1,585 research outputs found
Algorithms for Testing Monomials in Multivariate Polynomials
This paper is our second step towards developing a theory of testing
monomials in multivariate polynomials. The central question is to ask whether a
polynomial represented by an arithmetic circuit has some types of monomials in
its sum-product expansion. The complexity aspects of this problem and its
variants have been investigated in our first paper by Chen and Fu (2010),
laying a foundation for further study. In this paper, we present two pairs of
algorithms. First, we prove that there is a randomized time
algorithm for testing -monomials in an -variate polynomial of degree
represented by an arithmetic circuit, while a deterministic
time algorithm is devised when the circuit is a formula, here is a given
prime number. Second, we present a deterministic time algorithm for
testing multilinear monomials in
polynomials, while a randomized algorithm is given for these
polynomials. The first algorithm extends the recent work by Koutis (2008) and
Williams (2009) on testing multilinear monomials. Group algebra is exploited in
the algorithm designs, in corporation with the randomized polynomial identity
testing over a finite field by Agrawal and Biswas (2003), the deterministic
noncommunicative polynomial identity testing by Raz and Shpilka (2005) and the
perfect hashing functions by Chen {\em at el.} (2007). Finally, we prove that
testing some special types of multilinear monomial is W[1]-hard, giving
evidence that testing for specific monomials is not fixed-parameter tractable
Uses of randomness in computation
Random number generators are widely used in practical algorithms. Examples
include simulation, number theory (primality testing and integer
factorization), fault tolerance, routing, cryptography, optimization by
simulated annealing, and perfect hashing. Complexity theory usually considers
the worst-case behaviour of deterministic algorithms, but it can also consider
average-case behaviour if it is assumed that the input data is drawn randomly
from a given distribution. Rabin popularised the idea of "probabilistic"
algorithms, where randomness is incorporated into the algorithm instead of
being assumed in the input data. Yao showed that there is a close connection
between the complexity of probabilistic algorithms and the average-case
complexity of deterministic algorithms. We give examples of the uses of
randomness in computation, discuss the contributions of Rabin, Yao and others,
and mention some open questions. This is the text of an invited talk presented
at "Theory Day", University of NSW, Sydney, 22 April 1994.Comment: An old Technical Report, not published elsewhere. 14 pages. For
further details see http://wwwmaths.anu.edu.au/~brent/pub/pub147.htm
Optimized Entanglement Purification
We investigate novel protocols for entanglement purification of qubit Bell
pairs. Employing genetic algorithms for the design of the purification circuit,
we obtain shorter circuits achieving higher success rates and better final
fidelities than what is currently available in the literature. We provide a
software tool for analytical and numerical study of the generated purification
circuits, under customizable error models. These new purification protocols
pave the way to practical implementations of modular quantum computers and
quantum repeaters. Our approach is particularly attentive to the effects of
finite resources and imperfect local operations - phenomena neglected in the
usual asymptotic approach to the problem. The choice of the building blocks
permitted in the construction of the circuits is based on a thorough
enumeration of the local Clifford operations that act as permutations on the
basis of Bell states
Pseudorandomness for Multilinear Read-Once Algebraic Branching Programs, in any Order
We give deterministic black-box polynomial identity testing algorithms for
multilinear read-once oblivious algebraic branching programs (ROABPs), in
n^(lg^2 n) time. Further, our algorithm is oblivious to the order of the
variables. This is the first sub-exponential time algorithm for this model.
Furthermore, our result has no known analogue in the model of read-once
oblivious boolean branching programs with unknown order, as despite recent work
there is no known pseudorandom generator for this model with sub-polynomial
seed-length (for unbounded-width branching programs).
This result extends and generalizes the result of Forbes and Shpilka that
obtained a n^(lg n)-time algorithm when given the order. We also extend and
strengthen the work of Agrawal, Saha and Saxena that gave a black-box algorithm
running in time exp((lg n)^d) for set-multilinear formulas of depth d. We note
that the model of multilinear ROABPs contains the model of set-multilinear
algebraic branching programs, which itself contains the model of
set-multilinear formulas of arbitrary depth. We obtain our results by
recasting, and improving upon, the ideas of Agrawal, Saha and Saxena. We phrase
the ideas in terms of rank condensers and Wronskians, and show that our results
improve upon the classical multivariate Wronskian, which may be of independent
interest.
In addition, we give the first n^(lglg n) black-box polynomial identity
testing algorithm for the so called model of diagonal circuits. This model,
introduced by Saxena has recently found applications in the work of Mulmuley,
as well as in the work of Gupta, Kamath, Kayal, Saptharishi. Previously work
had given n^(lg n)-time algorithms for this class. More generally, our result
holds for any model computing polynomials whose partial derivatives (of all
orders) span a low dimensional linear space.Comment: 38 page
Essentially optimal interactive certificates in linear algebra
Certificates to a linear algebra computation are additional data structures
for each output, which can be used by a---possibly randomized---verification
algorithm that proves the correctness of each output. The certificates are
essentially optimal if the time (and space) complexity of verification is
essentially linear in the input size , meaning times a factor
, i.e., a factor with
. We give algorithms that compute essentially optimal certificates for the
positive semidefiniteness, Frobenius form, characteristic and minimal
polynomial of an dense integer matrix . Our certificates can be
verified in Monte-Carlo bit complexity , where
is the bit size of the integer entries, solving an open problem in
[Kaltofen, Nehring, Saunders, Proc.\ ISSAC 2011] subject to computational
hardness assumptions. Second, we give algorithms that compute certificates for
the rank of sparse or structured matrices over an abstract field,
whose Monte Carlo verification complexity is matrix-times-vector products
arithmetic operations in the field. For example, if the
input matrix is sparse with non-zero entries, our rank
certificate can be verified in field operations. This extends also
to integer matrices with only an extra factor. All our
certificates are based on interactive verification protocols with the
interaction removed by a Fiat-Shamir identification heuristic. The validity of
our verification procedure is subject to standard computational hardness
assumptions from cryptography
Dynamic Ordered Sets with Exponential Search Trees
We introduce exponential search trees as a novel technique for converting
static polynomial space search structures for ordered sets into fully-dynamic
linear space data structures.
