We consider a new construction of locality-sensitive hash functions for Hamming space that is
covering
in the sense that is it guaranteed to produce a collision for every pair of vectors within a given radius
r
. The construction is
efficient
in the sense that the expected number of hash collisions between vectors at distance
cr
, for a given
c
>1, comes close to that of the best possible data independent LSH without the covering guarantee, namely, the seminal LSH construction of Indyk and Motwani (STOC’98). The efficiency of the new construction essentially
matches
their bound when the search radius is not too large—e.g., when
cr
=
o
(log (
n
)/ log log
n
), where
n
is the number of points in the dataset, and when
cr
= log (
n
)/
k
, where
k
is an integer constant. In general, it differs by at most a factor ln (4) in the exponent of the time bounds. As a consequence, LSH-based similarity search in Hamming space can avoid the problem of false negatives at little or no cost in efficiency.
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