NDLI: Bit-Probe Lower Bounds for Succinct Data Structures

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Viola, Emanuele
Copyright Year	2012
Abstract	We prove lower bounds on the redundancy necessary to represent a set $S$ of objects using a number of bits close to the information-theoretic minimum $\log_2 \|S\|$, while answering various queries by probing few bits. Our main results are as follows: (i) To represent $n$ ternary values $t \in \{0,1,2\}^n$ in terms of $u$ bits $b \in \{0, 1\}^u$ while accessing a single value $t_i \in \{0,1,2\}$ by probing $q$ bits of $b$, one needs $u \geq (\log_2 3)n + n/2^{O(q)}$. This matches an exciting representation by Pǎtraşcu (FOCS 2008), later refined with Dodis and Thorup (STOC 2010), where $u \leq (\log_2 3)n + n/2^{\Omega(q)}$. We also note that results on logarithmic forms imply the lower bound $u \geq (\log_2 3)n + n/\log^{O(1)} n$ if we access $t_i$ by probing one cell of $\log n$ bits. (ii) To represent sets of size $n/3$ from a universe of $n$ elements in terms of $u$ bits $b \in \{0, 1\}^u$ while answering membership queries by probing $q$ bits of $b$, one needs $u \geq \log_2 \binom{n}{n/3} + n/2^{O(q)} - \log n$. Both results hold even if the probe locations are determined adaptively. Ours are the first lower bounds for these fundamental problems; we obtain them by drawing on ideas used in a lower bound for locally decodable codes by Shaltiel and the author [SIAM J. Comput., 39 (2010), pp. 3122--3154].
Starting Page	1593
Ending Page	1604
Page Count	12
File Format	PDF
ISSN	00975397
DOI	10.1137/090766619
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	6 (Special Section on the Forty-First Annual ACM Symposium on Theory of Computing (STOC 2009))
Volume Number	41
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2012-12-18
Access Restriction	Subscribed
Subject Keyword	succinct data structure Computational difficulty of problems logarithmic forms cell-probe Data structures bit-probe membership lower bound nonbinary array
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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