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The NPcompleteness column: an ongoing guide
 JOURNAL OF ALGORITHMS
, 1987
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book "Computers and Intractability: A Guide to the Theory of NPCompleteness," W. H. Freem ..."
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This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book "Computers and Intractability: A Guide to the Theory of NPCompleteness," W. H. Freeman & Co., New York, 1979 (hereinafter referred to as "[G&J]"; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, crossreferences will be given to that book and the list of problems (NPcomplete and harder) presented there. Readers who have results they would like mentioned (NPhardness, PSPACEhardness, polynomialtimesolvability, etc.) or open problems they would like publicized, should
Fast parallel circuits for the quantum Fourier transform
 PROCEEDINGS 41ST ANNUAL SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS’00)
, 2000
"... We give new bounds on the circuit complexity of the quantum Fourier transform (QFT). We give an upper bound of O(log n + log log(1/ε)) on the circuit depth for computing an approximation of the QFT with respect to the modulus 2 n with error bounded by ε. Thus, even for exponentially small error, our ..."
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Cited by 70 (1 self)
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We give new bounds on the circuit complexity of the quantum Fourier transform (QFT). We give an upper bound of O(log n + log log(1/ε)) on the circuit depth for computing an approximation of the QFT with respect to the modulus 2 n with error bounded by ε. Thus, even for exponentially small error, our circuits have depth O(log n). The best previous depth bound was O(n), even for approximations with constant error. Moreover, our circuits have size O(n log(n/ε)). We also give an upper bound of O(n(log n) 2 log log n) on the circuit size of the exact QFT modulo 2 n, for which the best previous bound was O(n 2). As an application of the above depth bound, we show that Shor’s factoring algorithm may be based on quantum circuits with depth only O(log n) and polynomialsize, in combination with classical polynomialtime pre and postprocessing. In the language of computational complexity, this implies that factoring is in the complexity class ZPP BQNC, where BQNC is the class of problems computable with boundederror probability by quantum circuits with polylogarithmic depth and polynomial size. Finally, we prove an Ω(log n) lower bound on the depth complexity of approximations of the
ON THRESHOLD CIRCUITS AND POLYNOMIAL COMPUTATION
"... A Threshold Circuit consists of an acyclic digraph of unbounded fanin, where each node computes a threshold function or its negation. This paper investigates the computational power of Threshold Circuits. A surprising relationship is uncovered between Threshold Circuits and another class of unbound ..."
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Cited by 60 (1 self)
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A Threshold Circuit consists of an acyclic digraph of unbounded fanin, where each node computes a threshold function or its negation. This paper investigates the computational power of Threshold Circuits. A surprising relationship is uncovered between Threshold Circuits and another class of unbounded fanin circuits which are denoted Finite Field ZP (n) Circuits, where each node computes either multiple sums or products of integers modulo a prime P (n). In particular, it is proved that all functions computed by Threshold Circuits of size S(n) n and depth D(n) can also be computed by ZP (n) Circuits of size O(S(n) log S(n)+nP (n) log P (n)) and depth O(D(n)). Furthermore, it is shown that all functions computed by ZP (n) Circuits of size S(n) and depth D(n) can be computed by Threshold Circuits of size O ( 1 2 (S(n) log P (n)) 1+) and depth O ( 1 5 D(n)). These are the main results of this paper. There are many useful and quite surprising consequences of this result. For example, integer reciprocal can be computed in size n O(1) and depth O(1). More generally, any analytic function with a convergent rational polynomial power series (such as sine, cosine, exponentiation, square root, and logarithm) can be computed within accuracy 2,nc, for any constant c, by Threshold Circuits of
Uniform ConstantDepth Threshold Circuits for Division and Iterated Multiplication
, 2002
"... this paper. 2.1. Circuit Classes We begin by formally defining the three circuit complexity classes that will concern us here. These are given by combinatorial restrictions on the circuits of the family. We will then define the uniformity restrictions we will use. Finally, we will give the equival ..."
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Cited by 59 (12 self)
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this paper. 2.1. Circuit Classes We begin by formally defining the three circuit complexity classes that will concern us here. These are given by combinatorial restrictions on the circuits of the family. We will then define the uniformity restrictions we will use. Finally, we will give the equivalent formulations of uniform circuit complexity classes in terms of descriptive complexity classes
Fast Parallel Absolute Irreducibility Testing
 J. Symbolic Comput
, 1985
"... We present a fast parallel deterministic algorithm for testing multivariate integral polynomials for absolute irreducibility, that is irreducibility over the complex numbers. More precisely, we establish that the set of absolutely irreducible integral polynomials belongs to the complexity class NC o ..."
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Cited by 32 (7 self)
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We present a fast parallel deterministic algorithm for testing multivariate integral polynomials for absolute irreducibility, that is irreducibility over the complex numbers. More precisely, we establish that the set of absolutely irreducible integral polynomials belongs to the complexity class NC of Boolean circuits of polynomial size and logarithmic depth. Therefore it also belongs to the class of sequentially polynomialtime problems. Our algorithm can be extended to compute in parallel one irreducible complex factor of a multivariate integral polynomial. However, the coefficients of the computed factor are only represented modulo a not necessarily irreducible polynomial specifying a splitting field. A consequence of our algoithm is that multivariate polynomials over finite fields can be tested for absolute irreducibility in deterministic sequential polynomial time in the size of the input. We also obtain a sharp bound for the last prime p for which, when taking an absolutely irreducible integral polynomial modulo p, the polynomial's irreducibility in the algebraic closure of the finite field order p is not preserved.
SpaceEfficient Deterministic Simulation of Probabilistic Automata
, 1993
"... Given a description of a probabilistic automaton (onehead probabilistic finite automaton or probabilistic Turing machine) and an input string x of length n, we ask how much space does a deterministic Turing machine need in order to decide the acceptance of the input string by that automaton? The q ..."
