BruteForce a ed ats

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???????µ?? Brute-Force ?a? ??e??d??? ??a??t?s?
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?e??????

• ???????µ?? t?p?? Brute-Force
• ?a?ade??µata
• ??a??t?s??
• ?a????µ?s??
• ???s??ste?a s?µe?a
• Convex hull
• ?e?t?st?p???s?
• Knapsack problem
• ???ß??µata ????es?? (Assignment)

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???????µ?? Brute-Force

• Se a?t? t?? ?at?????a eµp?pt??? ???? ??
  a??????µ?? ?? ?p???? p??????? ape??e?a? ap? t?
  d?at?p?s? t?? p??ß??µat?? ?a? t?? ???sµ? t??
  d?af???? e?????? p?? s??ep????ta? ap? t? f?s?
  t?? p??ß??µat??.
• ????? Ge???? ????d??
• ?a p?e?sta p??ß??µata µe ??s? ????? µ?a ??s? p??
  eµp?pte? se a?t? t?? ?at?????a
• ?p??e? ?a e??a? p??? ?a?? ?a? p?a?t??? µ???d??
  ??a p??ß??µata p?? ? e?s?d?? e??a? µ????? ?
  µesa??? µe??????.

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???ß??µata ??a??t?s??

• ??s?d?? ??sta A0,,n-1, ??e?d? ?
• ???d?? ? ??s? t?? ??e?d??? st? ??sta (e??
  ?p???e?)

Search(A, K) i0 While iltn and Ai!
K ii1 If iltn return i return -1
Search2(A, K) i0 An K While Ai!
K ii1 If iltn return i return -1

• ?a??te?? ?e??pt?s?
• ?e???te?? ?e??pt?s?

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???ß??µata ?a????µ?s??

• ??s?d?? ??sta A0,,n-1
• ???d?? ?a????µ?µ??? ??sta A0,,n-1

sort(A) for i0 to n-2 min i for ji1
to n-1 if Aj lt Amin min
j swap(Ai,Amin)
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Selection Sort
sort(A0n-1) for i0 to n-2 min i for
ji1 to n-1 if Aj lt Amin min
j swap(Ai,Amin)
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?p?d?s? Selection Sort
sort(A0n-1) for i0 to n-2 min i for
ji1 to n-1 if Aj lt Amin min
j swap(Ai,Amin)

• ?? ?a ?p?????s??µe
• ?a??te??, ?e???te?? ? µ?s? pe??pt?s?

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??a??t?s? se???? ?a?a?t???? (string)

• ??s?d?? ??ste? T0,,n-1, s0,,k-1, kltn
• ???d?? T?s? st?? ?p??a eµfa???eta? t? s st?? T

stringMatch(T0n-1,s0k) for i0 to
n-k j0 while sjTij jj1 if jk
return i return -1

• ?e???te?? pe??pt?s?

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???ß??µa e??es?? t?? d?? p??s??ste??? s?µe???

• ?ed?µ???? µ?a? ??sta? µe n s?µe?a P0,,Pn-1
  ß?e?te p??a d?? s?µe?a ????? t?? µ????te??
  ap?stas?.
• ?p?stas? µeta?? d?? s?µe??? se ???? D d?ast?se??
  
• G?a pa??de??µa se d?sd??stat? ????,

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???s??ste?a S?µe?a
ClosestPoints(P0n-1) dmin infinity for i0
to n-2 for ji1 to n-1 d
sqrt((x0-x1)2(y0-y1)2) if d lt dmin
dmin d p1i, p2j return p1, p2
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???ß??µa e??es?? t?? ???t?? ????? p??
eµpe????e?e? ??a s????? ap? s?µe?a

• ???t?? (convex) ?????
• ??a? ????? e??a? ???t?? e?? ??a ???e d?? s?µe?a
  p?? a?????? st?? ????, ??a ta s?µe?a p??
  ß??s???ta? st? tµ?µa t?? e??e?a? p?? e???e? ta
  d?? s?µe?a a???e? ep?s?? st?? ????
• G?a pa??de??µa se d?sd??stat? ????,

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???ß??µa e??es?? t?? ???t?? ????? p??
eµpe????e?e? ??a s????? ap? s?µe?a

• ?ed?µ???? µ?a? ??sta? µe n s?µe?a P1,,Pn
  ß?e?te t? µ????te?? ???t? ???? p?? eµpe????e?e?
  ??a ta s?µe?a P1,,Pn (convex hull).
• Te???µa ?? convex hull e??? s?????? ap? s?µe?a
  P1,,Pn (ngt2) e??a? ??a p??????? t?? ?p???? ??
  ????e? e??a? ??p??? ap? ta s?µe?a t?? s??????
  P1,,Pn.
• G?a pa??de??µa se d?sd??stat? ????,

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Convex Hull
ConvexHall(P1n) for i1 to n-1 for ji1 to
n-1 k1 PgetNextPoint while
(kltn-2) and kk1 PgetNextPoint
if kn-2 (Pi,Pj) on boundary
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Ge?µet??a
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?e?t?st?p???s?

