Here are a few variants and extensions of Bloom filters:

1. Counting Bloom Filters

Counting Bloom filters extend the standard Bloom filter by using an array of counters instead of a simple bit array. This modification allows for the deletion of elements from the filter, as each counter can be incremented or decremented when elements are added or removed, respectively. However, this added functionality comes at the cost of increased storage space and complexity.

2. Compressed Bloom Filters

Compressed Bloom filters aim to reduce the storage overhead of Bloom filters by compressing the underlying bit array. Several compression techniques, such as run-length encoding or Golomb coding, can be applied to achieve a more compact representation of the filter. However, these techniques may introduce additional computational overhead during insertion and query operations.

3. Spectral Bloom Filters

Spectral Bloom filters are designed to estimate the frequency of elements in a dataset. This variant uses multiple standard Bloom filters in parallel, each representing a different frequency range. By analyzing the presence of an element across these filters, the frequency of the element can be approximated. Spectral Bloom filters can be useful in applications such as data mining or network traffic analysis.

4. Scalable Bloom Filters

Scalable Bloom filters address the issue of dynamically growing datasets by automatically adjusting the filter's size and parameters as the number of elements increases. This variant maintains a series of Bloom filters, each with different parameters, and new filters are added as required. Scalable Bloom filters can maintain a target false positive rate while accommodating an unpredictable number of elements.

5. Cuckoo Filters

Cuckoo filters are a more recent variant of Bloom filters, designed to provide similar functionality with improved space efficiency and support for element removal. Cuckoo filters use a combination of cuckoo hashing and a compact fingerprint representation of elements to achieve these benefits. In many scenarios, cuckoo filters can outperform standard Bloom filters in terms of space efficiency and overall performance.