 1

A new multi-dimensional file structure has been developed in the course of a project to devise ways of improving the support for interactive queries to database and knowledge bases. Christened the 'BANG' file - a Balanced And Nested Grid - the new structure is of the 'grid file' type, but is fundamentally different from previous grid file designs in that it does not share their common underlying properties. It has a tree-structured directory which has the self-balancing property of a B-tree and which, in contrast to previous designs, always expands at the same rate as the data, whatever the form of the data distribution. Its partitioning strategy both accurately reflects the clustering of points in the data space, and is flexible enough to adapt gracefully to changes in the distribution.