Proc. 21st Annual European Symposium on Algorithms, Sophia Antipolis, France, volume 8125 of Lecture Notes in Computer Science, pages 61-72, Springer

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2 Proc. 21st Aual Europea Symposium o Algorithms, Sophia Atipolis, Frace, volume 8125 of Lecture Notes i Computer Sciece, pages 61-72, Spriger Verlag, Berli, 2013, / _6

3 Kadidat Bachelor Semester 1 Semester 2 Itroductio to Programmig Itroductio to Databases Implemetatio ad Applicatios of Databases Foudatios of Algorithms ad Data Structures << Kassogram >> Programmig Laguages Semester 3 Software Egieerig ad Architecture Iteractive Systems Semester 4 Computer Architecture, Networks ad Operatig Systems Experimetal Systems Developmet Calculus β Liear algebra Itroductio to Probability Theory ad Statistics Computability ad Logic Semester 5 Compilatio Distributed Systems ad Security Machie Learig Semester 6 Bachelor Project Philosophy of Iformatio Techology Optimizatio Semester 7 Computatioal Geometry: Theory ad Experimetatio Program Aalysis ad Verificatio Semester 8 Advaced Data Structures Laguage-based Security Semester 9 Theory of Algorithms ad Computatioal Complexity Fuctioal Programmig Buildig the Iteret of Thigs with P2P ad Cloud Computig Augmeted Reality Advaced Augmeted Reality Project Advaced Data Maagemet ad Aalysis Semester 10 Master Thesis Datalogi, obligatorisk Matematik støttefag Valgfrie

4 Proc. 21st Aual Europea Symposium o Algorithms, Sophia Atipolis, Frace, volume 8125 of Lecture Notes i Computer Sciece, pages 61-72, Spriger Verlag, Berli, 2013, / _6

5 I er her Bachelor Jakob Costatios Gerth Lars Kadidat Ph.d. Postdoc Adjukt Lektor Professor Aarhus B.Sc M.Sc Ph.d B.Sc. Athe M.Sc. Athe Ph.d. Eidhove Postdoc Aarhus ( Græsk militær) Samarbejde med Økoiformatik og Biodiversitet gruppe ved Biosciece Aarhus M.Sc Ph.d Postdoc (Saarbrücke) Postdoc Lektor Professor Aarhus B.Sc M.Sc Ph.d Postdoc (Duke) Adjukt (Duke) Professor Ceterleder

6 Rød: 64% = Grø:70% Blå: 44%

7 Iput x billede M 1 Output M 2 M 3 M d M, hvor M d størrelse d x d x C.E. Woodcock ad A.H. Strahler. The Factor of Scale i Remote Sesig. Remote Sesig of Eviromets, 21: , 1987 M 1 (iput) 2 x 2 M 2 3 x 3 M 3 4 x 4 5 x 5 6 x 6 7 x M 7 4 M 5 M 6 M 7 8 x 8 9 x 9

d=1 Samlet bredde 1 d = d=1 = 1 + 2 + 3 + + d + + 1 + = d 1 + න 1 1 x dx = 1 + l (x) 1 = 1+l

8 d=1 Samlet bredde 1 d = d=1 = d = d 1 + න 1 1 x dx = 1 + l (x) 1 = 1+l () 1 1/x 1/2 x x 2 3 x 3 4 x 4 5 x 5 6 x 6 7 x 7 8 M 1 M 2 M 3 M 4 M 5 M 6 M 7

9 Samlet størrelse = d=1 d 2 = d 1 = d=1 d න 1 x 2 dx = x /x 2 1/2 x x 2 3 x 3 4 x 4 5 x 5 6 x 6 7 x 7 8 M 1 M 2 M 3 M 4 M 5 M 6 M 7

10 M 2 M 3 M 1 M 4 M 5 M 6 M 7 M 8 M 9 M 10 M 11 M 12 M 13 M 14 M 15 M 16 M 17 M 18 M 19 M 20 M 21 M 22 M 23 M 24 M 25 M 26 M 27 M 28 M 29 M 30 M 31 M 32 M 33 Iput x Output x

11 i*d d M d j*d M d i d j Algoritme 1 for d = 2 to for i = 0 to /d -1 for j = 0 to /d -1 sum = 0 for r = 0 to d-1 for k = 0 to d-1 sum = sum + M 1 [i*d+r, j*d+k] M d [i, j] = sum/d 2 d /d /d M 1 tilgås 3 gage = 49

12 Algoritme 1 for d = 2 to for i = 0 to /d -1 Tid 3 for j = 0 to /d -1 sum = 0 for r = 0 to d-1 for k = 0 to d-1 sum = sum + M 1 [i*d+r, j*d+k] M d [i, j] = sum/d 2 Iput og output størrelse 2, me Algoritme 1 tager tid 3. Ka ma løse problemet hurtigere?

