CSE 613: Parallel Programming. Lecture 10 ( Cache Performance of Divide-and-Conquer Algorithms )
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1 CSE 613: Parallel Programmig Leture 10 ( Cahe Performae of Divide-ad-Coquer Algorithms ) Rezaul A. Chowdhury Deartmet of Comuter Siee SUN Stoy Brook Srig 2012
2 Moder Sigle Core Mahies Memory Hierarhy Cost of memory aess deeds o whether it s a (ahe)hitor miss Data i ahe may get evitedto make sae for ew data items Good erformae requires high loality i memory aesses
3 Multiores Itel Core Duo, Itel eo AMD Otero, AMD Athlo Itel Itaium 2 Itel Nehalem, AMD Bareloa
4 Assumtios Two Level Memory Hierarhy A sigle level of ahe (ahes) of size C(eah) oeted to a mai memory of ubouded size ad blok size B LRU (Least Reetly Used) Cahe Relaemet Poliy Whe a ew blok must be brought ito the ahe, but the ahe is full, the the ahe blok that was aessed least reetly is evited to make sae for the ew blok Automati Cahe Relaemet Doe automatially by the OS or the hardware Fully Assoiative Cahes A blok brought ito the ahe from memory a reside aywhere i the ahe
5 Parallel Cahig Model: Distributed Cahes 1 2 CPU CPU CPU rivate ahe (size = C) blok trasfer (size = B) rivate ahe (size = C) blok trasfer (size = B) mai memory rivate ahe (size = C) blok trasfer (size = B) Cofiguratio: roessig elemets a rivate aheof size Cfor eah roessig elemet a arbitrarily large global shared memory blok trasfer sizeb ( betwee ahes ad memory ) Cahe Performae Measure: umber of blok trasfers aross all ahes
6 Parallel Cahig Model: Shared Cahe 1 2 CPU CPU CPU shared ahe (size = C) blok trasfer (size = B) mai memory Cofiguratio: roessig elemets a shared aheof size C a arbitrarily large global shared memory C B, where Cis blok trasfer size Cahe Performae Measure: umber of blok trasfersbetwee the ahe ad the memory
7 Loality of Referee Satial Loality If a artiular memory loatio is aessed at a artiular time, the it is likely that earby memory loatios will also be aessed i the ear future. Take advatage of the blok size Bto load all memory loatios i the same blok ito the ahe whe a artiular memory loatio i that blok is aessed. Temoral Loality If a artiular memory loatio is aessed at a artiular time, the it is likely that the same memory loatio will be aessed agai i the ear future. Take advatage of the ahe size Cto retai memory loatios already loaded ito the ahe for future referees.
8 Iterative Matrix Multiliatio z = x y ij ik kj k= 1 z z z z z z z z z 1 2 x11 x12 x1 = x21 x22 x 2 x x x 1 2 y y y y y y y y y 1 2 Iter-MM ( Z,, ) {,, Z are matries, where is a ositive iteger } 1. for i 1 to do 2. for j 1 to do 3. Z[ i ][ j ] 0 4. for k 1 to do 5. Z[ i ][ j ] Z[ i ][ j ] + [ i ][ k ] [ k ][ j ]
9 Iterative Matrix Multiliatio Iter-MM ( Z,, ) {,, Z are matries, where is a ositive iteger } 1. for i 1 to do 2. for j 1 to do 3. Z[ i ][ j ] 0 4. for k 1 to do 5. Z[ i ][ j ] Z[ i ][ j ] + [ i ][ k ] [ k ][ j ] z z z z z z z z z 1 2 store i row-major order x11 x12 x1 = x21 x22 x 2 x x x 1 2 store i row-major order y y y y y y y y y 1 2 Eah iteratio of the forloo i lie 3 iurs Ο ahe misses. Cahe-omlexity of Iter-MM, Q Ο.
10 Iterative Matrix Multiliatio Iter-MM ( Z,, ) {,, Z are matries, where is a ositive iteger } 1. for i 1 to do 2. for j 1 to do 3. Z[ i ][ j ] 0 4. for k 1 to do 5. Z[ i ][ j ] Z[ i ][ j ] + [ i ][ k ] [ k ][ j ] z z z z z z z z z 1 2 store i row-major order x11 x12 x1 = x21 x22 x 2 x x x 1 2 store i olum-major order y y y y y y y y y 1 2 Eah iteratio of the forloo i lie 3 iurs Ο 1 Cahe-omlexity of Iter-MM, Q Ο 1 ahe misses. Ο.
