Tight bounds on computing error-correcting codes by bounded-depth circuits with arbitrary gates

Gál, Anna; Hansen, Kristoffer Arnsfelt; Koucký, Michal; Pudlák, Pavel; Viola, Emanuele

doi:10.1145/2213977.2214023

Cited by 7 publications

(4 citation statements)

References 29 publications

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“…Here the lower bound situation is a little better; Alon, Karchmer and Wigderson [2] showed in 1990 that a particular family of matrices requires Ω(n log n) wires for linear circuits in this model. This has recently been improved by Gál et al [11] who have proven that a concrete infinite family of matrices require Ω n log n log log n 2 wires when computed in depth 2. Recently Drucker [10] gave a survey of the strategies used for proving lower bounds on wire complexity for general (not necessarily linear) Boolean operators in bounded depth, and the limitations of these.…”

Section: Introduction and Known Resultsmentioning

confidence: 99%

Cancellation-free circuits: An approach for proving superlinear lower bounds for linear Boolean operators

Boyar,

Find

2012

Preprint

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We continue to study the notion of cancellation-free linear circuits. We show that every matrix can be computed by a cancellationfree circuit, and almost all of these are at most a constant factor larger than the optimum linear circuit that computes the matrix. It appears to be easier to prove statements about the structure of cancellation-free linear circuits than for linear circuits in general. We prove two nontrivial superlinear lower bounds. We show that a cancellation-free linear circuit computing the n × n Sierpinski gasket matrix must use at least 1 2 n log n gates, and that this is tight. This supports a conjecture by Aaronson. Furthermore we show that a proof strategy for proving lower bounds on monotone circuits can be almost directly converted to prove lower bounds on cancellation-free linear circuits. We use this together with a result from extremal graph theory due to Andreev to prove a lower bound of Ω(n 2−ǫ ) for infinitely many n × n matrices for every ǫ > 0 for. These lower bounds for concrete matrices are almost optimal since all matrices can be computed with O n 2 log n gates.

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Section: Introduction and Known Resultsmentioning

confidence: 99%

Cancellation-free circuits: An approach for proving superlinear lower bounds for linear Boolean operators

Boyar,

Find

2012

Preprint

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“…It is a long-standing open problem to exhibit explicit matrices requiring super-linear size ⊕-circuits. No such lower bounds are known even for log-depth circuits, and the only successes are in the case of bounded depth [2,9], [16, §13.5]. This, together with Fact 1, makes it particularly difficult to prove lower bounds on Gap ⊕/∨ .…”

Section: Related Workmentioning

confidence: 99%

Separating OR, SUM, and XOR circuits

Göös

Järvisalo

Kaski

et al. 2016

Journal of Computer and System Sciences

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Given a boolean n × n matrix A we consider arithmetic circuits for computing the transformation x ↦ Ax over different semirings. Namely, we study three circuit models: monotone OR-circuits, monotone SUM-circuits (addition of non-negative integers), and non-monotone XOR-circuits (addition modulo 2). Our focus is on separating OR-circuits from the two other models in terms of circuit complexity: We show how to obtain matrices that admit OR-circuits of size O(n), but require SUM-circuits of size Ω(n3/2/log2n).We consider the task of rewriting a given OR-circuit as a XOR-circuit and prove that any subquadratic-time algorithm for this task violates the strong exponential time hypothesis.

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“…Depth-2 circuits computing explicit linear operators are of particular interest. Currently, the best lower bound for an explicit linear operator is the recent Θ(n(lg n/ lg lg n) 2 ) bound of Gál et al [8] for circuits that compute error correcting codes. Another interesting question is whether general circuits are more powerful than linear circuits for computing linear operators.…”

Section: Theorem 3 ([10]mentioning

confidence: 99%

Adapt or Die: Polynomial Lower Bounds for Non-Adaptive Dynamic Data Structures

Brody,

Larsen

2012

Preprint

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In this paper, we study the role non-adaptivity plays in maintaining dynamic data structures. Roughly speaking, a data structure is non-adaptive if the memory locations it reads and/or writes when processing a query or update depend only on the query or update and not on the contents of previously read cells. We study such non-adaptive data structures in the cell probe model. This model is one of the least restrictive lower bound models and in particular, cell probe lower bounds apply to data structures developed in the popular word-RAM model. Unfortunately, this generality comes at a high cost: the highest lower bound proved for any data structure problem is only polylogarithmic. Our main result is to demonstrate that one can in fact obtain polynomial cell probe lower bounds for non-adaptive data structures.To shed more light on the seemingly inherent polylogarithmic lower bound barrier, we study several different notions of non-adaptivity and identify key properties that must be dealt with if we are to prove polynomial lower bounds without restrictions on the data structures.Finally, our results also unveil an interesting connection between data structures and depth-2 circuits. This allows us to translate conjectured hard data structure problems into good candidates for high circuit lower bounds; in particular, in the area of linear circuits for linear operators. Building on lower bound proofs for data structures in slightly more restrictive models, we also present a number of properties of linear operators which we believe are worth investigating in the realm of circuit lower bounds.

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Tight bounds on computing error-correcting codes by bounded-depth circuits with arbitrary gates

Cited by 7 publications

References 29 publications

Cancellation-free circuits: An approach for proving superlinear lower bounds for linear Boolean operators

Cancellation-free circuits: An approach for proving superlinear lower bounds for linear Boolean operators

Separating OR, SUM, and XOR circuits

Adapt or Die: Polynomial Lower Bounds for Non-Adaptive Dynamic Data Structures

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