13 M S B E D C A = Algoritme S for i = -1 to -1 S[i,-1] = 0, S[-1,i] = 0 Tid 2 for i = 0 to -1 for j = 0 to -1 S[i, j] = M 1 [i,j] + S[i-1, j] + S[i, j-1] S[i-1, j-1] A B C D E

14 S k r Algoritme 2 for d = 2 to Tid 2 for i = 0 to /d -1 for j = 0 to /d -1 r = i * d + d + 1, k = j * d + d + 1 M d [i,j] = (S[r,k] S[r d,k] S[r,k d] + S[r d,k d])/d 2 S tilgås 4 2 gage M d i d d j /d /d

15 Algoritme 2 for d = 2 to for i = 0 to /d -1 for j = 0 to /d -1 r = i * d + d + 1, k = j * d + d + 1 M d [i,j] = (S[r,k] S[r d,k] S[r,k d] + S[r d,k d])/d 2 Tid 2 Algoritme 2 tager optimal tid 2

16 Algoritme 2 bruger de fysiske hukommelse uhesigtsmæssigt

17 Leovo ThikStatio P710 leovo.com

2 x CPU 2 x Itel Xeo E5-2643 v4-processor 12 x hukommelse 12 x 32 GigaBytes

18 2 x CPU 2 x Itel Xeo E v4-processor 12 x hukommelse 12 x 32 GigaBytes = 384 x 10 9 bytes 4 x harddiske 4 x 4 TeraBytes = 16 x bytes leovo.com

19 leovo.com

Itel Xeo E5-2643 v4-processor Trasistors 3,200,000,000 The umber of CPU cores 6 The umber of threads 12 Level 1 istructio cache 6 x 32 KB 8-way set associative Level 1 data

20 Itel Xeo E v4-processor Trasistors 3,200,000,000 The umber of CPU cores 6 The umber of threads 12 Level 1 istructio cache 6 x 32 KB 8-way set associative Level 1 data cache 6 x 32 KB 8-way set associative Level 2 cache 6 x 256 KB 8-way set associative Level 3 cache 20 MB 20-way set associative shared Cache lie size 64 bytes Data width 64 bit

21 Xeo-E V3

22 Hukommelse (RAM) Harddisk Hukommelseshierarki Beregiger (ALU) Bus L1 L2 L3 CPU Cache størrelse Tilgagstid Blokstørrelse 32 KB 5 s 64 bytes 256 KB 20 s 64 bytes 20 MB 30 s 64 bytes 32 GB 60 s 64 bytes 1 TB s bytes

23 S Cache blokstørrelse = 4 Algoritme 2 og Cache d Algoritme 2 for d = 2 to Tid 2 for i = 0 to /d -1 for j = 0 to /d -1 r = i * d, k = j * d M d [i,j] = (S[r,k] S[r d,k] S[r,k d] + S[r d,k d])/d 2 Hver tilgag til S ka være i e y blok M

24 Forskigsområde algoritmedesig udytter at data kommer i blokke hardware parametree må ikke idgå i algoritme Matteo Frigo, Charles E. Leiserso, Harald Prokop, Sridhar Ramachadra. Cache-Oblivious Algorithms. ACM Trasactios o Algorithms, 8(1), Article No. 4, 2012.

25 Primtal Defiitio Et heltal p er et primtal hvis og ku hvis 1 og p går op i p 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, Primtalssætige Der er < 1.26 l primtal mellem 2 og

26 Primtalsfaktoriserig Defiitio Opskrivig af et heltal x som produkt af primtal = 2 * 2 * 5 * 7 * 7 * 7 * 11 = 2 2 * 5 *7 3 * 11

27 M 2 M 3 M 1 M 4 M 5 M 6 M 7 M 8 M 9 M 10 M 11 M 12 M 13 M 14 M 15 M 16 M 17 M 18 M 19 M 20 M 21 M 22 M 23 M 24 M 25 M 26 M 27 M 28 M 29 M 30 M 31 M 32 M 33 Iput x Output x Ide Kostruer M d ud fra M d/p, hvor p midste primfaktor i d

28 Cache Oblivious Algoritme M 1 x iput Algoritme S sca S 1 1 Iput 2 Output 2 Sum S i scaer getage scaiger σ primtal p p 2 π(p) er atal primtal p Primtalssætige π(p) c alle primtal S 2 sca M 2 S 3 sca M 3 S 5 sca M S 4 sca M 4 3 M S 6 M 10 6 sca S 15 M 15 sca S 9 M S 10 3

29 Sekuder / millioer celler Eksperimetelle Resultater Algoritme 2 Cache aware Arge, Haverkort, Tsirogiais 2012 Cache oblivious millioer celler

30 Opsummerig Eksempel på datalogisk forskig (med biologisk avedelse) Kombiatio af teori og praksis Matematik et cetralt værktøj x 2 x 2 3 x 3 4 x 4 5 x 5 6 x 6 7 x 7 8 M 1 M 2 M 3 M 4 M 5 M 6 M 7

31 Kadidat Bachelor Semester 1 Semester 2 Itroductio to Programmig Itroductio to Databases Implemetatio ad Applicatios of Databases Foudatios of Algorithms ad Data Structures << Kassogram >> Programmig Laguages Semester 3 Software Egieerig ad Architecture Iteractive Systems Semester 4 Computer Architecture, Networks ad Operatig Systems Experimetal Systems Developmet Calculus β Liear algebra Itroductio to Probability Theory ad Statistics Computability ad Logic Semester 5 Compilatio Distributed Systems ad Security Machie Learig Semester 6 Bachelor Project Philosophy of Iformatio Techology Optimizatio Semester 7 Computatioal Geometry: Theory ad Experimetatio Program Aalysis ad Verificatio Semester 8 Advaced Data Structures Laguage-based Security Semester 9 Theory of Algorithms ad Computatioal Complexity Fuctioal Programmig Buildig the Iteret of Thigs with P2P ad Cloud Computig Augmeted Reality Advaced Augmeted Reality Project Advaced Data Maagemet ad Aalysis Semester 10 Master Thesis Datalogi, obligatorisk Matematik støttefag Valgfrie

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