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Algorithmic complexity is discussed as a computational counterpart to the second law of thermodynamics. It is shown that algorithmic complexity, which is a measure of randomness, sets limits on the thermodynamic cost of computations and casts a new light on the limitations of Maxwell's demon. Algorithmic complexity can also be used to define distance between binary strings.
Quantitative measurements for algorithmic complexity of discrete function
Complexity algorithms provide information about datasets which is radically different from classical moment statistics. Instead of focusing on the divergences from central values, they quantify other characteristics such as order, pattern repetitions, or the existence of attractors. However, those analyses must be done with the proper statistical treatment, which is, unfortunately, not always the case. In this contribution, I provide an example of the hazards of applying complexity measures without sufficient care by correcting a previously published analysis that aimed to quantify the complexity of climate. I clarify some misconceptions about the use of Sample Entropy and revise the incorrect assessments and conclusions drawn from the previous misapplication of the methods.
The purpose is to document research to develop strategies for concurrent processing of complex algorithms in data driven architectures. The problem domain consists of decision-free algorithms having large-grained, computationally complex primitive operations. Such are often found in signal processing and control applications. The anticipated multiprocessor environment is a data flow architecture containing between two and twenty computing elements. Each computing element is a processor having local program memory, and which communicates with a common global data memory. A new graph theoretic model called ATAMM which establishes rules for relating a decomposed algorithm to its execution in a data flow architecture is presented. The ATAMM model is used to determine strategies to achieve optimum time performance and to develop a system diagnostic software tool. In addition, preliminary work on a new multiprocessor operating system based on the ATAMM specifications is described.
This paper presents a Petri-net model suitable for analyzing the concurrent processing of computationally complex algorithms. The decomposed operations are to be processed in a multiple processor, data driven architecture. Of particular interest is the application of the model to both the description of the data/control flow of a particular algorithm, and to the general specification of the data driven architecture. A candidate architecture is also presented.
The results of ongoing research directed at developing a graph theoretical model for describing data and control flow associated with the execution of large grained algorithms in a spatial distributed computer environment is presented. This model is identified by the acronym ATAMM (Algorithm/Architecture Mapping Model). The purpose of such a model is to provide a basis for establishing rules for relating an algorithm to its execution in a multiprocessor environment. Specifications derived from the model lead directly to the description of a data flow architecture which is a consequence of the inherent behavior of the data and control flow described by the model. The purpose of the ATAMM based architecture is to optimize computational concurrency in the multiprocessor environment and to provide an analytical basis for performance evaluation. The ATAMM model and architecture specifications are demonstrated on a prototype system for concept validation.
Research directed at developing a graph theoretical model for describing data and control flow associated with the execution of large grained algorithms in a special distributed computer environment is presented. This model is identified by the acronym ATAMM which represents Algorithms To Architecture Mapping Model. The purpose of such a model is to provide a basis for establishing rules for relating an algorithm to its execution in a multiprocessor environment. Specifications derived from the model lead directly to the description of a data flow architecture which is a consequence of the inherent behavior of the data and control flow described by the model. The purpose of the ATAMM based architecture is to provide an analytical basis for performance evaluation. The ATAMM model and architecture specifications are demonstrated on a prototype system for concept validation.
The paper considers a methodology on the analysis and design of a minimum mean-square error criterion linear system incorporating a tapped delay line (TDL) where all the full-precision multiplications in the TDL are constrained to be powers of two. A linear equalizer based on the dispersive and additive noise channel is presented. This microprocessor implementation with optimized power of two TDL coefficients achieves a system performance comparable to the optimum linear equalization with full-precision multiplications for an input data rate of 300 baud.
The performance modeling and enhancement for periodic execution of large-grain, decision-free algorithms in data flow architectures is examined. Applications include real-time implementation of control and signal processing algorithms where performance is required to be highly predictable. The mapping of algorithms onto the specified class of data flow architectures is realized by a marked graph model called ATAMM (Algorithm To Architecture Mapping Model). Performance measures and bounds are established. Algorithm transformation techniques are identified for performance enhancement and reduction of resource (computing element) requirements. A systematic design procedure is described for generating operating conditions for predictable performance both with and without resource constraints. An ATAMM simulator is used to test and validate the performance prediction by the design procedure. Experiments on a three resource testbed provide verification of the ATAMM model and the design procedure.
Performance modeling and performance enhancement for periodic execution of large-grain, decision-free algorithms in data flow architectures are discussed. Applications include real-time implementation of control and signal processing algorithms where performance is required to be highly predictable. The mapping of algorithms onto the specified class of data flow architectures is realized by a marked graph model called algorithm to architecture mapping model (ATAMM). Performance measures and bounds are established. Algorithm transformation techniques are identified for performance enhancement and reduction of resource (computing element) requirements. A systematic design procedure is described for generating operating conditions for predictable performance both with and without resource constraints. An ATAMM simulator is used to test and validate the performance prediction by the design procedure. Experiments on a three resource testbed provide verification of the ATAMM model and the design procedure.
