p es at

1
???????µ?? ??p?? ?e??s?? ???ß??µat??
2
?e??????

• ???????µ?? ??p?? ?e??s?? ???ß??µat?? (Decrease
  and Conquer)
• ?e??s? ?at? µ?a sta?e?? (decrease by a constant)
• ?e??s? ?at? ??a p?s?st? (decrease by a constant
  factor)
• ?e??s? ?at? µetaß??t? µ??e??? (variable-size-decre
  ase)

3
???????µ?? ?e??s?? ???ß??µat?? ?at? µ?a Sta?e??

• Ge???? ?e??d?????a
• ??t? ?a ??s??µe t? p?? d?s???? p??ß??µa µe??????
  n, ?????µe t? p?? e????? p??ß??µa µe?????? n-m.
• S?????? t? m1.
• ?p? t? ??s? t?? µ????te??? p??ß??µat?? µp????µe
  ?a ß???µe t? ??s? t?? p??ß??µat?? µe?????? n.
• ?p?? ?a??de??µa ?p?????ste t? s????t?s? an

4
???ß??µata ?a????µ?s??

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

• InsertionSort
• ?as??? ?d?a

5
???ß??µata ?a????µ?s??

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

insertionsort(A0,,n-1) for i0 to n-1 new
Ai ji-1 while jgt0 and Ajgt
new Aj1 Aj j j-1 Aj1 new
6
?p?d?s? InsertionSort()
insertionsort(A0,,n-1) for i0 to n-1 new
Ai ji-1 while jgt0 and Ajgt
new Aj1 Aj j j-1 Aj1 new
G?at? ?a ???s?µ?p???se? ??p???? t?
InsertionSort()

• ?e???te?? ?e??pt?s?

• ?a??te?? ?e??pt?s?

7
??e?e???s? G??f?? (Tree Traversal)

• ?ed?µ???? e??? ???f??, p?? µp??e? ??p???? ?a
  ep?s?efte? ????? t??? ??µß??? t?? ???f??.
• ??t? e??a? ??a p??ß??µa t? ?p??? pa???s???eta?
  s???? se ???f???.
• ?p??e? ?a ?p?????? p????? ??se??. Ta µe?et?????
  d?? t??p?? p?? eµp?pt??? ??t? ap? t?? ?at?????a
  µe??s?? t?? p??ß??µat?? ?at? µ?a sta?e?? (??a).
• ? ßas??? ?d?a e??a? ?t? t? p?? d?s???? p??ß??µa
  e?e?e???s?? ???f?? µe n ??µß??? sp??e? se ??a
  p??ß??µa ??a n-1 ??µß??? af?? p??ta ep?s?eft??µe
  t?? aµ?s?? ep?µe?? ??µß?.
• ?e t? ?d?? s?ept??? ?a µp????se ??p???? ?a
  ?atat??e? t??? a??????µ??? a?t??? ??t? ap? t??
  ?at?????a t?? greedy a??????µ??.

8
Depth-First Search (DFS) ?a? Breadth-First Search
(BFS)

• ? Ge???? p??s????s? t?? DFS e??a? af?? ep?s?efte?
  ??a ??µß?, ?a ???e? ??a t?? aµ?s?? ep?µe??
  ?e?t????? ??µß?.
• ???? ??????a ? a??????µ?? ap?µa????eta? ap? ??a
  ??µß?, p??? ep?s?efte? ????? t??? ?e?t??e?.
• ???p??e?ta? a?ad??µ??? ? µe st??ßa.
• ? ?e???? p??s????s? t?? BFS e??a? ?a ep?s?efte?
  ????? t??? ?e?t??e? e??? ??µß?? p???
  ap?µa?????e?.
• ???p??e?te µe ????
• ?? d?? a??????µ?? ????? t?? ?d?a ap?d?s?
  d?af????? st?? se??? µe t?? ?p??a ?a ep?s?eft???
  t??? ??µß???.

9
DFS
10
DFS ?a??de??µa
dfs(v) v.visited true for w 1 to wn if
w.visited false dfs(w)
11
BFS
BFS(GltV,Egt) for all v in V if v.visited
false bfs(v)
bfs(v) v.visited true while Q not
empty for w 1 to wn if w.visited
false w.visited true Q.enqueue(w) Q.
dequeue(Q.head)
12
DFS ?a??de??µa
bfs(v) v.visited true while Q not
empty for w 1 to wn if w.visited
false w.visited true Q.enqueue(w) Q.
dequeue(Q.head)
13
??p??????? ?a????µ?s?

