Succinct filters for sets of unknown sizes

The membership problem asks to maintain a set S ⊆ [u], supporting insertions and membership queries, i.e., testing if a given element is in the set. A data structure that computes exact answers is called a dictionary. When a (small) false positive rate ε is allowed, the data structure is called a filter. The space usages of the standard dictionaries or filters usually depend on the upper bound on the size of S, while the actual set can be much smaller. Pagh, Segev and Wieder [28] were the first to study filters with varying space usage based on the current |S|. They showed in order to match the space with the current set size n = |S|, any filter data structure must use (1 − o(1))n(log(1/ε) + (1 − O(ε)) log log n) bits, in contrast to the well-known lower bound of N log(1/ε) bits, where N is an upper bound on |S|. They also presented a data structure with almost optimal space of (1 + o(1))n(log(1/ε) + O(log log n)) bits provided that n > u^0.001, with expected amortized constant insertion time and worst-case constant lookup time. In this work, we present a filter data structure with improvements in two aspects: it has constant worst-case time for all insertions and lookups with high probability; it uses space (1 + o(1))n(log(1/ε) + log log n) bits when n > u^0.001, achieving optimal leading constant for all ε = o(1). We also present a dictionary that uses (1 + o(1))n log(u/n) bits of space, matching the optimal space in terms of the current size, and performs all operations in constant time with high probability.