# A Dynamic Space-Efficient Filter with Constant Time Operations

A dynamic dictionary is a data structure that maintains sets of cardinality at most n from a given universe and supports insertions, deletions, and membership queries. A filter approximates membership queries with a one-sided error that occurs with probability at most ϵ. The goal is to obtain dynamic filters that are space-efficient (the space is 1+o(1) times the information-theoretic lower bound) and support all operations in constant time with high probability. One approach to designing filters is to reduce to the retrieval problem. When the size of the universe is polynomial in n, this approach yields a space-efficient dynamic filter as long as the error parameter ϵ satisfies log(1/ϵ) = ω(loglog n). For the case that log(1/ϵ) = O(loglog n), we present the first space-efficient dynamic filter with constant time operations in the worst case (whp). In contrast, the space-efficient dynamic filter of Pagh, Pagh, Rao (SODA 2005) supports insertions and deletions in amortized expected constant time. Our approach employs the classic reduction of Carter et al. (STOC 1978) on a new type of dictionary construction that supports random multisets.

READ FULL TEXT