Set query is a fundamental problem in computer systems. Plenty of applications rely on the query results of membership, association, and multiplicity. A traditional method that addresses such a fundamental problem is derived from Bloom filter. However, such methods may fail to support element deletion, require additional filters or apriori knowledge, making them unamenable to a high-performance implementation for dynamic set representation and query. In this paper, we envision a novel sketch framework that is multi-functional, non-parametric, space efficient, and deletable. As far as we know, none of the existing designs can guarantee such features simultaneously. To this end, we present a general shifting framework to represent auxiliary information (such as multiplicity, association) with the offset. Thereafter, we specify such design philosophy for a hash table horizontally at the slot level, as well as vertically at the bucket level. Theoretical and experimental results jointly demonstrate that our design works exceptionally well with three types of set queries under small memory.

Sketches are widely deployed to represent network flows to support complex flow analysis. Typical sketches usually employ hash functions to map elements into a hash table or bit array. Such sketches still suffer from potential weaknesses upon throughput, flexibility, and functionality. To this end, we propose Ark filter, a novel sketch that stores the element information with either of two candidate buckets indexed by the quotient or remainder between the fingerprint and filter length. In this way, no further hash calculations are required for future queries or reallocations. We further extend the Ark filter to enable capacity elasticity and more functionalities (such as frequency estimation and top-$k$query). Comprehensive experiments demonstrate that, compared with Cuckoo filter, Ark filter has$2.08\times$,$1.34\times$, and$1.68\times$throughput of deletion, insertion, and hybrid query, respectively; compared with Quotient filter, Ark filter has$4.55\times$,$1.74\times$, and$22.12\times$throughput of deletion, insertion, and hybrid query, respectively; compared with Bloom filter, Ark filter has$2.55\times$and$2.11\times$throughput of insertion and hybrid query, respectively.

Cuckoo filter (CF) and its variants are emerging as replacements of Bloom filters in various networking and distributed systems to support efficient set representation and membership testing. Cuckoo filters store item fingerprints directly with two candidate buckets and a reallocation scheme is implemented to mitigate the bucket overflow problem for higher space utilization. Such a reallocation scheme, once triggered, however, can be time-consuming. This shortcoming makes the existing CFs not applicable for insertion-intensive scenarios such as online applications wherein the items join and leave frequently. To this end, in this paper, we propose the Vertical Cuckoo filter (VCF) which extends the standard Cuckoo filter by providing more candidate buckets to each item. Another challenging issue with such a design is how to ensure that the candidate buckets can be indexed by each other such that no additional hash computation and item access are necessary during fingerprint reallocation. Therefore, we present the vertical hashing, which indexes the candidate buckets with the fingerprint and given bitmasks. We further generalize and improve the VCF by realizing$k$(≥ 4) candidate buckets and avoiding unnecessary computation. The comprehensive experiments indicate that VCF outperforms its same kinds in terms of space utilization and insertion throughput, with a slight compromise of lookup speed.