Abraham Waksman

A Permutation Network

In this paper the construction of a switching network capable of <italic>n</italic>!-permutation of its <italic>n</italic> input terminals to its <italic>n</italic> output terminals is described. The building blocks for this network are binary cells capable of permuting their two input terminals to their two output terminals. The number of cells used by the network is <<italic>n</italic> · log<subscrpt>2</subscrpt> <italic>n</italic> - <italic>n</italic> + 1> = ∑<supscrpt><italic>n</italic></supscrpt> <subscrpt><italic>k</italic>=1</subscrpt> <log<subscrpt>2</subscrpt> <italic>k</italic>>. It could be argued that for such a network this number of cells is a lower bound, by noting that binary decision trees in the network can resolve individual terminal assignments only and not the partitioning of the permutation set itself which requires only <log<subscrpt>2</subscrpt> <italic>n</italic>!> = <∑<supscrpt><italic>n</italic></supscrpt> <subscrpt><italic>k</italic>=1</subscrpt> log<subscrpt>2</subscrpt> <italic>k</italic>> binary decisions. An algorithm is also given for the setting of the binary cells in the network according to any specified permutation.

Cellular Interconnection Arrays

Abstract—A class of networks is described that has the capability of permuting in an arbitrary manner a set of n digital input lines onto a set of n digital output lines. The circuitry of the networks is arranged in cellular form, i. e., in a two-dimensional iterative pattern with mainly local intercell connections, where the basic cell behaves as a reversing switch with a single memory flip-flop. Various network forms are described, differing in the number of cells needed, in the shape of the array, and in the length and regularity of intercell connections. Also discussed are some ways of setting up the array to achieve a desired permutation.

On Winograd's Algorithm for Inner Products

This correspondence demonstrates an improvement on Winogards algorithm for inner products as applied to the multiplication of two matrices.

On the Consecutive Retrieval Property in File Organization

The consecutive retrieval (CR) property has been defined as the property of a query that can be answered by the retrieval of consecutive records in a file. In this correspondence we add some observations about boundary conditions to Goshs study of the CR property.

A Model of Replication

A one-dimensional array of finite-state machines is being considered as a model for sequence replication. The authors consider the initial state of the first k machines in the array as representing the sequence of k symbols to be replicated along the array. A construction scheme is developed which allows for such replication to take place. It is also shown that the speed of replication approaches synchronous speed.

The Interface problem in Interactive systems

On-line computing implies a man-machine dialogue. It could be looked at as a communication channel with built-in constraints. We see a need to study in a systematic way how such constraints modify human behavior. Such studies will facilitate the needed specification for effective strategies in man-machine communication.

R-70-31 A Generalized Firing Squad Problem

This work is a solution to the generalization of the by now classical problem of Minsky, namely, the firing squad problem.

On the Complexity of Inversions

The inversion process which entails the interchange of two adjacent elements in a list is fundamental to most practical sorting algorithms.Consequently, one can utilize the inversions associated with a given permutation as its measure of complexity. To this end a recurrence relation is established.

Corrigendum: `` A Permutation Network''

Theorem 1 and Corollaries t and 2 (pp. 159, 160) are correct, as egm be verified from the given constructive proof (p. HiD. However, the proof given for Theorem 1 is in error, since it does not run through all subsets of the n elements. T h e author credits Jim Turner with the discovery of the error.