Mathematical Foundations of Parallel Computing

Mathematical Foundations of Parallel Computing

Valentin V. Voevodin

Language: English

Pages: 362

ISBN: 2:00362151

Format: PDF / Kindle (mobi) / ePub


Parallel implementation of algorithms involves many difficult problems. In particular among them are round-off analysis, the way to convert sequential programmes and algorithms into parallel mode, the choice of appropriate or optimal computer architect and so on. To solve the stumbling blocks of these problems it is necessary to know the structure of algorithms very well. The book treats the mathematical mechanism that permits us to investigate structures of both sequential and parallel algorithms. This mechanism allows us to recognize and explain the relations between different methods of constructing parallel algorithms, methods to analyze round-off errors, methods to optimize memory traffic, methods to work out the fastest implementation for a given parallel computer and other methods attending the joint investigation of algorithms and computers.

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that we s u c h model m a c h i n e c a n be e x p l i c i t l y generalized s c h e d u l e s . Of c o u r s e , never require any d e t a i l e d The m o d e l m a c h i n e w i l l i t will will consider. p r o v e t o be I t turns out specified f o r the set o f a l l retain i t s properties f o r any subset o f t h a t s e t . Consider a graph o f a l g o r i t h m . able t o perform the required We i n s e r t a f u n c t i o n a l u n i t t h a t i s operation assume

u t d e v i c e s . f o r algorithm do n o t t a k e on t h e delays t o o long t o complete on f u n c t i o n a l make u s e o f t h e f o r m u l a functional (10.1) On t h e execution i s continuous units are (10.1). i n the seto f s c h e d u l e s and does n o t change i f a l l components o f a s c h e d u l e a r e s h i f ted b y o n e a n d t h e same n u m b e r , i . e . T ( A o t ) = Tit) f o r a l l A. Therefore we c a n assume m i n f . = 0 w h e n e v e r we n e e

Singapore • New Jersey • London • Hong Kong Published by World Scientific Publishing Co. Pie. Lid. P O B o x 128. FarrerRoad, Singapore 9128 USA office: Suite IB, 1060 Main Street, River Edge, NI07661 UK office: 73 Lynton Mead, Toiteridge. London N20 SDH M A T H E M A T I C A L FOUNDATIONS OF P A R A L L E L C O M P U T I N G Copyright © 1992 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any

l y we h a v e i i = KulsTdO+maxtmin i the f i r s t v, v ) = i Now we o b t a i n . i i riultflv). i a 96 m i n f m a x u.,max v.) J J i i = m i n ( T ( u ) + m i n u., T ( v ) + m i n v . ) * i i s min(T(u)®r(v)*min i u .,r(u)@T{v)+min ' i IT,) = T ( u ) ® i ( v ) f - m i n ( m i n u ^ . m i n ?. 1. i i Now we o b t a i n t h e s e c o n d o f t h e i n e q u a l i t i e s r(u®v) = max(min(u . , v . ) ) a m i n ( m a x u.,max v.) 1 i i ' (10.5): m i n ( m i n ( u , v )) £

computed of at evaluating i s independent heavy i s proportional function. mode. R o u g h l y , evaluation of additions, a given the benefit of this makes +n-2p+2. k k=p+l have L Note t o compute i s not greater than n L operations required a function point function both In four i s that o f the dimension. an a r r a y o f d a t a this function a number of and operations i t s not times. t h e c o s t o f g r a d i e n t com- The o v e r a l l memory i s

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