8,987 research outputs found
Probabilistic communication complexity over the reals
Deterministic and probabilistic communication protocols are introduced in
which parties can exchange the values of polynomials (rather than bits in the
usual setting). It is established a sharp lower bound on the communication
complexity of recognizing the -dimensional orthant, on the other hand the
probabilistic communication complexity of its recognizing does not exceed 4. A
polyhedron and a union of hyperplanes are constructed in \RR^{2n} for which a
lower bound on the probabilistic communication complexity of recognizing
each is proved. As a consequence this bound holds also for the EMPTINESS and
the KNAPSACK problems
The Galois Complexity of Graph Drawing: Why Numerical Solutions are Ubiquitous for Force-Directed, Spectral, and Circle Packing Drawings
Many well-known graph drawing techniques, including force directed drawings,
spectral graph layouts, multidimensional scaling, and circle packings, have
algebraic formulations. However, practical methods for producing such drawings
ubiquitously use iterative numerical approximations rather than constructing
and then solving algebraic expressions representing their exact solutions. To
explain this phenomenon, we use Galois theory to show that many variants of
these problems have solutions that cannot be expressed by nested radicals or
nested roots of low-degree polynomials. Hence, such solutions cannot be
computed exactly even in extended computational models that include such
operations.Comment: Graph Drawing 201
Some Speed-Ups and Speed Limits for Real Algebraic Geometry
We give new positive and negative results (some conditional) on speeding up
computational algebraic geometry over the reals: (1) A new and sharper upper
bound on the number of connected components of a semialgebraic set. Our bound
is novel in that it is stated in terms of the volumes of certain polytopes and,
for a large class of inputs, beats the best previous bounds by a factor
exponential in the number of variables. (2) A new algorithm for approximating
the real roots of certain sparse polynomial systems. Two features of our
algorithm are (a) arithmetic complexity polylogarithmic in the degree of the
underlying complex variety (as opposed to the super-linear dependence in
earlier algorithms) and (b) a simple and efficient generalization to certain
univariate exponential sums. (3) Detecting whether a real algebraic surface
(given as the common zero set of some input straight-line programs) is not
smooth can be done in polynomial time within the classical Turing model (resp.
BSS model over C) only if P=NP (resp. NP<=BPP). The last result follows easily
from an unpublished result of Steve Smale.Comment: This is the final journal version which will appear in Journal of
Complexity. More typos are corrected, and a new section is added where the
bounds here are compared to an earlier result of Benedetti, Loeser, and
Risler. The LaTeX source needs the ajour.cls macro file to compil
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