11 Blok Matrix Multiliatio ahe ( size = C ) C/3 C/3 C/3 m m Z C/3 m C/3 m C/3 m m = Blok-MM(,, Z ) 1. for i 1 to / m do 2. for j 1 to / m do 3. for k 1 to / m do 4. Iter-MM( ik, kj, Z ij )
12 m m Blok Matrix Multiliatio Blok-MM(,, Z ) 1. for i 1 to / m do 2. for j 1 to / m do 3. for k 1 to / m do 4. Iter-MM( ik, kj, Z ij ) Choose 3, so that ik, kj adz ij just fit ito the ahe. The lie 4 iurs Θ 1 ahe misses. Cahe-omlexity of Blok-MM [assumig a tall ahe, i.e., ] Θ Θ Θ Θ ( Otimal: Hog & Kug, STOC 81 )
13 m m Blok Matrix Multiliatio Blok-MM(,, Z ) 1. for i 1 to / m do 2. for j 1 to / m do 3. for k 1 to / m do 4. Iter-MM( ik, kj, Z ij ) Choose 3, so that ik, kj adz ij just fit ito the ahe. Otimal for ay algorithm that erforms the oeratios give by the The lie 4 iurs Θ 1 followig defiitio of matrix multiliatio: ahe misses. Cahe-omlexity z of ij Blok-MM xiky kj[assumig a tall ahe, i.e., ] Θ = k = 1 Θ Θ Θ ( Otimal: Hog & Kug, STOC 81 )
14 s Multile Levels of Cahe s Blok-MM(,, Z ) 1. for i 1 to / s do 2. for j 1 to / s do 3. for k 1 to / s do 4. Iter-MM( ik, kj, Z ij )
15 t s Multile Levels of Cahe t s Blok-MM(,, Z ) 1. for i 1 1 to / s do 2. for j 1 1 to / s do 3. for k 1 1 to / s do 4. for i 2 1 to s / t do 5. for j 2 1 to s / t do 6. for k 2 1 to s / t do 7. Iter-MM( ( i1 k 1 ) i2 k 2, ( k1 j 1 ) k2 j 2, (Z i1 j 1 ) i2 j 2 )
16 t s Multile Levels of Cahe t s Oe Parameter Per Cahig Level! Blok-MM(,, Z ) 1. for i 1 1 to / s do 2. for j 1 1 to / s do 3. for k 1 1 to / s do 4. for i 2 1 to s / t do 5. for j 2 1 to s / t do 6. for k 2 1 to s / t do 7. Iter-MM( ( i1 k 1 ) i2 k 2, ( k1 j 1 ) k2 j 2, (Z i1 j 1 ) i2 j 2 )
17 Parallel Reursive MM Par-Re-MM ( Z,, ) {,, Z are matries, where = 2 k for iteger k 0 } 1. if = 1 the 2. Z Z + 3. else 4. saw Par-Re-MM ( Z 11, 11, 11 ) 5. saw Par-Re-MM ( Z 12, 11, 12 ) 6. saw Par-Re-MM ( Z 21, 21, 11 ) 7. Par-Re-MM ( Z 21, 21, 11 ) 8. sy 9. saw Par-Re-MM ( Z 11, 12, 21 ) 10. saw Par-Re-MM ( Z 12, 12, 22 ) 11. saw Par-Re-MM ( Z 21, 22, 21 ) 12. Par-Re-MM ( Z 22, 22, 22 ) 13. sy 14. edif Work: Θ 1, 1, 8 2 Θ 1,. Sa: Θ [ MT Case 1 ] Θ 1, 1, 2 2 Θ 1,. Θ [ MT Case 1 ] Parallel Ruig Time: Ο Parallelism: Ο Θ Additioal Sae: Θ 1
18 Parallel Reursive MM o a Sigle Core Par-Re-MM ( Z,, ) {,, Z are matries, where = 2 k for iteger k 0 } 1. if = 1 the 2. Z Z + 3. else 4. saw Par-Re-MM ( Z 11, 11, 11 ) 5. saw Par-Re-MM ( Z 12, 11, 12 ) 6. saw Par-Re-MM ( Z 21, 21, 11 ) 7. Par-Re-MM ( Z 21, 21, 11 ) 8. sy 9. saw Par-Re-MM ( Z 11, 12, 21 ) 10. saw Par-Re-MM ( Z 12, 12, 22 ) 11. saw Par-Re-MM ( Z 21, 22, 21 ) 12. Par-Re-MM ( Z 22, 22, 22 ) 13. sy Cahe Comlexity: Ο,, 8 2,. Θ 1, whe Ω 14. edif
19 Parallel Reursive MM o Distributed Cahes Par-Re-MM ( Z,, ) {,, Z are matries, where = 2 k for iteger k 0 } 1. if = 1 the 2. Z Z + 3. else 4. saw Par-Re-MM ( Z 11, 11, 11 ) 5. saw Par-Re-MM ( Z 12, 11, 12 ) 6. saw Par-Re-MM ( Z 21, 21, 11 ) 7. Par-Re-MM ( Z 21, 21, 11 ) 8. sy 9. saw Par-Re-MM ( Z 11, 12, 21 ) 10. saw Par-Re-MM ( Z 12, 12, 22 ) 11. saw Par-Re-MM ( Z 21, 22, 21 ) 12. Par-Re-MM ( Z 22, 22, 22 ) 13. sy 14. edif Cahe Comlexity: 4 Assumtio:if 1, the is evely distributed amog the simultaeously sawed futios., 1, 8 2,. Θ, whe Ω
