eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-18
28:1
28:12
10.4230/LIPIcs.MFCS.2020.28
article
Improved Explicit Data Structures in the Bit-Probe Model Using Error-Correcting Codes
Dey, Palash
1
Radhakrishnan, Jaikumar
2
Velusamy, Santhoshini
3
Department of Computer Science and Engineering, Indian Institute of Technology Kharagpur, India
School of Technology and Computer Science, Tata Institute of Fundamental Research, Mumbai, India
School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA
We consider the bit-probe complexity of the set membership problem: represent an n-element subset S of an m-element universe as a succinct bit vector so that membership queries of the form "Is x ∈ S" can be answered using at most t probes into the bit vector. Let s(m,n,t) (resp. s_N(m,n,t)) denote the minimum number of bits of storage needed when the probes are adaptive (resp. non-adaptive). Lewenstein, Munro, Nicholson, and Raman (ESA 2014) obtain fully-explicit schemes that show that
s(m,n,t) = 𝒪((2^t-1)m^{1/(t - min{2⌊log n⌋, n-3/2})}) for n ≥ 2,t ≥ ⌊log n⌋+1 .
In this work, we improve this bound when the probes are allowed to be superlinear in n, i.e., when t ≥ Ω(nlog n), n ≥ 2, we design fully-explicit schemes that show that
s(m,n,t) = 𝒪((2^t-1)m^{1/(t-{n-1}/{2^{t/(2(n-1))}})}),
asymptotically (in the exponent of m) close to the non-explicit upper bound on s(m,n,t) derived by Radhakrishan, Shah, and Shannigrahi (ESA 2010), for constant n.
In the non-adaptive setting, it was shown by Garg and Radhakrishnan (STACS 2017) that for a large constant n₀, for n ≥ n₀, s_N(m,n,3) ≥ √{mn}. We improve this result by showing that the same lower bound holds even for storing sets of size 2, i.e., s_N(m,2,3) ≥ Ω(√m).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol170-mfcs2020/LIPIcs.MFCS.2020.28/LIPIcs.MFCS.2020.28.pdf
Set membership
Bit-probe model
Fully-explicit data structures
Adaptive data structures
Error-correcting codes