A fast parallel algorithm for determining all roots of a polynomial with real roots

Ben-Or, Michael; Feig, Ephraim; Kozen, Dexter; Tiwari, Prashant

doi:10.1145/12130.12165

Cited by 27 publications

(21 citation statements)

References 15 publications

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“…Positive eigenspace computations for Hermitian operators can be approximated with high accuracy using the fact that polynomial root approximation is in NC [28,130,31].…”

Section: Pspace and Bounded-depth Circuitsmentioning

confidence: 99%

“…For instance, one may create the state 28) apply W 1 to the first register of this state together with |ψ , and then finally apply a controlled-NOT gate (and a permutation of the ordering of the qubits) to obtain (5.26). Given that the above transformation Q can be performed efficiently, one may recover |φ 0 (ψ) using the quantum rewinding lemma.…”

Section: Example: Graph Isomorphismmentioning

confidence: 99%

“…An important result in the field of hardness of approximation states that the unentangled-prover class is powerful enough to capture all problems in NEXP, meaning that the inclusion NEXP ⊆ ⊕MIP a,b (2, 2) (6.27) holds for a specific choice of constants 0 < b < a < 1. In contrast, allowing entanglement between the provers reduces the verifier's decision power (under the assumption that PSPACE is properly contained in NEXP): 28) which holds for any 0 ≤ b < a ≤ 1 separated by at least an inverse polynomial. Thus, the introduction of entanglement has the effect of collapsing the verifier's ability to decide problems, from NEXP to PSPACE.…”

Section: Xor Gamesmentioning

confidence: 99%

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Quantum Proofs

Vidick

Watrous

2016

FNT in Theoretical Computer Science

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Quantum information and computation provide a fascinating twist on the notion of proofs in computational complexity theory. For instance, one may consider a quantum computational analogue of the complexity class \class{NP}, known as QMA, in which a quantum state plays the role of a proof (also called a certificate or witness), and is checked by a polynomial-time quantum computation. For some problems, the fact that a quantum proof state could be a superposition over exponentially many classical states appears to offer computational advantages over classical proof strings. In the interactive proof system setting, one may consider a verifier and one or more provers that exchange and process quantum information rather than classical information during an interaction for a given input string, giving rise to quantum complexity classes such as QIP, QSZK, and QMIP* that represent natural quantum analogues of IP, SZK, and MIP. While quantum interactive proof systems inherit some properties from their classical counterparts, they also possess distinct and uniquely quantum features that lead to an interesting landscape of complexity classes based on variants of this model. In this survey we provide an overview of many of the known results concerning quantum proofs, computational models based on this concept, and properties of the complexity classes they define. In particular, we discuss non-interactive proofs and the complexity class QMA, single-prover quantum interactive proof systems and the complexity class QIP, statistical zero-knowledge quantum interactive proof systems and the complexity class \class{QSZK}, and multiprover interactive proof systems and the complexity classes QMIP, QMIP*, and MIP*.Comment: Survey published by NOW publisher

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“…Positive eigenspace computations for Hermitian operators can be approximated with high accuracy using the fact that polynomial root approximation is in NC [28,130,31].…”

Section: Pspace and Bounded-depth Circuitsmentioning

confidence: 99%

Section: Example: Graph Isomorphismmentioning

confidence: 99%

Section: Xor Gamesmentioning

confidence: 99%

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Quantum Proofs

Vidick

Watrous

2016

FNT in Theoretical Computer Science

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“…In the case where the input polynomial, f, is totally real (i.e., has only real roots), the notion of a ''splitting point'' is used in order to find the required geometrically balanced division of the root set [BFKT86,P89,R93b,and N94]. It is the purpose of this section to introduce the idea of a splitting set of points for general complex polynomials.…”

Section: Splitting Sets and Centered Pointsmentioning

confidence: 99%

An Efficient Algorithm for the Complex Roots Problem

Neff

Reif

1996

Journal of Complexity

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Given a univariate polynomial f (z) of degree n with complex coefficients, whose norms are less than 2 m in magnitude, the root problem is to find all the roots of f (z) up to specified precision 2 ϪȐ . Assuming the arithmetic model for computation, we provide an algorithm which has complexity O(n log 5 n log B), where b ϭ ϩ Ȑ, and ϭ max͕n, m͖. This improves on the previous best known algorithm of Pan for the problem, which has complexity O(n 2 log 2 n log(m ϩ Ȑ)). A remarkable property of our algorithm is that it does not require any assumptions about the root separation of f, which were either explicitly, or implicitly, required by previous algorithms. Moreover it also has a work-efficient parallel implementation. We also show that both the sequential and parallel implementations of the algorithm work without modification in the Boolean model of arithmetic. In this case, it follows from root perturbation estimates that we need only specify ϭ n(B ϩ log n ϩ 3) bits of the binary representations of the real and imaginary parts of each of the coefficients of f. We also show that by appropriate rounding of intermediate values, we can bound the number of bits required to represent all complex numbers occurring as intermediate quantities in the computation. The result is that we can restrict the numbers we use in every basic arithmetic operation to those having real and imaginary parts with at most bits, where ϭ ȏ ϩ 2n 2 ϩ 3(n ϩ log n ϩ 1) ϩ n 2 (1 ϩ 2 log n) ϩ log log n 0885-064X/96 $18.00Copyright © 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. 82NEFF AND REIF and ȏ ϭ Ȑ 2n ϩ n ϩ log log n ϩ 5.Thus, in the Boolean model, the overall work complexity of the algorithm is only increased by a multiplicative factor of M() (where M() ϭ O((log ) log log ) is the bit complexity for multiplication of integers of length ). The key result on which the algorithm is based, is a new theorem of Coppersmith and Neff relating the geometric distribution of the zeros of a polynomial to the distribution of the zeros of its high order derivatives. We also introduce several new techniques (splitting sets and ''centered'' points) which hinge on it. We also observe that our root finding algorithm can be efficiently parallelized to run in parallel time O(log 6 n log B) using n processors.

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“…Virtually all known parallel algorithms for weighted optimization and algebraic problems fit inside the model. Examples include fast parallel algorithms for solving linear systems [6], minimum weight spanning trees [14], shortest paths [14], global min-cuts in weighted, undirected graphs [13], blocking flows and max-flows [9,21], approximate computation of roots of polynomials [2,18], sorting algorithms [14] and several problems in computational geometry [20]. In constrast to Boolean circuits where no lower bounds are known for unbounded depth circuits, our result gives a lower bound for a natural problem in a natural model of computation.…”

Section: Introductionmentioning

confidence: 99%

A lower bound for the shortest path problem

Mulmuley

Shah

Proceedings 15th Annual IEEE Conference on Computational Complexity

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We show that the Shortest Path Problem cannot be solved in o(log n) time on an unbounded fan-in PRAM without bit operations using poly(n) processors even when the bit-lengths of the weights on the edges are restricted to be of size O(log 3 n). This shows that the matrix-based repeated squaring algorithm for the Shortest Path Problem is optimal in the unbounded fan-in PRAM model without bit operations.

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A fast parallel algorithm for determining all roots of a polynomial with real roots

Cited by 27 publications

References 15 publications

Quantum Proofs

Quantum Proofs

An Efficient Algorithm for the Complex Roots Problem

A lower bound for the shortest path problem

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