20 The Logest Commo Subsequee (LCS) Problem A subsequeeof a sequee is obtaied by deletig zero or more symbols from. Examle: = abba Z = ba obtaied by deletig the 1 st a ad the 2 d b from A Logest Commo Subsequee (LCS)of two sequee ad is a sequee Z that is a subsequee of both ad, ad is the logest amog all suh subsequees. Give ad, the LCS roblemasks for suh a Z.
![j > 0 x i = y j, max { [ i, j 1], [ i 1, j] }](/docs-images/92/109468554/images/21-2.jpg "otherwise.")
21 The Logest Commo Subsequee (LCS) Problem Give: = x 1 x 2 x ad = y 1 y 2 y Fills u a array [0, 0 ] usig the followig reurree. 0 if i = 0 j = 0, [ i, j] = [ i 1, j 1] + 1 if i, j > 0 x i = y j, max { [ i, j 1], [ i 1, j] } otherwise. y 1 y 2 y 3 y 4 y 5 y 6 y 7 y 8 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 [ i, j ] Traebak Path (LCS) [, ] = legth of LCS Loal Deedey: value of eah ell deeds oly o values of adjaet ells.
22 The Logest Commo Subsequee (LCS) Problem The lassi ( iterative ) serial LCS DPrus i Θ time, uses Θ sae, ad iurs Θ ahe misses. Ay algorithm usig Θ sae must iur Ω ahe misses. Hee i order to redue the ahe omlexity of the LCS algorithm from Θ we must first redue its sae usage below Θ.
23 Sequetial Cahe-effiiet LCS Algorithm Q [1, 1 ] = 2 q Q traebak ath
24 Sequetial Cahe-effiiet LCS Algorithm Q [1, 1 ] = 2 q Q traebak ath
25 Sequetial Cahe-effiiet LCS Algorithm Q [1, 1 ] = 2 q Q traebak ath
26 Sequetial Cahe-effiiet LCS Algorithm Q [1, 1 ] = 2 q Q traebak ath
27 Sequetial Cahe-effiiet LCS Algorithm Q [1, 1 ] = 2 q Q traebak ath
28 Sequetial Cahe-effiiet LCS Algorithm Q [1, 1 ] = 2 q 1. Deomose Q: Slit Q ito four quadrats. Q 11 Q 12 Q 21 Q 22 Q traebak ath
29 Sequetial Cahe-effiiet LCS Algorithm Q [1, 1 ] = 2 q 1. Deomose Q: Slit Q ito four quadrats. 2. Forward Pass ( Geerate Boudaries ): Geerate the right ad the bottom boudaries of the quadrats reursively. ( of at most 3 quadrats ) Q 11 Q 12 Q 21 Q 22 Q traebak ath
30 Sequetial Cahe-effiiet LCS Algorithm Q [1, 1 ] = 2 q 1. Deomose Q: Slit Q ito four quadrats. 2. Forward Pass ( Geerate Boudaries ): Geerate the right ad the bottom boudaries of the quadrats reursively. ( of at most 3 quadrats ) Q 11 Q 12 Q 21 Q 22 Q traebak ath
31 Sequetial Cahe-effiiet LCS Algorithm Q [1, 1 ] = 2 q 1. Deomose Q: Slit Q ito four quadrats. 2. Forward Pass ( Geerate Boudaries ): Geerate the right ad the bottom boudaries of the quadrats reursively. ( of at most 3 quadrats ) Q 11 Q 12 Q 21 Q 22 Q traebak ath
32 Sequetial Cahe-effiiet LCS Algorithm Q [1, 1 ] = 2 q 1. Deomose Q: Slit Q ito four quadrats. 2. Forward Pass ( Geerate Boudaries ): Geerate the right ad the bottom boudaries of the quadrats reursively. ( of at most 3 quadrats ) Q 11 Q 12 Q 21 Q 22 Q traebak ath
33 Sequetial Cahe-effiiet LCS Algorithm Q [1, 1 ] = 2 q 1. Deomose Q: Slit Q ito four quadrats. 2. Forward Pass ( Geerate Boudaries ): Geerate the right ad the bottom boudaries of the quadrats reursively. ( of at most 3 quadrats ) Q 11 Q 12 Q 21 Q 22 Q traebak ath
34 Sequetial Cahe-effiiet LCS Algorithm Q [1, 1 ] = 2 q 1. Deomose Q: Slit Q ito four quadrats. 2. Forward Pass ( Geerate Boudaries ): Geerate the right ad the bottom boudaries of the quadrats reursively. ( of at most 3 quadrats ) Q 11 Q 12 Q 21 Q 22 Q traebak ath
35 Sequetial Cahe-effiiet LCS Algorithm Q [1, 1 ] = 2 q 1. Deomose Q: Slit Q ito four quadrats. 2. Forward Pass ( Geerate Boudaries ): Geerate the right ad the bottom boudaries of the quadrats reursively. ( of at most 3 quadrats ) 3. Bakward Pass ( Extrat Traebak Path Fragmets ): Extrat fragmets of the traebak ath from the quadrats reursively. ( from at most 3 quadrats ) Q 11 Q 12 Q 21 Q 22 Q traebak ath
36 Sequetial Cahe-effiiet LCS Algorithm Q [1, 1 ] = 2 q 1. Deomose Q: Slit Q ito four quadrats. 2. Forward Pass ( Geerate Boudaries ): Geerate the right ad the bottom boudaries of the quadrats reursively. ( of at most 3 quadrats ) 3. Bakward Pass ( Extrat Traebak Path Fragmets ): Extrat fragmets of the traebak ath from the quadrats reursively. ( from at most 3 quadrats ) Q 11 Q 12 Q 21 Q 22 Q traebak ath
37 Sequetial Cahe-effiiet LCS Algorithm Q [1, 1 ] = 2 q 1. Deomose Q: Slit Q ito four quadrats. 2. Forward Pass ( Geerate Boudaries ): Geerate the right ad the bottom boudaries of the quadrats reursively. ( of at most 3 quadrats ) 3. Bakward Pass ( Extrat Traebak Path Fragmets ): Extrat fragmets of the traebak ath from the quadrats reursively. ( from at most 3 quadrats ) Q 11 Q 12 Q 21 Q 22 Q traebak ath
38 Sequetial Cahe-effiiet LCS Algorithm Q [1, 1 ] = 2 q 1. Deomose Q: Slit Q ito four quadrats. 2. Forward Pass ( Geerate Boudaries ): Geerate the right ad the bottom boudaries of the quadrats reursively. ( of at most 3 quadrats ) 3. Bakward Pass ( Extrat Traebak Path Fragmets ): Extrat fragmets of the traebak ath from the quadrats reursively. ( from at most 3 quadrats ) Q 11 Q 12 Q 21 Q 22 Q traebak ath
39 Sequetial Cahe-effiiet LCS Algorithm Q [1, 1 ] = 2 q 1. Deomose Q: Slit Q ito four quadrats. 2. Forward Pass ( Geerate Boudaries ): Geerate the right ad the bottom boudaries of the quadrats reursively. ( of at most 3 quadrats ) 3. Bakward Pass ( Extrat Traebak Path Fragmets ): Extrat fragmets of the traebak ath from the quadrats reursively. ( from at most 3 quadrats ) Q 11 Q 12 Q 21 Q 22 Q traebak ath
40 Sequetial Cahe-effiiet LCS Algorithm Q [1, 1 ] = 2 q 1. Deomose Q: Slit Q ito four quadrats. 2. Forward Pass ( Geerate Boudaries ): Geerate the right ad the bottom boudaries of the quadrats reursively. ( of at most 3 quadrats ) Q 11 Q Bakward Pass ( Extrat Traebak Path Fragmets ): Extrat fragmets of the traebak ath from the quadrats reursively. ( from at most 3 quadrats ) 4. Comose Traebak Path: Combie the ath fragmets. Q 21 Q 22 Q traebak ath
41 Cahe omlexity: Cahe Performae Ο 1,, Ο 1,. where is the ahe omlexity of reursive boudary geeratio (i the forward ass ): Ο 1,, 4 2 Ο 1,. Ο 1 Substitutig, Ο 1
42 Cahe omlexity: Cahe Performae Ο 1,, Ο 1,. where is the ahe omlexity of reursive boudary geeratio (i the forward ass ): Ο 1,, 4 2 Ο 1,. Ο 1 Substitutig, Ο 1 otimal
43 Parallel Cahe-effiiet LCS Algorithm for Distributed Cahes
44 Parallel Cahe-effiiet Boudary Comutatio ( PAR-BOUNDAR ) Q [1, 1 ] > C
45 Parallel Cahe-effiiet Boudary Comutatio ( PAR-BOUNDAR ) Q [1, 1 ] > C
46 Parallel Cahe-effiiet Boudary Comutatio ( PAR-BOUNDAR ) Q [1, 1 ] > C
47 Parallel Cahe-effiiet Boudary Comutatio ( PAR-BOUNDAR ) Q [1, 1 ] > C 1. Deomose Q: Slit Q ito 2 submatries of size ( / ) ( / ) eah.
