  A Bloom filter is a memory-efficient data structure for approximate
membership queries used in numerous fields of computer science. Recently,
learned Bloom filters that achieve better memory efficiency using machine
learning models have attracted attention. One such filter, the partitioned
learned Bloom filter (PLBF), achieves excellent memory efficiency. However,
PLBF requires a $O(N^3k)$ time complexity to construct the data structure,
where $N$ and $k$ are the hyperparameters of PLBF. One can improve memory
efficiency by increasing $N$, but the construction time becomes extremely long.
Thus, we propose two methods that can reduce the construction time while
maintaining the memory efficiency of PLBF. First, we propose fast PLBF, which
can construct the same data structure as PLBF with a smaller time complexity
$O(N^2k)$. Second, we propose fast PLBF++, which can construct the data
structure with even smaller time complexity $O(Nk\log N + Nk^2)$. Fast PLBF++
does not necessarily construct the same data structure as PLBF. Still, it is
almost as memory efficient as PLBF, and it is proved that fast PLBF++ has the
same data structure as PLBF when the distribution satisfies a certain
constraint. Our experimental results from real-world datasets show that (i)
fast PLBF and fast PLBF++ can construct the data structure up to 233 and 761
times faster than PLBF, (ii) fast PLBF can achieve the same memory efficiency
as PLBF, and (iii) fast PLBF++ can achieve almost the same memory efficiency as
PLBF. The codes are available at https://github.com/atsukisato/FastPLBF .
