ΚΕΦΑΛΑΙΟ 5 Το Πρόβλημα της Συνάντησης Πολλών Πρακτόρων

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1 k 2 n k n k n n k n k k S S

2 k 2 n O(n) O(k n) O(kn) O( n) ) O(k n) O(n) O( n) O(n) O( k) O(n k) O( k) O( n n n k n k > 2 Ω( n + k) k n n k k n n n/2 S = d 1,..., d k m > 1 j 1 m, j k k S

3 S O(k n) k n k k k d 1,..., d k S d 1,..., d k d 1,..., d k d k,..., d 1 MA i MA j MA i MA j MA i MA i MA j MA i MA j O(k n) O(n)

4 O( n) O( n) S O( n) c = 1 active = 1 inactive = 0 inactive c d c c = 1 c = 2 d c c + 1 k d i d i > d c active = 0 d i d i > d c inactive = k 1 c = k 1 inactive < k 1 c = c + 1 inactive = 0 5 O( n) O(kn) O(k n) ( n) O(k n) p 1,..., p r r r i=1 p i > n

5 O(k n) 1 p r r i=1 p i > n p i = p 1 = 2 α = k p i α d 1,..., d α p i d 1,..., d α p i d α,..., d 1 p i (d 1,..., d α/a ) a p i (d 1,..., d α/a ) 0 p i < p r p i = p i+1 α = a 6 p 2 < p r p i < p p i α i d 1,..., d αi (d 1,..., d αi /a) a p i a α i d 1,..., d αi /a p a t k σ 1, σ 2,..., σ t p 1, p 2,..., p O ) O(k n) ( n n n p i p r p i

6 9 p = p r a r k σ r i i = 1,..., a r p p p r σ r i i = 1,..., a r r i=1 p i r i=1 p i > n a r k σ r i i i = 1,..., a r n a r σ r i i a r n (k, n) = g > 1 S 9 S S p i p r 7 p i = p r S O(k n) r r r i=1 p i > n r n O(rn) r ϵ O( n ) O ( n n n n ) n k n (k, n) = 1 k k n k S i i

7 k (k, n) = 1 k k active = 1 count = 0 d 1, d 2, d 3 count count = k 1 d 2 > d 1 d 2 d 3 active = 0 3 (k, n) = 1 k k O( n) O(n) k (k, n) = 1 k k O( k) O(n) k (k, n) = 1 k k O( k) O(n) (k, n) = 1 k k k (k, n) = 1 k k O( k) O(n k)

8 k (k, n) = 1 k k k 1 k 0 k m 1, m 2, m 3 m 1 = k 1 m 2 > m 1 m 2 m 3 active = 0 5 n

9 k (k, n) = 1 k k O(n k) k d 1, d 2, d 3 k count count = k 1 d 2 > d 1 k d 2 d 3 k active = 0 4 R t t > t R t > t R R R v v

10 A B E D C A ((2, 3, 3, 1, 3) (5)) B ((3, 3, 1, 3, 2) (3)) C ((3, 1, 3, 2, 3) (0)) A {((2, 3, 3, 1, 3), (5)), ((3, 1, 3, 3, 2), (7))} a b a, b > 1 ((a 1,..., a r ) (b 1,..., b s )) a i b j u 1 u 1, u 2, u 3,..., u r u 1 a i i < r u i u i+1 a r u r u 1 u 1, u 2, u 3,..., u r u v1,..., u vs b i u 1 u vi u 1

11 A B E D C A ((2, 3, 3, 1, 3) (5, 9)) B ((3, 3, 1, 3, 2) (3, 7)) C ((3, 1, 3, 2, 3) (0, 4)) A {((2, 3, 3, 1, 3), (5, 9)), ((3, 1, 3, 3, 2), (3, 7))} F A B C E D A (2, 3, 1, 2, 3, 1) A D {(2, 3, 1, 2, 3, 1), (1, 3, 2, 1, 3, 2)} B E {(3, 1, 2, 3, 1, 2), (2, 1, 3, 2, 1, 3)} C F {(1, 2, 3, 1, 2, 3), (3, 2, 1, 3, 2, 1)} (a 1,..., a r ) C = (a 1,..., a r ) p C S

