scholarly journals The Complexity of Constructing Evolutionary Trees Using Experiments

2001 ◽  
Vol 8 (1) ◽  
Author(s):  
Gerth Stølting Brodal ◽  
Rolf Fagerberg ◽  
Christian N. S. Pedersen ◽  
Anna Östlin
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Optimal Time ◽  
Evolutionary Trees ◽  
Dynamic Algorithm ◽  
Running Time ◽  
Small Height

<p>We present tight upper and lower bounds for the problem of constructing evolutionary trees in the experiment model. We describe an algorithm which constructs an evolutionary tree of n species in time O(n d logd n) using at most n |d/2| (log2|d/2|−1 n + O(1)) experiments for d > 2, and<br />at most n(log n + O(1)) experiments for d = 2, where d is the degree of the tree. This improves the previous best upper bound by a factor Theta(log d). For d = 2 the previously best algorithm with running time O(n log n) had a bound of 4n log n on the number of experiments. By an explicit adversary argument, we show an <br />Omega(nd logd n) lower bound, matching our upper bounds and improving the previous best lower bound<br />by a factor Theta(logd n). Central to our algorithm is the construction and maintenance of separator trees of small height. We present how to maintain separator trees with height log n + O(1) under the insertion of new nodes in amortized time O(log n). Part of our dynamic algorithm is an algorithm for computing a centroid tree in optimal time O(n).</p><p>Keywords: Evolutionary trees, Experiment model, Separator trees, Centroid tree, Lower bounds</p>

scholarly journals On the Rank of $p$-Schemes

10.37236/3097 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Fateme Raei Barandagh ◽  
Amir Rahnamai Barghi
Keyword(s):  
Lower Bound ◽  
Prime Number ◽  
Lower Bounds ◽  
Upper Bound ◽  
Upper And Lower Bounds ◽  
Minimum Rank

Let $n>1$ be an integer and $p$ be a prime number. Denote by $\mathfrak{C}_{p^n}$ the class of non-thin association $p$-schemes of degree $p^n$. A sharp upper and lower bounds on the rank of schemes in $\mathfrak{C}_{p^n}$ with a certain order of thin radical are obtained. Moreover, all schemes in this class whose rank are equal to the lower bound are characterized and some schemes in this class whose rank are equal to the upper bound are constructed. Finally, it is shown that the scheme with minimum rank in $\mathfrak{C}_{p^n}$ is unique up to isomorphism, and it is a fusion of any association $p$-schemes with degree $p^n$.

scholarly journals When is non-trivial estimation possible for graphons and stochastic block models?‡

2017 ◽  
Vol 7 (2) ◽  
pp. 169-181
Author(s):  
Audra McMillan ◽  
Adam Smith
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Upper And Lower Bounds ◽  
Random Networks ◽  
Block Models

Abstract Block graphons (also called stochastic block models) are an important and widely studied class of models for random networks. We provide a lower bound on the accuracy of estimators for block graphons with a large number of blocks. We show that, given only the number $k$ of blocks and an upper bound $\rho$ on the values (connection probabilities) of the graphon, every estimator incurs error ${\it{\Omega}}\left(\min\left(\rho, \sqrt{\frac{\rho k^2}{n^2}}\right)\right)$ in the $\delta_2$ metric with constant probability for at least some graphons. In particular, our bound rules out any non-trivial estimation (that is, with $\delta_2$ error substantially less than $\rho$) when $k\geq n\sqrt{\rho}$. Combined with previous upper and lower bounds, our results characterize, up to logarithmic terms, the accuracy of graphon estimation in the $\delta_2$ metric. A similar lower bound to ours was obtained independently by Klopp et al.

