???a

1
???aµ???? ?????aµµat??µ??
2
?????ep??a??pt?µe?a ?p?p??ß??µata

• ?p?? ? ??a??es? ?a? ????a???a, ? ???aµ????
  ?????aµµat?sµ?? s??d???e? ??se?? se
  ?p?p??ß??µata.
• ? ???aµ???? ?????aµµat?sµ?? ???s?µ?p??e?ta?
  ?????? ?ta? ?? ??se?? t?? ?d??? ?p?p??ß??µ?t??
  ??e?????ta? p????? f????.

3
?et???taS ?e?ta
4
?et???ta? ??sta

• ???ß??µa ???es? t?? e????st?? p?????? ??µ?sµ?t??
  ap? ??a set ??µ?sµ?t?? ?ste ?a ft?s??µe se
  s???e???µ??? p?s?.
• G?a 31 ?ept?
• 1010101 ?
• 20101 (ß??t?st?)
• ??? t? ?????µe

5
?a??de??µa Set ??µ?sµ?t??

• Ta ?e???s??µe t? e??? set ??µ?sµ?t??
• 1 5 10 21 25
• ?st? ?t? ? st???? µa? e??a? ta 63

6
??a ?p?? ??s?

• ??e?a??µaste p??ta ??a ??µ?sµa 1, d?af??et???
  de? ?a ?p???e? ??s? ??a t? 1 ?ept?.
• G?a ?a ?????µe K ?ept?
• ?? ?p???e? ??a ??µ?sµa t?? ? ?ept??, t?te t? ??a
  ??µ?sµa e??a? t? e????st?
• ??af??et???, ??a ???e i lt K,
• ???es? t?? e????st?? p?????? ??µ?sµ?t?? ??a i
  ?ept?
• ???es? t?? e????st?? p?????? ??µ?sµ?t?? ??a Ki
  ?ept?
• ?p????? t?? i p?? e?a??st?p??e? t? ?????sµa

7
???? ??t?st???? ??s?

• ?e?????µe t? p??ß??µa a?ad??µ??? ep??????ta? t?
  1? ??µ?sµa ?a? ?pe?ta ?????ta? ??a t? p?s? p??
  ap?µ??e?
• G?a 63
• ??a 1 ??µ?sµa s?? t? ß??t?st? ??s? ??a 62
• ??a 5 ??µ?sµa s?? t? ß??t?st? ??s? ??a 58
• ??a 10 ??µ?sµa s?? t? ß??t?st? ??s? ??a 53
• ??a 21 ??µ?sµa s?? t? ß??t?st? ??s? ??a 42
• ??a 25 ??µ?sµa s?? t? ß??t?st? ??s? ??a 38
• ?p????? t?? ß??t?st?? ??s?? ap? t?? 5 pa?ap???
• ??t? ??a 62 a?ad??µ?? ????µe µ??? 5
• a??? pa?aµ??e? a???ß?...

8
???aµ???? ?????aµµat?sµ??

• ?d?a ??se ??a 1 ?ept?, ?pe?ta ??a 2 ?ept?,
  ?pe?ta ??a 3 ?ept? ???., µ???? t? p?s? p??
  ep???µ??µe.
• ?p????e?se ???e ??s? se ??a? p??a?a!
• G?a ???e ??? p?s? N, ?p?????se ??a ta d??at?
  ?e??? p??????µe??? ??se?? p?? e??a? ?sa µe N
• ?a??de??µa ??a 13 ?ept?,
• ?????? ?????µe ??a 1, 2, 3, ..., 12
• ?pe?ta ep??????µe t?? ?a??te?? ??s? µeta??
• 1 ??s? ??a 12
• 5 ??s? ??a 8
• 10 ??s? ??a 3

??s? ????? ??a te???? p?s? ? ?a? d?af??et????
??µ?sµ?t?? ?
9
S?????s? µe ??a??es? ?a? ????a???a

