Operations Research -

1. Jacek Błażewicz Operations Research - bridging gaps between Manufacturing and Biology

2. Presentation of our region

3. Presentation of our region

4. Presentation of our region

5. Siegen

6. Nodes Arcs GRAPHS One of the main concepts used in Computer Science and Operations Research. Used to present different processes.

7. Jan Węglarz ` Poznań Jacek Błażewicz

8. Siegen Poznań Erwin Pesch

9. Clausthal-Zellerfeld Poznań Klaus Ecker Siegen

10. Clausthal-Zellerfeld Poznań Saarbrücken Günter Schmidt Siegen

12. Erwin Pesch Poznań Jacek Błażewicz Małgorzata Sterna Siegen

13. Redmond Poznań Siegen

14. Redmond Poznań Livermore Siegen

15. Erwin Pesch Redmond Siegen Poznań Livermore Jacek Błażewicz

16. Bartosz Nowierski Bartosz Nowierski Bartosz Nowierski Erwin Pesch Redmond Siegen Łukasz Szajkowski Łukasz Szajkowski Poznań Livermore Łukasz Szajkowski Jacek Błażewicz

17. FLEXIBLE MANUFACTURING SYSTEM

18. HPC CENTER in POZMAN

19. Scheduling problems (deterministic) A set of mprocessors P1, P2, ..., Pm A set of n tasks T1, T2, ..., Tn Each task is characterized by - processing time - pj Precedence constraints TiTj

20. Preemptions P1 Tj Tl P2 Tk Tj t 0 Cj Cmax Criterion - Cmax = max{Cj}

21. Partial order Ti TjTypes of precedence graphs Independenttasks Dependent taskstask – on – node chains in-trees opposing forest out-trees TiTj

22. general graphs task – on - arc uniconnected activity network uan 2 T1 T4 1 4 T3 T2 T5 3

23. 2 T1 T4 1 4 T3 T2 T5 3 Pm│pmtn,uan │Cmax a) Uniquely ordered event nodes.

24. b) An example of a simple uniconnected activity network (a) and the corresponding precedence graph (b). T1 T4 T3 T2 T5

25. NowLPformulation: Minimize Subject to  j=1,2,...,n xj≥0 ComplexityK = O(nm)-a number of variables, thus for a fixed m the problem can be solved inpolynomial time[Khachiyan, Karmarkar]. [J.Błażewicz, W.Cellary, R.Słowiński, J.Węglarz, 77]

26. In practice: Polynomial time = easy (solvable in practice) NP-hard = difficult (not solvable in practice)

27. Theorem 1 Let G be an activity network (task-on-arc graph). G is uniconnected if and only if G has a Hamiltonian path.

28. Original graph G Hamiltonian Precedence graph H ?

29. Molecular biology • Chemical foundations of life • Information coded in chemical molecules Computational biology

30. Problems Methods Problems Methods Operations Research Molecular Biology

33. DNA recognition Human genome • pairs of bases • 3% nucleotides coding an information

35. Human genome 3000 books (valid information 90 books) 1 cell bacteria 20 books Some flies 5000books

36. Analyzed structures • One dimensional structures Analysis of DNA chains (and an information they carry on) • Two dimensional structures Analysis (and recognition) of substructures formed by consecutive subchains (e.g. Α-helix, β-harmony) • Three dimensional structures Analysis of 3-dimensional helix (NMR experiment) . . . . . A C G A T G C G A

37. One dimensional structures • Reading DNA chains • Understanding an information contained in DNA • sequence alignment • finding motifs in sequences • assigning functions to subsequences (or motifs)

38. Levels • Sequencing • up to 700 nucleotides • combinatorial exact methods • Assembling • up to 1000000 nucleotides • heuristics • Mapping • greater than 1000000 nucleotides • search in data bases

39. Genetic linkage map (works on 107-108 bp range) Chromosome Assembling (works on 105-106 bp range) Clones Sequencing (works on 103-104 bp range) CGGACACCGACGTCATTCTCATGTGCTTCTCGGCACA The different scales at which the human genome is studied

40. A A C A C G A C G T Round 1 A C G T A C G T A C G T A C G T A C G T A C G A A C Round 2 Hybridization Experiment 1. Making a DNA chip

41. A C G T A C G T Round 3 ... and so on ... DNA chip A A A A 44 – 0.0016 cm2 48 – 0.4096 cm2 410 – 6.5536 cm2 Full library of tetranucleotides 0,4mm 25m site per probe 0,4mm AAAA AACA AAGA AAAC AACC AAGC AAAT AACG AAGG AAAT AACT AAGT ACAA ACCA

42. . . . . . . . Hybridization Experiment –cont. 2. Hybridization reaction DNA chip TCCACTG... Many labeled copies of an original sequence 3. Reading results Fluorescence image of the chip Spectrum – a set of oligonucleotides complementary to fragments of original sequence spectrum

43. A hybridization reactionbetween a probe of known sequence (l-mer) and an unknown sequence (n-mer): n-mer - . . . A A C T A G A C C T . . . l-mer - G A T C T A A sequence complementary to the probe exists in the target

44. ACT AAC CTA CCT TAG AGA ACC GAC DNA sequencing without errors The original sequence: AACTAGACCT Spectrum = {AAC,ACT,CTA,TAG,AGA,GAC,ACC,CCT} (Two possible solutions: AACTAGACCT,AACCTAGACT) • Lysov (1988) A graph is based on l-mers (graph H) Finding a Hamiltonian path – NP-hard

45. Pevzner (1989) AAC AA AC A graph based on (l-1)-mers (graph G): AC AA CT TA AG CC GA A problem of equivalence A problem of uniqueness Finding an Eulerian path – polynomially solvable

46. Equivalence problem The above class of directed labeled graphs – DNA graphs. Characterization and recognition of these graphs and finding conditions for which the above transformation is possible. J.Błażewicz, A.Hertz, D.Kobler, D.de Werra, On some properties of DNA graphs, Discrete Applied Math., 1999.

47. Definition The directed line graphH = (V,U)of graph G = (X,V) is thegraph with vertex set V and such that there is an arc from vertex xto vertex yinHif and only if the terminal endpoint of arc x in G is the initial endpoint of arc y inG. Graph G – Pevzner graph Directed line graph H – Lysov graph

48. Theorem 2 Let Hbe the directed line-graph of a graph G. Then there is an Eulerianpath in G if and only if there is a Hamiltonian pathin H.

49. Back to scheduling. Original graph G Hamiltonian Its directed line-graph H ?

50. Theorem 3 Original graph G uan Hamiltonian Its directed line-graph H interval order. J.Błażewicz, D.Kobler European Journal of Operational Research, 2002