scholarly journals The Cell Probe Complexity of Succinct Data Structures

2003 ◽  
Vol 10 (44) ◽  
Author(s):  
Anna Gál ◽  
Peter Bro Miltersen
Keyword(s):  
Data Structure ◽  
Lower Bound ◽  
Lower Bounds ◽  
Query Time ◽  
Polynomial Evaluation ◽  
Trade Off ◽  
Static Data ◽  
Structure Problem ◽  
Natural Data ◽  
Query Algorithm

In the cell probe model with word size 1 (the bit probe model), a static data structure problem is given by a map f : {0,1}^n * {0,1}^m -> {0,1}, where {0,1}^n is a set of possible data to be stored, {0,1}^m is a set of possible queries (for natural problems, we have m << n) and f(x,y) is the answer to question y about data x.<br /> <br />A solution is given by a representation phi : {0,1}^n -> {0,1}^s and a query algorithm q so that q(phi(x), y) = f(x,y). The time t of the query algorithm is the number of bits it reads in phi(x).<br /> <br />In this paper, we consider the case of <em>succinct</em> representations where s = n + r for some <em>redundancy</em> r << n. For a boolean version of the problem of polynomial evaluation with preprocessing of coefficients, we show a lower bound on the redundancy-query time trade-off of the form <br />(r + 1) t >= Omega(n/log n).<br /> In particular, for very small redundancies r, we get an almost optimal lower bound stating that the query algorithm has to inspect almost the entire data structure (up to a logarithmic factor). We show similar lower bounds for problems satisfying a certain combinatorial property of a coding theoretic flavor. Previously, no omega(m) lower bounds were known on t in the general model for explicit functions, even for very small redundancies.<br /> <br />By restricting our attention to <em>systematic</em> or <em>index</em> structures phi satisfying phi(x) = x · phi*(x) for some map phi* (where · denotes concatenation) we show similar lower bounds on the redundancy-query time trade-off for the natural data structuring problems of Prefix Sum and Substring Search.

scholarly journals On Data Structures and Asymmetric Communication Complexity

1994 ◽  
Vol 1 (41) ◽  
Author(s):  
Peter Bro Miltersen
Keyword(s):  
Data Structure ◽  
Lower Bound ◽  
Lower Bounds ◽  
Data Structures ◽  
Communication Complexity ◽  
Reduction Techniques ◽  
Static Data ◽  
Asymmetric Case ◽  
Asymmetric Communication

In this paper we consider two party communication complexity when the input sizes of the two players differ significantly, the ``asymmetric'' case. Most of previous work on communication complexity only considers the total number of bits sent, but we study tradeoffs between the number of bits the first player sends and the number of bits the second sends. These types of questions are closely related to the complexity of static data structure problems in the cell probe model. We derive two generally applicable methods of proving lower bounds, and obtain several applications. These applications include new lower bounds for data structures in the cell probe model. Of particular interest is our ``round elimination'' lemma, which is interesting also for the usual symmetric communication case. This lemma generalizes and abstracts in a very clean form the ``round reduction'' techniques used in many previous lower bound proofs.<em> </em>

scholarly journals Encoding range minima and range top-2 queries

2014 ◽  
Vol 372 (2016) ◽  
pp. 20130131 ◽  
Author(s):  
Pooya Davoodi ◽  
Gonzalo Navarro ◽  
Rajeev Raman ◽  
S. Srinivasa Rao
Keyword(s):  
Data Structure ◽  
Lower Bounds ◽  
Query Time ◽  
Constant Time ◽  
Worst Case ◽  
Asymptotically Optimal ◽  
Random Array ◽  
Case Data ◽  
Natural Way

We consider the problem of encoding range minimum queries (RMQs): given an array A [1.. n ] of distinct totally ordered values, to pre-process A and create a data structure that can answer the query RMQ( i , j ), which returns the index containing the smallest element in A [ i .. j ], without access to the array A at query time. We give a data structure whose space usage is 2 n + o ( n ) bits, which is asymptotically optimal for worst-case data, and answers RMQs in O (1) worst-case time. This matches the previous result of Fischer and Heun, but is obtained in a more natural way. Furthermore, our result can encode the RMQs of a random array A in 1.919 n + o ( n ) bits in expectation, which is not known to hold for Fischer and Heun’s result. We then generalize our result to the encoding range top-2 query (RT2Q) problem, which is like the encoding RMQ problem except that the query RT2Q( i , j ) returns the indices of both the smallest and second smallest elements of A [ i .. j ]. We introduce a data structure using 3.272 n + o ( n ) bits that answers RT2Qs in constant time, and also give lower bounds on the effective entropy of the RT2Q problem.

