Dynamic Dictionaries for Multisets and Counting Filters with Constant Time Operations

We resolve the open problem posed by Arbitman, Naor, and Segev [FOCS 2010] of designing a dynamic dictionary for multisets in the following setting: (1) The dictionary supports multiplicity queries and allows insertions and deletions to the multiset. (2) The dictionary is designed to support multisets of cardinality at most n (i.e., including multiplicities). (3) The space required for the dictionary is (1+o(1))·nlogun+Θ(n) bits, where u denotes the cardinality of the universe of the elements. This space is 1 + o(1) times the information-theoretic lower bound for static dictionaries over multisets of cardinality n if u= ω(n). (4) All operations are completed in constant time in the worst case with high probability in the word RAM model. A direct consequence of our construction is the first dynamic counting filter (i.e., a dynamic data structure that supports approximate multiplicity queries with a one-sided error) that, with high probability, supports operations in constant time and requires space that is 1 + o(1) times the information-theoretic lower bound for filters plus O(n) bits. The main technical component of our solution is based on efficiently storing variable-length bounded binary counters and its analysis via weighted balls-into-bins experiments in which the weight of a ball is logarithmic in its multiplicity.