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Cited by 18 (4 self)
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Given a description of a probabilistic automaton (onehead probabilistic finite automaton or probabilistic Turing machine) and an input string x of length n, we ask how much space does a deterministic Turing machine need in order to decide the acceptance of the input string by that automaton? The question is interesting even in the case of onehead oneway probabilistic finite automata (PFA). We call (rational) stochastic languages (S ? rat ) the class of languages recognized by PFA's whose transition probabilities and cutpoints (i.e. recognition thresholds) are rational numbers. The class S ? rat contains contextsensitive languages that are not context free, but on the other hand there are contextfree languages not included in S ? rat . Our main results are as follows: ffl The (proper) inclusion of S ? rat in Dspace(log n), which is optimal (i.e. S ? rat 6ae Dspace(o(log n))). The previous upper bounds were Dspace(n) [Dieu 1972], [Wang 1992] and Dspace(log n log log n)...
Constantdepth circuits for arithmetic in finite fields of characteristic two
 IN PROCEEDINGS OF THE 23RD INTERNATIONAL SYMPOSIUM ON THEORETICAL ASPECTS OF COMPUTER SCIENCE (STACS), LECTURE NOTES IN COMPUTER SCIENCE
, 2006
"... We study the complexity of arithmetic in finite fields of characteristic two, F2n. We concentrate on the following two problems: • Iterated Multiplication: Given α1, α2,...,αt ∈ F2 n, compute α1 · α2 · · ·αt ∈ F2 n. • Exponentiation: Given α ∈ F2 n and a tbit integer k, compute αk ∈ F2 n. ..."
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Cited by 16 (9 self)
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We study the complexity of arithmetic in finite fields of characteristic two, F2n. We concentrate on the following two problems: • Iterated Multiplication: Given α1, α2,...,αt ∈ F2 n, compute α1 · α2 · · ·αt ∈ F2 n. • Exponentiation: Given α ∈ F2 n and a tbit integer k, compute αk ∈ F2 n.
OPTIMAL SIZE INTEGER DIVISION CIRCUITS
, 1988
"... Division is a fundamental problem for arithmetic and algebraic computation. This paper describes Boolean circuits (of bounded fanin) for integer division ( nding reciprocals) that have size O(M (n)) and depth O(log n log log n), where M(n) is the size complexity ofO(log n) depth integer multiplicat ..."
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Cited by 11 (2 self)
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Division is a fundamental problem for arithmetic and algebraic computation. This paper describes Boolean circuits (of bounded fanin) for integer division ( nding reciprocals) that have size O(M (n)) and depth O(log n log log n), where M(n) is the size complexity ofO(log n) depth integer multiplication circuits. Currently, M(n) isknown to be O(n log n log log n), but any improvement in this bound that preserves circuit depth will be re ected by a similar improvement in the size complexity of our division algorithm. Previously, no one has been able to derive a division circuit with size O(n log c n) for any c, and simultaneous depth less than (log 2 n). The circuit families described in this paper are logspace uniform; that is, they can be constructed by a deterministic Turing machine in space O(log n). The results match the bestknown depth bounds for logspace uniform circuits, and are optimal in size. The general method of highorder iterative formulas is of independent interest as a way of efciently using parallel processors to solve algebraic problems. In particular, this algorithm implies that any rational function can be evaluated in these complexity bounds. As an introduction to highorder iterative methods a circuit is rst presented for nding polynomial reciprocals (where the coe cients come from an arbitrary ring, and ring operations are unit cost in the circuit) in size O(PM(n)) and depth O(log n log log n), where PM(n) is the size complexity of optimal depth polynomial multiplication.
Complexity Of Parallel Arithmetic Using The Chinese Remainder Representation
, 1995
"... In this thesis, we show that all of the fundamental arithmetic operations in the Chinese remainder representation can be performed in O(log n) time using O(n^O(1)) boolean processors. In particular, we present an O(log n) depth, O(n^O(1)) node, uniform circuit family for division, thereby, showing t ..."
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Cited by 8 (2 self)
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In this thesis, we show that all of the fundamental arithmetic operations in the Chinese remainder representation can be performed in O(log n) time using O(n^O(1)) boolean processors. In particular, we present an O(log n) depth, O(n^O(1)) node, uniform circuit family for division, thereby, showing the division, iterated product, and powering problems to be in NC¹. We also present NC¹ circuits for a wide variety of other arithmetic operations in the Chinese remainder representation, such as addition, multiplication, computation of the inverse, base extension, and conversion to and from binary.
Bits and Relative Order from Residues, Space Efficiently
 Information Processing Letters
, 1994
"... . For each k, let P k be the product of the first k primes. By the Chinese remainder theorem, each integer in the interval [0; P k ) is determined by its residues modulo these k primes. We address the problems of spaceefficiently computing the bits and the relative order of such numbers from their ..."
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. For each k, let P k be the product of the first k primes. By the Chinese remainder theorem, each integer in the interval [0; P k ) is determined by its residues modulo these k primes. We address the problems of spaceefficiently computing the bits and the relative order of such numbers from their residues. Introduction For each k, let P k be the product of the first k primes, p 1 ! p 2 ! \Delta \Delta \Delta ! p k . By the Chinese remainder theorem, each integer in the interval [0; P k ) is determined by its residues modulo these k primes. 1 This fact is easily exploited to yield a spaceefficient test for equality of two such numbers. We show how to exploit it for an equally spaceefficient determination of which of two such numbers is larger, and we address the more general problem of spaceefficiently computing the bits of such a number from its residues. The best we can hope for is space O(S k ) for S k = log log P k , because it takes that much space just to write down k or...