• ??ste t? p?? ??t? p??ß??µa ße?t?st?p???s??

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?e?t?st?p???s?
minimize(f,x0,x1,y0,y1,n) d(x1-x0)/(n-1) e(y1
-y0)/(n-1) min infinity for i0 to n-1 for
j0 to n-1 if f(x0id, y0je)ltmin min
f(x0id, y0je) xmx0id ym
y0je return min, xm, ym
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???ß??µa t?? ?e???de??? ????t? (Traveling
Salesman Problem (TSP))

• ??a? p???t?? ???e? ?a ep?s?efte? n p??e?? ?a? ?a
  ep?st???e? st? s?µe?? ap? ?p?? ?e????se µe t? p??
  ??????? t??p?.
• ?e? p??pe? ?a ep?s?efte? µ?a p??? pe??ss?te?e?
  ap? µ?a f????
• ??e?te t? p?? s??t?µ? Hamiltonian Circuit.
• Hamiltonian Circuit e??a? ??a µ???p?t? t? ?p???
  ep?s??pteta? ????? t??? ??µß??? t?? ???f?? ap?
  µ?a f??? ?a? ep?st??fe? st? s?µe?? ap? ?p??
  ?e????se.

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???ß??µa t?? ?e???de??? ????t? (Traveling
Salesman Problem (TSP))

• ???ste ??a ta p??a?? µ???p?t?a
• ??at?ste e?e??? µe t? ?aµ???te?? ??st??

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Knapsack Problem

• ??a? ??e?ß?t?? p????aµµat??e? t?? ep?µe?? t??
  pe??p?te?a. St? sa??d?? t?? µp??e? ?a µetaf??e?
  µ???? n ????.
• ?p?????? m a?t??e?µe?a ta ?p??a µp??e? ?a t??
  ???s?µe?s???
• ???e a?t??e?µe?? ?????e? wi ????
• ? ???s?µ?t?ta t?? ???e a?t??e?µ???? d??eta? ap?
  ??a de??t? yi.
• ?????ste t?? ??e?ß?t? ?a ß?e? p?? ?a ?eµ?se? t?
  sa??d?? t?? ?ts? ?ste ?a µe??st?p???se? t?
  s??????? ???s?µ?t?ta ???? t?? a?t??e?µ???? p??
  ?a p??e? µa?? t??.
• ?a?ade??µata sta ?p??a t? p?? p??? p??ß??µa
  efa?µ??e?

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Knapsack Problem ?a??de??µa

• ?? sa??d?? µp??e? ?a ????se? 10 ????
• ?p?????? 4 a?t??e?µe?a
• ????? 7, ???s?µ?t?ta 42
• ????? 3, ???s?µ?t?ta 12
• ????? 4, ???s?µ?t?ta 40
• ????? 5, ???s?µ?t?ta 25
• ???? ?? p??a??? s??d?asµ??

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???ß??µa ????es?? (Assignment)

• ?p?????? n f??t?t?? ?a? n p??ß??µata. G?a t??
  ?a??te?? ?ata??µ? t?? f??t?? e??as?a? ? ???e
  f??t?t?? ?a ??se? ??a ap? ta p??ß??µata.
• ??? ? f??t?t?? i ep??e???se? t? p??ß??µa j t?te
  ?a p??e? bij ßa?µ???
• ??? p??pe? ?a ?ata?eµ????? ta p??ß??µata ?ts?
  ?ste ?a µe??st?p????e? ? ßa?µ?? ????????? t??
  t????
• ???a pa?ade??µata sta ?p??a efa?µ??e? a?t? t?
  p??ß??µa

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?a??de??µa ???ß??µat?? ????es??

• ?p???ste p?? ?p?????? 4 f??t?t?? ?a? 4 p??ß??µata
  ?a? ?? ßa?µ?? fa????ta? p?? ??t?.

• ?? d????sµa f1, f2, f3, f4 a?t?p??s?pe?e? t?
  f??t?t? p?? ?a ??se? t? p??ß??µa 1, 2, 3, ?a? 4
  a?t?st???a
• ??e?te ??e? t?? p??a??? a?a??se?? (permutations)
  ?a? ?p?????ste t? s??????? ßa?µ?
• ??se? ?? p??a??? a?a??se?? ?p??????