• St? ??a??es? ?a? ????a???a ??a p??ß??µa
  d?a??e?ta? se ?p?p??ß??µata t?? ?p???? ? ??s?
  s??d???eta? ??a t?? ??s? st? a????? p??ß??µa
• ???a? ?????? a??????µ?? µe f??? ap? ep??? p??? ta
  ??t? (top-down)
• ??a a??????µ?? d??aµ???? p????aµµat?sµ?? ep???e?
  µ???? p??ß??µata ta ?p??a ta s??d???e? ??a ?a
  ß?e? t? ??s? se µe?a??te?a
• ?p??e? ?a?e?? ?a t?? s?efte? sa? µ?a ap? ??t?
  p??? ta ep??? d?ad??as?a

10
? ???? t?? ?e?t?st?t?ta?

• ? d??aµ???? p????aµµat?sµ?? e??a? µ?a te????? ??a
  e??es? ß??t?st?? ??s??
• ? a??? t?? ße?t?st?t?ta? efa?µ??eta? a? ?
  ß??t?st? ??s? se ??a p??ß??µa pe????e? ß??t?ste?
  ??se?? se ??a ta ?p?p??ß??µata

?a??de??µa ???ß??µa ????? ??a N ?ept? µe t?
µ????te?? p????? ??µ?sµ?t?? ??te ?p???e? ??a
??µ?sµa t?? N ?ept?? ? ?? s????? t?? ??µ?sµ?t??
??a t? ß??t?st? ??s? ??a N ?ept? d?a??e?ta? se
d?? µ? ?e?? s????a n1 ?a? n2 ?? ??p??? ap? ta
?p?s????a n1 ? n2, µp??e? ?a ???e? µe ????te?a
??µ?sµata, t?te p??fa??? ?a? ??a N ?ept? µp??e?
?a ???e? µe ????te?a ??µ?sµata - ?t?p?
11
? ???? t?? ?e?t?st?t?ta?

• ? a??? t?? ße?t?st?t?ta? de? ??e? ?t?
• ?? ??e?? ß??t?ste? ??se?? se ??a ta
  ?p?p??ß??µata...
• ...t?te µp????? ?a s??d?ast??? se µ?a ß??t?st?
  ??s?
• ?a??de??µa
• ? ß??t?st? ??s? ??a 7 e??a? 511, ?a?
• ? ß??t?st? ??s? ??a 6 e??a? 51, a???
• ? ß??t?st? ??s? ??a 13 de? e??a? 51151

???? ?p???e? ??p???? t??p?? d?a??es?? t?? 13 se
?p?s????a µe ß??t?ste? ??se?? (p.?. 11 2) p??
d??e? ß??t?st? ??s? ??a 13 ??a ? a??? t??
ße?t?st?t?ta? ?s??e? ??a a?t? t? p??ß??µa
12
? ???? t?? ?e?t?st?t?ta?

• ?st? k µ?a e?d??µes? ????f? se ??a e????st? ap? i
  se j µ???p?t? i, a, b, , k, l, m, , j . ??
  µ???p?t? i, a, b, , k ?a p??pe? ?a e??a?
  e????st? ap? i se k, ?a? t? µ???p?t? k, l, m, ,
  j ?a p??pe? ?a e??a? e????st? ap? k se j

k
i
j
l
a
b
m
13
?a??de??µa ?p?t???a? ?? ?a???te?? ?p?? ????p?t?

• ?? µa???te?? ap?? µ???p?t? (????? ???????) ap?
  t? A st? D e??a? t? A B C D
• ?µ??, t? ?p?µ???p?t? A B de? e??a? t? µa???te??
  µ???p?t? ap? t? A st? B (t? A C B e??a?
  µa???te??)
• ? a??? t?? ße?t?st?t?ta? de? ??a??p??e?ta? se
  a?t? t? p??ß??µa
• ??a, t? p??ß??µa a?t? de? µp??e? ?a ???e? µe
  d??aµ??? p????aµµat?sµ?

