Algorithms and Data Structures

1. Algorithms and Data Structures Lecture 1

2. Agenda: • Course Overview • Goals of the Course • Syllabus • Notation • Basics of Algorithms • Algorithms Evaluation

3. Course Overview: • Lectures, 32 hours, 16 weeks • Practical classes, 32 hours, 16 weeks • Grading consists of home works and examination • Examination (up to 80%) • Homework assignments (up to 20%) • 3 Homework assignments • Each homework consists of several small tasks

4. Milestones: • Home Work I Completes by March 1, 2002 • Home Work II Completes by April 1, 2002 • Home Work III Completes by May 1, 2002 • Examination Date is NA

5. Contacts: • Dmitri Brodski, Lecturer: dimetr@previo.ee • Mike Pikkov, Mentor for practical classes: pikkov@previo.ee • Course main WWW, lectures, homework assignments, announces: www.itcollege.ee/~dmitri • Practice materials, homework results: www.itcollege.ee/~mike

6. Organization: • Classes are not obligatory, you are free whether to visit or not • Home assignments could be obtained from WWW • Accomplished assignments should be sent by e-mail to a practice mentor: pikkov@previo.ee • Name, Group, IT College ID, Assignment ID

7. Environment: • Every standard “C” compiler is acceptable for training • MS VC 6.0 will be used during practical classes

8. Goals of the Course: • Give an overview of various classical algorithms • Give an overview of algorithm construction and evaluation • Give an overview of various data structures and their implementations in “C” language • Develop rational and creative approach to given tasks

9. Literature: • Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest Introduction to Algorithms • Jüri Kiho, Algoritmid ja Andmesruktuurid,Tartu 1994(1st edition), Tartu 1997 (2nd edition),Tartu Ülikool

10. Syllabus: • Basics of Algorithms • Algorithms Evaluation • Algorithms Construction • Data Structures • Sorting Algorithms • Graph Processing • Text Processing • Algorithms on Plane Geometry • Some Simple Mathematical Algorithms

11. Basics of Algorithms • Definition: Algorithm is a formally described computational procedure consisting of zero or more elementary steps. • Elementary step = evaluation, assignment, loop iteration and others. • Every algorithm works on some source data (INPUT) and produces some resulting data (OUTPUT); it is a function: OUTPUT = F(INPUT). • Every algorithm solves some computational problem what should be formally specified in terms of its INPUT and OUTPUT; e.g. sorting task INPUT: set of N numbers <a1, a2, … , an > OUTPUT: rearranged set <a’1, a’2, … , a’n > where a’1=< a’2=< … =< a’n.

12. Basics of Algorithms • Definition: Algorithms is correct if for any acceptable (for the problem) INPUT it always completes its execution and produces an OUTPUT that complies with the specification of the problem. • If algorithm is correct (regarding the given problem specification), we say that it solves the problem. • Sometimes incorrect algorithms may be useful if we need to solve a problem for some particular INPUT; usage of incorrect algorithms is a quite rare practice.

13. Algorithms Evaluation • Why do we analyze and evaluate various algorithms? • Definition: running time of an algorithm is a number of elementary operations performed by the algorithm for the given INPUT. • Each line of code has a value in terms of elementary operations, let’s denote it cj,where j – index of line. • Most common evaluation purpose is to determine a function describing how running time depends on the INPUT size; such a function is usually denoted as T(n) (time on n), n – characterizes size of INPUT

14. Algorithms Evaluation

15. Algorithms Evaluation

16. Algorithms Evaluation • T(n) = c1(n+1)+c2n+c3k+c5(n-k) • c3= c5, let’s use c3 instead of c5 • T(n) = c1(n+1)+c2n+c3k+c3(n-k) • T(n) = c1n+c2n+c3k+c3n-c3k+c1 • T(n) = c1n+c2n+c3n+c1 • T(n) = n(c1+c2+c3)+c1 • Let’s denote a = c1+c2+c3, b = c1 • T(n) = an+b, linear function

17. Algorithms Evaluation • Sometimes running time depends on kind of INPUT data; e.g. running time of sorting algorithm may depend on INPUT data, not only size of the INPUT; e.g. INPUT data may be already sorted or not sorted at all. • There are three measurable quantities available: best-case running time, worst-case running time and average-case running time. • Best-case running time – lowest value of running time that algorithm can spend handling INPUT of the size N. • Worst-case running time – highest value of running time that algorithm can spend handling INPUT of size N.

18. Algorithms Evaluation • Average-case running time – is mostly expected (or predicted) average time needed to handle an INPUT of size N; usually depends on probability distribution for particular type of INPUT data. • Best- and Worst- case running times may differ greatly. • Worst-case running time is most interesting measurable quantity, because ....

19. Algorithms Evaluation • If we know WCRT we may guaranty that the algorithm completes its execution in a certain time limit, for INPUT of given size. • In practice, worst-cases are highly probable, e.g. database query for non-existent element. • In practice, ACRT is very close to WCRT.

20. Algorithms Evaluation

21. Algorithms Evaluation

22. Algorithms Evaluation • BCRT(n) = c1+c2+c4+c5 • Let’s denote c = c1+c2+c4+c5 • BCRT(n) = c • c - some constant value for INPUT of any size • BCRT(n) in this case does not depend on ‘n’ • BCRT = C – constant function

23. Algorithms Evaluation

24. Algorithms Evaluation • WCRT(n) = c1(n+1)+c2n • WCRT(n) = c1n+c2n+c1 • WCRT(n) = n(c1+c2)+c1 • Let’s denote a = c1+c2, b = c1 • WCRT(n) =an+b, linear function