48 Parallel Cahe-effiiet Boudary Comutatio ( PAR-BOUNDAR ) Q [1, 1 ] > C 1. Deomose Q: Slit Q ito 2 submatries of size ( / ) ( / ) eah. 2. Geerate Boudaries: I iteratio i [1, 2-1], solve all submatries o the i-th forward diagoal i arallel usig the sequetial ahe-oblivious algorithm. For eah ell also omute: the ell o the iut boudary where the traebak ath through the give ell itersets.
49 Parallel Cahe-effiiet Boudary Comutatio ( PAR-BOUNDAR ) Q [1, 1 ] > C 1. Deomose Q: Slit Q ito 2 submatries of size ( / ) ( / ) eah. 2. Geerate Boudaries: I iteratio i [1, 2-1], solve all submatries o the i-th forward diagoal i arallel usig the sequetial ahe-oblivious algorithm. For eah ell also omute: the ell o the iut boudary where the traebak ath through the give ell itersets.
50 Parallel Cahe-effiiet Boudary Comutatio ( PAR-BOUNDAR ) Q [1, 1 ] > C 1. Deomose Q: Slit Q ito 2 submatries of size ( / ) ( / ) eah. 2. Geerate Boudaries: I iteratio i [1, 2-1], solve all submatries o the i-th forward diagoal i arallel usig the sequetial ahe-oblivious algorithm. For eah ell also omute: the ell o the iut boudary where the traebak ath through the give ell itersets.
51 Parallel Cahe-effiiet Boudary Comutatio ( PAR-BOUNDAR ) Q [1, 1 ] > C 1. Deomose Q: Slit Q ito 2 submatries of size ( / ) ( / ) eah. 2. Geerate Boudaries: I iteratio i [1, 2-1], solve all submatries o the i-th forward diagoal i arallel usig the sequetial ahe-oblivious algorithm. For eah ell also omute: the ell o the iut boudary where the traebak ath through the give ell itersets.
52 Parallel Cahe-effiiet Boudary Comutatio ( PAR-BOUNDAR ) Q [1, 1 ] > C 1. Deomose Q: Slit Q ito 2 submatries of size ( / ) ( / ) eah. 2. Geerate Boudaries: I iteratio i [1, 2-1], solve all submatries o the i-th forward diagoal i arallel usig the sequetial ahe-oblivious algorithm. For eah ell also omute: the ell o the iut boudary where the traebak ath through the give ell itersets.
53 Parallel Cahe-effiiet Boudary Comutatio ( PAR-BOUNDAR ) Q [1, 1 ] > C 1. Deomose Q: Slit Q ito 2 submatries of size ( / ) ( / ) eah. 2. Geerate Boudaries: I iteratio i [1, 2-1], solve all submatries o the i-th forward diagoal i arallel usig the sequetial ahe-oblivious algorithm. For eah ell also omute: the ell o the iut boudary where the traebak ath through the give ell itersets.
54 Parallel Cahe-effiiet Boudary Comutatio ( PAR-BOUNDAR ) Q [1, 1 ] > C 1. Deomose Q: Slit Q ito 2 submatries of size ( / ) ( / ) eah. 2. Geerate Boudaries: I iteratio i [1, 2-1], solve all submatries o the i-th forward diagoal i arallel usig the sequetial ahe-oblivious algorithm. For eah ell also omute: the ell o the iut boudary where the traebak ath through the give ell itersets.
