Straight-line Algorithms Research Articles

On Scalability of Routing Tables in Dense Flat-Label Wireless Networks

Consider a large wireless ad hoc network that facilitates communication between random pairs of network nodes. This paper investigates the size of routing tables as the number of nodes in the network increases. A routing protocol in a flat-label network is information-efficient if the amount of information at individual nodes required to route packets does not increase with the network size. It is shown that the shortest-path and straight-line routing algorithms are not information-efficient, i.e., these protocols can be implemented only when nodes' memory increases with the network size. On the other hand, it is established that there exists an information-efficient algorithm that routes packets correctly even if each node in the network is capable of storing information on a fixed number of destinations only.

On the Complexity of Bilinear Forms over Associative Algebras

Let F be a field, and let $q(\alpha ) = q_1^{d_1 } (\alpha ) \cdots q_1^{d_k } (\alpha ) \in F[\alpha ]$ be a polynomial of degree n, where $q_1 (\alpha ) \cdots q_k (\alpha )$ are distinct irreducible polynomials. Let $y(\alpha ),y_1 (\alpha ), \ldots ,y_r (\alpha ),x_1 (\alpha ), \ldots ,x_s (\alpha )$ be $(n - 1)$-degree polynomials with distinct nonscalar coefficients. The authors show the following: the number of nonscalar multiplications/divisions required to compute the coefficients of $x_1 (\alpha ),y(\alpha )\bmod q(\alpha )$ for $i = 1, \ldots ,s$ by straight line algorithms is $s(2n - k)$. If H is a $s \times r$- matrix with entries from F, then the number of nonscalar multiplications/ divisions required to compute the coefficients of $(x_1 (\alpha ), \ldots ,x_s (\alpha ))H(y_1 (\alpha ), \ldots ,y_r (\alpha ))^T \bmod q(\alpha )$ by straight line algorithms is equal to $(2n - k)$ rank $(H)$. All the above systems satisfy the direct sum conjecture strongly. The above results also hold for some other algebras that are direct sums of local algebras, such as commutative algebras and division algebras.

Inverting Polynomials and Formal Power Series

The problem of inverting a general polynomial of degree $d > 0$, the inverse being a formal power series with coefficients $z_0 ,z_1 ,z_2 , \ldots $ is considered. It is shown that computing $z_0 , \ldots ,z_n $ can be carried out in $n + (2d - 1)\lceil {\log _2 n} \rceil + 2$ nonscalar operations over any infinite field. A lower bound of $n + 1$ for all fields follows from standard linear-independence arguments. (The model of computation consists of straight-line algorithms.)For fields of characteristic 0, it is also shown that computing $z_{n + 1} , \ldots ,z_{n + s} $ for $n \geqslant 0$, $s \geqslant 1$, requires at least $s + \min {{(n + 1,d)} / {2 - 1}}$ nonscalar operations. For the case of inverting a general formal power series a corresponding lower bound of $s + {{(n + 1)} / {2 - 1}}$ exists. In particular, computing the nth coefficient of the inverse of a general formal power series requires at least ${n / 2}$ nonscalar operations (the coefficients are numbered from zero).

Multiplication of Polynomials over Finite Fields

The authors prove the $2.5 n - o(n)$ lower bound on the number of multiplications/divisions required to compute the coefficients of the product of two polynomials of degree n over a finite field by means of straight-line algorithms.

A simple algorithm for the estimation of the spectral radiation distribution on a horizontal surface, based on global radiation measurements

A simple algorithm is presented for the evaluation of the spectral distribution of the solar radiation on a horizontal surface as a function of the measured total solar irradiance (global radiation). The method, valid for industrial applications (agriculture, aging of materials, etc.), is based on a triangular representation of the spectrum. Straight line algorithms between 300 and 465 nm and 465 and 900 nm are used, which are functions of the global radiation and mean adjustment coefficients. Our method has been tested on a complete year of spectral observations at eight wavelengths, as well as on typical spectral distributions for clear and overcast skies.

Addition Requirements for Rational Functions

A notion of rank or independence for arbitrary sets of rational functions is developed, which bounds from below the number of additions and subtractions required of all straight-line algorithms which compute those functions. This permits a uniform derivation of the best lower bounds known for a number of familiar sets of rational functions. The result is proved without the use of substitution arguments. This not only provides an interesting contrast to standard approaches for arithmetic lower bounds, but also allows the algebraic setting to be somewhat generalized. Keywords: additions, algorithms, analysis of algorithms, arithmetic complexity, computational complexity, dimensionality, lower bounds, matrix multiplication, optimality, polynomials, rational functions.