49,817 research outputs found
Solving -SUM using few linear queries
The -SUM problem is given input real numbers to determine whether any
of them sum to zero. The problem is of tremendous importance in the
emerging field of complexity theory within , and it is in particular open
whether it admits an algorithm of complexity with . Inspired by an algorithm due to Meiser (1993), we show
that there exist linear decision trees and algebraic computation trees of depth
solving -SUM. Furthermore, we show that there exists a
randomized algorithm that runs in
time, and performs linear queries on the input. Thus, we show
that it is possible to have an algorithm with a runtime almost identical (up to
the ) to the best known algorithm but for the first time also with the
number of queries on the input a polynomial that is independent of . The
bound on the number of linear queries is also a tighter bound
than any known algorithm solving -SUM, even allowing unlimited total time
outside of the queries. By simultaneously achieving few queries to the input
without significantly sacrificing runtime vis-\`{a}-vis known algorithms, we
deepen the understanding of this canonical problem which is a cornerstone of
complexity-within-.
We also consider a range of tradeoffs between the number of terms involved in
the queries and the depth of the decision tree. In particular, we prove that
there exist -linear decision trees of depth
Searching for Globally Optimal Functional Forms for Inter-Atomic Potentials Using Parallel Tempering and Genetic Programming
We develop a Genetic Programming-based methodology that enables discovery of
novel functional forms for classical inter-atomic force-fields, used in
molecular dynamics simulations. Unlike previous efforts in the field, that fit
only the parameters to the fixed functional forms, we instead use a novel
algorithm to search the space of many possible functional forms. While a
follow-on practical procedure will use experimental and {\it ab inito} data to
find an optimal functional form for a forcefield, we first validate the
approach using a manufactured solution. This validation has the advantage of a
well-defined metric of success. We manufactured a training set of atomic
coordinate data with an associated set of global energies using the well-known
Lennard-Jones inter-atomic potential. We performed an automatic functional form
fitting procedure starting with a population of random functions, using a
genetic programming functional formulation, and a parallel tempering
Metropolis-based optimization algorithm. Our massively-parallel method
independently discovered the Lennard-Jones function after searching for several
hours on 100 processors and covering a miniscule portion of the configuration
space. We find that the method is suitable for unsupervised discovery of
functional forms for inter-atomic potentials/force-fields. We also find that
our parallel tempering Metropolis-based approach significantly improves the
optimization convergence time, and takes good advantage of the parallel cluster
architecture
Progress on Polynomial Identity Testing - II
We survey the area of algebraic complexity theory; with the focus being on
the problem of polynomial identity testing (PIT). We discuss the key ideas that
have gone into the results of the last few years.Comment: 17 pages, 1 figure, surve
- …