7,952 research outputs found

    Strong ETH Breaks With Merlin and Arthur: Short Non-Interactive Proofs of Batch Evaluation

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    We present an efficient proof system for Multipoint Arithmetic Circuit Evaluation: for every arithmetic circuit C(x1,,xn)C(x_1,\ldots,x_n) of size ss and degree dd over a field F{\mathbb F}, and any inputs a1,,aKFna_1,\ldots,a_K \in {\mathbb F}^n, \bullet the Prover sends the Verifier the values C(a1),,C(aK)FC(a_1), \ldots, C(a_K) \in {\mathbb F} and a proof of O~(Kd)\tilde{O}(K \cdot d) length, and \bullet the Verifier tosses poly(log(dKF/ε))\textrm{poly}(\log(dK|{\mathbb F}|/\varepsilon)) coins and can check the proof in about O~(K(n+d)+s)\tilde{O}(K \cdot(n + d) + s) time, with probability of error less than ε\varepsilon. For small degree dd, this "Merlin-Arthur" proof system (a.k.a. MA-proof system) runs in nearly-linear time, and has many applications. For example, we obtain MA-proof systems that run in cnc^{n} time (for various c<2c < 2) for the Permanent, #\#Circuit-SAT for all sublinear-depth circuits, counting Hamiltonian cycles, and infeasibility of 00-11 linear programs. In general, the value of any polynomial in Valiant's class VP{\sf VP} can be certified faster than "exhaustive summation" over all possible assignments. These results strongly refute a Merlin-Arthur Strong ETH and Arthur-Merlin Strong ETH posed by Russell Impagliazzo and others. We also give a three-round (AMA) proof system for quantified Boolean formulas running in 22n/3+o(n)2^{2n/3+o(n)} time, nearly-linear time MA-proof systems for counting orthogonal vectors in a collection and finding Closest Pairs in the Hamming metric, and a MA-proof system running in nk/2+O(1)n^{k/2+O(1)}-time for counting kk-cliques in graphs. We point to some potential future directions for refuting the Nondeterministic Strong ETH.Comment: 17 page

    Synthesis and Optimization of Reversible Circuits - A Survey

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    Reversible logic circuits have been historically motivated by theoretical research in low-power electronics as well as practical improvement of bit-manipulation transforms in cryptography and computer graphics. Recently, reversible circuits have attracted interest as components of quantum algorithms, as well as in photonic and nano-computing technologies where some switching devices offer no signal gain. Research in generating reversible logic distinguishes between circuit synthesis, post-synthesis optimization, and technology mapping. In this survey, we review algorithmic paradigms --- search-based, cycle-based, transformation-based, and BDD-based --- as well as specific algorithms for reversible synthesis, both exact and heuristic. We conclude the survey by outlining key open challenges in synthesis of reversible and quantum logic, as well as most common misconceptions.Comment: 34 pages, 15 figures, 2 table

    Faster all-pairs shortest paths via circuit complexity

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    We present a new randomized method for computing the min-plus product (a.k.a., tropical product) of two n×nn \times n matrices, yielding a faster algorithm for solving the all-pairs shortest path problem (APSP) in dense nn-node directed graphs with arbitrary edge weights. On the real RAM, where additions and comparisons of reals are unit cost (but all other operations have typical logarithmic cost), the algorithm runs in time n32Ω(logn)1/2\frac{n^3}{2^{\Omega(\log n)^{1/2}}} and is correct with high probability. On the word RAM, the algorithm runs in n3/2Ω(logn)1/2+n2+o(1)logMn^3/2^{\Omega(\log n)^{1/2}} + n^{2+o(1)}\log M time for edge weights in ([0,M]Z){}([0,M] \cap {\mathbb Z})\cup\{\infty\}. Prior algorithms used either n3/(logcn)n^3/(\log^c n) time for various c2c \leq 2, or O(Mαnβ)O(M^{\alpha}n^{\beta}) time for various α>0\alpha > 0 and β>2\beta > 2. The new algorithm applies a tool from circuit complexity, namely the Razborov-Smolensky polynomials for approximately representing AC0[p]{\sf AC}^0[p] circuits, to efficiently reduce a matrix product over the (min,+)(\min,+) algebra to a relatively small number of rectangular matrix products over F2{\mathbb F}_2, each of which are computable using a particularly efficient method due to Coppersmith. We also give a deterministic version of the algorithm running in n3/2logδnn^3/2^{\log^{\delta} n} time for some δ>0\delta > 0, which utilizes the Yao-Beigel-Tarui translation of AC0[m]{\sf AC}^0[m] circuits into "nice" depth-two circuits.Comment: 24 pages. Updated version now has slightly faster running time. To appear in ACM Symposium on Theory of Computing (STOC), 201

    Optimizing construction of scheduled data flow graph for on-line testability

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    The objective of this work is to develop a new methodology for behavioural synthesis using a flow of synthesis, better suited to the scheduling of independent calculations and non-concurrent online testing. The traditional behavioural synthesis process can be defined as the compilation of an algorithmic specification into an architecture composed of a data path and a controller. This stream of synthesis generally involves scheduling, resource allocation, generation of the data path and controller synthesis. Experiments showed that optimization started at the high level synthesis improves the performance of the result, yet the current tools do not offer synthesis optimizations that from the RTL level. This justifies the development of an optimization methodology which takes effect from the behavioural specification and accompanying the synthesis process in its various stages. In this paper we propose the use of algebraic properties (commutativity, associativity and distributivity) to transform readable mathematical formulas of algorithmic specifications into mathematical formulas evaluated efficiently. This will effectively reduce the execution time of scheduling calculations and increase the possibilities of testability

    Efficient parallel binary decision diagram construction using Cilk

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    Thesis (S.B. and M.Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2000.Includes bibliographical references (leaves 44-45).by David B. Berman.S.B.and M.Eng
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