sqrt2DSH

sqrt2DSH is a term used in theoretical discussions to denote a hypothetical class of two-dimensional, scale-adjusted hash schemes. The name suggests incorporating a square-root factor, sqrt(2), in conjunction with a two-dimensional input to produce a hashed representation. There is no widely accepted formal definition, and different authors describe different constructions under the same label.

Definition and construction: In a common informal description, an input vector x = (x1, x2) is first scaled by sqrt(2) and then discretized onto a grid of cell size g. The discretized coordinates may be offset by random or fixed shifts, yielding y = (floor((sqrt(2) * x1)/g + r1), floor((sqrt(2) * x2)/g + r2)). A hash function h maps y to a bucket index. Variants may apply multiple such grids and combine results, as in locality-sensitive hashing schemes. The exact choice of g, the offsets, and the hash function define a concrete realization of sqrt2DSH.

Properties: sqrt2DSH does not imply a single formal guarantee. It is intended to provide approximate locality, so that points near in the plane tend to collide in the same or neighboring buckets, with performance depending on the grid parameters and hash function. It is sensitive to rotation and coordinate scaling and does not universally preserve distances.

Applications and variants: Proposed uses include fast approximate nearest-neighbor search in two-dimensional datasets, pattern recognition on planar features, and lightweight embedding for geographic information systems. Variants differ in grid construction, hashing function, and whether multiple grids are combined or probabilistically sampled.

History and reception: The label sqrt2DSH appears mainly in informal discussions and project notes rather than in standard textbooks or peer-reviewed literature. Because there is no canonical definition, practitioners typically specify a concrete construction before applying the method.

See also: Locality-sensitive hashing, grid hashing, hashing, dimensionality reduction.