  MinHash and SimHash are the two widely adopted Locality Sensitive Hashing
(LSH) algorithms for large-scale data processing applications. Deciding which
LSH to use for a particular problem at hand is an important question, which has
no clear answer in the existing literature. In this study, we provide a
theoretical answer (validated by experiments) that MinHash virtually always
outperforms SimHash when the data are binary, as common in practice such as
search.
  The collision probability of MinHash is a function of resemblance similarity
($\mathcal{R}$), while the collision probability of SimHash is a function of
cosine similarity ($\mathcal{S}$). To provide a common basis for comparison, we
evaluate retrieval results in terms of $\mathcal{S}$ for both MinHash and
SimHash. This evaluation is valid as we can prove that MinHash is a valid LSH
with respect to $\mathcal{S}$, by using a general inequality $\mathcal{S}^2\leq
\mathcal{R}\leq \frac{\mathcal{S}}{2-\mathcal{S}}$. Our worst case analysis can
show that MinHash significantly outperforms SimHash in high similarity region.
  Interestingly, our intensive experiments reveal that MinHash is also
substantially better than SimHash even in datasets where most of the data
points are not too similar to each other. This is partly because, in practical
data, often $\mathcal{R}\geq \frac{\mathcal{S}}{z-\mathcal{S}}$ holds where $z$
is only slightly larger than 2 (e.g., $z\leq 2.1$). Our restricted worst case
analysis by assuming $\frac{\mathcal{S}}{z-\mathcal{S}}\leq \mathcal{R}\leq
\frac{\mathcal{S}}{2-\mathcal{S}}$ shows that MinHash indeed significantly
outperforms SimHash even in low similarity region.
  We believe the results in this paper will provide valuable guidelines for
search in practice, especially when the data are sparse.
