Performance Analysis of Divide and Conquer Algorithms for the WHT
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1 Performace Aalyss of Dvde ad Coquer Algorhms for he WHT Jeremy Johso Mha Furs, Pawel Hczeko, Hug-Je Huag Dep. of Compuer Scece Drexel Uversy
2 Movao O moder maches operao cou s o always he mos mpora performace merc. Effecve ulzao of he memory herarchy, ppelg, ad Isruco Level Parallelsm s mpora, ad s o easy o deerme such ulzao from source code. Auomac Performace Tug ad Archecure Adapao Geerae ad Tes FFT, Marx Mulplcao, Expla performace dsrbuo LACSI 6 Auomac Tug of Lbrares ad Applcaos
3 Oule Space of WHT Algorhms WHT Package ad Performace Dsrbuo Performace Model Isruco Cou Cache LACSI 6 Auomac Tug of Lbrares ad Applcaos
4 Walsh-Hadamard Trasform y WHT N x, N WHT... WHT WHT WHTN WHT WHT WHT LACSI 6 Auomac Tug of Lbrares ad Applcaos
5 LACSI 6 Auomac Tug of Lbrares ad Applcaos Facorg he WHT Marx Facorg he WHT Marx AC ΒD (Α Β(C D A Β (Α Ι(Ι Β A (Β C (A Β C I m Ι Ι m WHT WHT WHT (WHT Ι (Ι WHT
6 LACSI 6 Auomac Tug of Lbrares ad Applcaos Recursve Algorhm Recursve Algorhm (WHT (WHT I (I (I (WHT (WHT I (I (I WHT WHT
7 LACSI 6 Auomac Tug of Lbrares ad Applcaos Ierave Algorhm Ierave Algorhm (WHT (WHT I (I (I WHT WHT I (I (I WHT WHT
8 WHT Algorhms Recursve Ierave Geeral N ( ( WHT WHT IN / I WHT / N ( WHT I WHT I where (, L I WHT I WHT L L N LACSI 6 Auomac Tug of Lbrares ad Applcaos
9 WHT Implemeao N N * N * *N N M x WHT N *x x (x(b,x(bs, x(b(m-s b,s Implemeao(esed loop RN; S; for,, RR/N for j,,r- for k,,s- x jns k, S WHTN x jns k, S SS* N ; ( I - WHT I WHT LACSI 6 Auomac Tug of Lbrares ad Applcaos
10 Paro Trees Lef Recursve 3 Ierave 9 3 Rgh Recursve 3 Balaced LACSI 6 Auomac Tug of Lbrares ad Applcaos
11 LACSI 6 Auomac Tug of Lbrares ad Applcaos Number of Algorhms Number of Algorhms >,, T T T L L 6.8, / ( ( 8 8 T( ( T( ( 6 3/ 3 T T( T T( Θ α α z z z z z z z z z z z z z z L
12 Oule WHT Algorhms WHT Package ad Performace Dsrbuo Performace Model Isruco Cou Cache LACSI 6 Auomac Tug of Lbrares ad Applcaos
13 WHT Package Püschel & Johso (ICASSP Allows easy mplemeao of ay of he possble WHT algorhms Paro ree represeao W(small[] spl[w(, W( ] Tools Measure rume of ay algorhm Measure hardware eves (coupled wh PCL/PAPI Search for good mplemeao Dyamc programmg Evoluoary algorhm LACSI 6 Auomac Tug of Lbrares ad Applcaos
14 Algorhm Comparso Recursve/Ierave Rume Rec &Bal/I Isruco Cou rao.e.8e.6e.e.e.e 8.E- 6.E-.E-.E-.E WHT sze(^ r/ rr/ lr/ bal/ Rec& I /Bes Rume Small/I Rume rao.e.e 8.E 6.E.E.E r/b r3/b /b 3/b b/b rao.e.e 8.E 6.E.E.E.E I_/r r_/r.e WHT sze(^ WHT sze(^ LACSI 6 Auomac Tug of Lbrares ad Applcaos
15 Cache Mss Daa Recursve vs. Ierave Normalzed o Bes Recursve vs. Ierave.E.6E Rao Alg Tme/Bes Tme.E 8.E 6.E.E.E Recursve Tme Ierave Tme Rao Recursve/Ierave.E.E.E 8.E- 6.E-.E-.E- Isrucos L Daa Cache Mss es L Daa Cache Mss es.e.e 7 3 sze sze 6 9 Ierave vs. Bes Recursve vs. Bes 9.E 6.E Rao Ierave/Bes 8.E 7.E 6.E 5.E.E 3.E.E.E Isrucos L Daa Cache Msses L Cache Msses Rao Recursve/Ierave 5.E.E 3.E.E.E Isrucos L Daa Cache Mss es L Cache Msses.E.E sze LACSI 6 Auomac Tug of Lbrares ad Applcaos sze
16 Hsogram ( 6,, samples Wde rage performace despe equal umber of arhmec operaos ( flops Peum III vs. UlraSPARC II LACSI 6 Auomac Tug of Lbrares ad Applcaos
17 Oule WHT Algorhms WHT Package ad Performace Dsrbuo Performace Model Isruco Cou Cache LACSI 6 Auomac Tug of Lbrares ad Applcaos