The Heine-Borel theorem in extended basic logic

1949 ◽  
Vol 14 (1) ◽  
pp. 9-15 ◽  
Author(s):  
Frederic B. Fitch
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Fundamental Theorem ◽  
Upper Bounds ◽  
Basic Logic ◽  
Real Numbers ◽  
Consistent Theory

A demonstrably consistent theory of real numbers has been outlined by the writer in An extension of basic logic1 (hereafter referred to as EBL). This theory deals with non-negative real numbers, but it could be easily modified to deal with negative real numbers also. It was shown that the theory was adequate for proving a form of the fundamental theorem on least upper bounds and greatest lower bounds. More precisely, the following results were obtained in the terminology of EBL: If С is a class of U-reals and is completely represented in Κ′ and if some U-real is an upper bound of С, then there is a U-real which is a least upper bound of С. If D is a class of (U-reals and is completely represented in Κ′, then there is a U-real which is a greatest lower bound of D.

scholarly journals ON THE POSSIBLE RANKS AMONG MATRICES WITH A GIVEN PATTERN

2010 ◽  
Vol 02 (03) ◽  
pp. 363-377 ◽  
Author(s):  
CHARLES R. JOHNSON ◽  
YULIN ZHANG
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Null Space ◽  
Upper And Lower Bounds ◽  
Minimum Rank

Given are tight upper and lower bounds for the minimum rank among all matrices with a prescribed zero–nonzero pattern. The upper bound is based upon solving for a matrix with a given null space and, with optimal choices, produces the correct minimum rank. It leads to simple, but often accurate, bounds based upon overt statistics of the pattern. The lower bound is also conceptually simple. Often, the lower and an upper bound coincide, but examples are given in which they do not.

BOUNDS ON THE PRICE OF STABILITY OF UNDIRECTED NETWORK DESIGN GAMES WITH THREE PLAYERS

2011 ◽  
Vol 12 (01n02) ◽  
pp. 1-17 ◽  
Author(s):  
VITTORIO BILÒ ◽  
ROBERTA BOVE
Keyword(s):  
Lower Bound ◽  
Network Design ◽  
Lower Bounds ◽  
Upper Bound ◽  
Cost Allocation ◽  
Upper And Lower Bounds ◽  
Price Of Stability ◽  
Undirected Network ◽  
Basic Case

After almost seven years from its definition,2 the price of stability of undirected network design games with fair cost allocation remains to be elusive. Its exact characterization has been achieved only for the basic case of two players2,7 and, as soon as the number of players increases, the gap between the known upper and lower bounds becomes super-constant, even in the special variants of multicast and broadcast games. Motivated by the intrinsic difficulties that seem to characterize this problem, we analyze the already challenging case of three players and provide either new or improved bounds. For broadcast games, we prove an upper bound of 1.485 which exactly matches a lower bound given in Ref. 4; for multicast games, we show new upper and lower bounds which confine the price of stability in the interval [1.524; 1.532]; while, for the general case, we give an improved upper bound of 1.634. The techniques exploited in this paper are a refinement of those used in Ref. 7 and can be easily adapted to deal with all the cases involving a small number of players.

Lower bounds to eigenvalues

1969 ◽  
Vol 47 (17) ◽  
pp. 1877-1879 ◽  
Author(s):  
Maurice Cohen ◽  
Tova Feldmann
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Hamiltonian Operator ◽  
Upper And Lower Bounds ◽  
Quantum Mechanical ◽  
Classical Procedure ◽  
Ritz Procedure ◽  
Reasonable Choice ◽  
Lowest Eigenvalue

The classical procedure of Weinstein has been employed to obtain rigorous upper and lower bounds to the eigenvalues E of a quantum mechanical Hamiltonian operator H. The new bounds represent an improvement over Weinstein's bounds for any reasonable choice of variational trial function. In the case of the lowest eigenvalue E0, for which the Rayleigh–Ritz procedure gives the optimum upper bound, the new lower bound is an improvement over the lower bound formula of Stevenson and Crawford.

Quantum collision-resistance of non-uniformly distributed functions: upper and lower bounds

2018 ◽  
Vol 18 (15&16) ◽  
pp. 1332-1349
Author(s):  
Ehsan Ebrahimi ◽  
Dominique Unruh
Keyword(s):  
Lower Bound ◽  
Uniform Distribution ◽  
Lower Bounds ◽  
Upper Bounds ◽  
Query Complexity ◽  
Upper And Lower Bounds ◽  
Collision Resistance ◽  
Quantum Query Complexity ◽  
Quantum Query

We study the quantum query complexity of finding a collision for a function f whose outputs are chosen according to a non-uniform distribution D. We derive some upper bounds and lower bounds depending on the min-entropy and the collision-entropy of D. In particular, we improve the previous lower bound by Ebrahimi Targhi et al. from \Omega(2^{k/9}) to \Omega(2^{k/5}) where k is the min-entropy of D.

