Traditionally if one wants to sketch streams for Jaccard similarity hashing, one finds the minimum element in each of $k$ permutation for comparison purposes, and then takes number_of_collisions / $k$ as an estimate of Jaccard similarity between target streams. (There is additional work on how to compute these k-hashes more efficiently, or use a single bit from each, etc, but let's not get into that.)
My question is why not find the $k$ minimum elements instead in a single permutation? Especially since reduces the computation cost of $k$ hashes to just $1$. Is the required size of $k$ much larger, or does the analysis break?