18 WHT Implemeao N N * N N N M x WHT N *x x (x(b,x(bs, x(b(m-s b,s Implemeao(esed loop RN; S; for,, RR/N for j,,r- for k,,s- x jns k, S WHTN x jns k, S SS* N ; ( I - WHT I WHT LACSI 6 Auomac Tug of Lbrares ad Applcaos
19 Isruco Cou Model IC( 3 α A( β ( α l L 3 l A l ( A( umber of calls o WHT procedure α umber of srucos ousde loops A l ( Number of calls o base case of sze l α l umber of srucos base case of sze l L umber of eraos of ouer (, mddle (, ad ouer (3 loop β umber of srucos ouer (, mddle (, ad ouer (3 loop body LACSI 6 Auomac Tug of Lbrares ad Applcaos
20 Small[].fle gcc_compled.:.ex "s_.c".verso ".".alg.globl apply_small.ype apply_small: movl 8(%esp,%edx //load srde S o EDX movl (%esp,%eax //load x array's base address o EAX fldl (%eax // s(r7x[] fldl (%eax,%edx,8 //s(r6x[s] fld %s( //s(r5x[] fadd %s(,%s // R5x[]x[S] fxch %s( //s(r5x[],s(r7x[]x[s] fsubp %s,%s( //s(r6x[s]-x[]????? fxch %s( //s(r6x[]x[s],s(r7x[s]-x[] fspl (%eax //sore x[]x[]x[s] fspl (%eax,%edx,8 //sore x[]x[]-x[s] re LACSI 6 Auomac Tug of Lbrares ad Applcaos
21 LACSI 6 Auomac Tug of Lbrares ad Applcaos Recurreces Recurreces leaf a, A(..., A( A( leaf a, (..., ( (..., ( (..., ( ( L L L L L L L 3... l l l l l A leaves umber of where, ( ν ν
22 Hsogram usg Isruco Model (P3 α l, α l 3, ad α l 6 α 7 β 8, β 8, ad β LACSI 6 Auomac Tug of Lbrares ad Applcaos
23 Cache Model Dffere WHT algorhms access daa dffere paers All algorhms wh he same se of leaf odes have he same umber of memory accesses Cou msses for accesses o daa array Parameerzed by cache sze, assocavy, ad block sze smulae usg program races (resrc o daa vecor accesses Aalyc formula? LACSI 6 Auomac Tug of Lbrares ad Applcaos
24 Blocked Access 3 WHT 6 ( WHT ( WHT I 8 ( I I 8 ( I ( WHT WHT I 8 ( I WHT LACSI 6 Auomac Tug of Lbrares ad Applcaos
25 Ierleaved Access 3 WHT 6 (( WHT ( WHT I 8 ( I I ( I 8 WHT WHT I ( I 8 WHT LACSI 6 Auomac Tug of Lbrares ad Applcaos
26 Cache Smulaor 3 3 memory accesses C, A, B (8, C, A, B (8, 8 C, A, B (7, 88 Ierave vs. Recursve (9 memory accesses C, A, B (8, LACSI 6 Auomac Tug of Lbrares ad Applcaos
27 Cache Msses as a Fuco of Cache Sze C C 3 C C 5 LACSI 6 Auomac Tug of Lbrares ad Applcaos
28 LACSI 6 Auomac Tug of Lbrares ad Applcaos Formula for Cache Msses Formula for Cache Msses M(L,W N,R Number of msses for (Ι L WHT N Ι R...,,... ( leaf s a f / f,, ( M 3 N M C W W N N W N r r N l r N l
29 Closed Form M(L,W N,R Number of msses for (Ι L WHT N Ι R M(,W_, 3(-c* k* C c, k umber of pars he rghmos c posos c 3, Ierave k 3 Balaced k Rgh Recursve 3 k LACSI 6 Auomac Tug of Lbrares ad Applcaos
30 Summary of Resuls ad Fuure Work Isruco Cou Model m, max, expeced value, varace, lmg dsrbuo Cache Model Drec mapped (closed form soluo, dsrbuo, expeced value, ad varace Combe models Exed cache formula o clude A ad B Use as heursc o lm search ad predc performace LACSI 6 Auomac Tug of Lbrares ad Applcaos
31 Sposors Work suppored by DARPA (DSO, Appled & Compuaoal Mahemacs Program, OPAL, hrough gra maaged by research gra DABT admsered by he Army Drecorae of Coracg, DESA: Iellge HW-SW Complers for Sgal Processg Applcaos, ad NSF ITR/NGS #35687: Iellge HW/SW Complers for DSP. LACSI 6 Auomac Tug of Lbrares ad Applcaos
Performance Analysis of Divide and Conquer Algorithms for the WHT
Performace Aalyss of Dvde ad Coquer Algorhms for he WHT CS 50 Hgh Performace Compug Jeremy Johso Mha Furs, Pawel Hczeko, Hug-Je Huag Dep. of Compuer Scece Drexel Uversy www.spral.e Movao O moder maches
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