• ???ß??µa ?ed?µ???? e??? ?ate?????µe??? ???f??,
  ?p???e? t??p?? ?a ta????µ?s??µe ????? t???
  ??µß??? ?ts? ?ste e?? ?p???e? a?µ? ap? t?? ??µß?
  u st?? v, st?? ta????µ?µ??? se???, ? u ?a
  eµfa???eta? p??? ap? t?? v
• ?a?ade??µata

14
??p??????? ?a????µ?s?

• Te???µa ?? p??ß??µa t?? t?p???????? ta????µ?s??
  ??e? ??s? e?? ?a? µ???? a? ? ?ate?????µe???
  ???f?? p?? pe?????fe? t? p??ß??µa de? ??e?
  ??????? (?ate?????µe??? ??????? ???f?? directed
  acyclic graph (dag)).

• ?fa?µ???? t?p???????? ta????µ?s??
• ???apa?t??µe?a µa??µata
• ?????p????aµµat?sµ?? (scheduling)
• ??e??as??? se ?p?????st?
• ???as??? ??a t?? s?µp????s? µe????? ?????.
• ??e?d??e?µ??e? ??se??
• Critical Path Method (CPM)
• PERT (Program Evaluation and Review)

15
??s? µe DFS
dfs(v) v.visited true for w 1 to wn if
w.visited false dfs(w)
?
E
C
F
D
B
16
??s? µe DFS
dfs(v) v.visited true for w 1 to wn if
w.visited false dfs(w)
?
E
C
F
D
B
17
??p??????? ?a????µ?s? ??s? ßas?sµ??? se µe??s?
?at? ??a

• ?as??? ?d?a
• Se ???e ß?µa ??????µe ??a ????? se ??a ???f? ?
  ?p???? e??a? ?at? ??a µ????te??? ap? ?t? ?
  p??????µe???.
• ?p? t?? µe?a??te?? ???f?, afa????µe ??a ??µß?
  st?? ?p??? de? ?ata???e? p??? t?? ?aµ?? a?µ?
  ?a??? ?a? ??e? t?? a?µ?? p?? ?e?????? ap? t? e?
  ???? ??µß?.
• ?a??de??µa

18
??µ??????a ???? t?? p??a??? ?p?s?????? ?a?
permutations

• ???ß??µa S??µat?ste ??a ta p??a?? ?p?s????a
  (power set) p?? µp????µe ?a d?µ??????s??µe ap?
  ??a s????? n st???e???.
• ???ß??µa S??µat?ste ??e? t?? p??a??? d?at??e??
  (permutations) t?? ?p??e? µp????µe ?a
  d?µ??????s??µe ap? ??a s????? n st???e???.
• ?p? t? f?s? t??? ta p??ß??µata a?t? de? µp?????
  ?a ????? ap?d?t??? ??s?! G?at?
• ?? d??f???? s??d?asµ?? ?a? d?at??e??
  d?µ????????ta? se???a?? ?p?ta? p????? f????
  e??a? ep???µ?t? ?a µ?? ?p?????? µe???e? e?a??a???
  ap? t? ??a ?p?s????? st? ep?µe?? ? t? µ?a d??ta??
  st?? ep?µe??.

19
Power Set
?????sµa n st???e??? ?p?? t? ???e st???e?? µp??e?
?a p??e? t? t?µ? 0 (e?? t? st???e?? de?
pe???aµß??eta? st? ?p?s?????) ?a? 1 (e??
pe???aµß??eta?)

• ???a?? ?p?s????a ??a n5

• ??sa p??a?? ?p?s????a

20
Permutations
?????sµa n st???e??? ?p?? t? ???e st???e?? µp??e?
?a p??e? t? t?µ? e??? ap? ta st???e?a t?? s??????.

• ???a?? permutations ??a n5

• ??sa p??a?? ?p?s????a

21
Permutations ???????µ?? ßas?sµ???? se µe??s?
?at? ??a

• ?p???ste p?? ?????µe p?? ?a d?µ??????s??µe ??a ta
  permutations ??a n-1, p?? µp????µe ?a
  d?µ??????s??µe ta permutations ??a n
• ?p?? t?p??et??µe t? ??? st???e?? µeta?? ???? t??
  st???e??? ap? t?? ?p?????se? d?at??e??

