Algorithms and Data Structures

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Algorithms and Data Structures

• Lecture 1

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Agenda

• Course Overview
• Goals of the Course
• Syllabus
• Notation
• Basics of Algorithms
• Algorithms Evaluation

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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

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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

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Contacts

• Dmitri Brodski, Lecturer dimetr_at_previo.ee
• Mike Pikkov, Mentor for practical classes
  pikkov_at_previo.ee
• Course main WWW, lectures, homework assignments,
  announces www.itcollege.ee/dmitri
• Practice materials, homework results
  www.itcollege.ee/mike

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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_at_previo.ee
• Name, Group, IT College ID, Assignment ID

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Environment

• Every standard C compiler is acceptable for
  training
• MS VC 6.0 will be used during practical classes

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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

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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

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Syllabus

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

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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 lta1, a2, , an gt OUTPUT rearranged
  set lta1, a2, , an gt where a1lt a2lt lt
  an.

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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.

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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, lets 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

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Algorithms Evaluation
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Algorithms Evaluation
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Algorithms Evaluation

• T(n) c1(n1)c2nc3kc5(n-k)
• c3 c5, lets use c3 instead of c5
• T(n) c1(n1)c2nc3kc3(n-k)
• T(n) c1nc2nc3kc3n-c3kc1
• T(n) c1nc2nc3nc1
• T(n) n(c1c2c3)c1
• Lets denote a c1c2c3, b c1
• T(n) anb, linear function

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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.

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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 ....

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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.

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Algorithms Evaluation
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Algorithms Evaluation
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Algorithms Evaluation

• BCRT(n) c1c2c4c5
• Lets denote c c1c2c4c5
• 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

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Algorithms Evaluation
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Algorithms Evaluation

• WCRT(n) c1(n1)c2n
• WCRT(n) c1nc2nc1
• WCRT(n) n(c1c2)c1
• Lets denote a c1c2, b c1
• WCRT(n) anb, linear function