How many hash functions are required in a minhash algorithm

Question

I am keen to try and implement minhashing to find near duplicate content. http://blog.cluster-text.com/tag/minhash/ has a nice write up, but there the question of just how m

Answer 1

One way to generate 200 hash values is to generate one hash value using a good hash algorithm and generate 199 values cheaply by XORing the good hash value with 199 sets of random-looking bits having the same length as the good hash value (i.e. if your good hash is 32 bits, build a list of 199 32-bit pseudo random integers and XOR each good hash with each of the 199 random integers).

Do not simply rotate bits to generate hash values cheaply if you are using unsigned integers (signed integers are fine) -- that will often pick the same shingle over and over. Rotating the bits down by one is the same as dividing by 2 and copying the old low bit into the new high bit location. Roughly 50% of the good hash values will have a 1 in the low bit, so they will have huge hash values with no prayer of being the minimum hash when that low bit rotates into the high bit location. The other 50% of the good hash values will simply equal their original values divided by 2 when you shift by one bit. Dividing by 2 does not change which value is smallest. So, if the shingle that gave the minimum hash with the good hash function happens to have a 0 in the low bit (50% chance of that) it will again give the minimum hash value when you shift by one bit. As an extreme example, if the shingle with the smallest hash value from the good hash function happens to have a hash value of 0, it will always have the minimum hash value no matter how much you rotate the bits. This problem does not occur with signed integers because minimum hash values have extreme negative values, so they tend to have a 1 at the highest bit followed by zeros (100...). So, only hash values with a 1 in the lowest bit will have a chance at being the new lowest hash value after rotating down by one bit. If the shingle with minimum hash value has a 1 in the lowest bit, after rotating down one bit it will look like 1100..., so it will almost certainly be beat out by a different shingle that has a value like 10... after the rotation, and the problem of the same shingle being picked twice in a row with 50% probability is avoided.