14
???µ?????S? ???aS??? µe ?a??
15
???µ?????s? ???as??? µe ????

• ? e??as?a j ?e???? st?? sj, te?e???e? st?? fj,
  ?a? ??e? t?µ? vj.
• ??? e??as?e? e??a? s?µßat?? a? de?
  ep??a??pt??ta?.
• St???? e??es? µe??st?? ß????? ?p?s????? s?µßat??
  e??as???.

a
b
c
d
e
f
g
h
??????
0
1
2
3
4
5
6
7
8
9
10
16
???µ?????s? ???as??? µe ????

• ?????µe se se??? t?? e??as?e? µe ß?s? t? ?????
  ????????s?? f1?f2?...?fn.
• p(j) µ???st?? de??t?? i lt j ?ts? ?ste ? e??as?a
  i ?a e??a? s?µßat? µe j.
• p.?. p(8) 5, p(7) 3, p(2) 0.

1
2
3
4
5
6
7
8
??????
0
1
2
3
4
5
6
7
8
9
10
11
17
??a ??ad??? ?p?????

• OPT(j) t?µ? t?? ß??t?st?? ??s?? st? p??ß??µa
  p?? ap?te?e?ta? ap? t?? e??as?e? 1, 2, ..., j.
• ?e??pt?s? 1 ? OPT ep????e? t?? e??as?a j.
• ???d?? vj
• ?e? µp????µe ?a ???s?µ?p???s??µe t?? as?µßate?
  e??as?e? p(j)1, p(j)2, ..., j1
• ???t?st? ??s? ??a t? e?ap?µe??a? p??ß??µa 1, 2,
  ..., p(j)
• ?e??pt?s? 2 ? OPT de? ep????e? t?? e??as?a j.
• Ta pe???aµß??e? t? ß??t?st? ??s? st?? ?p????pe?
  e??as?e? 1, 2, ..., j-1

???t?st? ?p?p??ß??µa
18
??a?t??t??? ??a??t?s?
??s?d?? n, s1,,sn , f1,,fn ,
v1,,vn ?a????µ?se t?? e??as?e? µe ß?s? t? ?????
????????s?? f1 ? f2 ? ... ? fn. ?p?????se
ta p(1), p(2), , p(n) ?p?????se-OPT(j) if
(j 0) return 0 else return
max(vjCompute-OPT(p(j)),Compute-OPT(j-1))
19
??a?t??t??? ??a??t?s?

• ? a?ad??µ???? a??????µ?? ap?t?????e? µ?a? ?a?
  p??pe? ?a epa?a- ?p??????e? µe ?d?a ?p?p??ß??µata
  ? e??et???? a??????µ??.
• St?? pa?a??t? pe??pt?s? ? p???p????t?ta ??????
  e?f???eta? ap? µ?a a???????a Fibonacci (d??ad?
  e??et???).

1
2
3
4
5
p(1) 0, p(j) j-2
20
? ?e????? t?? ?p?µ??µ??e?s?? (Memoization)

• ?p????e?s? t?? ap?te??sµat?? ???e ?p?p??ß??µat??.
  ??a??t?s? ?ta? ta ??e?a??µaste.

??s?d?? n, s1,,sn , f1,,fn ,
v1,,vn ?a????µ?se t?? e??as?e? µe ß?s? t? ?????
????????s?? f1 ? f2 ? ... ? fn. ?p?????se
ta p(1), p(2), , p(n) for j 1 to n Mj
empty M0 0 M-Compute-OPT(j) if (Mj is
empty) Mjmax(vjM-Compute-OPT(p(j)),
M-Compute-OPT(j-1)) return Mj
?p?µ??µ??e?s? se p??a?a
21
?????? ??t??es??

