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SalingerO  =JOptimal Speedup on a Low-Degree Multi-Core Parallel Architecture (LoPRAM)KK$CAlejandro Salinger Cheriton School of Computer Science University of Waterloo Joint work with Alejandro Lpez-Ortiz and Reza Dorrigiv6P;8j 4Multicore Challenge$PRAM model will no longer accurately reflect the architecture on which algorithms are executed. PRAM facilitates design and analysis, however: Unrealistic. Difficult to derive work-optimal algorithms for (n) processors. 2, 4, or 8 cores per chip: low degree parallelism. Thread-based parallelism.BNMNL] Multicore Challenge$ Design a model such that: Reflects available degree of parallelism. Multi-threaded. Easy theoretical analysis. Easy to program. :gU"q 4 The LoPRAM Model$Number of cores is not a constant: modeled as O(log n). Similar to bit-level parallelism, w = O(log n)-bit word. LoPRAM: PRAM with p = O(log n) processors running in MIMD mode. Concurrent Read Exclusive Write (CREW). Simplest form: high-level thread-based parallelism. Semaphores and automatic serialization available and transparent to programmer. p = O(log n) but not p = (log n).qZ ZZ ( w  ' PAL-threads$( $ovoid mergeSort(int numbers[], int temp[], int array_size) { m_sort(numbers, temp, 0, array_size - 1); } void m_sort(int numbers[], int temp[], int left, int right) { int mid = (right + left) / 2; if (right > left) { m_sort(numbers, temp, left, mid); m_sort(numbers, temp, mid+1, right); merge(numbers, temp, left, mid+1, right); } }hpP  Y  4  ' PAL-threads$( $void mergeSort(int numbers[], int temp[], int array_size) { m_sort(numbers, temp, 0, array_size - 1); } void m_sort(int numbers[], int temp[], int left, int right) { int mid = (right + left) / 2; if (right > left) { palthreads { // do in parallel if possible m_sort(numbers, temp, left, mid); m_sort(numbers, temp, mid+1, right); } // implicit join merge(numbers, temp, left, mid+1, right); } }hP  Z  4  )Work-Optimal Algorithms: Divide & Conquer**$TRecursive divide-and-conquer algorithms with time given by: By the master theorem:&U<Divide & Conquer$ Divide & Conquer$&Parallel Master theorem in the LoPRAM:Matrix Multiplication$ Dynamic programming"QGeneric parallel algorithm that exploits the parallelism of execution in the DAG.RR Conclusions "Computers have a small number of processors. 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=1:Bvisible*o3>+B#style.visibility<*&l%(D' =-o6Bwipe(up)*<3<*&lD3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*-l%(D' =-o6Bwipe(up)*<3<*-lD3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*.l%(D' =-o6Bwipe(up)*<3<*.lD3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*5l%(D' =-o6Bwipe(up)*<3<*5lD3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*6l%(D' =-o6Bwipe(up)*<3<*6lD3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*=l%(D' =-o6Bwipe(up)*<3<*=lD3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*>l%(D' =-o6Bwipe(up)*<3<*>lD ' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$l%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*+l%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*3l%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*;l%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,l%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4l%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*+B#style.visibility<*?l%(D' =%(D' =%(D!' =4@BBBB%(D' =,54*3>Bfillcolor=@BPB<*lD' =1:B solid*a3>Bfill.type<*lD' =1:B true*]3>Bfill.on<*lD!' =4@BBBB%(D' =,54*3>Bfillcolor=@BPB<*lD' =1:B solid*a3>Bfill.type<*lD' =1:B true*]3>Bfill.on<*lD!' =4@BBBB%(D' =,54*3>Bfillcolor=@BPB<*lD' =1:B solid*a3>Bfill.type<*lD' =1:B true*]3>Bfill.on<*lD!' =4@BBBB%(D' =,54*3>Bfillcolor=@BPB<* lD' =1:B solid*a3>Bfill.type<* lD' =1:B true*]3>Bfill.on<* lD!' =4@BBBB%(D' =,54*3>Bfillcolor=@BPB<*$lD' =1:B solid*a3>Bfill.type<*$lD' =1:B true*]3>Bfill.on<*$lD!' =4@BBBB%(D' =,54*3>Bfillcolor=@BPB<*+lD' =1:B solid*a3>Bfill.type<*+lD' =1:B true*]3>Bfill.on<*+lD!' =4@BBBB%(D' =,54*3>Bfillcolor=@BPB<*3lD' =1:B solid*a3>Bfill.type<*3lD' =1:B true*]3>Bfill.on<*3lD!' =4@BBBB%(D' =,54*3>Bfillcolor=@BPB<*;lD' =1:B solid*a3>Bfill.type<*;lD' =1:B true*]3>Bfill.on<*;l+8+0+El0 + U 0 0(  ~  s * `     s *@ `<$ 0    0A?P ,$D 0  0A ?X 4,$D 0H  0޽h ? 33 , TIMING|1.1|9.1p ___PPT10P ..^+vkD '  = @B D ' = @BA?%,( < +O%,( < +De' =%(D)' =%(D' =ABBB@B0B%())))?D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(De' =%(D)' =%(D' =ABBB@B0B%())))?D' =1:Bvisible*o3>+B#style.visibility<*>U%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(+8+0+0 + U 0 @p(  ~  s *Te(hM   X  0A?@ R  s *A C rH  0޽h ? 33___PPT10i.=\\+D='  = @B +  U 0 B(  ~  s *+e(hM   ~  s *+ `C  X  0A?0`V 5  60 ` ,$ 0 IIf we assume parallel merging the third case becomes Tp(n) = (f (n)/p). 6J7 a  <d7 M,$ 0 AOptimal speedup [i.e. Tp(n) = T(n)/p ] so long as p = O(log n). BA>H  0޽h ? 33. TIMING|16.4|6.2T___PPT104.++~zD'  = @B DS' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(+p+0+0 ++0+0 + U 0 (  x  c $Ee(hM   V  0,IG 0T(n)=7T(n/2)+O(n2) T(n)=O(n2.8) Tp(n)=O(n2.8/p)l1     6  |  C TA<matrixmult_speedup_sch4_colorH  0޽h ? 33___PPT10i.o+D='  = @B + U 0  (  r  S ^    r  S ^Hs   p  C HA0mcm_speedup_2neighboursp8H  0޽h ? 33___PPT10i.= 0+D='  = @B +Q  U 0 P(  r  S @e(hM     S HA `C<$ 0  H  0޽h ? 339 1 ___PPT10 ..F2+[W_D'  = @B D\' = @BA?%,( < +O%,( < +D' =%(D)' =%(D' =ABBB@B0B%())))?D' =1:Bvisible*o3>+B#style.visibility<*.%(D' =%(D)' =%(D' =ABBB@B0B%())))?D' =1:Bvisible*o3>+B#style.visibility<*.v%(D' =%(D)' =%(D' =ABBB@B0B%())))?D' =1:Bvisible*o3>+B#style.visibility<*v%(+8+0+0 +  0 (  X  C      S (9 3    H  0ηo~ ? 3380___PPT10.M%rlN wA,s7yGY  Arial Courier NewDefault DesignCustom DesignKOptimal Speedup on a Low-Degree Multi-Core Parallel Architecture (LoPRAM)Multicore ChallengeMulticore ChallengeThe LoPRAM Model( PAL-threads( PAL-threads*Work-Optimal Algorithms: Divide & ConquerDivide & ConquerDivide & ConquerMatrix MultiplicationDynamic programming Conclusions  Fonts UsedDesign Template Slide Titles *_C0Alejandro SalingerAlejandro Salinger  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~Root EntrydO)PicturesCurrent UserSummaryInformation(DUPowerPoint Document(~CDocumentSummaryInformation8