72 2 2 2 2.24 2 s t, 2,..., 0 s t a, b,..., k t s, 2,..., 0 a, b,..., k s t 0 ts 0 ( 2.25) 2.24 2 ½ ¾ ½¼ x j = x 2c = x 3e = x 4s = x 5g = x 6i = x 7d = x 8h = x 9f = x 0k = x ta = x tb = x ts = 9 2.26 9 a b 7 8 2.27 (cover) M (jmj) C (jcj) jcj jmj 2 M C jc j = jm j
2.6. 73 2.25 2 Ø ½ ¾ ½¼ 2.26 Ø ½ ¾ ½¼ 2.27 2 3 4 5 6 7 8 7/2 7/2 7/6 7/9 7/2 7/2 7/5 7/20 7/3 7/9 7/3 7/ 7/9 7/4 7/9 7/23 2 8 7 3 8 3 5 4
74 2 2.28 () 2 () - 2 2 3 4 5 6 2 3 4 5 6 7-2 3 4-3 8-8 9 - - - - 4 5 6 7 5 6 7 8 0 7 8 8-8 [-] i j i j 2 2 i j i j 2 8 9-8 0 i(i=,..., 8) i (i =,..., 8) 09
2.7. 75 min x 09 s.t. x 8 + x 9 = x 25 + x 26 + x 27 + x 28 + x 29 = x 38 + x 39 = x 45 + x 46 + x 47 + x 49 + x 49 = x 59 = x 67 + x 68 + x 69 = x 78 + x 79 = x 89 = x 0 = x 02 = x 03 = x 04 = x 25 + x 45 + x 05 = x 26 + x 46 + x 06 = x 47 + x 67 + x 07 = x 8 + x 28 + x 38 + x 68 + x 78 + x 08 = 8 i= x i9 + x 09 = 8 8 j= x 0j + x 09 = 8 x ij 0 ij x 8 = 0 x 9 = x 25 = x 26 = 0 x 27 = 0 x 28 = 0 x 29 = 0 x 38 = 0 x 39 = x 45 = 0 x 46 = x 47 = 0 x 49 = 0 x 49 = 0 x 59 = x 67 = x 68 = 0 x 69 = 0 x 78 = x 79 = 0 x 89 = x 0 = x 02 = x 03 = x 04 = x 05 = 0 x 06 = 0 x 07 = 0 x 08 = 0 x 09 = 4 x ij = 2 54 66 77 8 4 6 4 6 7 (3) 4 7 n 2 M M n M 2.7
76 2 2.29 3 <3> <2> <5> <2> <4> 5 <0> s <8> 2 4 <7> <> 6 <9> t 2.7. 2.29 s t t s ts ( 2.30 ) x ts () 2.30 3 <3> <2> <5> <2> <4> 5 <0> s <8> 2 4 <7> <> 6 <9> t []
2.7. 77 max x ts s.t. x ts + x s + x s2 = 0 x s + x 3 + x 4 = 0 x s2 + x 26 = 0 x 3 + x 35 = 0 x 4 + x 45 + x 46 = 0 x 35 x 45 + x 5t = 0 x 46 x 26 + x 6t = 0 x 5t x 6t + x ts = 0 x s 2, x s2 8 x 3 5, x 4 2 x 26 7, x 35 3 x 45 4, x 46 x 5t 0, x 6t 9 x ij 0, ij E (2.9) x s = 5 x s2 = 7 x 3 = 3 x 4 = 2 x 26 = 7 x 35 = 3 x 45 = 2 x 46 = 0 x 5t = 5 x 6t = 7 x ts = 2 2.7.2 (2.9) (2.9) min 2ξ s + 8ξ s2 + 5ξ 3 + 2ξ 4 + 7ξ 26 + 3ξ 35 +4ξ 45 + ξ 46 + 0ξ 56 + 0ξ 5t + 9ξ 6t s.t. λ s + λ t λ s λ + ξ s 0 λ s λ 2 + ξ s2 0 λ λ 3 + ξ 3 0 λ λ 4 + ξ 4 0 λ 2 λ 6 + ξ 26 0 λ 3 λ 5 + ξ 35 0 λ 4 λ 5 + ξ 45 0 λ 4 λ 6 + ξ 46 0 λ 5 λ t + ξ 5t 0 λ 6 λ t + ξ 6t 0 ξ ij 0, ij E (2.0)