• ? ??????? p???p????t?ta e??a? O(nlogn).
• ?a????µ?s? ?? p??? ????? ????????s?? O(nlogn)
• ?p?????sµ?? p(?) O(nlogn) µe ta????µ?s? ??
  p??? t? ????? ??a????.
• M-Compute-OPT(j) ???e ???s? apa?te? O(1) ?????
  ?a?
• e?te ep?st??fe? t?? ?p?????sa t?µ? Mj
• e?te ?eµ??e? ??a ?e?? t?? Mj ?a? e?te?e? 2
  a?ad??µ???? ???se??
• ?et???? p???d?? ? ?eµ?t?? ?e???? t?? M.
• ?????? ? 0, e?? ?s??e? p??ta ? ? n
• ?????e? t?? ? ?at? 1 ? t? p??? 2n a?ad??µ????
  ???se??.
• ? ?????? ??a t?? M-Compute-OPT(n) e??a? O(n). ?

22
???es? ??s??

• ???s???µe t?? t?µ? a??? t? ???eta? t?? ?d?a t?
  ??s?
• ?????µe µ?a p??epe?e??as?a.
• a?ad??µ???? ???se?? ? n ? O(n)

Run M-Compute-Opt(n) Run Find-Solution(n) Find-S
olution(j) if (j 0) output nothing
else if (vj Mp(j) gt Mj-1) print j
Find-Solution(p(j)) else
Find-Solution(j-1)
23
???aµ???? ?????aµµat?sµ?? ?pa?a??pt???

• ?ed?p?????µe t?? a?ad??µ?.

??s?d?? n, s1,,sn , f1,,fn ,
v1,,vn ?a????µ?se t?? e??as?e? µe ß?s? t? ?????
????????s?? f1 ? f2 ? ... ? fn ?p?????se ta
p(1), p(2), , p(n) Iterative-Compute-OPT
M0 0 for j 1 to n Mj max(vj
Mp(j), Mj-1)
24
???? ??s? Sta ??µ?sµata

•  

25
???????µ?? ?p?????sµ?? ??µ??
26
???a e??a? ta ??µ?sµata
27
?a??de??µa
???a e??a? ta ??µ?sµata
 
0 1 2 3 4 5 6 7 8 9 10 11
0
p?s?
4
2
2
1
3
3
4
1
5
1
2
??µ.
-
1
1
1
1
1
1
5
1
1
10
1
??µ?sµa p?? ???s?µ?p??????e
28
1. ?etaßat??? ??e??t?t?ta2. µ????te?a µ???pat?a
µeta?? ???? t?? ?e????
29
???????µ?? Warshall

• ?p??????e? t? µetaßat??? ??e?st?t?ta µ?a? s??s??,
  d??. ??a ta µ???p?t?a e??? ?ate?????µe??? ???f??
• ?a??de??µa

0 0 1 0 1 1 1 1 0 0 0 0 1 1 1 1
0 0 1 0 1 0 0 1 0 0 0 0 0 1 0 0
30
???????µ?? Warshall ?d?a

• ????a ?d?a ?p???e? ??a µ???p?t? µeta?? t??
  ????f?? i ?a? j a?
• ?p???e? a?µ? ap? t?? i st?? j,
• ?p???e? µ???p?t? ap? t?? i st?? j µ?s? t??
  ????f?? 1
• ?p???e? µ???p?t? ap? t?? i st?? j µ?s? t??
  ????f?? 1 ?/?a? 2
• ...
• ?p???e? µ???p?t? ap? t?? i st?? j µ?s?
  ?p??asd?p?te ap? t?? ???e? ????f??

