k?????? (Robertson-Seymour ???????)

1
????????????????
?? ??

• ???? ????????????

?????????? 2012 ? 7? 13?
2
????????
3
????

• ?????????
• ??????????
• 2??????
• k?????? (Robertson-Seymour ???????)
• ??????
• ????????
• ??
• ?????(????????)

4
???????? (Min-Sum)
Input ??? (s1, t1),, (sk, tk), ?? e ???
l(e) Find ???? P1,, Pk (Pi si ? ti )
s.t. ?????????

• ????????
• ?????????????????

5
???????? (Min-Sum)
Input ??? (s1, t1),, (sk, tk), ?? e ???
l(e) Find ???? P1,, Pk (Pi si ? ti )
s.t. ?????????
s1

• ????????
• ?????????????????

t1
s2
t2
s3
t3
???????

• NP???????????? NP??
• ??????????????? ?????
• ??????,s1,..., sk ?????,t1,..., tk ?????

????? (Colin de Verdière Schrijver, 2008)
6
????????????
????? ?????
k ?? NP-?? ????? (?????) ????? (????) ????? ???? (?????)
k ?? NP-?? NP-??
k ????
7
Min-Sum ??????????
??????

• ??? ???????????????
• ???? ??????????????????

??????????
???????

• ??? ?????????????
• ???? ?????????? ????

8
Min-Sum ??????????
??????

• ??? ???????????????
• ???? ??????????????????

??????????
???????

• ??? ?????????????
• ???? ?????????? ????

???????????

• ???????????????
• ??????,s1,..., sk ?????,t1,..., tk ?????

(Colin de Verdière Schrijver, 2008)
?? ??????? ??????????
9
Min-Sum ??????????
??????
???????

• ??? ???????????????
• ???? ??????????????????

• ?????
• ??????

???
??????????
???????

• ??? ?????????????
• ???? ?????????? ????

???????????
???????????

• ???????????????
• ??????,s1,..., sk ?????,t1,..., tk ?????

(Colin de Verdière Schrijver, 2008)
?? ??????? ??????????
10
Min-Sum ????????????
(K. and Sommer, 2009)
??????????
???????? ?????? ??2???? ?????k2 ? Min-Sum
??????? ????? ????
s1
s1
s2
t1
t1
t2
t2
s2
???
s1
t1
s2
t2
???????
Colin de Verdière Schrijver
11
Min-Sum ??????????

• ?????
• ??????

???
???????????

• ??????
• ??????

???????? ?????????
??????? ?????????
J. Erickson and A. Nayyeri Shortest
non-crossing walks in the plane, SODA 2011
12
Shortest non-crossing walks
Input ??? (s1, t1),, (sk, tk), ?? e ???
l(e) Find ??????? walk P1,, Pk (Pi si ?
ti ) s.t. walk ??????????
t1
s1
t2
s1
t2
s1
t1
t2
t1
s2
s2
s2
OK
NG
NG
?
s2
t1
s1
t2
13
Shortest non-crossing walks
Input ??? (s1, t1),, (sk, tk), ?? e ???
l(e) Find ??????? walk P1,, Pk (Pi si ?
ti ) s.t. walk ??????????
t1
s1
t2
s1
t2
s1
t1
t2
t1
s2
s2
s2
OK
NG
NG
?
s2
t1
s1
t2
14
?? (Erickson-Nayyeri, 2009)
?????? h ???????????, 2 n log k ?????????
O(h2)
?
obstacle
s2
t1
s1
h 2
t2
15
obstacle
?? (Erickson-Nayyeri, 2009)
?????? h ???????????, 2 n log k ?????????
O(h2)
????

• h 1 ??? ???????????

• h 2 ??? O(n log k) ??

(Takahashi-Suzuki-Nishizeki, 1992)

• h ?? ??? NP-hard

(Bastert-Fekete, 1998)
?
obstacle
s2
t1
s1
h 2
t2
16
obstacle
?? (Erickson-Nayyeri, 2009)
?????? h ???????????, 2 n log k ?????????
O(h2)
?????
?????? Obstacle ??? ??????????????
?
obstacle
s2
t1
s1
h 2
t2
17
obstacle
?? (Erickson-Nayyeri, 2009)
?????? h ???????????, 2 n log k ?????????
O(h2)
??
????? walk ???,???????????
?????????
ST-walks
s2
s2
s1
s1
t3
t3
t2
t2
t1
t1
s3
s3
18
??
obstacle

• ? i ????, si ? ti ???????????,
• Shortest non-crossing walks ???????
• ????????.
• (???????????????)

19
??????????????
??????? ???
20
??????????????
??????? ???
?? homotopy class ????????
??
2?? walk a, ß ??? homotopy class
????? h 0,10,1 ? (?? - obstacle)

• h(0, ? ) a, h(1, ? ) ß
• h(? , 0) a(0) ß(0), h(? , 1) a(1) ß(1)

s.t.
a
a
obstacle
ß
ß
??
??
21
??????
??
Shortest non-crossing walks ????????
? walk ?,?? homotopy class ?????
??
a ??? homotopy class ??????????, ????????????.
s3
s3
t3
t3
P3
P3
P2
P2
t2
s2
t2
s2
a
s1
t1
a
s1
t1
P1
22
??????
??
Shortest non-crossing walks ????????
? walk ?,?? homotopy class ?????
??
a ??? homotopy class ??????????, ????????????.
?? homotopy class ?????????
?????????????
s3
s3
t3
t3
P3
P3
P2
P2
t2
s2
t2
s2
a
s1
t1
a
s1
t1
P1
23
??????