12 A E B D C A (2, 2, 4, 2, 2) A {(2, 2, 4, 2, 2), (2, 2, 4, 2, 2)} B C (2, 4, 2, 2, 2) (4, 2, 2, 2, 2) B E {(2, 4, 2, 2, 2), (2, 2, 2, 4, 2)} C D {(4, 2, 2, 2, 2), (2, 2, 2, 2, 4)} A B C I D H E G F A (2, 2, 1, 2, 2, 1, 2, 2, 1) 3 A, C, D, F, G, I {(2, 2, 1, 2, 2, 1, 2, 2, 1), (1, 2, 2, 1, 2, 2, 1, 2, 2)} B, E, H {(2, 1, 2, 2, 1, 2, 2, 1, 2), (2, 1, 2, 2, 1, 2, 2, 1, 2)} S S

13 R ((a 1,..., a r ), (b 1,..., b s )) R u 1 R {((a 1,..., a r ), (b 1,..., b s )), ((a r, a r 1,..., a 1 ), (n b s,..., n b 1 ))} R R {((a 1,..., a r ), (0, b 2,..., b s )), ((a r, a r 1,..., a 1 ), (0, n b s,..., n b 2 ))} R {((a 1,..., a r ), (b 1,..., b s )), ((a r, a r 1,..., a 1 ), (n b s,..., n b 1 ))} {(a 1,..., a r ), (a r, a r 1,..., a 1 )} 9 1,..., 9 1, 2, 4 R 1 {(1, 2, 6), (6, 2, 1)}

14 A B E D C A (3, 3, 3, 2, 1) A, B, C, D, E {(3, 3, 3, 2, 1), (1, 2, 3, 3, 3)} {(3, 3, 2, 1, 3), (3, 1, 2, 3, 3)} {(3, 2, 1, 3, 3), (3, 3, 1, 2, 3)} {(2, 1, 3, 3, 3), (3, 3, 3, 1, 2)} {(1, 3, 3, 3, 2), (2, 3, 3, 3, 1)}

15 C p C C C

16 a, b {(a 1,..., a r ), (a r, a r 1,..., a 1 )} {(b 1,..., b r ), (b r, b r 1,..., b 1 )} a a + = (a 1,..., a r ) b b = (b r, b r 1,..., b 1 ) a w a + u w+1 b w b u 2 u 1 u 1 a a b a 1 = b r = a w = b r w+1 a r = b 1 j j a + a 1, a w, a r j b b r, b r w+1, b 1 (a 1,..., a w,..., a r ) = (b r,..., b r w+1,..., b 1 ) C C C = (a 1,..., a r ) (a 1,..., a p ) (a 1,..., a p,..., a r ) = (a p+1,..., a r, a 1,..., a p ) (a 1,..., a p ) a + = (a 1,..., a p,..., a r ) a a u 1 (a p+1,..., a r, a 1,..., a p ) b b u p+1 a, b C

17 C a, b a a + = (a 1,..., a r ) a = (a r, a r 1,..., a 1 ) b b + = (b 1,..., b r ) b = (b r, b r 1,..., b 1 ) (a 1,..., a r ) = (b r, b r 1,..., b 1 ) (a 1,..., a r ) = (b 1,..., b r ) (a 1,..., a r ) = (b r, b r 1,..., b 1 ) a w a + u w+1 b a w = b r w b u 2 u 1 u 1 a b r w+1 = a 1 a s w+1 a 2 + a 1 a w a m w+r a 2 + a w a r a 1 = b r = a w = b r w+1 a s, a m s m b a r = b 1 j j a + a 1, a s, a w, a m, a r j b b r, b r s+1, b r w+1, b r m+1, b 1 (a 1,..., a s,..., a w,..., a m,..., a r ) = (b r,..., b r s+1,..., b r w+1,..., b r m+1,..., b 1 ). a s u s+1 a m u m+1 (a 1,..., a r ) = (b 1,..., b r ) a b c c + = a + b a w a + u w+1 b a w+1 = b 1 = a 1 a w+2 = b 2 = a 2 a w+i = b i = a i 1 i w a mw+i = b i = a i mw + i < r (a 1,..., a w ) r w r = mw + x 1 x < w a r = a mw+x = b x = a x a r+w x = a w x a r+w x = a (m+1)w = b mw a r+w x+1 = b mw+1 = b 1 = a 1 a r+w x+i = b mw+i = b i = a i a w x a w a w x+1

