167 research outputs found
Probabilistic Polynomials and Hamming Nearest Neighbors
We show how to compute any symmetric Boolean function on variables over
any field (as well as the integers) with a probabilistic polynomial of degree
and error at most . The degree
dependence on and is optimal, matching a lower bound of Razborov
(1987) and Smolensky (1987) for the MAJORITY function. The proof is
constructive: a low-degree polynomial can be efficiently sampled from the
distribution.
This polynomial construction is combined with other algebraic ideas to give
the first subquadratic time algorithm for computing a (worst-case) batch of
Hamming distances in superlogarithmic dimensions, exactly. To illustrate, let
. Suppose we are given a database
of vectors in and a collection of query vectors
in the same dimension. For all , we wish to compute a
with minimum Hamming distance from . We solve this problem in randomized time. Hence, the problem is in "truly subquadratic"
time for dimensions, and in subquadratic time for . We apply the algorithm to computing pairs with maximum
inner product, closest pair in for vectors with bounded integer
entries, and pairs with maximum Jaccard coefficients.Comment: 16 pages. To appear in 56th Annual IEEE Symposium on Foundations of
Computer Science (FOCS 2015
Sparser Johnson-Lindenstrauss Transforms
We give two different and simple constructions for dimensionality reduction
in via linear mappings that are sparse: only an
-fraction of entries in each column of our embedding matrices
are non-zero to achieve distortion with high probability, while
still achieving the asymptotically optimal number of rows. These are the first
constructions to provide subconstant sparsity for all values of parameters,
improving upon previous works of Achlioptas (JCSS 2003) and Dasgupta, Kumar,
and Sarl\'{o}s (STOC 2010). Such distributions can be used to speed up
applications where dimensionality reduction is used.Comment: v6: journal version, minor changes, added Remark 23; v5: modified
abstract, fixed typos, added open problem section; v4: simplified section 4
by giving 1 analysis that covers both constructions; v3: proof of Theorem 25
in v2 was written incorrectly, now fixed; v2: Added another construction
achieving same upper bound, and added proof of near-tight lower bound for DKS
schem
Faster all-pairs shortest paths via circuit complexity
We present a new randomized method for computing the min-plus product
(a.k.a., tropical product) of two matrices, yielding a faster
algorithm for solving the all-pairs shortest path problem (APSP) in dense
-node directed graphs with arbitrary edge weights. On the real RAM, where
additions and comparisons of reals are unit cost (but all other operations have
typical logarithmic cost), the algorithm runs in time
and is correct with high probability.
On the word RAM, the algorithm runs in time for edge weights in . Prior algorithms used either time for
various , or time for various
and .
The new algorithm applies a tool from circuit complexity, namely the
Razborov-Smolensky polynomials for approximately representing
circuits, to efficiently reduce a matrix product over the algebra to
a relatively small number of rectangular matrix products over ,
each of which are computable using a particularly efficient method due to
Coppersmith. We also give a deterministic version of the algorithm running in
time for some , which utilizes the
Yao-Beigel-Tarui translation of circuits into "nice" depth-two
circuits.Comment: 24 pages. Updated version now has slightly faster running time. To
appear in ACM Symposium on Theory of Computing (STOC), 201
Higher Order Approximation to the Hill Problem Dynamics about the Libration Points
An analytical solution to the Hill problem Hamiltonian expanded about the
libration points has been obtained by means of perturbation techniques. In
order to compute the higher orders of the perturbation solution that are needed
to capture all the relevant periodic orbits originated from the libration
points within a reasonable accuracy, the normalization is approached in complex
variables. The validity of the solution extends to energy values considerably
far away from that of the libration points and, therefore, can be used in the
computation of Halo orbits as an alternative to the classical
Lindstedt-Poincar\'e approach. Furthermore, the theory correctly predicts the
existence of the two-lane bridge of periodic orbits linking the families of
planar and vertical Lyapunov orbits.Comment: 28 pages, 8 figure
Families with infants: a general approach to solve hard partition problems
We introduce a general approach for solving partition problems where the goal
is to represent a given set as a union (either disjoint or not) of subsets
satisfying certain properties. Many NP-hard problems can be naturally stated as
such partition problems. We show that if one can find a large enough system of
so-called families with infants for a given problem, then this problem can be
solved faster than by a straightforward algorithm. We use this approach to
improve known bounds for several NP-hard problems as well as to simplify the
proofs of several known results.
For the chromatic number problem we present an algorithm with
time and exponential space for graphs of average
degree . This improves the algorithm by Bj\"{o}rklund et al. [Theory Comput.
Syst. 2010] that works for graphs of bounded maximum (as opposed to average)
degree and closes an open problem stated by Cygan and Pilipczuk [ICALP 2013].
For the traveling salesman problem we give an algorithm working in
time and polynomial space for graphs of average
degree . The previously known results of this kind is a polyspace algorithm
by Bj\"{o}rklund et al. [ICALP 2008] for graphs of bounded maximum degree and
an exponential space algorithm for bounded average degree by Cygan and
Pilipczuk [ICALP 2013].
For counting perfect matching in graphs of average degree~ we present an
algorithm with running time and polynomial
space. Recent algorithms of this kind due to Cygan, Pilipczuk [ICALP 2013] and
Izumi, Wadayama [FOCS 2012] (for bipartite graphs only) use exponential space.Comment: 18 pages, a revised version of this paper is available at
http://arxiv.org/abs/1410.220
Fast Large Scale Structure Perturbation Theory using 1D FFTs
The usual fluid equations describing the large-scale evolution of mass
density in the universe can be written as local in the density, velocity
divergence, and velocity potential fields. As a result, the perturbative
expansion in small density fluctuations, usually written in terms of
convolutions in Fourier space, can be written as a series of products of these
fields evaluated at the same location in configuration space. Based on this, we
establish a new method to numerically evaluate the 1-loop power spectrum (i.e.,
Fourier transform of the 2-point correlation function) with one-dimensional
Fast Fourier Transforms. This is exact and a few orders of magnitude faster
than previously used numerical approaches. Numerical results of the new method
are in excellent agreement with the standard quadrature integration method.
This fast model evaluation can in principle be extended to higher loop order
where existing codes become painfully slow. Our approach follows by writing
higher order corrections to the 2-point correlation function as, e.g., the
correlation between two second-order fields or the correlation between a linear
and a third-order field. These are then decomposed into products of
correlations of linear fields and derivatives of linear fields. The method can
also be viewed as evaluating three-dimensional Fourier space convolutions using
products in configuration space, which may also be useful in other contexts
where similar integrals appear.Comment: 10+4 pages, published versio
Symbolic computation of moments of sampling distributions
By means of the notion of umbrae indexed by multisets, a general method to
express estimators and their products in terms of power sums is derived. A
connection between the notion of multiset and integer partition leads
immediately to a way to speed up the procedures. Comparisons of computational
times with known procedures show how this approach turns out to be more
efficient in eliminating much unnecessary computation.Comment: 21 pages, 7 table
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