R0 0 0 1 0 1 0 0 1 0 0 0 0 0 1 0
0
R1 0 0 1 0 1 0 1 1 0 0 0 0 0 1 0 0
R2 0 0 1 0 1 0 1 1 0 0 0 0 1 1 1 1
R3 0 0 1 0 1 0 1 1 0 0 0 0 1 1 1 1
R4 0 0 1 0 1 1 1 1 0 0 0 0 1 1 1 1
31
???????µ?? Warshall S??s?
St? k-?st? st?d?? ß??s???µe a? ?p???e? ??a
µ???p?t? µeta?? t?? ????f?? i, j ???s?µ?p????ta?
µ??? t?? ????f?? 1,,k
 
k
i
k-?st? st?d??
j
32
?a??de??µa
0 0 1 0 1 0 0 1 0 0 0 0 0 1 0 0
R(0)
33
???????µ?? Warshall - ?e?d???d??a?
Algorithm Warshall(A1..n,1..n) R(0)?A for k?1
to n do for i?1 to n do for j?1 to n do
R(k)i,j ? R(k-1)i,j or R(k-1)i,k and
R(k-1)k,j return R(k) ???????
ap?te?esµat???t?ta T(n3) ??????
ap?te?esµat???t?ta T(n2) epa?a???s?µ?p???s?
?????
34
S??t?µ?te?a ????p?t?a ?eta?? ???? t?? ?e????
(APSP)

• ?a ß???µe ta s??t?µ?te?a µ???p?t?a µeta?? ????
  t?? ????f?? e??? ????sµ???? ???f??
• ?d?a ?a ß???µe t? ??s? µ?s? µ?a? se???? p??????
  d(0), d(1), ???s?µ?p????ta? ??a a?????
  ?p?s????? ????f?? ?? e?d??µese?
• ?a??de??µa

 
35
???????µ?? Floyd - ?a??de??µa
0 8 3 8 2 0 8 8 8 7 0 1 6 8 8 0
0 8 3 8 2 0 5 8 8 7 0 1 6 8 9
0
D(0)
D(1)
0 8 3 8 2 0 5 8 9 7 0 1 6 8 9
0
0 10 3 4 2 0 5 6 9 7 0 1 6 16 9 0
0 10 3 4 2 0 5 6 7 7 0 1 6 16 9 0
D(2)
D(3)
D(4)
36
???????µ?? Floyd

• Algorithm Floyd(W1..n,1..n)
• D?W
• for k?1 to n do
• for i?1 to n do
• for j?1 to n do
• Di,j?min(Di,j,Di,kDk,j)
• return D
• ??????? ?a? ?????? ap?te?esµat???t?ta
• ??te de? d???e?e??

37
?? ???ß??µa t?? S?k?d???
38
?? ???ß??µa t?? Sa??d???

• ?????ta? n a?t??e?µe?a µe ß??? w1,w2,,wn, a??e?
  v1,v2,,vn ?a? ????t???t?ta s???? W. ?a ß?e?e? t?
  p???t?µ?te?? ?p?s????? t?? a?t??e?µ???? p??
  ?????? st? s???
• ??s? µe e?a?t??t??? a?a??t?s?
• ?st? ?t? t? Vi,j d??e? t? ß??t?st? ??s? ??a ta
  p??ta i a?t??e?µe?a ?a? ????t???t?ta j.
• S??p?? e??a? ? ?p?????sµ?? t?? Vn,W

 
39
?a??de??µa
W 1
n 1
???t?st? ??s? 4,3 t?µ? 22 18 40
W 11
40
S??a?t?se?? ???µ??

• ??a µe?????t?µa t?? d??aµ???? p????aµµa-t?sµ??
  e??a? ?t? ep???e? ?p?p??ß??µata p?? de?
  ??e?????ta? te????
• ??a e?a??a?t??? te????? e??a? ? s??d?asµ?? t??
  top-down µe t?? bottom-up p??s????s? (d??.
  a?ad??µ? s?? ??a p??a?a p??s?????? ap?te?esµ?t??)

41
?a??de??µa
W 1
1
1
1
1
1
1
1
7
7
7
7
n 1
18
7
25
40
7
40
 
???t?st? ??s? 4,3 t?µ? 22 18 40
W 11
42
?p?d?s?

• ?????? ?(nW).
• ?e? e??a? p??????µ???? (??at?)
• ?e?d?-p??????µ????
• ?? p??ß??µa ap?fas?? t?? Sa??d??? e??a?
  NP-p???e?.
• ???s????s? ?p???e? ??a? p??????µ???? a??????µ??
  ?(n3/e) p?? ß??s?e? µ?a ??s? t? p??? e µa???? ap?
  t? ß??t?st?.

