3,615 research outputs found
New Acceleration of Nearly Optimal Univariate Polynomial Root-findERS
Univariate polynomial root-finding has been studied for four millennia and is
still the subject of intensive research. Hundreds of efficient algorithms for
this task have been proposed. Two of them are nearly optimal. The first one,
proposed in 1995, relies on recursive factorization of a polynomial, is quite
involved, and has never been implemented. The second one, proposed in 2016,
relies on subdivision iterations, was implemented in 2018, and promises to be
practically competitive, although user's current choice for univariate
polynomial root-finding is the package MPSolve, proposed in 2000, revised in
2014, and based on Ehrlich's functional iterations. By proposing and
incorporating some novel techniques we significantly accelerate both
subdivision and Ehrlich's iterations. Moreover our acceleration of the known
subdivision root-finders is dramatic in the case of sparse input polynomials.
Our techniques can be of some independent interest for the design and analysis
of polynomial root-finders.Comment: 89 pages, 5 figures, 2 table
Algebraic Methods in the Congested Clique
In this work, we use algebraic methods for studying distance computation and
subgraph detection tasks in the congested clique model. Specifically, we adapt
parallel matrix multiplication implementations to the congested clique,
obtaining an round matrix multiplication algorithm, where
is the exponent of matrix multiplication. In conjunction
with known techniques from centralised algorithmics, this gives significant
improvements over previous best upper bounds in the congested clique model. The
highlight results include:
-- triangle and 4-cycle counting in rounds, improving upon the
triangle detection algorithm of Dolev et al. [DISC 2012],
-- a -approximation of all-pairs shortest paths in
rounds, improving upon the -round -approximation algorithm of Nanongkai [STOC 2014], and
-- computing the girth in rounds, which is the first
non-trivial solution in this model.
In addition, we present a novel constant-round combinatorial algorithm for
detecting 4-cycles.Comment: This is work is a merger of arxiv:1412.2109 and arxiv:1412.266
Algorithms and lower bounds for de Morgan formulas of low-communication leaf gates
The class consists of Boolean functions
computable by size- de Morgan formulas whose leaves are any Boolean
functions from a class . We give lower bounds and (SAT, Learning,
and PRG) algorithms for , for classes
of functions with low communication complexity. Let
be the maximum -party NOF randomized communication
complexity of . We show:
(1) The Generalized Inner Product function cannot be computed in
on more than fraction of inputs
for As a corollary, we get an average-case lower bound for
against .
(2) There is a PRG of seed length that -fools . For
, we get the better seed length . This gives the first
non-trivial PRG (with seed length ) for intersections of half-spaces
in the regime where .
(3) There is a randomized -time SAT algorithm for , where In particular, this implies a nontrivial
#SAT algorithm for .
(4) The Minimum Circuit Size Problem is not in .
On the algorithmic side, we show that can be
PAC-learned in time
New Developments in Quantum Algorithms
In this survey, we describe two recent developments in quantum algorithms.
The first new development is a quantum algorithm for evaluating a Boolean
formula consisting of AND and OR gates of size N in time O(\sqrt{N}). This
provides quantum speedups for any problem that can be expressed via Boolean
formulas. This result can be also extended to span problems, a generalization
of Boolean formulas. This provides an optimal quantum algorithm for any Boolean
function in the black-box query model.
The second new development is a quantum algorithm for solving systems of
linear equations. In contrast with traditional algorithms that run in time
O(N^{2.37...}) where N is the size of the system, the quantum algorithm runs in
time O(\log^c N). It outputs a quantum state describing the solution of the
system.Comment: 11 pages, 1 figure, to appear as an invited survey talk at MFCS'201
New Structured Matrix Methods for Real and Complex Polynomial Root-finding
We combine the known methods for univariate polynomial root-finding and for
computations in the Frobenius matrix algebra with our novel techniques to
advance numerical solution of a univariate polynomial equation, and in
particular numerical approximation of the real roots of a polynomial. Our
analysis and experiments show efficiency of the resulting algorithms.Comment: 18 page
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