325: paapt d

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??? 325 ?pa?a??pt???? ????d??

• ??d?s??? ???st?? ?a?a???t??

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?pa?a??pt???? ????d??

• ?? e??a? a??????µ??
• ???????µ?? e??a? µ?a se??? ap? saf? (unambiguous)
  ß?µata ta ?p??a ?????? ??a p??ß??µa.

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???????µ??

• Saf? ß?µata (e?t????)
• ??s?d?? ?? e?d?? ?a? e???? t?? ded?µ???? e?s?d??
  ?a p??pe? ep?s?? ?a e??a? saf?
• ? a??????µ?? µp??e? ?a pe????afe? µe p??????
  t??p??? (d?a??aµµat???, ?e?d?-??d??a?)
• ?p??e? ?a ?p?????? p????? a??????µ?? p?? ??????
  ??a p??ß??µa
• ?p??e? ?a ????? p??? d?af??et??? ap?d?s?
  (ta??t?ta ep???s?? ? apa?t?se?? µ??µ??)

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??? 325 ?pa?a??pt???? ????d??.

• A. Levitin, Introduction to the Design and
  Analysis of Algorithms, 2nd Ed.
• ?e?????? µa??µat??
• ?p?p??s?ete? ?????f???e?
• www.eng.ucy.ac.cy/christos/courses/ECE325

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• ?e? ?p???e? (?a? de? µp??e? ?a ?p???e?) ??a?
  ???????µ?? p?? ?a ???e? ??a ta p??ß??µata!

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?a??de??µa

• ??e?te t? µ???st? ????? d?a???t? (greatest common
  divisor) µeta?? d?? a???a??? a???µ?? m, n.
• ??s?d??
• ???d?? z // µ???st?? ?????? d?a???t??

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???????µ?? 1

• Find the prime factors of m
• Identify all common factors that appear in both
• Compute the product of the common factors

• ?a??de??µa

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???????µ?? 1

• Find the prime factors of m
• Find the prime factors of n
• Identify all common factors that appear in both
• Compute the product of the common factors

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???????µ?? 2

• t minm, n
• If m mod t 0 Goto 3, Else Goto to 4
• If n mod t 0 return t
• Set t t-1 and Goto 2.

• ?a??de??µa m60, n 24

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???????µ?? 3 (???????µ?? t?? ????e?d?)

• gcd(m, n)
• If n0 Return m
• gcd(n,m mod n)

• ?a??de??µa m60, n 24

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????? ?p???s?? ???ß??µ?t?? µe ???????µ????
?e??d???

• ?ata???s? t?? p??ß??µat??
• ???a? ad??at? ?a ??sete ??a p??ß??µa e?? de? t?
  ?ata??ßete!
• ???pe? ?a ???ste? epa???ß?? t? s????? ???? t??
  p??a??? e?s?d??.
• ???at?t?te? t?? µ??a??? p?? ?a ??se? t?
  p??ß??µa
• ??? ?a ???e? st? ???? t?te de? µp??e? ?a ??e?
  pe??ss?te?a ap? µe????? de??de? ß?µata.
• ??a? ?p?????st?? de? ??e? ?pe??? ?p?????st???
  ??a??t?ta!

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????? ?p???s?? ???ß??µ?t?? µe ???????µ????
?e??d???

• ????ß?? ??s? ? ??s? ?at? p??s????s? (exact or
  approximate algorithm)
• ?e???? p??ß??µata de? µp????? ?a ?????? a???ß??
• ??a a???ß?? ??s? µp??e? ?a s??ep??eta? te??st??
  ?p?????st??? f??t? se a?t??es? µe µ?a
  p??se???st??? ??s? ? ?p??a µp??e? ?a e??a?
  ?p?????st??? p??? p?? ap?d?t???
• ? s??d?asµ?? p??se???st???? ??se?? µp??e? ?a µa?
  d?se? t?? a???ß? ??s?

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????? ?p???s?? ???ß??µ?t?? µe ???????µ????
?e??d???

• ???te??p???s?
• S?ed?asµ?? t?? a??????µ?? ep???s?? t??
  p??ß??µat??
• ?e????af? t?? a??????µ??
• ?????s? t?? ????t?ta? t?? a??????µ??
• G?a ??a ta p??a?? ded?µ??a e?s?d??
• ?a??µat??? epa???? (mathematical induction)

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????? ?p???s?? ???ß??µ?t?? µe ???????µ????
?e??d???

• ?????s? t?? ap?d?s?? t?? a??????µ??
• ?pa?t?se?? µ??µ??
• ?p??t?ta (simplicity)
• ?p??? a??????µ?? ?ata?????ta? ?a? ???p?????ta?
  e?????te?a
• ??a? a??????µ?? p?? ?aµß??e? ?p??? ??e? t??
  p??a??? e?s?d??? µp??e? ?a e??a? p??? p??
  p???p????? ap? ??p???? ???? ? ?p???? µp??e? ?a
  a???e? se????a ta ?p??a ?a µ?? eµfa?????ta? p???
  s????.
• ???p???s?

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??p?? ???ß??µ?t??

