Knuth's Algorithm Research Articles

Performance of efficient balanced codes

The problem of appraising the spectral performance of codes based on a new algorithm for generating zero-disparity codewords presented by D.E. Knuth (1986) is addressed. In order to get some insight into the efficiency of Knuth's construction technique, the authors evaluate the spectral properties of its code streams. The structure of Knuth codes allows the derivation a simple expression for (an approximation to) the sum of variance of these codes. This quantity plays a key role in the spectral performance characterization of DC-balanced codes. The authors evaluate this expression and compare the sum variance of Knuth codes with the sum variance of the polarity bit codes for fixed redundancy. Under the premise that the sum variance can serve as a quantity to judge the width of the spectral notch, the authors conclude that codes based on Knuth's algorithm offer less spectral suppression than polarity bit codes with the same redundancy. >

Life: Nasty, brutish, and short

@ as well as to the right) in order to give back the simple matrix result. In contrast, the SHARP APL approach doesn't require the Sharp equivalent of an enclosed array, and its subsequent disclosure, since it uses the generalization of scalar extension called cell extension, which is built-in to the rank operator.This was the state of things until early 1986. I was reasonably happy with what I had. Then I chanced to read (for reasons having nothing to do with the game of Life) Donald Knuth's book on his typefont designing system, Metafont [Kn86]. Among the data types in the METAFONT language is the picture. Looked at from the APL point of view, a METAFONT picture is a numeric matrix. The elements of a picture describe the way a plane surface is marked, that is, they describe a picture. Pictures can be added or subtracted, and can be shifted, reflected, and rotated by multiples of 90 degrees. In other words, METAFONT has stumbled through a back door into the same area where APL has held sway all alone for so many years. The book gives the following Exercise 13.24:In John Conway's “Game of Life,” pixels are said to be either alive or dead. Each pixel is in contact with eight neighbors. The live pixels in the (n+1)st generation are those who were dead and had exactly three live neighbors in the nth generation, or those who were alive and had exactly two or three live neighbors in the nth generation. Write a short METAFONT program that displays successive generations on your screen.Turning to the answers section of the book, I found Answer 13.24:13.24 (We assume that currentpicture initially has some configuration in which all pixel values are zero or one; one means “alive.”)picture v; def c = currentpicture enddef; forever: v := c; showit; addto c also c shifted left + c shifted right; addto c also c shifted up + c shifted down; addto c also c - v; cull c keeping (5, 7); endfor.(It is wise not to waste too much computer time watching this program.)I was impressed by this algorithm, because it reduced the number of vertical and horizontal rotations required from the eight that I had used in Nbr to three. Racing to my APL machine I translated Knuth's METAFONT algorithm into APL. I also began counting tokens at this point. The APL version of Knuth's algorithm used 49 tokens. ∇z ← LifeKnuth on n ← o o ← o + (1 p o) + - 1 p o o ← o + (1 t o) + - 1 t o o ← o + o - n z ← o ∈ 5 6 7 dThis APL algorithm isn't fully equivalent to Knuth's solution, since his METAFONT shifted primitive shifts in zeros, whereas the APL rotate primitive uses nondestructive cyclic rotation. By ensuring that the argument is bordered by an edge of zeros all around, the APL and METAFONT programs will be completely equivalent.The workings of this algorithm may not be immediately evident. What is happening was expressed as follows [Hu86]: in the pattern x x x x

The Characters of the Infinite Symmetric Group and Probability Properties of the Robinson–Schensted–Knuth Algorithm

Connections between the Robinson–Schensted–Knuth algorithm, random infinite Young tableaux, and central indecomposable measures are investigated. A generalization of the RSK algorithm leads to a combinatorial interpretation of extended Schur functions. Applications are given to Ulam’s problem on longest increasing subsequences and to a law of large numbers for representations. An analogous theory for other graphs is discussed.

Circularity Testing of Attribute Grammars Requires Exponential Time: A Simpler Proof

Jazayeri, Ogden and Rounds have shown that the high time complexity of Knuth's algorithm for testing attribute grammars for circularity is no accident. It was proved that there is a constant c>0 such that any deterministic Turing Machine which correctly tests for circularity must run for more than 2 ^(cn/log n) steps on infinitely many attribute grammars (AGs) (the size of an AG is the number of symbols required to write it down). The proof was rather complex; the purpose of this note is to provide a simpler one.

An extension of knuth's algorithm for parsing context-free languages

AN EXTENSION of Knuth's algorithm for parsing context-free languages is considered. The algorithm proposed is not a left-sided parsing algorithm. It is shown how, by slightly changing the grammar, it is in many cases possible to attain unique parsing by means of an extension of Knuth's algorithm.

A practical method for constructing LR ( k ) processors

A practical method for constructing LR( k ) processors is developed. These processors are capable of recognizing and parsing an input during a single no-backup scan in a number of steps equal to the length of the input plus the number of steps in its derivation. The technique presented here is based on the original method described by Knuth, but decreases both the effort required to construct the processor and the size of the processor produced. This procedure involves partitioning the given grammar into a number of smaller parts. If an LR( k ) processor can be constructed for each part (using Knuth's algorithm) and if certain conditions relating these individual processors are satisfied, then an LR( k ) processor for the entire grammar can be constructed for them. Using this procedure, an LR(1) parser for ALGOL has been obtained.