scholarly journals A New Trade-off for Deterministic Dictionaries

2000 ◽  
Vol 7 (4) ◽  
Author(s):  
Rasmus Pagh
Keyword(s):  
Data Structure ◽  
Linear Space ◽  
Unit Cost ◽  
Query Time ◽  
Instruction Set ◽  
Trade Off ◽  
Word Size ◽  
Membership Queries ◽  
Standard Instruction ◽  
The Universe

We consider dictionaries over the universe U = {0, 1}^w on a unit-cost<br />RAM with word size w and a standard instruction set. We present<br />a linear space deterministic dictionary with membership queries in<br />time (log log n)^O(1) and updates in time (log n)^O(1), where n is the size<br />of the set stored. This is the first such data structure to simultaneously<br />achieve query time (log n)^o(1) and update time O(2 ^sqrt log n) ).

IMPROVED POINTER MACHINE AND I/O LOWER BOUNDS FOR SIMPLEX RANGE REPORTING AND RELATED PROBLEMS

2013 ◽  
Vol 23 (04n05) ◽  
pp. 233-251 ◽  
Author(s):  
PEYMAN AFSHANI
Keyword(s):  
Data Structure ◽  
Lower Bound ◽  
Computational Geometry ◽  
Lower Bounds ◽  
Data Structures ◽  
Dimensional Space ◽  
Block Size ◽  
External Memory ◽  
Geometric Transformation ◽  
The Right

We investigate one of the fundamental areas in computational geometry: lower bounds for range reporting problems in the pointer machine and the external memory models. We develop new techniques that lead to new and improved lower bounds for simplex range reporting as well as some other geometric problems. Simplex range reporting is the problem of storing n points in the d-dimensional space in a data structure such that the k points that lie inside a query simplex can be found efficiently. This is one of the fundamental and extensively studied problems in computational geometry. Currently, the best data structures for the problem achieve Q(n) + O(k) query time using [Formula: see text] space in which the [Formula: see text] notation either hides a polylogarithmic or an nε factor for any constant ε > 0, (depending on the data structure and Q(n)). The best lower bound on this problem is due to Chazelle and Rosenberg who showed any pointer machine data structure that can answer queries in O(nγ + k) time must use Ω(nd-ε-dγ) space. Observe that this bound is a polynomial factor away from the best known data structures. In this article, we improve the space lower bound to [Formula: see text]. Not only this bridges the gap from polynomial to sub-polynomial, it also offers a smooth trade-off curve. For instance, for polylogarithmic values of Q(n), our space lower bound almost equals Ω((n/Q(n))d); the latter is generally believed to be the “right” bound. By a simple geometric transformation, we also improve the best lower bounds for the halfspace range reporting problem. Furthermore, we study the external memory model and offer a new simple framework for proving lower bounds in this model. We show that answering simplex range reporting queries with Q(n)+O(k/B) I/Os requires [Formula: see text]) space or [Formula: see text] blocks, in which B is the block size.

A Lower Bound for Schur Numbers and Multicolor Ramsey Numbers

10.37236/1188 ◽  
1994 ◽  
Vol 1 (1) ◽  
Author(s):  
Geoffrey Exoo
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Ramsey Numbers

For $k \geq 5$, we establish new lower bounds on the Schur numbers $S(k)$ and on the k-color Ramsey numbers of $K_3$.

scholarly journals A Lower Bound for the Query Phase of Contraction Hierarchies and Hub Labels and a Provably Optimal Instance-Based Schema

Algorithms ◽  
2021 ◽  
Vol 14 (6) ◽  
pp. 164
Author(s):  
Tobias Rupp ◽  
Stefan Funke
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Real World ◽  
Road Network ◽  
Upper And Lower Bounds ◽  
Bounded Degree ◽  
Shortest Path Routing ◽  
Graph Classes ◽  
Speed Up ◽  
To Come

We prove a Ω(n) lower bound on the query time for contraction hierarchies (CH) as well as hub labels, two popular speed-up techniques for shortest path routing. Our construction is based on a graph family not too far from subgraphs that occur in real-world road networks, in particular, it is planar and has a bounded degree. Additionally, we borrow ideas from our lower bound proof to come up with instance-based lower bounds for concrete road network instances of moderate size, reaching up to 96% of an upper bound given by a constructed CH. For a variant of our instance-based schema applied to some special graph classes, we can even show matching upper and lower bounds.

scholarly journals On Computing Multilinear Polynomials Using Multi- r -ic Depth Four Circuits

2021 ◽  
Vol 13 (3) ◽  
pp. 1-21
Author(s):  
Suryajith Chillara
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Complexity Measure ◽  
Natural Question ◽  
Multilinear Polynomial ◽  
Arithmetic Circuit ◽  
Theory Of Computing ◽  
Small Constant ◽  
Multilinear Polynomials ◽  
The Individual

