Construction and Analysis of Efficient Algorithms

1. Construction and Analysis of Efficient Algorithms Introduction Autumn 2019,Juris Vīksna

2. Timetable Regular lecture times: Wednesdays14.30 - 16.00 413. aud. Fridays 12.30 - 14.00 413. aud. It is likely that few lectures will be rescheduled (hopefully, not too many). The dates and times of these (an of replacement lectures) will be announced when known.

3. Outline • Motivation and some examples • Subjects covered in the course • Requirements • Textbooks • Other practical information

4. Motivation Is there a sufficiently fast algorithm for a given problem? Problem 1a For a given pair of vertices in connected undirected graph find the length of shortest path between these vertices. Problem 1b For a given pair of vertices in connected undirected graph find the length of longest non-intersecting path between these vertices.

5. Problem 1a Problem 1a For a given pair of vertices in connected undirected graph find the length of shortest path between these vertices.

6. Problem 1a Problem 1a For a given pair of vertices in connected undirected graph find the length of shortest path between these vertices. There is an algorithm that solves this problem in time less than const · (|V| + |E|)

7. Problem 1a

8. Problem 1a d = 0

9. Problem 1a d = 1

10. Problem 1a d = 2

11. Problem 1a d = 3

12. Problem 1a d = 4

13. Problem 1a d = 5  OK

14. Problem 1b Problem 1b For a given pair of vertices in connected undirected graph find the length of longest non-intersecting path between these vertices.

15. Problem 1b Problem 1b For a given pair of vertices in connected undirected graph find the length of longest non-intersecting path between these vertices. There is not known an algorithm that solves this problem within polynomial time. (And it is widely believed that such algorithm does not exist.)

16. Problem 1c Problem 1c For a given pair of vertices in connected undirected graph find a non-intersecting path between these vertices consisting of even number of edges.

17. Problem 2a Problem 2a For a given undirected graph find an Euler cycle (if such exists). Euler cycle is defined as a closed path that contains every edge exactly once.

18. Eulerian path (cycle) problem Eulerian path (cycle) problem For a given graph find a path (cycle) that visits every edge exactly once (or show that such path does not exist).

19. Problem 2a Problem 2a For a given undirected graph find an Euler cycle (if such exists). Euler cycle is defined as a closed path that contains every edge exactly once. There is an algorithm that solves this problem in time less than const · (|V| + |E|)

20. Eulerian path (cycle) problem Eulerian cycle exists if and only if each of graph vertices has even degree. Moreover, there is a simple linear time algorithm for finding Eulerian cycle. Eulerian path (cycle) problem For a given graph find a path (cycle) that visits every edge exactly once (or show that such path does not exist).

21. Problem 2b Problem 2b For a given undirected graph find a Hamiltonian cycle (if such exists). Hamiltonian cycle is defined as a closed path that contains every vertex exactly once.

22. Hamiltonian path (cycle problem) Hamiltonian path (cycle) problem For a given graph find a path (cycle) that visits every vertex exactly once (or show that such path does not exist).

23. Hamiltonian path (cycle problem) Hamiltonian path (cycle) problem For a given graph find a path (cycle) that visits every vertex exactly once (or show that such path does not exist).

24. Hamiltonian path (cycle problem)

25. Problem 2b Problem 2b For a given undirected graph find a Hamiltonian cycle (if such exists). Hamiltonian cycle is defined as a closed path that contains every vertex exactly once. There is not known an algorithm that solves this problem within polynomial time. (And it is widely believed that such algorithm does not exist.)

26. Genome sequence assembly

27. Genome sequence assembly

28. Sequence assembly problem Ok, let us assume that we have these hybridizations. How can we reconstruct the initial DNA sequence from them? Affymetrix GeneChip W.Bains, C.Smith (1988) A novel method for nucleic acid sequence determination.Journal of theoretical biology .Vol. 135:3, 303-307.

29. Sequence assembly problem Ok, let us assume that we have these hybridizations. How can we reconstruct the initial DNA sequence from them?

30. SBH – Hamiltonian path approach

31. SBH – Hamiltonian path approach

32. SBH – Eulerian path approach

33. An Eulerian path approach to SBH problem PNAS (Proceedings of the National Academy of Sciences), 2001

34. SBH and de Bruijn graphs (~ 1950 :)

35. Using de Bruijn graphs for NGS data assembly

36. Motivation How to compute the time of computation required for a given algorithm? Does the given algorithm work correctly? Problem 3 Given two strings P and T over the same alphabet , determine whether P occurs as a substring in T.

37. Problem 3 SimpleMatcher(string P, string T) n  length[T] m  length[P] for s  0 to n  m do if P[1...m] = T[s+1 ... s+m] then print s Time = const ·n · m

38. Problem 3 KnuthMorrisPrattMatcher(string P, string T) n  length[T] m  length[P]   PrefixFunction(P) q  0 for i  1 to n do while q > 0 & P[q+1]  T[i] do q  [q] if P[q+1] = T[i] then q  q + 1 if q = m then print i  m q  [q]

39. Problem 3 PrefixFunction(string P) m  length[P] [1]  0 k  0 for q  2 to m do while k > 0 & P[k+1]  P[q] do k  [q] if P[q+1] = P[q] then k  k + 1 [q]  k return 

40. Problem 3 KnuthMorrisPrattMatcher(string P, string T) n  length[T] m  length[P]   PrefixFunction(P) q  0 for i  1 to n do while q > 0 & P[q+1]  T[i] do q  [q] if P[q+1] = T[i] then q  q + 1 if q = m then print i  m q  [q] T(n,m) = TP(m) + n TWhile(m) TP(m) = const · m T(n,m) const ·n · m It can be shown that T(n,m) = const · (n + m)

41. Knuth-Morris-Pratt Algorithm - Idea T = gadjama gramma berida P = gaga

42. Subjects covered in the course • Models of computation and complexity of algorithms • Complexity analysis • Data structures • Sorting algorithms • Algorithms on graphs • String searching algorithms • Dynamical programming • Computational geometry • Algorithms for arithmetical problems • NP completeness

43. Requirements • Two short-term homeworksMost likely will be given in October-November-DecemberStrict (or almost) one week deadline20% each • Programming assignmentProblem to be announced in second half of OctoberNo deadline – must be submitted before the exam20% • Exam40% • To qualify for grade 10 you may be asked to copewith some additional question(s)/problem(s)

44. Academic honesty You are expected to submit only your own work! Sanctions: Receiving a zero on the assignment (in no circumstances a resubmission will be allowed) No admission to the exam and no grade for the course

45. Textbooks Thomas H.Cormen Charles E.Leiserson Ronald L.Rivest Clifford Stein Introduction to Algorithms The MIT Press, 2009 (3rd ed, 1st ed 1990)

46. Textbooks Harry R.Lewis Larry Denenberg Data Structures and their Algorithms Harper Collins Publishers, 1991

47. Textbooks Steven S.Skiena The Algorithm Design Manual Springer Verlag, 2008 (2nd ed, 1st ed 1997)

48. Textbooks Alfred V.Aho, John E.Hopcroft, Jeffrey D.Ullman The Design and Analysis of Computer Algorithms Addison-Wesley Publishing Company, 1976

49. Textbooks David Harel, Algorithmics: the spirit of computing Addison-Wesley Publishing Company, 2004 (3rd ed)

50. Textbooks Dan Gusfield Algorithms on Strings Trees and Sequences Cambridge University Press, 1997