78 2 (2.0) λ i (i = s,, 2,..., 6, t) λ i (i = s,, 2,..., 6, t) ξ ij (ij E) δ λ i = λ i + δ, i = s,, 2,..., 6, t λ i (i = s,, 2,..., 6, t) ξ ij (ij E) λ s = 0 (2.0) (λ s, λ,..., λ t, ξ s,..., ξ 6t ) λ λ 4 + ξ 4 > 0 (ξ ij ) (2.0) min 2ξ s + 8ξ s2 + 5ξ 3 + 2ξ 4 + 7ξ 26 + 3ξ 35 +4ξ 45 + ξ 46 + 0ξ 56 + 0ξ 5t + 9ξ 6t s.t. λ s + λ t = λ s λ + ξ s = 0 λ s λ 2 + ξ s2 = 0 λ λ 3 + ξ 3 = 0 λ λ 4 + ξ 4 = 0 λ 2 λ 6 + ξ 26 = 0 λ 3 λ 5 + ξ 35 = 0 λ 4 λ 5 + ξ 45 = 0 λ 4 λ 6 + ξ 46 = 0 λ 5 λ t + ξ 5t = 0 λ 6 λ t + ξ 6t = 0 λ s = 0 ξ ij 0, ij E (2.) s t : s 3 5 t 2 : s 4 5 t 3 : s 4 6 t 4 : s 2 6 t P = {s, 3, 35, 5t} P 2 = {s, 4, 45, 5t} P 3 = {s, 4, 46, 6t} P 4 = {s2, 26, 6t}
2.8. (Shortest Path Problem) 79 λ s λ + ξ s = 0 λ λ 3 + ξ 3 = 0 λ 3 λ 5 + ξ 35 = 0 λ 5 λ t + ξ 5t = 0 λ s + λ s = 0 λ t = ij P ξ ij λ t = 0 ij P ξ ij = min 2ξ s + 8ξ s2 + 5ξ 3 + 2ξ 4 + 7ξ 26 + 3ξ 35 s.t. +4ξ 45 + ξ 46 + 0ξ 56 + 0ξ 5t + 9ξ 6t ξ ij =, k =, 2, 3, 4 ij P k ξ ij 0, ij E (2.2) () (2.0) : 2 λ s = λ = λ 2 = λ 3 = λ 4 = 0 λ 5 = 0 λ 6 = 0 λ t = 0 ξ 3 = 0 ξ 4 = ξ 26 = ξ 35 = ξ 45 = 0 ξ 46 = 0 ξ 56 = 0 ξ 5t = 0 ξ 6t = 0 ξ s = 0 ξ s2 = 0 2.2 7 3 5 64543 ( ) 2.8 (Shortest Path Problem)
80 2 2 A B 2.3? 2.3 A B ¾ ½ ½ ½¼ ¾ ¾ ¾ A B 2.5 ( ) LP (Dijkstra) V = {s,, 2,..., n} E ij E d ij > 0 s j [] :
2.8. (Shortest Path Problem) 8 v s = 0 v j =, M = {s} N = j V, j s 2 M i v i = min j M {v j} 3 M N N N {i} M M/{i} 4 v j j V/M ij E v i + d ij < v j 5 v j v i + d ij M M {j} M = 2 d j s j 2.3 2.32, (dynamic programming, DP) 2 V =, 2,..., n, t t ( 2.3 6) i t n k J k (i) k = 0,,..., n J k (i) 3 { J k (i) = min j=,...,n [d ij + J k+ (j)], k = 0,,..., n 2; i =,..., n J n (i) = a it, i =,..., n (2.3) d ij ij d ii = 0 k = n 0 (2.3) () DP 2.3 2 DP 3 (2.3)
82 2 2.32 step v = 0, v j =, j = 2, 3, 4, 5, 6 M = {}, N = { } step 2 min j M {v j } = v = 0 step 3 M = { }, N = {} step 4 2: min{v 0 + d 2, v 2 } = min{0 +, } = 3: min{v 0 + d 3, v 3 } = min{0 + 7, } = 7 M = {2, 3} step 5 M step 2 2 step 2 min j M {v j } = min{v 2, v 3 } = min{, 7} = 2 step 3 M = {3}, N = {, 2} step 4 3: min{v 2 + d 23, v 3 } = min{ + 3, 7} = 4 4: min{v 2 + d 24, v 4 } = min{ + 6, } = 7 5: min{v 2 + d 25, v 5 } = min{ + 0, } = M = {3, 4, 5} step 5 M step 2 3 step 2 min j M {v j } = min{v 3, v 4, v 5 } = min{4, 7, }=4 3 step 3 M = {4, 5}, N = {, 2, 3} step 4 4: min{v 3 + d 34, v 4 } = min{4 + 2, 7} = 6 5: min{v 3 + d 35, v 5 } = min{4 + 8, } = M = {4, 5} step 5 M step 2 4 step 2 min j M {v j } = min{v 4, v 5 } = min{6, }=6 4 step 3 M = {5}, N = {, 2, 3, 4} step 4 5: min{v 4 + d 45, v 5 } = min{6 + 2, } = 8 6: min{v 4 + d 46, v 6 } = min{6 + 5, } = M = {5} step 5 M step 2 5 step 2 min j M {v j } = min{v 5, v 6 } = min{8, }=8 5 step 3 M = {6}, N = {, 2, 3, 4, 5} step 4 6: min{v 5 + d 56, v 6 } = min{8 + 2, } = 0 M = {6} step 5 M step 2 6 step 2 min j M {v j } = min{v 6 } = 0 6 step 3 M = { }, N = {, 2, 3, 4, 5, 6} step 4 M = { } step 5 M = v = 0, v 2 =, v 3 = 4, v 4 = 6, v 5 = 8, v 6 = 0 : 2 3 4 5 6
2.8. (Shortest Path Problem) 83 k = 4 ( t ) J 4 (4) = 5 J 4 (5) = 2 k = 3 (2 t ) J 3 (5) = min {d 5j + J 4 (j)} = d 55 + J 4 (5) = 0 + 2 = 2 j=,...,n J 3 (4) = min{d 44 + J 4 (4), d 45 + J 4 (5)} = {0 + 5, 2 + 2} = 4 J 3 (3) = min{d 34 + J 4 (4), d 35 + J 4 (5)} = {2 + 5, 8 + 2} = 7 J 3 (2) = min{d 24 + J 4 (4), d 25 + J 4 (5)} = {6 + 5, 0 + 2} = k = 2 (3 t ) J 2 (5) = min{d 55 + J 3 (5)} = 0 + 2 = 2 J 2 (4) = min{d 44 + J 3 (4), d 45 + J 3 (5)} = {0 + 4, 2 + 2} = 4 J 2 (3) = min{d 33 + J 3 (3), d 34 + J 3 (4), d 35 + J 3 (5)} = {0 + 7, 2 + 4, 8 + 2} = 6 J 2 (2) = min{d 22 + J 3 (2), d 23 + J 3 (3), d 24 + J 3 (4), d 25 + J 3 (5)} = {0 +, 3 + 7, 6 + 4, 0 + 2} = 0 J 2 () = min{d 2 + J 3 (2), d 3 + J 3 (3)} = min{ +, 7 + 7} = 2 k = (4 t ) J (5) = min{d 55 + J 2 (5)} = 0 + 2 = 2 J (4) = min{d 44 + J 2 (4), d 45 + J 2 (5)} = {0 + 4, 2 + 2} = 4 J (3) = min{d 33 + J 2 (3), d 34 + J 2 (4), d 35 + J 2 (5)} = {0 + 6, 2 + 4, 8 + 2} = 6 J (2) = min{d 22 + J 2 (2), d 23 + J 2 (3), d 24 + J 2 (4), d 25 + J 2 (5)} = {0 + 0, 3 + 6, 6 + 4, 0 + 2} = 9 J () = min{d + J 2 (), d 2 + J 2 (2), d 3 + J 2 (3)} = min{0 + 2, + 0, 7 + 6} = k = 0 (5 t ) J 0 (5) = min{d 55 + J (5)} = 0 + 2 = 2 J 0 (4) = min{d 44 + J (4), d 45 + J (5)} = {0 + 4, 2 + 2} = 4 J 0 (3) = min{d 33 + J (3), d 34 + J (4), d 35 + J (5)} = {0 + 6, 2 + 4, 8 + 2} = 6 J 0 (2) = min{d 22 + J (2), d 23 + J (3), d 24 + J (4), d 25 + J (5)} = {0 + 9, 3 + 6, 6 + 4, 0 + 2} = 9 J 0 () = min{d + J (), d 2 + J (2), d 3 + J (3)} = min{0 +, + 9, 7 + 6} = 0 2.33 J k (i) 2 k = 0 i = 2 2( 2) 3 4 5 6 6 (0)
84 2 2.33 DP i 6 2 2 2 2 2 5 4 4 4 4 5 4 6 6 6 7 3 9 9 0 0 2 0 0 2 3 4 k 2.3 4 200 8 00 2 80 4 8 2 3 8 4 2 0.5 ( ) 5 2300 [] 2.34 0 4 = 2300 + 4 0.5 200 0 8 = 2300 + 8 0.5 (200 + 00) 0 2 = 2300 + 2 0.5 (200 + 00 + 80) 4 8 = 2300 + 8 0.5 00 4 2 = 2300 + 2 0.5 (00 + 80) 8 2 = 2300 + 2 0.5 80