The Cell Probe Complexity of Succinct Data Structures

Gál, Anna; Miltersen, Peter Bro

doi:10.1007/3-540-45061-0_28

Cited by 42 publications

(61 citation statements)

References 26 publications

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“…In this model, the array A must be represented as is, but in addition the data structure may store an index of sublinear size, which the query algorithm can examine at no cost. See [GM03,Mil05,GRR08,Gol07] for increasingly tight lower bounds in this model. Note, however, that in the systematic model, the best achievable redundancy with query time t is n t·poly log n , i.e.…”

Section: Introductionmentioning

confidence: 99%

Cell-Probe Lower Bounds for Succinct Partial Sums

Pătraşcu¹,

Viola²

2010

Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms

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The partial sums problem in succinct data structures asks to preprocess an array A[1 . . n] of bits into a data structure using as close to n bits as possible, and answer queries of the formThe problem has been intensely studied, and features as a subroutine in a number of succinct data structures.We show that, if we answer Rank(k) queries by probing t cells of w bits, then the space of the data structure must be at least n + n/w O(t) bits. This redundancy/probe trade-off is essentially optimal: Patrascu [FOCS'08] showed how to achieve n + n (w/t) Ω(t) bits. We also extend our lower bound to the closely related Select queries, and to the case of sparse arrays.

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Section: Introductionmentioning

confidence: 99%

Cell-Probe Lower Bounds for Succinct Partial Sums

Pătraşcu¹,

Viola²

2010

Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms

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“…Setting t = O(1), we get redundancy n/ log O(1) n with constant query time. For a growing body of problems, we have strong redundancy lower bounds of Ω(n/t) [GM03], Ω(n/t 2 ) [Gol09], or n/ log O(t) n [PV10]. Combining this with the non-locality of known optimal encodings (e.g., arithmetic coding described in Section 1.3), one may conjecture that any representation of a vector will require some nontrivial trade-off between the redundancy and query time.…”

Section: Representing a Vectormentioning

confidence: 99%

Changing base without losing space

Dodis

Pătraşcu

Thorup

2010

Proceedings of the Forty-Second ACM Symposium on Theory of Computing

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We describe a simple, but powerful local encoding technique, implying two surprising results: 1. We show how to represent a vector of n values from Σ using ⌈n log 2 Σ⌉ bits, such that reading or writing any entry takes O(1) time. This demonstrates, for instance, an "equivalence" between decimal and binary computers, and has been a central toy problem in the field of succinct data structures. Previous solutions required space of n log 2 Σ + n/ log O(1) n bits for constant access. 2. Given a stream of n bits arriving online (for any n, not known in advance), we can output a prefix-free encoding that uses n + log 2 n + O(log log n) bits. The encoding and decoding algorithms only require O(log n) bits of memory, and run in constant time per word. This result is interesting in cryptographic applications, as prefix-free codes are the simplest counter-measure to extensions attacks on hash functions, message authentication codes and pseudorandom functions. Our result refutes a conjecture of [Maurer, Sjödin 2005] on the hardness of online prefix-free encodings.

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“…Miltersen [Mil95] proves that the trivial algorithm (with query complexity n) is essentially optimal when the field size is exponentially large and the data structure is limited to polynomial size, and he conjectures that this lower bound holds for smaller fields as well (this is in an algebraic model that does not permit the modular operations we employ). Finally, Gál and Miltersen [GM07] show a lower bound of Ω(n/ log n) on the product of the additive redundancy (in the data structure size) and the query complexity, thus exhibiting a tradeoff that rules out low query complexity when the data structure is required to be very small (i.e., significantly smaller than 2n).…”

Section: A Data Structure For Polynomial Evaluationmentioning

confidence: 99%

Fast Modular Composition in any Characteristic

Kedlaya

Umans

2008

2008 49th Annual IEEE Symposium on Foundations of Computer Science

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We give an algorithm for modular composition of degree n univariate polynomials over a finite field F q requiring n 1+o(1) log 1+o(1) q bit operations; this had earlier been achieved in characteristic n o(1) by Umans (2008). As an application, we obtain a randomized algorithm for factoring degree n polynomials over F q requiring (n 1.5+o(1) + n 1+o(1) log q) log 1+o(1) q bit operations, improving upon the methods of von zur Gathen & Shoup (1992) and Kaltofen & Shoup (1998). Our results also imply algorithms for irreducibility testing and computing minimal polynomials whose running times are best-possible, up to lower order terms. As in Umans (2008), we reduce modular composition to certain instances of multipoint evaluation of multivariate polynomials. We then give an algorithm that solves this problem optimally (up to lower order terms), in arbitrary characteristic. The main idea is to lift to characteristic 0, apply a small number of rounds of multimodular reduction, and finish with a small number of multidimensional FFTs. The final evaluations are then reconstructed using the Chinese Remainder Theorem. As a bonus, we obtain a very efficient data structure supporting polynomial evaluation queries, which is of independent interest.Our algorithm uses techniques which are commonly employed in practice, so it may be competitive for real problem sizes. This contrasts with previous asymptotically fast methods relying on fast matrix multiplication.

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The Cell Probe Complexity of Succinct Data Structures

Abstract: Abstract. We show lower bounds in the cell probe model for the redundancy/query time tradeoff of solutions to static data structure problems.

Cited by 42 publications

References 26 publications

Cell-Probe Lower Bounds for Succinct Partial Sums

Cell-Probe Lower Bounds for Succinct Partial Sums

Changing base without losing space

Fast Modular Composition in any Characteristic

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