• STEP 1
• ???? homotopy class ???????,
• ??? ST-walks O ???

STEP 2 O ??????????,???????????
STEP 3 ???? ST-walks ???????????
24
??????
(1) ??? 2 ????
O(h2)

• STEP 1
• ???? homotopy class ???????,
• ??? ST-walks O ???

(2) ????????
STEP 2 O ??????????,???????????
STEP 3 ???? ST-walks ???????????
25
??

• s ??2???????

O ?1, , ?k ??? ST-walks
? ?i ? s ??? 22h-2 ?????????
26
??

• s ??2???????

O ?1, , ?k ??? ST-walks
? ?i ? s ??? 22h-2 ?????????
s2
t3
?????????...
s1
???? s1, , sh-1 ???? h ?? obstacle faces ???
s1
t2
s2
t1
s3
27
??

• s ??2???????

O ?1, , ?k ??? ST-walks
? ?i ? s ??? 22h-2 ?????????
s2
t3
?????????...
s1
???? s1, , sh-1 ???? h ?? obstacle faces ???
s1
t2
s2
t1
s3
s2
si ????????,si , si, ti ? ?????????? ? ???
t3
s1
s2
s1
t2
t1
? ??? 2 ??
O(h2)
s2
s1
s3
?
28
??

• s ??2???????

O ?1, , ?k ??? ST-walks
? ?i ? s ??? 22h-2 ?????????
s2
t3
?????????...
s1
O(h)
???? s1, , sh-1 ???? h ?? obstacle faces ???
2O(h) (2O(h))
s1
t2
s2
t1
??????
?????
s3
si ????????,si , si, ti ? ?????????? ? ???
? ??? 2 ??
O(h2)
?
29
??

• ??? n ??????????????,
• ?????????????? 2O(n) ??
• ????????.

30
walks ???
? ?? walk ??????
s2
t3
s1
homotopy class ?????? ? walk ??????????
s1
t2
s2
t1
s3
?
31
walks ???
? ?? walk ??????
s2
t3
s1
homotopy class ?????? ? walk ??????????
s1
t2
s2
t1
s3
???,s3?s1?s2?t3 ??, ??? 3 ???????????
???? 2O(h) n
?
32
??????(??)
(1) ??? 2 ????
O(h2)

• STEP 1
• ???? homotopy class ???????,
• ??? ST-walks O ???

(2) ????????
STEP 2 O ??????????,???????????
STEP 3 ???? ST-walks ???????????
33
??

• s ??2???????

O ?1, , ?k ??? ST-walks
? ?i ? s ??? 22h-2 ?????????
???????????.
?????,?2k ?????????? ???
??
X(s, O) s ????? walk ?????? index ?
X(s, O) ????? even
?? index ???????
34
??1
X(s, O) ???? even ?????????

• s ??2???????

O ?1, , ?k ??? ST-walks
X(s, O) s ????? walk ?????? index ?
35
??1
X(s, O) ???? even ?????????
??
even ????????,walks ?????? ???? ST-walks
?????????.
????????? 1 ? 4 5 ? 8
9 ? 12
8
3
1
3
3
1
1
5
s
4
2
2
6
4
4
2
7

• s ??2???????

O ?1, , ?k ??? ST-walks
X(s, O) s ????? walk ?????? index ?
36
??1
X(s, O) ???? even ?????????
??
even ????????,walks ?????? ???? ST-walks
?????????.
????????? 1 ? 4 5 ? 8
9 ? 12
8
3
1
3
3
1
1
5
s
4
2
2
6
4
4
2
7
8
3
1
3
3
1
1
5
s
?? ???????,?????????????
4
2
2
6
4
4
2
7
37
??2
?? 2k ??? X(s, O) ?,??? even ???????
?? ??2 ?????

• s ??2???????

O ?1, , ?k ??? ST-walks
X(s, O) s ????? walk ?????? index ?
X(s, O) ????? even
?? index ???????
38
??

• s ??2???????

O ?1, , ?k ??? ST-walks
? ?i ? s ??? 22h-2 ?????????
?????,?2k ?????????? ???
??1
X(s, O) ???? even ?????????
??2
?? 2k ??? X(s, O) ?,??? even ???????
??1,??2??,????.
39
??????(??)
(1) ??? 2 ????
O(h2)

• STEP 1
• ???? homotopy class ???????,
• ??? ST-walks O ???

(2) ????????
STEP 2 O ??????????,???????????
STEP 3 ???? ST-walks ???????????
40
??????(??)
?? (Erickson-Nayyeri, 2009)
?????? h ???????????, 2 n log k ?????????
(1) ??? 2 ????
O(h2)
O(h2)

• STEP 1
• ???? homotopy class ???????,
• ??? ST-walks O ???

(2) ????????
STEP 2 O ??????????,???????????
STEP 3 ???? ST-walks ???????????
41
???????????

• ????????
• ?????????????????
• ??
• ?????
• ?????? ???????????????

42
???????????? (Min-Max)
Input ??? (s1, t1),, (sk, tk), ?? e ???
l(e) Find ???? P1,, Pk (Pi si ? ti )
s.t. ???????????
?????? NP?? (ex. k2, ?????, s1s2 ,
t1t2 ??NP??)

• ???
• k2, ?????
• ?? ?3 ?? NP??
• ?? ?2 ?? ?????

?? ??????????
??????????