This leads to an optimal bound of O(sqrt(log n/loglog n)) for searching and
updating a dynamic set of n integer keys in linear space. Here searching an
integer y means finding the maximum key in the set which is smaller than or
equal to y. This problem is equivalent to the standard text book problem of
maintaining an ordered set (see, e.g., Cormen, Leiserson, Rivest, and Stein:
Introduction to Algorithms, 2nd ed., MIT Press, 2001).
The best previous deterministic linear space bound was O(log n/loglog n) due
Fredman and Willard from STOC 1990. No better deterministic search bound was
known using polynomial space.
We also get the following worst-case linear space trade-offs between the
number n, the word length w, and the maximal key U < 2^w: O(min{loglog n+log
n/log w, (loglog n)(loglog U)/(logloglog U)}). These trade-offs are, however,
not likely to be optimal.
Our results are generalized to finger searching and string searching,
providing optimal results for both in terms of n.Comment: Revision corrects some typoes and state things better for
applications in subsequent paper
Algebra in Computational Complexity
At its core, much of Computational Complexity is concerned with combinatorial objects and structures. But it has often proven true that the best way to prove things about these combinatorial objects is by establishing a connection to a more well-behaved algebraic setting. Indeed, many of the deepest and most powerful results in Computational Complexity rely on algebraic proof techniques. The Razborov-Smolensky polynomial-approximation method for proving constant-depth circuit lower bounds, the PCP characterization of NP, and the Agrawal-Kayal-Saxena polynomial-time primality test are some of the most prominent examples. The algebraic theme continues in some of the most exciting recent progress in computational complexity. There have been significant recent advances in algebraic circuit lower bounds, and the so-called "chasm at depth 4" suggests that the restricted models now being considered are not so far from ones that would lead to a general result. There have been similar successes concerning the related problems of polynomial identity testing and circuit reconstruction in the algebraic model, and these are tied to central questions regarding the power of randomness in computation. Representation theory has emerged as an important tool in three separate lines of work: the "Geometric Complexity Theory" approach to P vs. NP and circuit lower bounds, the effort to resolve the complexity of matrix multiplication, and a framework for constructing locally testable codes. Coding theory has seen several algebraic innovations in recent years, including multiplicity codes, and new lower bounds. This seminar brought together researchers who are using a diverse array of algebraic methods in a variety of settings. It plays an important role in educating a diverse community about the latest new techniques, spurring further progress
Non-Interactive Statistically-Hiding Quantum Bit Commitment from Any Quantum One-Way Function
We provide a non-interactive quantum bit commitment scheme which has
statistically-hiding and computationally-binding properties from any quantum
one-way function. Our protocol is basically a parallel composition of the
previous non-interactive quantum bit commitment schemes (based on quantum
one-way permutations, due to Dumais, Mayers and Salvail (EUROCRYPT 2000)) with
pairwise independent hash functions. To construct our non-interactive quantum
bit commitment scheme from any quantum one-way function, we follow the
procedure below: (i) from Dumais-Mayers-Salvail scheme to a weakly-hiding and
1-out-of-2 binding commitment (of a parallel variant); (ii) from the
weakly-hiding and 1-out-of-2 binding commitment to a strongly-hiding and
1-out-of-2 binding commitment; (iii) from the strongly-hiding and 1-out-of-2
binding commitment to a normal statistically-hiding commitment. In the
classical case, statistically-hiding bit commitment scheme (by Haitner, Nguyen,
Ong, Reingold and Vadhan (SIAM J. Comput., Vol.39, 2009)) is also constructible
from any one-way function. While the classical statistically-hiding bit
commitment has large round complexity, our quantum scheme is non-interactive,
which is advantageous over the classical schemes. A main technical contribution
is to provide a quantum analogue of the new interactive hashing theorem, due to
Haitner and Reingold (CCC 2007). Moreover, the parallel composition enables us
to simplify the security analysis drastically
Purifying GHZ States Using Degenerate Quantum Codes
Degenerate quantum codes are codes that do not reveal the complete error
syndrome. Their ability to conceal the complete error syndrome makes them
powerful resources in certain quantum information processing tasks. In
particular, the most error-tolerant way to purify depolarized Bell states using
one-way communication known to date involves degenerate quantum codes. Here we
study three closely related purification schemes for depolarized GHZ states
shared among players by means of degenerate quantum codes and
one-way classical communications. We find that our schemes tolerate more noise
than all other one-way schemes known to date, further demonstrating the
effectiveness of degenerate quantum codes in quantum information processing.Comment: Significantly revised with a few new results added, 33 pages, 7
figure
In-materio neuromimetic devices: Dynamics, information processing and pattern recognition
The story of information processing is a story of great success. Todays'
microprocessors are devices of unprecedented complexity and MOSFET transistors
are considered as the most widely produced artifact in the history of mankind.
The current miniaturization of electronic circuits is pushed almost to the
physical limit and begins to suffer from various parasitic effects. These facts
stimulate intense research on neuromimetic devices. This feature article is
devoted to various in materio implementation of neuromimetic processes,
including neuronal dynamics, synaptic plasticity, and higher-level signal and
information processing, along with more sophisticated implementations,
including signal processing, speech recognition and data security. Due to vast
number of papers in the field, only a subjective selection of topics is
presented in this review
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