1. Construction and Analysis of Efficient Algorithms
Introduction
Autumn 2017, Juris Vīksna

2. Timetable Regular lecture times: Wednesdays 12.30 - 14.00 413. aud.
Fridays aud.
The lectures at the following dates will be rescheduled:
13.09.,
Some other changes are possible (but hopefully, not too many).
Replacement lectures most likely will be scheduled starting
from the second half of October.

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
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)
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)
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,

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 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

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
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 g a j m r b e i d g a j m r b e i d

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 homeworks Most likely will be given in October-November-December Strict (or almost) one week deadline 20% each
Programming assignment Problem to be announced in second half of October No deadline – must be submitted before the exam 20%
Exam 40%
To qualify for grade 10 you may be asked to cope with 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

51. Textbooks Michael R. Garey, David S. Johnson
Computers and Intractability –
A Guide to the Theory of
NP-Completeness
W. H. Freedman and Company,
1979

52. Web page http://susurs.mii.lu.lv/juris/courses/alg2017.html
It is expected to contain:
short summaries of lectures
announcements
power point presentations
homework and programming assignment problems
frequently asked questions (???)
your grades
other useful information
Course information available also at

53. Contact information
Juris Vīksna
Room 421, Rainis boulevard 29
phone: