pla85900.tsp.eps
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1 ( ) Tel: , Fax: tohru@nls.ics.saitama-u.ac.jp URL: tohru/ p.1/38
2 1. 1 p.2/38
3 (a) H16 (b) p.2/38
4 ? 1. y = f (x) or or 2. p.3/38
5 ? 1. y = f (x) or or 2. p.3/38
6 1.? y = f (x) or y = f (x 1, x 2,..., x n ) or 2. p.3/38
7 1.? y = f (x) or y = f (x 1, x 2,..., x n ) or 2. p.3/38
8 3? p.4/38
9 3? p.4/38
10 3? p.4/38
11 3? p.4/38
12 3 TDR TDR? p.4/38
13 3 TDR TDR TDL 52 TDL? p.4/38
14 3 TDR TDR TDL 52 TDS 35 TDL TDS? p.4/38
15 3 TDR TDR TDL 52 TDS 35 TDL TDS 87? p.4/38
16 3 TDR TDR TDL 52 TDS 35 TDL TDS 87? p.4/38
17 S 1 S!!! S! S 672( )!!! 672? p.5/38
18 S ? p.6/38
19 7 3 ( ) 3? p.7/38
20 ( ) 7 3 ( ) 3? p.7/38
21 TDR?! p.8/38
22 TDR? (Traveling Salesman Problem, TSP)! p.8/38
23 TDR? (Traveling Salesman Problem, TSP)! p.8/38
24 N N ( ) N ( ) G = (V, E) ( ) c : E R 1 ( ) p.9/38
25 N N ( ) p.10/38
26 TSP (att48) p.11/38
27 TSP (att532) p.12/38
28 TSP (usa13509) p.13/38
29 TSP (d15112) p.14/38
30 TSP (sw24978) p.15/38
31 TSP (pla85900) p.16/38
32 TSP 1. Solving Traveling Salesman Problem 2. TSPLIB ( ) 3. TSPBIB moscato/tspbib_home.html p.17/38
33 TSP? TSP TSP 1. ( ) ( ) VLSI ( ) = TSP p.18/38
34 TSP? TSP TSP 1. ( ) ( ) VLSI ( ) = TSP p.18/38
35 TSP? TSP TSP 1. ( ) ( ) VLSI ( ) = TSP p.18/38
36 TSP? TSP TSP 1. ( ) ( ) VLSI ( ) = TSP ( ) p.18/38
37 TSP?? = p.19/38
38 TSP?? = p.19/38
39 TSP?? = p.19/38
40 TSP?? = p.19/38
41 p.20/38
42 TSP?? = = N p.21/38
43 TSP?? = = N p.21/38
44 TSP?? = N = 13 = N p.21/38
45 TSP?? = N = 13 (N 1)!/2 = N p.21/38
46 TSP?? = N = 13 (N 1)!/2 = N p.21/38
47 (vs ) ; km ( CX ) p.22/38
48 (vs ) 80 k=0 2 k = = = ; km ( CX ) p.22/38
49 (vs ) 80 k=0 2 k = = = ; km ( CX ) p.22/38
50 (vs ) 80 k=0 2 k = = = ; = 40 km ( CX ) p.22/38
51 (vs ) 80 k=0 2 k = = = ; = 40 km ( CX ) p.22/38
52 TSP N! N! = 2πN ( N e ) N ( ( )) 12N N N + O 3 N 4 N! N N : N : N N (exponential time algorithm) p.23/38
53 TSP N (N 1)!/ , , , ,814, ,958, ,500, ,113,510, ,589,145, ,837,184, ,461,394,944, ,834,714,048, ,201,185,852,864, ,822,550,204,416, ,420,880,996,869,850,977,271,808,000, ,329,087,585,471,939,285,830,318,428,201,883,487,644,752,720,441,638,912,000,000,000, p.24/38
54 N N 2 N 3 N 5 2 N 3 N N! µ 1µ 100µ 1µ 59µ 3.6 m µ 8µ 3.2m 1 m µ 27µ 24.3m µ 64µ µ 125µ µ 1 m f : femto (10 15 ) p : pico (10 12 ) n : nano (10 9 ) µ : micro (10 6 ) P : Peta (10 15 ), T : Tera (10 12 ) G : Giga (10 9 ) ( ) 150 N = 20 N = 15 N = 12 :, 2000 p.25/38
55 TSP?? = N = 13 (N 1)!/2 = N TSP ( ) p.26/38
56 TSP?? = N = 13 (N 1)!/2 = N TSP N = 13 ( N = 17 ) p.26/38
57 TSP?? = N = 13 (N 1)!/2 = N TSP N = 13 ( N = 17 ) p.26/38
58 TSP?? = N = 13 (N 1)!/2 = N TSP N = 13 ( N = 17 ) p.26/38
59 (algorithm) (Abu Ja far Mohammed ibn Mûsâ al-khowârizmi) cf. (algebra) ( ) (countable) an algorithm for... ing p.27/38
60 (Polynomial time algorithm) N N (exponential time algorithm) N N p.28/38
61 (Polynomial time algorithm) N N (exponential time algorithm) N N p.28/38
62 (Polynomial time algorithm) N N (exponential time algorithm) N N p.28/38
63 (Polynomial time algorithm) N N (exponential time algorithm) N N p.28/38
64 (Polynomial time algorithm) N N (exponential time algorithm) N N p.28/38
65 (Polynomial time algorithm) N N (exponential time algorithm) N N p.28/38
66 (Polynomial time algorithm) N N (exponential time algorithm) N N p.28/38
67 ( ) (polynomial time algorithm) (exponential time algorithm) ( ) : (Polynomial Time Solvable) : (Nondeterministic Polynomial Time Solvable) 1 Time Solvable 2 3 p.29/38
