68,320 research outputs found
An Ω lower bound for computing the sum of even-ranked elements
Given a sequence A of 2n real numbers, the \ers\ problem asks for the sum of the n values that are at the even positions in the sorted order of the elements in A. We prove that, in the algebraic computation-tree model, this problem has time complexity \Theta(n \log n). This solves an open problem posed by Michael Shamos at the Canadian Conference on Computational Geometry in 2008
Algebraicity and transcendence of power series: combinatorial and computational aspects
DoctoralFrom ancient times, mathematicians are interested in the following question: is a given real number "algebraic" (that is, a root of a nonzero univariate polynomial with rational number coefficients), or is it "transcendental"? Although almost all real numbers are transcendental, it is notoriously difficult to actually prove, or disprove, the transcendence of a given constant. More recently, and especially with the advent of computers, different related questions arose: What is the "complexity" of a real number? How fast can one compute the first digits, or one single digit, of a (computable) real number? Can digits of algebraic numbers be computed faster than those of (computable) transcendental numbers? In this series of lectures, we will consider the (simpler) functional analogues of these questions: given a formal power series with rational number coefficients, decide whether it is algebraic (root of a nontrivial bivariate polynomial) or transcendental, and determine how fast can one compute its coefficients? We will first motivate these questions by presenting some examples of algebraic power series coming from combinatorics, with a focus on enumeration of lattice walks. Then we will discuss several methods that allow to discover and prove the nature (algebraic or transcendental) of a generating function, with an emphasis on an experimental mathematics approach combined with algorithmic methods such as Guess'n'Prove and Creative Telescoping. Finally, we will overview efficient algorithms for various operations on algebraic power series, including the computation of one or several selected terms
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
Noncomputable functions in the Blum-Shub-Smale model
Working in the Blum-Shub-Smale model of computation on the real numbers, we
answer several questions of Meer and Ziegler. First, we show that, for each
natural number d, an oracle for the set of algebraic real numbers of degree at
most d is insufficient to allow an oracle BSS-machine to decide membership in
the set of algebraic numbers of degree d + 1. We add a number of further
results on relative computability of these sets and their unions. Then we show
that the halting problem for BSS-computation is not decidable below any
countable oracle set, and give a more specific condition, related to the
cardinalities of the sets, necessary for relative BSS-computability. Most of
our results involve the technique of using as input a tuple of real numbers
which is algebraically independent over both the parameters and the oracle of
the machine
On the asymptotic and practical complexity of solving bivariate systems over the reals
This paper is concerned with exact real solving of well-constrained,
bivariate polynomial systems. The main problem is to isolate all common real
roots in rational rectangles, and to determine their intersection
multiplicities. We present three algorithms and analyze their asymptotic bit
complexity, obtaining a bound of \sOB(N^{14}) for the purely projection-based
method, and \sOB(N^{12}) for two subresultant-based methods: this notation
ignores polylogarithmic factors, where bounds the degree and the bitsize of
the polynomials. The previous record bound was \sOB(N^{14}).
Our main tool is signed subresultant sequences. We exploit recent advances on
the complexity of univariate root isolation, and extend them to sign evaluation
of bivariate polynomials over two algebraic numbers, and real root counting for
polynomials over an extension field. Our algorithms apply to the problem of
simultaneous inequalities; they also compute the topology of real plane
algebraic curves in \sOB(N^{12}), whereas the previous bound was
\sOB(N^{14}).
All algorithms have been implemented in MAPLE, in conjunction with numeric
filtering. We compare them against FGB/RS, system solvers from SYNAPS, and
MAPLE libraries INSULATE and TOP, which compute curve topology. Our software is
among the most robust, and its runtimes are comparable, or within a small
constant factor, with respect to the C/C++ libraries.
Key words: real solving, polynomial systems, complexity, MAPLE softwareComment: 17 pages, 4 algorithms, 1 table, and 1 figure with 2 sub-figure
The Cardinality of an Oracle in Blum-Shub-Smale Computation
We examine the relation of BSS-reducibility on subsets of the real numbers.
The question was asked recently (and anonymously) whether it is possible for
the halting problem H in BSS-computation to be BSS-reducible to a countable
set. Intuitively, it seems that a countable set ought not to contain enough
information to decide membership in a reasonably complex (uncountable) set such
as H. We confirm this intuition, and prove a more general theorem linking the
cardinality of the oracle set to the cardinality, in a local sense, of the set
which it computes. We also mention other recent results on BSS-computation and
algebraic real numbers
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