Space Complexity of Two Adaptive Bitprobe Schemes Storing Three Elements

We consider the following set membership problem in the bitprobe model - that of storing subsets of size at most three from a universe of size m, and answering membership queries using two adaptive bitprobes. Baig and Kesh [Mirza Galib Anwarul Husain Baig and Deepanjan Kesh, 2018] proposed a scheme for the problem which takes O(m^{2/3}) space. In this paper, we present a proof which shows that any scheme for the problem requires Omega(m^{2/3}) amount of space. These two results together settle the space complexity issue for this particular problem

Improved Explicit Data Structures in the Bit-Probe Model Using Error-Correcting Codes

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) = ?((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 ? ?(nlog n), n ? 2, we design fully-explicit schemes that show that s(m,n,t) = ?((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 n?, for n ? n?, 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) ? ?(?m)

Data structures for storing small sets in the bitprobe model

We study the following set membership problem in the bit probe model: given a set S from a finite universe U, represent it in memory so that membership queries of the form "Is x in S?" can be answered with a small number of bitprobes. We obtain explicit schemes that come close to the information theoretic lower bound of Buhrman et al. [STOC 2000, SICOMP 2002] and improve the results of Radhakrishnan et al. [ESA 2001] when the size of sets and the number of probes is small. We show that any scheme that stores sets of size two from a universe of size m and answers membership queries using two bitprobes requires space Ω(m4/7). The previous best lower bound (shown by Buhrman et al. using information theoretic arguments) was Ω(√m). The same lower bound applies for larger sets using standard padding arguments. This is the first instance where the information theoretic lower bound is found to be not tight for adaptive schemes. We show that any non-adaptive three probe scheme for storing sets of size two from a universe of size m requires Ω(√m) bits of memory. This extends a result of Alon and Feige [SODA 2009] to small sets

Space-Efficient Data Structures in the Word-RAM and Bitprobe Models

This thesis studies data structures in the word-RAM and bitprobe models, with an emphasis on space efficiency. In the word-RAM model of computation the space cost of a data structure is measured in terms of the number of w-bit words stored in memory, and the cost of answering a query is measured in terms of the number of read, write, and arithmetic operations that must be performed. In the bitprobe model, like the word-RAM model, the space cost is measured in terms of the number of bits stored in memory, but the query cost is measured solely in terms of the number of bit accesses, or probes, that are performed. First, we examine the problem of succinctly representing a partially ordered set, or poset, in the word-RAM model with word size Theta(lg n) bits. A succinct representation of a combinatorial object is one that occupies space matching the information theoretic lower bound to within lower order terms. We show how to represent a poset on n vertices using a data structure that occupies n^2/4 + o(n^2) bits, and can answer precedence (i.e., less-than) queries in constant time. Since the transitive closure of a directed acyclic graph is a poset, this implies that we can support reachability queries on an arbitrary directed graph in the same space bound. As far as we are aware, this is the first representation of an arbitrary directed graph that supports reachability queries in constant time, and stores less than n choose 2 bits. We also consider several additional query operations. Second, we examine the problem of supporting range queries on strings of n characters (or, equivalently, arrays of n elements) in the word-RAM model with word size Theta(lg n) bits. We focus on the specific problem of answering range majority queries: i.e., given a range, report the character that is the majority among those in the range, if one exists. We show that these queries can be supported in constant time using a linear space (in words) data structure. We generalize this result in several directions, considering various frequency thresholds, geometric variants of the problem, and dynamism. These results are in stark contrast to recent work on the similar range mode problem, in which the query operation asks for the mode (i.e., most frequent) character in a given range. The current best data structures for the range mode problem take soft-Oh(n^(1/2)) time per query for linear space data structures. Third, we examine the deterministic membership (or dictionary) problem in the bitprobe model. This problem asks us to store a set of n elements drawn from a universe [1,u] such that membership queries can be always answered in t bit probes. We present several new fully explicit results for this problem, in particular for the case when n = 2, answering an open problem posed by Radhakrishnan, Shah, and Shannigrahi [ESA 2010]. We also present a general strategy for the membership problem that can be used to solve many related fundamental problems, such as rank, counting, and emptiness queries. Finally, we conclude with a list of open problems and avenues for future work

Set Membership with Non-Adaptive Bit Probes

We consider the non-adaptive bit-probe complexity of the set membership problem, where a set S of size at most n from a universe of size m is to be represented as a short bit vector in order to answer membership queries of the form "Is x in S?" by non-adaptively probing the bit vector at t places. Let s_N(m,n,t) be the minimum number of bits of storage needed for such a scheme. In this work, we show existence of non-adaptive and adaptive schemes for a range of t that improves an upper bound of Buhrman, Miltersen, Radhakrishnan and Srinivasan (2002) on s_N(m,n,t). For three non-adaptive probes, we improve the previous best lower bound on s_N(m,n,3) by Alon and Feige (2009)