68 ( ) (polynomial time algorithm) (exponential time algorithm) ( ) P : (Polynomial Time Solvable) : (Nondeterministic Polynomial Time Solvable) 1 Time Solvable 2 3 p.29/38
69 ( ) (polynomial time algorithm) (exponential time algorithm) ( ) P : (Polynomial Time Solvable) NP : (Nondeterministic Polynomial Time Solvable) 1 Time Solvable 2 3 p.29/38
70 ( ) (polynomial time algorithm) (exponential time algorithm) ( ) P : (Polynomial Time Solvable) NP : (Nondeterministic Polynomial Time Solvable) 1 Not-Polynomial Time Solvable 2 3 p.29/38
71 ( ) (polynomial time algorithm) (exponential time algorithm) ( ) P : (Polynomial Time Solvable) NP : (Nondeterministic Polynomial Time Solvable) 1 Not-Polynomial Time Solvable 2 NP 3 NP p.29/38
72 1. (deterministic algorithm) 2. (nondeterministic algorithm)! p.30/38
73 p.31/38
74 p.31/38
75 p.31/38
76 p.31/38
77 P? NP? 1. ( P ) 2. = cf. p.32/38
78 P? NP? 1. ( P ) 2. = cf. p.32/38
79 P? NP? 1. ( P ) 2. = (P ) cf. p.32/38
80 P? NP? 1. ( P ) 2. = (P ) cf. p.32/38
81 P? NP? 1. ( P ) 2. = (P ) cf. p.32/38
82 P = NP P NP 1 P = NP P NP ( ) 2 P 3 ( ) (Clay Mathematics Institute) (Millennium Problems) p.33/38
83 P = NP P NP P = NP P NP 1 P NP ( ) 2 P 3 ( ) (Clay Mathematics Institute) (Millennium Problems) p.33/38
84 P = NP P NP P = NP P NP 1 P NP ( ) 2 P NP P 3 ( ) (Clay Mathematics Institute) (Millennium Problems) p.33/38
85 P = NP P NP P = NP P NP 1 P NP ( ) 2 P NP P 3 (+100 ) (Clay Mathematics Institute) (Millennium Problems) p.33/38
86 Millennium Problems In order to celebrate mathematics in the new millennium, the Clay Mathematics Institute of Cambridge, Massachusetts (CMI) has named seven Prize Problems. The Scientific Advisory Board of CMI selected these problems, focusing on important classic questions that have resisted solution over the years. The Board of Directors of CMI designated a 7 million prize fund for the solution to these problems, with 1 million allocated to each. During the Millennium Meeting held on May 24, 2000 at the Collége de France, Timothy Gowers presented a lecture entitled The Importance of Mathematics, aimed for the general public, while John Tate and Michael Atiyah spoke on the problems. The CMI invited specialists to formulate each problem. 1. Birch and Swinnerton-Dyer Conjecture 2. Hodge Conjecture 3. Navier-Stokes Equations 4. P vs NP 5. Poincaré Conjecture 6. Riemann Hypothesis 7. Yang-Mills Theory p.34/38
87 ( ) : sw24978@tsp CPU : 2-opt, 3-opt, λ-opt Lin-Kernighan p.35/38
88 1. (a) 2. (a) (b) (c) (d) P NP 3. (a) (b) p.36/38
89 OS Windows Windows setup UNIX? Makefile? configure? ( ) CG goo excite p.37/38
90 OS Windows Windows setup UNIX? Makefile? configure? ( ) CG goo excite p.37/38
91 OS Windows Windows setup UNIX? Makefile? configure? ( ) CG goo excite p.37/38
92 OS Windows Windows setup UNIX? Makefile? configure? ( ) CG goo excite p.37/38
93 : : 5F 506 : PDF tohru/lectures/ : saidai : ics p.38/38
1. 1 H18 p.2/37
( ) 338 8570 255 Tel : 048 858 3577, Fax : 048 858 3716 Email : tohru@nls.ics.saitama-u.ac.jp URL : http://www.nls.ics.saitama-u.ac.jp/ tohru H18 p.1/37 1. 1 H18 p.2/37 1. 1 2. (a) H17 (b) 50 90 H18 p.2/37
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量子計算モデル 参考資料 今井浩 東京大学情報理工学系研究科コンピュータ科学専攻 JST ERATO-SORST 量子情報システムアーキテクチャ 情報を処理する観点から どうして量子計算 通信を研究するのか? 量子計算によって 古典計算 通信では現実的に解けない問題を 実際に解く 今のコンピュータの高速化に量子力学が障害 その量子力学を肯定的に使って 今解けないものを解けるようにする 古典計算 確率計算の発展としての量子計算
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