In this article, we are interested in understanding the complexity of computing multilinear polynomials using depth four circuits in which the polynomial computed at every node has a bound on the individual degree of r ≥ 1 with respect to all its variables (referred to as multi- r -ic circuits). The goal of this study is to make progress towards proving superpolynomial lower bounds for general depth four circuits computing multilinear polynomials, by proving better bounds as the value of r increases. Recently, Kayal, Saha and Tavenas (Theory of Computing, 2018) showed that any depth four arithmetic circuit of bounded individual degree r computing an explicit multilinear polynomial on n O (1) variables and degree d must have size at least ( n / r 1.1 ) Ω(√ d / r ) . This bound, however, deteriorates as the value of r increases. It is a natural question to ask if we can prove a bound that does not deteriorate as the value of r increases, or a bound that holds for a larger regime of r . In this article, we prove a lower bound that does not deteriorate with increasing values of r , albeit for a specific instance of d = d ( n ) but for a wider range of r . Formally, for all large enough integers n and a small constant η, we show that there exists an explicit polynomial on n O (1) variables and degree Θ (log 2 n ) such that any depth four circuit of bounded individual degree r ≤ n η must have size at least exp(Ω(log 2 n )). This improvement is obtained by suitably adapting the complexity measure of Kayal et al. (Theory of Computing, 2018). This adaptation of the measure is inspired by the complexity measure used by Kayal et al. (SIAM J. Computing, 2017).

scholarly journals Lower bounds for the state complexity of probabilistic languages and the language of prime numbers

2020 ◽  
Vol 30 (1) ◽  
pp. 175-192
Author(s):  
NathanaËl Fijalkow
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Prime Numbers ◽  
Regular Languages ◽  
Automata Theory ◽  
Seminal Paper ◽  
Probabilistic Automata ◽  
State Complexity ◽  
Probabilistic Languages ◽  
Alternating Automata

Abstract This paper studies the complexity of languages of finite words using automata theory. To go beyond the class of regular languages, we consider infinite automata and the notion of state complexity defined by Karp. Motivated by the seminal paper of Rabin from 1963 introducing probabilistic automata, we study the (deterministic) state complexity of probabilistic languages and prove that probabilistic languages can have arbitrarily high deterministic state complexity. We then look at alternating automata as introduced by Chandra, Kozen and Stockmeyer: such machines run independent computations on the word and gather their answers through boolean combinations. We devise a lower bound technique relying on boundedly generated lattices of languages, and give two applications of this technique. The first is a hierarchy theorem, stating that there are languages of arbitrarily high polynomial alternating state complexity, and the second is a linear lower bound on the alternating state complexity of the prime numbers written in binary. This second result strengthens a result of Hartmanis and Shank from 1968, which implies an exponentially worse lower bound for the same model.

scholarly journals Encoding Two-Dimensional Range Top-k Queries

Algorithmica ◽  
2021 ◽  
Author(s):  
Seungbum Jo ◽  
Rahul Lingala ◽  
Srinivasa Rao Satti
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Total Order ◽  
Two Dimensional ◽  
Information Theoretic ◽  
Cartesian Tree ◽  
Dimensional Range

AbstractWe consider the problem of encoding two-dimensional arrays, whose elements come from a total order, for answering $${\text{Top-}}{k}$$ Top- k queries. The aim is to obtain encodings that use space close to the information-theoretic lower bound, which can be constructed efficiently. For an $$m \times n$$ m × n array, with $$m \le n$$ m ≤ n , we first propose an encoding for answering 1-sided $${\textsf {Top}}{\text {-}}k{}$$ Top - k queries, whose query range is restricted to $$[1 \dots m][1 \dots a]$$ [ 1 ⋯ m ] [ 1 ⋯ a ] , for $$1 \le a \le n$$ 1 ≤ a ≤ n . Next, we propose an encoding for answering for the general (4-sided) $${\textsf {Top}}{\text {-}}k{}$$ Top - k queries that takes $$(m\lg {{(k+1)n \atopwithdelims ()n}}+2nm(m-1)+o(n))$$ ( m lg ( k + 1 ) n n + 2 n m ( m - 1 ) + o ( n ) ) bits, which generalizes the joint Cartesian tree of Golin et al. [TCS 2016]. Compared with trivial $$O(nm\lg {n})$$ O ( n m lg n ) -bit encoding, our encoding takes less space when $$m = o(\lg {n})$$ m = o ( lg n ) . In addition to the upper bound results for the encodings, we also give lower bounds on encodings for answering 1 and 4-sided $${\textsf {Top}}{\text {-}}k{}$$ Top - k queries, which show that our upper bound results are almost optimal.

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