Complementary variational principles in perturbation theory

1968 ◽  
Vol 303 (1475) ◽  
pp. 503-509 ◽  
Keyword(s):  
Perturbation Theory ◽  
Lower Bound ◽  
Harmonic Oscillator ◽  
Lower Bounds ◽  
Upper Bound ◽  
Variational Principles ◽  
Upper And Lower Bounds ◽  
Quantum Mechanical ◽  
Simple Application ◽  
Mechanical Perturbation

Complementary upper and lower bounds are derived for second-order quantum-mechanical perturbation energies. The upper bound is equivalent to that of Hylleraas. The lower bound appears to be new, but reduces to that of Prager & Hirschfelder if a certain constraint is applied. A simple application to a perturbed harmonic oscillator is presented.

The number of solutions of the Erdős-Straus Equation and sums of k unit fractions

2019 ◽  
Vol 150 (3) ◽  
pp. 1401-1427
Author(s):  
Christian Elsholtz ◽  
Stefan Planitzer
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Rational Number ◽  
Residue Class ◽  
Upper Bounds ◽  
Log P ◽  
Number Of Solutions ◽  
Running Time ◽  
Unit Fractions ◽  
All Solutions

AbstractWe prove new upper bounds for the number of representations of an arbitrary rational number as a sum of three unit fractions. In particular, for fixed m there are at most ${\cal O}_{\epsilon }(n^{{3}/{5}+\epsilon })$ solutions of ${m}/{n} = {1}/{a_1} + {1}/{a_2} + {1}/{a_3}$. This improves upon a result of Browning and Elsholtz (2011) and extends a result of Elsholtz and Tao (2013) who proved this when m=4 and n is a prime. Moreover, there exists an algorithm finding all solutions in expected running time ${\cal O}_{\epsilon }(n^{\epsilon }({n^3}/{m^2})^{{1}/{5}})$, for any $\epsilon \gt 0$. We also improve a bound on the maximum number of representations of a rational number as a sum of k unit fractions. Furthermore, we also improve lower bounds. In particular, we prove that for given $m \in {\open N}$ in every reduced residue class e mod f there exist infinitely many primes p such that the number of solutions of the equation ${m}/{p} = {1}/{a_1} + {1}/{a_2} + {1}/{a_3}$ is $\gg _{f,m} \exp (({5\log 2}/({12\,{\rm lcm} (m,f)}) + o_{f,m}(1)) {\log p}/{\log \log p})$. Previously, the best known lower bound of this type was of order $(\log p)^{0.549}$.

scholarly journals Maximal prime gaps bounds

2021 ◽  
Author(s):  
Jan Feliksiak
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Estimation Error ◽  
The Other ◽  
Upper And Lower Bounds ◽  
Close Approximation ◽  
Implicit Constant ◽  
Error Cost ◽  
Prime Gaps

This paper presents research results, pertinent to the maximal prime gaps bounds. Four distinct bounds are presented: Upper bound, Infimum, Supremum and finally the Lower bound. Although the Upper and Lower bounds incur a relatively high estimation error cost, the functions representing them are quite simple. This ensures, that the computation of those bounds will be straightforward and efficient. The Lower bound is essential, to address the issue of the value of the lower bound implicit constant C, in the work of Ford et al (Ford, 2016). The concluding Corollary in this paper shows, that the value of the constant C does diverge, although very slowly. The constant C, will eventually take any arbitrary value, providing that a large enough N (for p <= N) is considered. The Infimum/Supremum bounds on the other hand are computationally very demanding. Their evaluation entails computations at an extreme level of precision. In return however, we obtain bounds, which provide an extremely close approximation of the maximal prime gaps. The Infimum/Supremum estimation error gradually increases over the range of p and attains at p = 18361375334787046697 approximately the value of 0.03.

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