Abstract
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 [Pagh et al., 2013] 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 (1o(1))n(log(1/ε)+(1O(ε))log log n) bits, in contrast to the wellknown 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 worstcase constant lookup time.
In this work, we present a filter data structure with improvements in two aspects:
 it has constant worstcase 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))nlog(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.
BibTeX  Entry
@InProceedings{liu_et_al:LIPIcs:2020:12486,
author = {Mingmou Liu and Yitong Yin and Huacheng Yu},
title = {{Succinct Filters for Sets of Unknown Sizes}},
booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
pages = {79:179:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771382},
ISSN = {18688969},
year = {2020},
volume = {168},
editor = {Artur Czumaj and Anuj Dawar and Emanuela Merelli},
publisher = {Schloss DagstuhlLeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12486},
URN = {urn:nbn:de:0030drops124867},
doi = {10.4230/LIPIcs.ICALP.2020.79},
annote = {Keywords: Bloom filters, Data structures, Approximate set membership, Dictionaries}
}
Keywords: 

Bloom filters, Data structures, Approximate set membership, Dictionaries 
Collection: 

47th International Colloquium on Automata, Languages, and Programming (ICALP 2020) 
Issue Date: 

2020 
Date of publication: 

29.06.2020 