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325: paapt d

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A. Levitin, 'Introduction to the Design and Analysis of Algorithms', 2nd Ed. ?e?????? a?? at?? ... Find the prime factors of m. Identify all common factors ... – PowerPoint PPT presentation

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Title: 325: paapt d

1
??? 325 ?pa?a??pt???? ????d??

??d?s??? ???st?? ?a?a???t??

2
?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.

3
???????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??)

4
??? 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

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

6
?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??

7
???????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

8
???????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

9
???????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

10
???????µ?? 3 (???????µ?? t?? ????e?d?)

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

?a??de??µa m60, n 24

11
????? ?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!

12
????? ?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?

13
????? ?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)

14
????? ?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?

15
??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)

16
???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?

17
???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?).

18
???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).

19
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?

20
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)

21
????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???!

22
???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??

23
???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??.

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