• ???ß??µata ?a????µ?s?? (sorting)
• ???ß??µata ??a??t?s?? (searching)
• ?pe?e??as?a se???? s?µß???? (string processing)
• ???ß??µata se ???f??? (graph problems)
• S??d?ast??? ???ß??µata (combinatorial problems)
• Ge?µet???? ???ß??µata (geometric problems)
• ????µ?t??? ???ß??µata (numerical problems)

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???ß??µata ?a????µ?s?? (sorting)

• ?ed?µ???? e??? s?????? ap? st???e?a ta????µ?ste
  ta se µ?a se??? µe ß?s? ??p??? ??e?d? (key).
• ?? ??e?d? ?a p??pe? ?a ??a??p??e? µ?a s??s? p??
  ?a ep?t??pe? t?? ta????µ?s?
• G?at? e??a? s?µa?t??? ?a d?at????µe ta????µ?µ??e?
  ??ste?
• ?p???e? ??a? µe????? a???µ?? ap? a??????µ???
  ta????µ?s??. G?at?
• ?d??t?te?
• ??sta?e?? (stable) a??????µ?? ta????µ?s??
• ???????µ?? p?? de? ??e?????ta? ep?p??s?et? µ??µ?

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???ß??µata ??a??t?s?? (searching)

• ??a??t?s? e??? ? pe??ss?t???? st???e???
  (a?t??e?µ????) se ??a s????? s?µf??a µe t? ??e?d?
  a?a??t?s??.
• ?????? f???? ta p??ß??µata a?a??t?s?? µp??e? ?a
  s??d?ast??? µe d?? ???a p??ß??µata
• ???ß??µata ße?t?st?p???s?? µp????? ?a d?at?p?????
  sa? p??ß??µata a?a??t?s??
• ??e?te ta st???e?a t?? s?????? ??a ta ?p??a µ?a
  s????t?s? pa???e? t? ß??t?st? t?µ? (e????st? ?
  µ???st?).

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???ß??µata se ???f??? (graph problems)

• G(V,E)
• ???s?µ?p?????ta? e????? ??a t? µ??te??p???s?
  p????? p??ß??µ?t??
• ???ß??µata ße?t?st?p???s?? µp????? ?a d?at?p?????
  sa? p??ß??µata se ???f??? s??d???ta? ta t??a µe
  ??p??? ??st?? ? ß???? (cost or weight).

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S??d?ast??? ???ß??µata (combinatorial problems)

• ???ß??µata p?? s?????? ??t??? t?? e??es? t??
  s??d?asµ?? a?t??e?µ???? p?? ??a??p????? ??p?????
  pe?????sµ??? (constraints) ?a? p?? ße?t?st?p?????
  ??p??a s????t?s? ??st???.
• ?a??de??µa
• ??a? ßa?????? p??pe? ?a pe??se? ??a ????, µ?a
  ?ats??a ?a? ??a µa????? ap? t?? µ?a ???? t??
  p?taµ?? st?? ???? ???s?µ?p????ta? µ?a ß???a st??
  ?p??a µp??e? ?a ????se? µ??? ??a a?t??e?µe??!
• ????? ?? pe?????sµ?? ?a? p?? µp??e? ?a t?
  ?ataf??e?

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Ge?µet???? ???ß??µata (geometric problems)

• ???????µ??? ??s? ?e?µet????? p??ß??µ?t??
• ?a p??ß??µata a?t? eµfa?????ta? s???? se d??f??e?
  efa?µ???? ??µp?t????, ??af????, t?µ???af?a? ??p.
• ?a?ade??µata
• ?e????? t?? p?? ???t???? s?µe??? (closest-pair
  problem)
• Convex hull problem (p??? t? µ????te?? µ???? e???
  f???t? p?? ?a eµpe????e?e? ??a s????? ap? s?µe?a)

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????µ?t??? ???ß??µata (Numerical problems)

• S?µpe???aµß????? ??a µe???? e???? p??ß??µ?t??
• ????? ap? a?t? ta p??ß??µata ?????ta? (se
  ?p?????st??) ?at? p??s????s? af?? ??a?
  ?p?????st?? ??e? pe?????sµ??? a???ße?a e?? ??
  ????t?? a???µ?? ??e?????ta? ?pe??? a???µ? ??f???.
• ? a???µ?t??? p??s????s? ???? t?? st???????p???s??
  a???µ?? µp??e? ?a d?µ??????se? p??? µe???a
  sf??µata ?d?a?te?a se epa?a??pt????? (iterative)
  ? a?ad??µ????? (recursive) a??????µ???!

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???ß??µata ?p?f?se?? ?a? ?e?t?st?p???s??

• ???ß??µata ?p?f?se?? (Decision Problems)
• ???ß??µata t?? t?p?? a?t?? e??a? d?at?p?µ??a µe
  t?t??? t??p? ?ste ? ap??t?s? pa???e? t? µ??f?
  (?a?/???, yes/no, 1/0).
• ?a??de??µa
• ?e?t??????? ???ß??µata (function problems)
• ???ß??µata sta ?p??a ? ap??t?s? de? e??a? t??
  µ??f?? ?a?/???
• ?a??de??µa
• ???ß??µata ße?t?st?p???s??

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???ß??µata ?p?f?se?? ?a? ?e?t?st?p???s??

• ???ß??µata ße?t?st?p???s??
• ?p???ste p?? X ap?te?e? t? s????? ???? t??
  p??a??? pe??pt?se?? (st??µ??t?p??) e???
  p??ß??µat??.
• C(X)?? e??a? ?p?s????? t?? ? µe ??a ta st???e?a
  p?? ??a??p????? ????? t??? pe?????sµ??? t??
  p??ß??µat??.