55 Parallel Cahe-effiiet Boudary Comutatio ( PAR-BOUNDAR ) Q [1, 1 ] > C 1. Deomose Q: Slit Q ito 2 submatries of size ( / ) ( / ) eah. 2. Geerate Boudaries: I iteratio i [1, 2-1], solve all submatries o the i-th forward diagoal i arallel usig the sequetial ahe-oblivious algorithm. For eah ell also omute: the ell o the iut boudary where the traebak ath through the give ell itersets.
56 Parallel Cahe-effiiet Boudary Comutatio ( PAR-BOUNDAR ) Q [1, 1 ] > C 1. Deomose Q: Slit Q ito 2 submatries of size ( / ) ( / ) eah. 2. Geerate Boudaries: I iteratio i [1, 2-1], solve all submatries o the i-th forward diagoal i arallel usig the sequetial ahe-oblivious algorithm. For eah ell also omute: the ell o the iut boudary where the traebak ath through the give ell itersets.
57 Parallel Cahe-effiiet Boudary Comutatio ( PAR-BOUNDAR ) Q [1, 1 ] > C 1. Deomose Q: Slit Q ito 2 submatries of size ( / ) ( / ) eah. 2. Geerate Boudaries: I iteratio i [1, 2-1], solve all submatries o the i-th forward diagoal i arallel usig the sequetial ahe-oblivious algorithm. For eah ell also omute: the ell o the iut boudary where the traebak ath through the give ell itersets.
58 Parallel Cahe-effiiet Boudary Comutatio ( PAR-BOUNDAR ) Q [1, 1 ] > C 1. Deomose Q: Slit Q ito 2 submatries of size ( / ) ( / ) eah. 2. Geerate Boudaries: I iteratio i [1, 2-1], solve all submatries o the i-th forward diagoal i arallel usig the sequetial ahe-oblivious algorithm. For eah ell also omute: the ell o the iut boudary where the traebak ath through the give ell itersets.
![Performae Bouds ( PAR-BOUNDAR ) Q [1, 1 ]](/docs-images/92/109468554/images/59-1.jpg "Parallel Time Comlexity: > C Ο Ο Cahe Comlexity:")
59 Performae Bouds ( PAR-BOUNDAR ) Q [1, 1 ] Parallel Time Comlexity: > C Ο Ο Cahe Comlexity: Ο Ο Ο 1 Ο
60 Parallel Cahe-effiiet Traebak Path ( PAR-TRACEBACK ) Q [1, 1 ] > C Q traebak ath
61 Parallel Cahe-effiiet Traebak Path ( PAR-TRACEBACK ) Q [1, 1 ] 1. Deomose Q: Slit Q ito four quadrats. > C Q 11 Q 12 Q 21 Q 22 Q traebak ath
62 Parallel Cahe-effiiet Traebak Path ( PAR-TRACEBACK ) Q [1, 1 ] 1. Deomose Q: Slit Q ito four quadrats. > C 2. Forward Pass ( Geerate Boudaries ): Geerate the right ad the bottom boudaries of all quadrats by allig PAR-BOUNDAR ( usig all roessors ). Q 11 Q 12 Q 21 Q 22 Q traebak ath
63 Parallel Cahe-effiiet Traebak Path ( PAR-TRACEBACK ) Q [1, 1 ] 1. Deomose Q: Slit Q ito four quadrats. > C 2. Forward Pass ( Geerate Boudaries ): Geerate the right ad the bottom boudaries of all quadrats by allig PAR-BOUNDAR ( usig all roessors ). Q 11 Q 12 Q 21 Q 22 Q traebak ath
64 Parallel Cahe-effiiet Traebak Path ( PAR-TRACEBACK ) Q [1, 1 ] 1. Deomose Q: Slit Q ito four quadrats. > C 2. Forward Pass ( Geerate Boudaries ): Geerate the right ad the bottom boudaries of all quadrats by allig PAR-BOUNDAR ( usig all roessors ). Q 11 Q 12 Q 21 Q 22 Q traebak ath
65 Parallel Cahe-effiiet Traebak Path ( PAR-TRACEBACK ) Q [1, 1 ] 1. Deomose Q: Slit Q ito four quadrats. > C 2. Forward Pass ( Geerate Boudaries ): Geerate the right ad the bottom boudaries of all quadrats by allig PAR-BOUNDAR ( usig all roessors ). Q 11 Q 12 Q 21 Q 22 Q traebak ath