Winograd's algorithm for computing the discrete Fourier transform is extended considerably for certain large transform lengths. This is accomplished by performing the cyclic convolution, required by Winograd's method, by a fast transform over certain complex integer fields. This algorithm requires fewer multiplications than either the standard fast Fourier transform or Winograd's more conventional algorithms.
A patched-grid algorithm for the analysis of complex configurations with an implicit, upwind-biased Navier-Stokes solver is presented. Through the use of a generalized coordinate transformation at the zonal interface between two or more blocks, the algorithm can be applied to highly stretched viscous grids and to arbitrarily-shaped patch boundaries. Applications are made to the SR71 reconnaissance aircraft in a high-altitude environment at a supersonic speed and to the F/A-18 forebody-strake configuration at subsonic, high-alpha conditions, in support of the NASA High-Alpha Research Program.
A patched-grid algorithm for the analysis of complex configurations with an implicit, upwind-biased Navier-Stokes solver is presented. Results from both a spatial-flux and a time-flux conservation approach to patching across zonal boundaries are presented. A generalized coordinate transformation with a biquadratic geometric element is used at the zonal interface in order to treat highly stretched viscous grids and arbitrarily-shaped zonal boundaries. Applications are made to the F-18 forebody-strake configuration at subsonic, high-alpha conditions. Computed surface flow patterns compare well with ground-based and flight-test results; the large effect of Reynolds number on the forebody flow-field is shown.
A patched-grid algorithm for the analysis of complex configurations with an implicit, upwind-biased Navier-Stokes solver is presented. Results from both a spatial-flux and a time-flux conservation approach to patching across zonal boundaries are presented. A generalized coordinate transformation with a biquadratic geometric element is used at the zonal interface in order to treat highly stretched viscous grids and arbitrarily-shaped zonal boundaries. Applications are made to the F-18 forebody-strake configuration at subsonic, high-alpha conditions. Computed surface flow patterns compare well with ground-based and flight-test results; the large effect of Reynolds number on the forebody flowfield is shown.
Winograd (1976) has developed a new class of algorithms which depend heavily on the computation of a cyclic convolution for computing the conventional DFT (discrete Fourier transform); this new algorithm, for a few hundred transform points, requires substantially fewer multiplications than the conventional FFT algorithm. Reed and Truong have defined a special class of finite Fourier-like transforms over GF(q squared), where q = 2 to the p power minus 1 is a Mersenne prime for p = 2, 3, 5, 7, 13, 17, 19, 31, 61. In the present paper it is shown that Winograd's algorithm can be combined with the aforementioned Fourier-like transform to yield a new algorithm for computing the DFT. A fast method for accurately computing the DFT of a sequence of complex numbers of very long transform-lengths is thus obtained.
This document describes exploratory research on a distributed, trajectory oriented approach for traffic complexity management. The approach is to manage traffic complexity based on preserving trajectory flexibility and minimizing constraints. In particular, the document presents metrics for trajectory flexibility; a method for estimating these metrics based on discrete time and degree of freedom assumptions; a planning algorithm using these metrics to preserve flexibility; and preliminary experiments testing the impact of preserving trajectory flexibility on traffic complexity. The document also describes an early demonstration capability of the trajectory flexibility preservation function in the NASA Autonomous Operations Planner (AOP) platform.
The development of a gridless computational fluid dynamics (CFD) method for the solution of the two-dimensional Euler and Navier-Stokes equations is described. The method uses only clouds of points and does not require that the points be connected to form a grid as is necessary in conventional CFD algorithms. The gridless CFD approach appears to resolve the problems and inefficiencies encountered with structured or unstructured grid methods. As a result, the method offers the greatest potential for accurately and efficiently solving viscous flows about complex aircraft configurations. The method is described in detail, and calculations are presented for standard Euler and Navier-Stokes cases to assess the accuracy and efficiency of the capability.
The problem of synthesizing transfer functions from frequency response measurements is considered. Given a complex vector representing the measured frequency response of a physical system, a transfer function of specified order is determined that minimizes the sum of the magnitude-squared of the frequency response errors. This nonlinear least squares minimization problem is solved by an iterative global descent algorithm of the Newton type which converges quadratically near the minimum. The unknown transfer function is expressed as a sum of second order rational polynomials, a parameterization that facilitates a numerically robust computer implementation. The algorithm is developed for single-input, single-output, causal, stable transfer functions.