43
??T????µµ??? ????????O? (sequence alignment)
44
?µ???t?ta S?µß???se????
e
?
?
s
t
e
s
?
?
-

• ??s? ?µ??e? e??a? ?? d?? s?µß???se????
• e??stes??
• e???stas??

?
?
?
s
t
a
s
?
?
e
6 a?a?t?st????e?, 1 ?e??
45
?p?stas? ?pe?e??as?a?

• ?fa?µ????
• ??s? ??a t?? Unix p???? diff
• ??a?????s? ?µ???a?
• ?p?????st??? ???????a
• ?p?stas? ?pe?e??as?a?. Levenshtein 1966,
  Needleman-Wunsch 1970
• ??st?? ?e??? ? - ??st?? a?a?t?st????a? ?pq
• ??st?? ???????µµ?s?? ?????sµa ??st???
  a?a?t?st?????? ?a? ?e???.

C
G
A
C
C
T
A
C
C
T
T
T
G
A
C
C
T
A
C
C
T
-
C
C
T
G
A
C
T
A
C
A
T
C
C
T
G
A
C
T
A
C
A
T
C
-
?TC ?GT ?AG 2?CA
2? ?CA
46
?? ???ß??µa

• ?a ß?e?te t?? e?a??st?? ??st??? e??????µµ?s?
  µeta?? d?? s?µß???se???? X x1x2...xm ?a? Y
  y1y2...yn.
• ??a e??????µµ?s? M e??a? ??a s????? ap?
  d?ateta?µ??a ?e??? xi-yj ?ts? ?ste ???e s?µß???
  ?a eµfa???eta? se t? p??? ??a ?e???? ?a? ?a µ??
  ????µe d?asta???se??
• ?a ?e??? xi-yj ?a? xi'-yj' d?asta??????ta? ?ta?
  ilti? ?a? jgtj?.

M x2-y1, x3-y2, x4-y3, x5-y4, x6-y6.
47
? ??ad??µ?

• OPT(i, j) e????st? ??st?? e??????µµ?s?? t??
  s?µß???se???? x1 x2 . . . xi ?a? y1 y2 . . . yj.
• ?e??pt?s? 1 ? OPT ft????e? t? ?e???? xi-yj
• ??st?? a?a?t?st????a? xi-yj e????st? ??st??
  e??????µµ?s?? t?? s?µß???se???? x1 x2 . . . xi-1
  ?a? y1 y2 . . . yj-1
• ?e??pt?s? 2(a) ? OPT de? ta?????e? t? xi
• ??st?? ?e??? ??a t? xi e????st? ??st??
  e??????µµ?s?? t?? x1 x2 . . . xi-1 ?a? y1 y2 . .
  . yj
• ?e??pt?s? 2(ß) ? OPT de? ta?????e? t? yj
• ??st?? ?e??? ??a t? yj e????st? ??st??
  e??????µµ?s?? t?? x1 x2 . . . xi ?a? y1 y2 . . .
  yj-1

48
? ???????µ??
Sequence-Alignment(m, n, x1x2...xm, y1y2...yn, ?,
?) for i 0 to m Mi, 0 i? for
j 0 to n M0, j j? for i 1 to m
for j 1 to n Mi, j min(?xi,
yj Mi-1, j-1, ?
Mi-1, j, ? Mi, j-1)
return Mm, n
49
?a??de??µa
?st? d1 ?a? ???e a?a?t?st????a ??st??e? 1 e??
???e a?t?st????a ??st??e? 0.
- e ? ? s t e s ? ?
- 0 1 2 3 4 5 6 7 8 9
e 1
? 2
? 3
? 4
s 5
t 6
a 7
s 8
? 9
? 10
1
2
7
0
3
4
5
8
6
1
2
7
1
3
8
4
5
6
1
2
7
2
3
8
4
5
6
2
3
1
2
6
7
3
4
5
3
4
2
1
5
6
2
3
4
4
5
3
2
4
5
1
2
3
5
4
4
6
3
2
2
5
3
6
5
3
7
4
3
3
4
2
7
6
2
8
5
3
4
4
3
8
7
3
9
6
2
5
5
4
50
??a O?a?? T?µa

