[3wolf]

A minor quibble with your use-case explanation: The advantage of a bloom filter isn't strictly time complexity. For example, a hash table would also have constant lookup time (best case), and would give a definitive answer on set membership. However, to store 1 million IPv6 addresses would take 16 MB. You can see very quickly that this would not scale very well to, say, a billion addresses stored in-memory on a laptop. With a bloom filter, we can shrink the amount of storage space required* while maintaining an acceptable, calculable false positive rate.

* IP addresses actually aren't a great use case for basic bloom filters, as they're fairly storage efficient to begin with, as opposed to a url for example. Taking your example, say we need to store 1 million IP addresses in our bloom filter and we're okay with a ~1% false positive rate. Well then, if we use a bloom filter with 2^23 bits (1 MB), the optimal number of hash functions is (2^23)/(10^6)*ln(2) = 6, yielding a false positive rate of (1 - exp(-6* 10^6 /2^23))^6 = ~1.8%. So we're using 6% of the storage space, but with a nearly 2% false positive rate.