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1 /2002/3() Joural of Software Vol3 No Z 2 23 ( ); 2 ( ); 3 ( 00080) jywa820@siacom http//cssjtueduc Z Z Z ; ; TP3 A [] [2] Z Z PaZ Z Z Z PaZPaZ f X YAFX f A f A FY f A x A fx ; ( ) (974 ) ; (93 ) ; (947 )

2 Z (divide ad coquer) (greedy) (brach ad boud) [2] DPs OMT PaZ Z 3 SRVRT arb [3] arb Z arb PaZ arb PaZ arb arb 2 Z arb ; ; SRVRT( ) DPs Library Tras-Rules Specificatio Refiemet Para-Desig Refiemet arb program Trasformatio Para-Program verificatio trasformatio PaZ laguage frame arb model frame arb PaZ arb Fig SRVRT parallel program develop model SRVRT Specificatio Z 2 Refiemet usig DPs PaZ 3 Verificatio 4 Refiemet usig laws PaZ arb arb 5 Trasformatio [3] arb

3 208 Joural of Software 20023() 3 2 SRVRT 3 (Parallel Divide & Coquer) Parallel Divide & Coquer SRVRT Parallel Divide & Coquer PaZ Parallel Divide & Coquer [DATARESULT] datas DATA result RESULT divide DATA F DATA merge F REAULT RESULT baseoperate DATA RESULT basedatas DATA BOOL D&C DATA RESULT D&C datas = if basedatas datas the baseoperate datas Else merge D&C divide datas Result=D&C data 32 SRVRT 2 SRVRT Specificatio Z DFT a seq complex b seq complex FFT seq complex seq complex k N w complex =#a=#b k i = 2 w = 0 < i < w j b( = b=fft a a( w ( i )( j ) 2 Refiemet usig DPs Parallel Divide & Coquer DFT DFT Parallel Divide & Coquer Parallel Divide & Coquer DFT a b Parallel Divide & Comquer datas result DATA seq complex RESULT seq complex Datas a Result b Parallel Divide & Coquer basedatasbaseoperatemergedivide D&C FFTbasedatasFFTbaseoperateFFTmergeFFTdivideFFTD&C FDT k i ( i )( j ) = #a = #b = 2 w = 0 < i < w b=fftd&c a j b( = a( w i =

4 Z 209 DFT FFTbasedatas FFTbaseoperate FFTbasedatas a = (=) FFTbaseoperate a = a > Parallel Divide & Coquer a a ( i )( j ) 2( i )( j ) j 2( i )( j ) j b( = a( w j b( = a (2i ) w + w a (2 w i = w b 2( i )( j ) j 2( i )( j ) j b( = a(2i ) w + w a(2 w i = i = 2( i )( j ) j 2( i )( j ) j b( j + ) = a(2i ) w w a(2 w FFTdivide FFTmerge a seq complex subdatas F seq complex a eve a odd seq complex FFTdivide seq complex F seq complex /2=#a odd =#a eve Subdatas=FFTdivide a Subdatas={a eve a odd } i a ( = a(2 a ( eve odd = a ( 2i ) Subresults F seq complex b eve b odd seq complex b seq complex FFTmerge F seq complex seq complex /2=#b odd =#b eve b=fftmerge subresults Subresults={b eve b odd } i i b( = b ( ) odd i + w beve( b( i + ) FFTD&C Parallel Divide & Coquer FFTD&C a = if FFTbasedatas a the FFTbaseoperate a Else FFTmerge FFTD & C FFTdivide a = b odd i ( w beve( 3 Verificatio DFT k i ( i )( j ) =#a=#b = 2 w = 0 < i < w b=fftd&c a j b( = a( w () () if = = w= FFTbaseoperate a=a() b()=a() () ; (2) If =2 k () w w k j b( = a( i = ( i )( j ) w k =2 k+ () ( ) 4 Refiemet usig laws PaZ arb arb PaZ arb i op i arball(n) call op i ed arb 2 arb itegern complexa(n)b(n)aeve(n/2)aodd(n/2)beve(n/2)bodd(n/2) complexw subroutie fftbaseoperate(a) b()=a() i = (2)