• ? a??????µ?? Johnson-Trotter d?µ?????e? ??a ta
  permutations ????? ?a p??pe? ?a d?µ??????se? ta
  n-1, n-2, 1, permutations ?a? se ???e ß?µa
  a??????? µ??? 2 st???e?a ??s?.

22
???????µ?? µe??s?? t?? p??ß??µat?? ?at? ??a
p?s?st?

• Ge???? ?e??d?????a
• ??t? ?a ??s??µe t? p?? d?s???? p??ß??µa µe??????
  n, ?????µe t? p?? e????? p??ß??µa µe?????? n/m.
• S?????? t? m2.
• ?p? t? ??s? t?? µ????te??? p??ß??µat?? µp????µe
  ?a ß???µe t? ??s? t?? p??ß??µat?? µe?????? n.
• ???? ap?d?t???? a??????µ?? (a?a??t?s? se d?ad???
  d??t??)!
• ?p?? ?a??de??µa ?p?????ste t? s????t?s? an

23
???p??? ??µ?sµa

• ?p???ste ?t? ??ete n ??µ?sµata e? t?? ?p??? t?
  ??a ???p???. ???e ??µ?sµa ?????e? w ??aµµ???a
  e?? t? ???p??? ?????e? qltw ??aµµ???a.
• ?p???ste p?? ??ete st? d???es? sa? µ?a ???a???.
• ??? µp??e?te ?a e?t?p?sete t? ???p??? ??µ?sµa
  ?????ta? ????te?a ap? n ????sµata
• ??sa ????sµata ??e???este

24
???p??? ??µ?sµa

• ??sa ????sµata ??e???este

findCoin(C1,,n) if(n l) return C fake
found ffloor(n/2) P1C1,,f
P2Cf1,,2f P3Cn-C2f X weight(P1,
P2) if X1 return findCoin(P1) P1 weights
more if X0 return P3 P1 P2 weigh the
same if X-1 return findCoin(P2) P2 weights
more
25
????ap?as?asµ?? ??e?a??? se ????? (Hardware)

• ??? µp??e? ??p???? ?a p???ap?as??se? d??
  a???a???? a???µ??? m, n

• ?p???ste ?t? t? n e??a? ??t??, t?te

• ?p???ste ?t? t? n e??a? pe??tt?, t?te

• ??p???? µp??e? ?a ???p???se? ??a p???ap?as?ast?
  µe µ??? a????sµata ?a? µetat?p?se??!

26
?p???s? µ? ??aµµ???? e??s?se?? µ?a? µetaß??t??

• ??? µp??e?te ?a ß?e?te t? ??s? t?? p?? ??t? µ?
  ??aµµ???? e??s?s?? st? d??st?µa a,b

• ?p???ste p?? µa? e?d?af????? µ??? ta p?? ??t?
  se????a

• ??e?te ??a a??????µ? p?? ?a p??se????e? t? ??s?

27
Bisection Method Divide-and-Conquer

• ???a? ????? ? a??????µ??
• ??te te?µat??e?

28
Bisection Method Divide-and-Conquer
solve(f(),a,b,e,iter,N) x(ab)/2 if (b-a)lte
or itergtN return x fxf(x) if fx0 return x
if fxf(a)gt0 return solve(f,x,b,e,iter1,N) r
eturn solve(f,a,x,e,iter1,N)

• ??? µetaß???eta? t? µ???st? d??at? sf??µa t??
  a??????µ?? ap? epa?????? se epa??????.

29
False Position Method Divide-and-Conquer

• ???s?s? t?? e??e?a? p?? pe??? ap? ta s?µe?a
  (a,fa), (b,fb)

• ?p???f?a ??s?

• St? s????e?a e??????µe t? t?µ? t?? s????t?s?? st?
  s?µe?? x ?a? s??e?????µe a?????a t?? a?a??t?s?.
• ? a??????µ?? a?t?? eµp?pte? st?? ?at?????a
  µe??s?? t?? p??ß??µat?? a??? ?at? µetaß??t?
  µ??e??? (variable size decrease).
• ?p??e?te ?a s?efte?te a?t?st???? t??p? µe t??
  ?p??? ?a µetaß??ete t? a??????µ? a?a??t?s?? se
  d?ad??? d??t?a (binary search)