Cuckoo filter

A cuckoo filter uses a ${\displaystyle n}$-way set-associative hash table based on cuckoo hashing to store the fingerprints of all items (every bucket of the hashtable can store up to ${\displaystyle n}$ entries). Particularly, the two potential buckets in the table for a given item ${\displaystyle x}$ required by cuckoo hashing are calculated by the following two hash functions (termed as partial-key cuckoo hashing[2]):

${\displaystyle h_{1}(x)={\text{hash}}(x)}$

${\displaystyle h_{2}(x)=h_{1}(x)\oplus {\text{hash}}({\text{fingerprint}}(x))}$

Applying the above two hash functions to construct a cuckoo hash table enables item relocation based only on fingerprints when retrieving the original item is impossible. As a result, when inserting a new item that requires relocating an existing item ${\displaystyle y}$, the other possible location ${\displaystyle j}$ in the table for this item ${\displaystyle y}$ kicked out from bucket ${\displaystyle i}$ is calculated by

${\displaystyle j=i\oplus {\text{hash}}({\text{fingerprint}}(y))}$

Based on partial-key cuckoo hashing, the hash table can achieve both highly-utilization (thanks to cuckoo hashing), and compactness because only fingerprints are stored. Lookup and delete operations of a cuckoo filter are straightforward. There are a maximum of two locations to check by ${\displaystyle h_{1}(x)}$ and ${\displaystyle h_{2}(x)}$. If found, the appropriate lookup or delete operation can be performed in ${\displaystyle O(1)}$ time. More theoretical analysis of cuckoo filters can be found in the literature.[3][4]

A cuckoo filter is similar to a Bloom filter in that they both are very fast and compact, and they may both return false positives as answers to set-membership queries:

• Space-optimal Bloom filters use ${\displaystyle 1.44\log _{2}(1/\epsilon )}$ bits of space per inserted key, where ${\displaystyle \epsilon }$ is the false positive rate. A cuckoo filter requires ${\displaystyle (\log _{2}(1/\epsilon )+2)/\alpha }$ where ${\displaystyle \alpha }$ is the hash table load factor, which can be ${\displaystyle 95.5\%}$ based on the cuckoo filter's setting. Note that the information theoretical lower bound requires ${\displaystyle \log _{2}(1/\epsilon )}$ bits for each item.
• On a positive lookup, a space-optimal Bloom filter requires a constant ${\displaystyle \log _{2}(1/\epsilon )}$ memory accesses into the bit array, whereas a cuckoo filter requires constant two lookups at most.
• Cuckoo filters have degraded insertion speed after reaching a load threshold, when table expanding is recommended. In contrast, Bloom filters can keep inserting new items at the cost of a higher false positive rate before expansion.

• A cuckoo filter can only delete items that are known to be inserted before.
• Insertion can fail and rehashing is required like other cuckoo hash tables. Note that the amortized insertion complexity is still ${\displaystyle O(1)}$.[5]

• Probabilistic Filters By Example – A tutorial comparing Cuckoo and Bloom filters.