66 Parallel Cahe-effiiet Traebak Path ( PAR-TRACEBACK ) Q [1, 1 ] 1. Deomose Q: Slit Q ito four quadrats. > C 2. Forward Pass ( Geerate Boudaries ): Geerate the right ad the bottom boudaries of all quadrats by allig PAR-BOUNDAR ( usig all roessors ). Q 11 Q 12 Q 21 Q 22 Q traebak ath
67 Parallel Cahe-effiiet Traebak Path ( PAR-TRACEBACK ) Q [1, 1 ] 1. Deomose Q: Slit Q ito four quadrats. > C 2. Forward Pass ( Geerate Boudaries ): Geerate the right ad the bottom boudaries of all quadrats by allig PAR-BOUNDAR ( usig all roessors ). Q 11 Q 12 Q 21 Q 22 Q traebak ath
68 Parallel Cahe-effiiet Traebak Path ( PAR-TRACEBACK ) Q [1, 1 ] 1. Deomose Q: Slit Q ito four quadrats. > C 2. Forward Pass ( Geerate Boudaries ): Geerate the right ad the bottom boudaries of all quadrats by allig PAR-BOUNDAR ( usig all roessors ). Q 11 Q 12 Q 21 Q 22 Q traebak ath
69 Parallel Cahe-effiiet Traebak Path ( PAR-TRACEBACK ) Q [1, 1 ] 1. Deomose Q: Slit Q ito four quadrats. > C 2. Forward Pass ( Geerate Boudaries ): Geerate the right ad the bottom boudaries of all quadrats by allig PAR-BOUNDAR ( usig all roessors ). Q 11 Q Bakward Pass ( Extrat Traebak Path Fragmets ): Q 21 Q 22 Q traebak ath
70 Parallel Cahe-effiiet Traebak Path ( PAR-TRACEBACK ) Q [1, 1 ] 1. Deomose Q: Slit Q ito four quadrats. > C 2. Forward Pass ( Geerate Boudaries ): Geerate the right ad the bottom boudaries of all quadrats by allig PAR-BOUNDAR ( usig all roessors ). Q 11 Q Bakward Pass ( Extrat Traebak Path Fragmets ): Q 21 Q 22 Q traebak ath
71 Parallel Cahe-effiiet Traebak Path ( PAR-TRACEBACK ) Q [1, 1 ] 1. Deomose Q: Slit Q ito four quadrats. > C 2. Forward Pass ( Geerate Boudaries ): Geerate the right ad the bottom boudaries of all quadrats by allig PAR-BOUNDAR ( usig all roessors ). Q 11 Q Bakward Pass ( Extrat Traebak Path Fragmets ): Q 21 Q 22 Q traebak ath
72 Parallel Cahe-effiiet Traebak Path ( PAR-TRACEBACK ) Q [1, 1 ] 1. Deomose Q: Slit Q ito four quadrats. > C 2. Forward Pass ( Geerate Boudaries ): Geerate the right ad the bottom boudaries of all quadrats by allig PAR-BOUNDAR ( usig all roessors ). Q 11 Q Bakward Pass ( Extrat Traebak Path Fragmets ): Q 21 Q 22 Q traebak ath
73 Parallel Cahe-effiiet Traebak Path ( PAR-TRACEBACK ) Q [1, 1 ] 1. Deomose Q: Slit Q ito four quadrats. > C 2. Forward Pass ( Geerate Boudaries ): Geerate the right ad the bottom boudaries of all quadrats by allig PAR-BOUNDAR ( usig all roessors ). Q 11 Q Bakward Pass ( Extrat Traebak Path Fragmets ): Extrat ath fragmets from Q 22, Q 12 ad Q 11 i arallel by allig PAR-TRACEBACK with / 3 roessors eah. Q 21 Q 22 Q traebak ath
74 Parallel Cahe-effiiet Traebak Path ( PAR-TRACEBACK ) Q [1, 1 ] 1. Deomose Q: Slit Q ito four quadrats. > C 2. Forward Pass ( Geerate Boudaries ): Geerate the right ad the bottom boudaries of all quadrats by allig PAR-BOUNDAR ( usig all roessors ). Q 11 Q Bakward Pass ( Extrat Traebak Path Fragmets ): Extrat ath fragmets from Q 22, Q 12 ad Q 11 i arallel by allig PAR-TRACEBACK with / 3 roessors eah. Q 21 Q 22 Q traebak ath
![Parallel Cahe-effiiet Traebak Path ( PAR-TRACEBACK ) Q [1, 1 ] 1. Deomose Q: Slit Q ito four quadrats. > C 2.](/docs-images/92/109468554/images/75-15.jpg "Forward Pass ( Geerate Boudaries ): Geerate the right ad the bottom boudaries of all quadrats by allig PAR-BOUNDAR ( usig all roessors ). Q 11 Q 12 3.")
