sqrtmur2

sqrtmur2 is a concept used in theoretical and educational discussions of hashed-space transforms. It combines a MurmurHash2–style hash with the square root operation to map integers into a compact, distributed range. It is not a standard mathematical function and implementations vary in the exact hash used and the output width.

Definition: Given a positive integer n and a hash function h with output in an integer range [0, M−1], sqrtmur2(n) is defined as floor(sqrt(h(n))). The hash function is assumed deterministic and independent of n’s arithmetic properties. A common choice uses a MurmurHash2–like function with 32- or 64-bit output. Example: if h(n) = (n * 2654435761) mod 2^32, then sqrtmur2(n) = floor(sqrt(h(n))).

Properties: The mapping is deterministic for a fixed h but not monotone with respect to n. It compresses the input space into the hash range and then into a smaller square-rooted range. The resulting values lie in 0..floor(sqrt(M−1)). The operation does not preserve order and is primarily used for dispersion rather than arithmetic exactness.

Computational aspects: Computing sqrtmur2(n) involves first evaluating h(n) and then computing an integer square root of the hash value. Time complexity is the sum of the hash computation and the square-root calculation, typically dominated by the hashing step. Memory usage is constant aside from any state needed by the hash function.

Applications: It is used in randomized algorithms, simulations, and data structures where a uniform-ish spread of identifiers into a compact bucket range is desirable, without relying on the full hash space. It should not be used where preserving order or exact arithmetic is required.

See also: MurmurHash2, integer square root, hash-based bucketing. References: foundational MurmurHash2 literature and general hashing texts.