FIT2017( 第 16 回情報科学技術フォーラム ) CF-011 Study on Visualization and Optimal Combination in Card Shuffling Hiroyasu Ide Takashi Okuda 1. UNO [1] [2] -[7] 7
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1 CF-011 Study on Visualization and Optimal Combination in Card Shuffling Hiroyasu Ide Takashi Okuda 1. UNO [1] [2] -[7] 7 2 [8] [8] [8] , Graduate School of Information Science and Technology, Aichi Prefectural University, Department of Information Science and Technology, Faculty of Information Science and Tehnology, Aichi Prefectural University 2. n i = {1, 2,, n} i j i = {1, 2,, n} π π : {1, 2,, n} {1, 2,, n} ( ) 1 2 n π = (1) j 1 j 2 j n [9] 3 52 n = 52 u 2.1. Hindu Shuffle u = 10 n = 10 u = 3 ( ) π = (2) [1] 2.2. Riffle Shuffle 2 n = 10 u = 5 ( ) π = (3)
2 2.3. Deal Shuffle 1 d 2 d 1 n = 10 d = 3 ( ) π = (4) {n d } d! m m S 1 Python n P S S i j t i,j 0 t i,j S 1 n t i,j (i = 1) A 5 (j = 5) t 1,5 1 S t i,j t i,j n T S T S = t 1,1 t 1,2 t 1,3... t 1,n t 2,1 t 2,2 t 2,3... t 2,n t 3,1 t 3,2 t 3,3... t 3,n t n,1 t n,2 t n,3... t n,n (5) i j t i,j /S p i,j 0 p i,j 1 p i,j n P S p 1,1 p 1,2 p 1,3... p 1,n P S = T p 2,1 p 2,2 p 2,3... p 2,n S S = p 3,1 p 3,2 p 3,3... p 3,n (6) p n,1 p n,2 p n,3... p n,n S n P S ϵ P S p i,j 1/n S Random p i,j P 1/n P ϵ p i,j 1/n ϵ P S Ω E E = p i,j 1 ϵ (7) n Ω ω E { E = p i,j 1 } n ϵ ω (8) E p(e) lim p(e) = 0 (9) S S S 10 4 (9) 1: 1 t i,j lim p(e) = 0 (10) S 104 ϵ (10) P 10 4 ϵ = (10) 100
3 p i,j P (8) (11) (13) p i,j 3 {1, 0, 1} 1 1/n ϵ p i,j 1 ϵ p i,j 1 n (11) p i,j 2 1/n ±ϵ p i,j 0 ϵ < p i,j 1 n < ϵ (12) p i,j 1/n 3 1/n ϵ p i,j 1 p i,j 1 ϵ (13) n p i,j {1} { } {0} { } { 1} { } p i,j P 10 4 n n 3 (6) P S 3 i j (i, j) = (1, 1) 3 w C C := w n 2 (14) C P 10 4 C = P N(10, 5/3) P (1)-(45) m C w m C (x, y) = (0, 0) m y = x m y = x m m 10 1/n ± ϵ m m = 45 C = 1.000(2704) 5 L C m = Gilbert-Shannon-Reeds [10][11] GSR GSR 2 a b a a/(a + b) b b/(a + b) 2 0 GSR P GSR 3(1) m = 1 2 3(2) m = 2 4 m = m 3(3) C 5 m 1/n ± ϵ 101
4 FIT2017 第 16 回情報科学技術フォーラム (1) m = (156) (2) m = (353) (3) m = (571) (4) m = (650) (5) m = (852) (6) m = (977) (7) m = (1120) (8) m = (1226) (9) m = (1303) (10) m = (1367) (11) m = (1449) (12) m = (1498) (13) m = (1568) (14) m = (1673) (15) m = (1719) (16) m = (1820) (17) m = (1900) (18) m = (1991) (19) m = (2079) (20) m = (2155) (21) m = (2252) (22) m = (2320) (23) m = (2379) (24) m = (2467) (25) m = (2523) (26) m = (2545) (27) m = (2585) (28) m = (2622) (29) m = (2646) (30) m = (2651) (31) m = (2676) (32) m = (2682) (33) m = (2689) (34) m = (2683) (35) m = (2698) (36) m = (2699) (37) m = (2694) (38) m = (2700) (39) m = (2697) (40) m = (2694) (41) m = (2703) (42) m = (2703) (43) m = (2701) (44) m = (2703) (45) m = (2704) 図 2: ヒンズー シャッフルにおける可視化と定量化 m = 1, 2,, 45 (1) m = (154) (2) m = (497) (3) m = (2254) (4) m = (2590) (5) m = (2661) (6) m = (2692) (7) m = (2704) (8) m = (2704) (9) m = (2704) 図 3: リフル シャッフルにおける可視化と定量化 m = 1, 2,, 9 (1) m = 1 (2) m = 2 (3) m = 3 (4) m = 4 (5) m = 5 (6) m = 6 (7) m = 7 (8) m = 8 (9) m = 9 図 4: ディール シャッフルにおける可視化と定量化 m = 1, 2,, 9 られる これ以降も m の増加とともに残っていた上 下部のラインも徐々に消滅していき m = 7 において はじめて C = 1.000(2704) となった 以上の結果から リフル シャッフル単独の試行はヒ ンズー シャッフルと比較して効率が良く m = 3 の とき C が急激に上昇し その後 m = 7 においてデッ クがランダムな状態となることがわかった なお 7 回 の試行でデックが良く混ざった というシミュレーショ ン結果は リフル シャッフルのカット オフ現象に関 する先行研究 [6][7] とも結果が一致している 4.3. ディール シャッフルの単独試行 ディール シャッフルにおけるパケットの配置箇所 d は 先行研究 [8] に合わせて d = 10 とする ここでカードを 配置した順番でそれぞれのパケットに k = {1, 2,, d} を割り振り 任意のパケットを dk と表記する このと き d 個のパケットを重ねる順序 についても先行研 究 [8] に合わせて {k の奇数順 } {k の偶数順 } と設 定した つまり d = 10 の場合においてパケット dk は {d1 d3 d5 d7 d9 d2 d4 d6 d8 d10 } と いう順序で重ね合わせることになる 102
5 5: m C P {n d } w = 0 4 m C = 5 m 3 m 4 y = x 5. 4 H R D A = {H, R, D} m r = {1, 2,, m} L r A m A m A m = {(L 1, L 2,, L m ) L 1 A L 2 A L m A} (15) (L 1, L 2 ) (L 2, L 1 ) m (L 1, L 2,, L m ) L 1 L 2 L m 6(2)HR (2)HR 5.1 A 2 (m = 2) 5.2 A 3 (m = 3) 5.3 A 4 (m = 4) 5.4 A 5 (m = 5) A 2 m = 2 A C (2)HR H R (4)RH C (2)HR 1/3 (3)HD (7)DH (6)RD (8)DR D C C A 2 D D L 1 = D L m = D m 3 A 2 (2)HR A A 3 m = 3 A (1)-(9) 6 H 7 2 (10)-(18) 6 R 7 3 (19)-(27) 6 D 4 H R D A 2 (7)HDH (8)HDR (16)RDH (17)RDR A 2 C (7)(16) C 0.9 A 2 D 4 L 2 D L 1 L m D C 6 7(19)-(27) C C = D C A 3 (7)HDH (16)RDH D A 4 m = 4 A (1)- (27) 7 H (28)-(54) 7 R 8 103
6 (1) HH 0.126(341) (2) HR 0.363(982) (3) HD 0.061(165) (4) RH 0.115(312) (5) RR 0.183(496) (6) RD 0.059(160) (7) DH 0.056(151) 6: A 2 (m = 2) 9 (8) DR 0.063(169) (9) DD (1) HHH 0.204(551) (2) HHR 0.659(1781) (3) HHD 0.126(341) (4) HRH 0.450(1218) (5) HRR 0.737(1992) (6) HRD 0.360(973) (7) HDH 0.918(2481) (8) HDR 0.810(2190) (9) HDD 0.058(156) (10) RHH 0.172(464) (11) RHR 0.584(1580) (12) RHD 0.118(319) (13) RRH 0.246(665) (14) RRR 0.828(2240) (15) RRD 0.184(497) (16) RDH 0.909(2458) (17) RDR 0.818(2213) (18) RDD 0.057(153) (19) DHH 0.127(343) (20) DHR 0.356(962) (21) DHD 0.059(160) (22) DRH 0.123(332) (23) DRR 0.186(504) (24) DRD 0.060(161) (25) DDH 0.067(181) (26) DDR 0.058(157) 7: A 3 (m = 3) 27 (27) DDD 7 9 (55)-(81) 7 D 4 H R D A 3 8 C (19)-(24) (46)-(51) L 2 = D {L 2, L 4 } = D C (7)(8) (16)(17) (34)(35) (43)(44) L 3 = D D 1 A 4 L 1 L 4 D 1 L 1 = D 7 9 L 4 = D A 3 C A 4 C C (16)HRDH (19)HDHH (20)HDHR 3 L 1 L 4 D 1 H (L 1 = H) D H A 4 A 3 A 5 A 4 (16)HRDH (19)HDHH (20)HDHR L 1 L 4 D 1 L 1 = H D H C C = A 5 m = 5 A C C C C = C w (1)HHDHH (2)HHDHR (3)HRDHH (4)HRDHR (5)HDHHR (6)HDHRR (7)HDHDH (8)HDHDR (9)RHDHH (10)RHDHR (11)RDHDH 3 C = 1.000(2704) S = S = S = S 10 3 C = 1.000(2704) S (0 S 10 3 )
7 FIT2017 第 16 回情報科学技術フォーラム (1) HHHH 0.246(664) (2) HHHR 0.726(1963) (3) HHHD 0.208(562) (4) HHRH 0.709(1917) (5) HHRR 0.900(2433) (6) HHRD 0.655(1771) (7) HHDH 0.962(2601) (8) HHDR 0.897(2426) (9) HHDD 0.133(359) (10) HRHH 0.532(1439) (11) HRHR 0.858(2320) (12) HRHD 0.448(1211) (13) HRRH 0.775(2095) (14) HRRR 0.954(2579) (15) HRRD 0.742(2006) (16) HRDH 0.987(2669) (17) HRDR 0.956(2586) (18) HRDD 0.361(976) (19) HDHH 0.989(2673) (20) HDHR 0.985(2663) (21) HDHD 0.919(2485) (22) HDRH 0.954(2579) (23) HDRR 0.942(2547) (24) HDRD 0.806(2180) (25) HDDH 0.281(759) (26) HDDR 0.512(1385) (27) HDDD 0.065(177) (28) RHHH 0.236(637) (29) RHHR 0.744(2013) (30) RHHD 0.181(490) (31) RHRH 0.688(1860) (32) RHRR 0.820(2217) (33) RHRD 0.598(1616) (34) RHDH 0.952(2573) (35) RHDR 0.879(2378) (36) RHDD 0.111(301) (37) RRHH 0.316(855) (38) RRHR 0.881(2383) (39) RRHD 0.251(678) (40) RRRH 0.871(2356) (41) RRRR 0.960(2597) (42) RRRD 0.821(2221) (43) RRDH 0.975(2637) (44) RRDR 0.923(2495) (45) RRDD 0.194(524) (46) RDHH 0.955(2581) (47) RDHR 0.972(2627) (48) RDHD 0.908(2456) (49) RDRH 0.946(2558) (50) RDRR 0.933(2523) (51) RDRD 0.816(2206) (52) RDDH 0.252(681) (53) RDDR 0.413(1117) (54) RDDD 0.059(160) (55) DHHH 0.205(553) (56) DHHR 0.659(1781) (57) DHHD 0.126(342) (58) DHRH 0.450(1216) (59) DHRR 0.742(2007) (60) DHRD 0.354(957) (61) DHDH 0.920(2487) (62) DHDR 0.814(2200) (63) DHDD 0.053(143) (64) DRHH 