5 20 Joural of Software 20023() ed subroutie fftbaseoperate subroutie fftdivide(a) arball (N/2) aeve(=a(2 aodd(=a(2i ) ed arb ed subroutie fftdivide subroutie fftmerge arball (N/2) b(=bodd(+w i beve( b(i+n/2)=bodd( w i beve( ed arb ed subroutie fftmerge recursive subroutie fft(a) if fftbasedatas(a) the call fftbaseoperate(a) else call fftdivide(a) arb call fft(aeve) call fft(aodd) ed arb call fftmerge(bevebodd) ed if ed subroutie fft PRAM [4] 5 Trasform arb ( [3]) 4 Z [5~7] SRVRT SRVRT PQE2000 Project [8] [9] [0] ( ) (programmig) (desig) SRVRT SRVRT PaZ arb

6 Z 2 Refereces [] Gamma E Helm R Johso R et al Desig Patters Elemets of Reusable Object-Orieted Software Addiso-Wesley 995 [2] Wa Jia-yi Su Yog-qiag Xue Ji-yu Expadig desig patters to parallel programmig I Che Jia Che Pig Meyor B eds Proceedigs of the 38th Iteratioal Coferece o Techology of Object Orieted Laguages ad Systems TOOLS 36 Los Alamitos CA IEEE Computer Society 2000 [3] Massigill Bera A Structured Approach to Parallel Programmig [PhD Thesis] Califoria Istitute of Techology 998 [4] Berma KA Berma K Paul J Fudametals of Sequetial ad Parallel Algorithms Bosto MA PWS Publishig Compay 997 [5] Digel J A trace-based refiemet calculus for shared-variable parallel programs I Haeberer AM ed Algobraic Methodology ad Software Techology Proceedigs of the 7th Iteratioal Coferece AMAST 98 LNCS 548 Berli Sprig-Verlag 999 [6] Bodeveix JP Filali M A framework for parallel program refiemet I Briksma E Cleavelad WR Larse KG et al eds Proceedigs of the Workshop o Tools ad Algorithms for the Costructio ad Aalysis of Systems Berli Sprig-Verlag 995 [7] Chi Wei-Nga Khoo Siau-Cheg Hu Zhe-jiag et al Derivig parallel codes via ivariats I Static Aalysis Proceedigs of the 7th Iteratioal Symposium SAS 2000 LNCS 824 Sprig-Verlag 2000 [8] Bacci B Daelutto M et al SkIE a heterogeeous eviromet for HPC applicatios Parallel Computig ~852 [9] Massigill BL Chady KM Parallel program archetypes I IPPS/SPDP Proceedigs of the 3th Iteratioal Parallel Processig Symposium 0th Symposium o Parallel ad Distributed Processig Los Alamitos CA IEEE Computer Society 999 http//resolverlibrarycaltechedu/caltechcstr997cs-tr [0] Goswami D Sigh A Preiss BR Buildig parallel applicatios usig desig patters I Advaces i Software Egieerig Topics i Comprehesio Evolutio ad Evaluatio New York Spriger-Verlag 2000 Refiemet from Z Specificatio to Parallel Program WAN Jia-yi 2 SUN Yog-qiag XUE Ji-yu 23 (Departmet of Computer Sciece ad Egieerig Shaghai Jiaotog Uiversity Shaghai Chia); 2 (Departmet of Computer Sciece Jiagxi Normal Uiversity Nachag Chia); 3 (Key Laboratory for Computer Sciece Istitute of Software The Chiese Academy of Scieces Beijig Chia) jywa820@siacom http//cssjtueduc Abstract I this paper a method of gettig parallel program from Z specificatio through refiemet of desig patters is proposed It expads parallel cocepts ito Z otatio Begiig with Z fuctioal specificatio it refies expaded desig patters step by step to get parallel desig ad succeed i obtaiig abstract parallel program through sematic-preservig trasformatios which ca be trasformed to parallel code fially This method is described i detail through a example Key words desig patters; refiemet; parallel program developmet Received March 6 200; accepted Jue Supported by the Natioal Natural Sciece Foudatio of Chia uder Grat No