TY - GEN

T1 - Succinct filters for sets of unknown sizes

AU - Liu, Mingmou

AU - Yin, Yitong

AU - Yu, Huacheng

N1 - Funding Information: Funding Mingmou Liu: Part of the research was done when Mingmou Liu was visiting the Princeton University. Mingmou Liu is supported by National Key R&D Program of China 2018YFB1003202 and NSFC under Grant Nos. 61722207 and 61672275. Yitong Yin: Yitong Yin is supported by National Key R&D Program of China 2018YFB1003202 and NSFC under Grant Nos. 61722207 and 61672275. Funding Information: Mingmou Liu: Part of the research was done when Mingmou Liu was visiting the Princeton University. Mingmou Liu is supported by National Key R&D Program of China 2018YFB1003202 and NSFC under Grant Nos. 61722207 and 61672275. Yitong Yin: Yitong Yin is supported by National Key R&D Program of China 2018YFB1003202 and NSFC under Grant Nos. 61722207 and 61672275. Publisher Copyright: © Mingmou Liu, Yitong Yin, and Huacheng Yu; licensed under Creative Commons License CC-BY 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020).

PY - 2020/6/1

Y1 - 2020/6/1

N2 - 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 > u0.001, 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 > u0.001, 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.

AB - 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 > u0.001, 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 > u0.001, 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.

KW - Approximate set membership

KW - Bloom filters

KW - Data structures

KW - Dictionaries

UR - http://www.scopus.com/inward/record.url?scp=85089348461&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85089348461&partnerID=8YFLogxK

U2 - https://doi.org/10.4230/LIPIcs.ICALP.2020.79

DO - https://doi.org/10.4230/LIPIcs.ICALP.2020.79

M3 - Conference contribution

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020

A2 - Czumaj, Artur

A2 - Dawar, Anuj

A2 - Merelli, Emanuela

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020

Y2 - 8 July 2020 through 11 July 2020

ER -