: The dynamic tree expression problem (DTEP) was defined in (Ma87). In this paper, efficient implementations of the DTEP algorithm are developed for the hypercube, butterfly, perfect shuffle and multi-dimensional mesh of trees families of networks.

We present a uniform method for obtaining efficient parallel algorithms for a rather large class of problems. The method is based on a logic programming model, and it derives its efficiency from fast parallel routines for the evaluation of expression trees.

methods

Network Implementations of the DTEP Algorithm

The Dynamic Tree Expression Problem

Given a vector of N elements, the perfect shuffle of this vector is a permutation of the elements that are identical to a perfect shuffle of a deck of cards. Elements of the first half of the vector are interlaced with elements of the second half in the perfect shuffle of the vector.

Parallel Processing with the Perfect Shuffle

Computational Aspects of Vlsi

Very high-speed computers may be classified as follows: 1) Single Instruction Stream-Single Data Stream (SISD) 2) Single Instruction Stream-Multiple Data Stream (SIMD) 3) Multiple Instruction Stream-Single Data Stream (MISD) 4) Multiple Instruction Stream-Multiple Data Stream (MIMD). "Stream," as used here, refers to the sequence of data or instructions as seen by the machine during the execution of a program. The constituents of a system: storage, execution, and instruction handling (branching) are discussed with regard to recent developments and/or systems limitations. The constituents are discussed in terms of concurrent SISD systems (CDC 6600 series and, in particular, IBM Model 90 series), since multiple stream organizations usually do not require any more elaborate components. Representative organizations are selected from each class and the arrangement of the constituents is shown.

Very high-speed computing systems

: Trees play a fundamental role in many computations, both for sequential as well as parallel problems. The classic paradigm applied to generate parallel algorithms in the presence of trees has been divide-conquer; finding a 1/3 - 2/3 separator and recursively solving the two subproblems. A now classic example is Brent's work on parallel evaluation of arithmetic expressions. This top-down approach has several complications, one of which is finding the separators. We define dynamic expression evaluation as the task of evaluating the expression with no free preprocessing. If we apply Brent's method, finding the separators seems to add a factor of log n to the running time. We give a bottom-up algorithm to handle trees. That is, all modifications to the tree are done locally. This bottom-up approach which we call CONTRACT has two major advantages over the top-down approach: (1) the control structure is straight forward and easier to implement facilitating new algorithms using fewer processors and less time; and (2) problems for which it was too difficult or too complicated to find polylog parallel algorithms are now easy.

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Parallel tree contraction and its application

Matrix multiplication algorithms for cube connected and perfect shuffle computers are presented. It is shown that in both these models two $n \times n$ matrices can be multiplied in $O(n/m + \log m)$ time when $n^2 m$, $1 \leqq m \leqq n$, processing elements (PEs) are available. When only $m^2 $, $1 \leqq m \leqq n$, PEs are available, two $n \times n$ matrices can be multiplied in $O(n^2/m + m(n/m)^{2.61} )$ time. It is shown that many graph problems can be solved efficiently using the matrix multiplication algorithms.

Network Implementations of the DTEP Algorithm

Citations

The Dynamic Tree Expression Problem

References

Parallel Processing with the Perfect Shuffle

Computational Aspects of Vlsi

Very high-speed computing systems

Parallel tree contraction and its application

Parallel Matrix and Graph Algorithms

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