• ?st? ?t? e?ste s?µß????? µ?a? eta??e?a?
  ?atas?e??? ??ßd?? ?a????, t??? ?p????? t???
  st???ete se ??? t? ???a. G?a ???e µ?a ap? t??
  ep?µe?e? n4 eßd?µ?de?, ??ete µ?a p??ß?ep?µe??
  pa?a???? si ????? ??ßd?? ?a???? (??a t?? eßd?µ?da
  i), ? ?p??a ?a p??pe? ?a µetafe??e? st?? te????
  t?? p?????sµ? ae??p??????. ? pa?a???? ???e
  eßd?µ?da? µp??e? ?a µetafe??e? µ?s? d??
  d?af??et???? eta??e??? ae??µetaf????, t?? ? ?a?
  ?.
• ? ? ??e??e? r e??? ??a ???e ???? (??a ??st??e?
  r?si ??a ?a µetaf??e? t?? pa?a???? t?? eßd?µ?da?
  i)
• ? ? ??e??e? a?? eßd?µ?da ??a sta?e?? p?s? c
  a?e?a?t?t?? s???????? ß?????. ?µ??, t? s?µß??a??
  µe t?? eta???a ? ?a p??pe? ?a e??a? d????e?a?
  a???ß?? tess???? s??e??µe??? eßd?µ?d??.
• ??a p????aµµa ??a t?? eta??e?a ?a???? e??a? ?
  ep????? µ?a? eta??e?a? ae??µetaf???? ??a ???e µ?a
  ap? t?? n eßd?µ?de? µe t?? pe?????sµ? ?t? ?ta?
  ep??????µe t?? eta??e?a ? ?a p??pe? t? ???e
  s?µß??a?? ?a af??? 4 eßd?µ?de?. ?at? a?t?? t??
  t??p?, p???????ta? t? p?s? c st?? eta??e?a ?
  µp????µe ??a 4 eßd?µ?de? ?a st?????µe t??
  pa?a???? ????? ??p??? pe?????sµ? st? ß????. ??
  ??st?? t?? p?????µµat?? e??a? ?s? µe t? s???????
  p?s? p??? ?a? t?? d?? eta??e?e? ? ?a? ? ??a t??
  µetaf??? t?? ?a???? ?a? t?? n eßd?µ?de?.
• ?a??de??µa ?st? r1, c10 ?a? ? a???????a
  pa?a????? e??a? ? 11, 9, 9, 12, 12, 12, 12, 9, 9,
  11. St? ß??t?st? p????aµµa ?a ep????ate t?? ? ??a
  t?? t?e?? p??te? eßd?µ?de?, t?? ? ??a t??
  ep?µe?e? t?sse??? ?a? t?? ? ??a t?? te?e?ta?e?
  t?e?? µe s??????? ??st?? 98.
• (?) ??ste ??s? ???aµ???? ?????aµµat?sµ??
  (a?ad??µ??? s??s?) ? ?p??a µe e?s?d? t?? pa?a????
  ???e eßd?µ?da? s1,s2,,sn ?a ep?st??fe? t?
  e????st? ??st?? ??a t? µetaf??? t??. ???a e??a? ?
  ??????? p???p????t?ta t?? a??????µ?? sa?
• (?) ?a efa?µ?sete t? s???e???µ??? a??????µ? ??a
  t?? e?s?d? t?? pa?ade??µat?? p?? ??e? d??e?
  pa?ap??? d????ta? t?? p??a?a d??aµ????
  p????aµµat?sµ?? ??a ???e ß?µa.