75 Parallel Cahe-effiiet Traebak Path ( PAR-TRACEBACK ) Q [1, 1 ] 1. Deomose Q: Slit Q ito four quadrats. > C 2. Forward Pass ( Geerate Boudaries ): Geerate the right ad the bottom boudaries of all quadrats by allig PAR-BOUNDAR ( usig all roessors ). Q 11 Q Bakward Pass ( Extrat Traebak Path Fragmets ): Extrat ath fragmets from Q 22, Q 12 ad Q 11 i arallel by allig PAR-TRACEBACK with / 3 roessors eah. Q 21 Q Comose Traebak Path: Combie the ath fragmets. Q traebak ath
76 Performae Bouds ( PAR-TRACEBACK ) Q [1, 1 ] Parallel Time Comlexity: 4 Ο > C Ο Q 11 Q 12 Cahe Comlexity: Ο 1 Ο Q 21 Q 22 Ο Q traebak ath
77 DP with Loal Deedeies Geeralizatio of the LCS Result Problem Time Sae Cahe-omlexity Parallel Time Classi Logest Commo Subsequee Pairwise Sequee Aligmet ( affie ga osts ) Θ ( 2 ) Θ ( ) Ο 2 BC Ο 2 B Gotoh, 1982 Ο 2 B Ο 2 + Media of three Sequees ( affie ga osts ) RNA Seodary Struture Preditio with Simle Pseudokots Θ ( 3 ) Θ ( 4 ) Θ ( 2 ) Ο Ο 3 B C 4 B C Kudse, 2003 Ο 3 B Akutsu, 2000 Ο 4 B Ο Ο log = sequee legth, C = ahe size, B = blok trasfer size, = #roessors 77
78 Performae of Cahe-effiiet Serial LCS Algorithms omared: The ahe-effiiet LCS algorithm ( CO ) Hirshberg s liear-sae LCS algorithm ( Hi ) Comutig Eviromet: Arhiteture Proessor Seed L1 Cahe ( B ) L2 Cahe ( B ) RAM Itel eo 3 GHz 8 KB ( 64 B ) 512 KB ( 64 B ) 4 GB AMD Otero 2.4 GHz 64 KB ( 64 B ) 1 MB ( 64 B ) 4 GB SUN Blade 1 GHz 64 KB ( 32 B ) 8 MB ( 512 B ) 1 GB
79 Ratio of Ruig Times o Radom Sequees (Hirshberg vs the Cahe-effiiet Algorithm)
80 Ratio of L1 Misses o Radom Sequees (Hirshberg vs the Cahe-effiiet Algorithm)
81 Cahe Performae of Divide-ad-Coquer Algorithms uder the Work-Stealig Sheduler
82 Series-Parallel DAG base ase serial omositio arallel omositio
83 Assumtios s ( fork / saw ) G 1 G 2 t ( joi / sy ) Two-way Divisio ( Saw ): Eah divisio geerates oly two subtasks. Serial Exeutio: The left (first) subtask geerated by a fork ode is always exeuted first. Parallel Exeutio: Oly the right (seod) subtask geerated by a fork ode a be stole. Drifted Nodes: I a arallel exeutio we say that a ode is driftedwhe it is exeuted o a differet roessig elemet tha its redeessor i the serial exeutio.
84 Observatios Observatio 1: Cosider two exeutios of a sequee of istrutios. Eah exeutio takes lae omletely o a sigle roessig elemet oeted to a ahe of size Cad blok size B. The the umber of ahe misses iurred by the two exeutios a differ by at most C / B. ( As uder LRU ahe relaemet oliy oly the first aess to eah of the C/ Bbloks a ause a ahe miss i oe exeutio that is ot a miss i the other ) Observatio 2: Eah steal a ause at most two odes to drift: the stole ode ad ossibly the joi ode with its siblig.
85 Imliatios If there are Ssuessful steals durig arallel exeutio the there will be at most 2 sequetial exeutio. additioal ahe misses omared to the Now suose a divide-ad-oquer algorithm iurs ahe misses o a serial mahie. The o a arallel mahie with arallel roessig elemets eah oeted to a ahe of size ad blok size, the total umber of ahe misses iurred: Ο Ο...
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