18 a w+1 = b 1 c u w x+1 a b a + c + a + S 1 S 2 a, b S 1 a + = (a 1,..., a r ) a b = (b r, b r 1,..., b 1 ) b a + = b a S 2 (a 1,..., a r ) = (a r, a r 1,..., a 1 ) (a 1,..., a r ) = (b 1,..., b r ) a, b S 2 c b S 2 (b 1,..., b r ) = (c r, c r 1,..., c 1 ) c = (c r, c r 1,..., c 1 ) c (a 1,..., a r ) = (c 1,..., c r ) S S S n n S

19 1 1 k > 1 k k = 2 k < k k C R R R k 1 k 1 k

20 t > 1 r t t r r + 1 t t t > 1 r R R R R R R r r + 1 r + 1 R R R R Single-Multiplicity-Gathering

21 v v v v v t Single-Multiplicity-Gathering

22 A, B A, B A, B A, B A, B M N Max M M N N M M 2 a N N 2 b M M 2 M N Max + 1 M M 2 a 1 M N M N Max + 1 a 1 M M 2 b N N 2 M N a 1 M M 2 b N N 2 M M 2 Single-Multiplicity-Gathering

23 Max a i M Max N M Max Max j 1; M 1 M; N 1 N; M 0 N; N 0 M; j j + 1; M j M j 1 M j 2 N j N j 1 N j 2 N N j M M j N N j ; M M j ; N N M M N N 2 M M 2 Odd-Gathering Rigid-Gathering

24 Single-Multiplicity-Gathering Rigid-Gathering (1) (2) Single-Multiplicity-Gathering Rigid-Gathering C = (a, a + 1, a 1, a + 1, a 1, a + 1, a) 7 a > 1 C

25 A A G a a B G a-1 a+1 B a+1 a+1 a+1 a+1 F E a-1 a+1 a-1 D C F E a-1 a+1 a-1 D C (i) (ii) a C = (a + 1, a + 1, a 1, a + 1, a 1, a + 1, a 1) C a + 1 C Rigid-Gathering C C C C C = (a, b 1,..., b s 1, b s, b s 1,..., b 1, a) C (a + 1, b 1,..., b s 1, b s, b s 1,..., b 1, a 1) C d p d 1 = a + 1 d p = a 1 C d p = d 1

26 C = (a 1,..., a r ) C {a 1,..., a r } a i C a i (a 1,..., a r ) Cf Cf Cf S C D S x C D y x Cf y y C, D x C, D x Cf Cf Cf Cf 4 C D 2 A E A B Cf Cf Cf C z f C C C C z = f 1 z = f + 1 C z + 1 z 1 z + 1 z 1 C C C f C f = z 1 f = z + 1 z 1 z + 1 C C C

27 C C C C C b(c) C C C C C b(c ) < b(c) z C f z = f 1 z = f + 1 f = z + 1 C z + 1 z 1 1 z 2 f C z 1 z 1 C C z C C z 1 z + 1 C C 2 C f = z 1 (C 1, C 2,... ) C i+1 C i i b(c i ) = 0 C b(c) = 0 C C C C b(c ) < b(c) b(c) = 0

28 C A C 1 C A C 1 Single-Multiplicity-Gathering C 1 C 1 C 2 C 1 C 2 C 1, C 2,... C i b(c i ) = 0 C C i C Single-Multiplicity-Gathering

29 n k k n p(n) n O( n) p(n) = Ω( n) n n Ω(n) 4 ) k = 2l k = 2l + 1 k > 2 k 4 3 Θ( n n

31 n > 1 1 n r

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