0.179(484) (65) DRHR 0.582(1574) (66) DRHD 0.116(313) (67) DRRH 0.243(656) (68) DRRR 0.828(2239) (69) DRRD 0.188(507) (70) DRDH 0.906(2451) (71) DRDR 0.821(2221) (72) DRDD 0.061(165) (73) DDHH 0.126(342) (74) DDHR 0.359(972) (75) DDHD 0.061(166) (76) DDRH 0.116(313) (77) DDRR 0.183(494) (78) DDRD 0.058(156) (79) DDDH 0.054(145) (80) DDDR 0.057(155) (81) DDDD 図 8: シャッフルの組み合わせ A4 (m = 4) における可視化と定量化 81 通り 表 1 に 11 通りの組み合わせにおける シャッフルの所 要時間 s 概算値 および S = 104 のシミュレーショ ンを 103 回試行した結果である w の内訳 を示す この 所要時間 s の算出には先行研究 [8] に従い 1 回の H を 3 秒 R を 7 秒 D を 30 秒と仮定した まず表 1 から 103 回すべての試行において C かつ w 2700 を満 たしている組み合わせは (3)HRDHH (7)HDHDH の 2 通りであることがわかる このうち (3)HRDHH がもっ とも良い結果となり 65.4%(S = 654) の頻度で w = 2704 となっている ただし (1)HHDHH (4)HRDHR 105
8 1: A 5 C 11 s[ ] S = w [ ] A 5 s (S ) w N/A (1)HHDHH (2)HHDHR (3)HRDHH (4)HRDHR (5)HDHHR (6)HDHRR (7)HDHDH (8)HDHDR (9)RHDHH (10)RHDHR (11)RDHDH (5)HDHHR (7)HDHDH 30 40% w = 2704 (3)HRDHH s D 11 D (3)HRDHH D 1 46 (3)HRDHH C D 1 HHRRR HRRRR RHRRR 3 C 0.996(2694) 0.995(2690) 0.995(2691) s HHRRR 30 D A 5 C = (3)HRDHH s C (3)HRDHH HHRRR (3)HRDHH 4 A 5 C = A 6 (m 6) HR(C = 0.363) 3 C 0.9 HDH RDH 4 C 0.98 HRDH HDHH HDHR 5 HRDHH(C = 1.000, s = 46) HHRRR(C = 0.996, s = 27) 5 (C = 1.000) [1] (2016) [2] D,Aldous., P,Diaconis. Shuffling cards and stopping times American Mathematical Monthly Vol.93, No.5, pp (1986) [3] P,Diaconis. Group representations in probability and statistics Institute of Mathematical Statistics Lecture Notes Monograph Series Vol.11(1988) [4] B,Mann. How many times should you shuffle a deck of cards? Probability and Stochastics Series pp (1995) [5] S,Assaf., P,Diaconis., K,Soundararajan. Riffle shuffles of a deck with repeated cards The Annals of Probability(Institute of Mathematical Statistics) Vol.34, No.2, pp (2006) [6] P,Diaconis. The cutoff phenomenon in finite Markov chains Proceedings of the National Academy of Sciences Vol.93, No.4, pp (1996) [7] P,Diaconis. The cutoff phenomenon for randomized riffle shuffles Wiley InterScience pp (2007) [8] TCG Vol.2011-GI-25 No.4 pp.1-8(2011) [9] kokai-koza/h23-kumagai.pdf [ ] [10] E,Gilbert. Theory of shuffling Technical memorandum Bell Laboratories(1955) [11] J,Reeds. Theory of shuffling Unpublished manuscript (1976) 106
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