# September 1972 - Vol. 15 No. 9

## Features

Cellular arrays for the solution of graph problems

A cellular array is a two-dimensional, checkerboard type interconnection of identical modules (or cells), where each cell contains a few bits of memory and a small amount of combinational logic, and communicates mainly with its immediate neighbors in the array. The chief computational advantage offered by cellular arrays is the improvement in speed achieved by virtue of the possibilities for parallel processing. In this paper it is shown that cellular arrays are inherently well suited for the solution of many graph problems. For example, the adjacency matrix of a graph is easily mapped onto an array; each matrix element is stored in one cell of the array, and typical row and column operations are readily implemented by simple cell logic. A major challenge in the effective use of cellular arrays for the solution of graph problems is the determination of algorithms that exploit the possibilities for parallelism, especially for problems whose solutions appear to be inherently serial. In particular, several parallelized algorithms are presented for the solution of certain spanning tree, distance, and path problems, with direct applications to wire routing, PERT chart analysis, and the analysis of many types of networks. These algorithms exhibit a computation time that in many cases grows at a rate not exceeding log2 n, where n is the number of nodes in the graph. Straightforward cellular implementations of the well-known serial algorithms for these problems require about n steps, and noncellular implementations require from n2 to n3 steps.

File organization: the consecutive retrieval property

The consecutive retrieval property is an important relation between a query set and record set. Its existence enables the design of an information retrieval system with a minimal search time and no redundant storage. Some important theorems on the consecutive retrieval property are proved in this paper. Conditions under which the consecutive retrieval property exists and remain invariant have been established. An outline for designing an information retrieval system based on the consecutive retrieval property is also discussed.

A new approach to automatic scanning of contour maps

The problem of automatic digitizing of contour maps is discussed. The structure of a general contour map is analyzed, and its topological properties are utilized in developing a new scanning algorithm. The problem of detection and recognition of contour lines is solved by a two color labeling method. It is shown that for maps containing normal contour lines only, it suffices to distinguish between so-called “even” and “odd” lines. The “tangency problem” involved in practical scanning is discussed, and a solution based on minimizing computer memory space and simplifying control program is suggested.

Automatic error analysis for determining precision

The problem considered is that of evaluating a rational expression to within any desired tolerance on a computer which performs variable-precision floating-point arithmetic operations. For example, the expression might be &pgr;/(&pgr; + 1/2 - e) √2), which is rational in the data &pgr;, e, √2. An automatic error analysis technique is given for determining, directly from the results of a trial low-precision interval arithmetic calculation, just how much precision and data accuracy are required to achieve a desired final accuracy. The techniques given generalize easily to the evaluation of many nonrational expressions.

Thinning algorithms on rectangular, hexagonal, and triangular arrays

In this report three thinning algorithms are developed: one each for use with rectangular, hexagonal, and triangular arrays. The approach to the development of each algorithm is the same. Pictorial results produced by each of the algorithms are presented and the relative performances of the algorithms are compared. It is found that the algorithm operating with the triangular array is the most sensitive to image irregularities and noise, yet it will yield a thinned image with an overall reduced number of points. It is concluded that the algorithm operating in conjunction with the hexagonal array has features which strike a balance between those of the other two arrays.

A comparison of floating point summation methods

In the June 1970 issue of Communications, Linz [1] proposed a method of pairwise summing of numbers to reduce accumulated roundoff error. Linz compared his method with only a simple recursive summation. The Linz method should have also been compared with a method published in the January 1965 issue by Kahan [2], which is more accurate.
This note compares the schemes by Linz and Kahan with the straight recursive summation. Comparisons of accuracy, speed and storage were carried out on an IBM 360/75 with various compiler optimization levels. (Linz' restriction to a binary exponent is not essential to his method.)

A controller for a braille terminal

Anderson and Rogers [1] have described a set of modifications which convert a standard model 33 tele-type into a device which produces embossed braille characters. Thus, blind computer users can communicate with a computer without the help of a sighted person to read the output. The disadvantage of this terminal alone is that either the applications program or the operating system of the host computer must have special software to map each output character into a sequence of three characters necessary to emboss its braille equivalent. Such software modification can be expensive and has to be done for each applications program or operating system the blind person wants to use.

On Foster’s information storage and retrieval using AVL trees

Foster [2] has proposed a method of constructing an AVL tree with depth (height) h and the number of items N being fixed and with the weighted sum of the items a maximum. The purpose of this note is to show that this method does not work in general.