離散数理工学 第 2回 数え上げの基礎:漸化式の立て方

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1 :48 ( ) (2) / 44

2 ( ) 1 (10/7) ( ) (10/14) 2 (10/21) 3 ( ) (10/28) 4 ( ) (11/4) 5 (11/11) 6 (11/18) 7 (11/25) ( ) (2) / 44

3 ( ) 8 (12/2) 9 (12/9) (12/16) 10 ( ) (1/6) 11 ( ) (1/13) 12 ( ) (1/20) ( ) (1/27) 13 ( ) (2/3) (2/17?) ( ) (2) / 44

4 ( ) (2) / 44

5 1 2 3 ( ) (2) / 44

6 (V, E) V E 2 V V 2 V = {1, 2, 3, 4, 5} E = {{1, 2}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {4, 5}} {2, 5} = {5, 2} ( ) ( ) (2) / 44

7 V = {1, 2, 3, 4, 5} E = {{1, 2}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {4, 5}} ( ) (2) / 44

8 G = (V, E) V G E G V G E G {u, v} E u, v v e v e u v u v V = {1, 2, 3, 4, 5} E = {{1, 2}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {4, 5}} 2, 3 {2, 3} {2, 3} ( ) (2) / 44

9 ( ) (2) / 44

10 ( ) (2) / 44

11 G = (V, E) G I V 2 u, v I {u, v} E ( ) (2) / 44

12 ( ) 22 ( ) (2) / 44

13 22 ( ) (2) / 44

14 G = (V, E) v V σ v {0, 1} σ v = 0 v σv = 1 v σ v = 1 v V = ( ) (2) / 44

15 P 1 P 2 P 3 P 4 P 5 P n ( ) (2) / 44

16 n ( ) (2) / 44

17 P 5 2 (A) = P 4 (B) = 2 P 3 + P 5 = P 4 + P 3 ( ) (2) / 44

18 P 5 2 (A) = P 4 (B) = 2 P 3 + P 5 = P 4 + P 3 ( ) (2) / 44

19 P 5 2 (A) = P 4 (B) = 2 P 3 + P 5 = P 4 + P 3 ( ) (2) / 44

20 P 5 2 (A) = P 4 (B) = 2 P 3 + P 5 = P 4 + P 3 ( ) (2) / 44

21 P 5 2 (A) = P 4 (B) = 2 P 3 + P 5 = P 4 + P 3 ( ) (2) / 44

22 P 5 2 (A) = P 4 (B) = 2 P 3 + P 5 = P 4 + P 3 ( ) (2) / 44

23 ( ) P n 2 ( n 3) (A) = P n 1 (B) = 2 P n 2 + n 3 P n = P n 1 + P n 2 ( ) (2) / 44

24 ( ) P n 2 ( n 3) (A) = P n 1 (B) = 2 P n 2 + n 3 P n = P n 1 + P n 2 ( ) (2) / 44

25 ( ) P n 2 ( n 3) (A) = P n 1 (B) = 2 P n 2 + n 3 P n = P n 1 + P n 2 ( ) (2) / 44

26 a n = P n 2 (n = 1 ) a n = 3 (n = 2 ) a n 1 + a n 2 (n 3 ) ( ) (2) / 44

27 P n P 2 (G n ) G 1 G 2 G 3 G 4 G 5 G n ( ) (2) / 44

28 P n P 2 G n 2 (A) = (B) = G k ( ) (2) / 44

29 P n P 2 G n 2 (A) = (B) = G k ( ) (2) / 44

30 P n P 2 G n 2 (A) = (B) = G k ( ) (2) / 44

31 P n P 2 (H n ) H 1 H 2 H 3 H 4 H 5 H n n 2 G n = H n + H n 1 ( ) (2) / 44

32 P n P 2 H n 2 (A) = (B) = n 2 H n = G n 1 + H n 1 ( ) (2) / 44

33 P n P 2 H n 2 (A) = (B) = n 2 H n = G n 1 + H n 1 ( ) (2) / 44

34 P n P 2 H n 2 (A) = (B) = 2 + G n 1.. H n 1.. n 2 H n = G n 1 + H n 1 ( ) (2) / 44

35 P n P 2 H n 2 (A) = (B) = 2 + G n 1.. H n 1.. n 2 H n = G n 1 + H n 1 ( ) (2) / 44

36 P n P 2 b n = G n c n = H n b n = c n = { 3 (n = 1 ) c n + c n 1 (n 2 ) { 2 (n = 1 ) b n 1 + c n 1 (n 2 ) ( ) (2) / 44

37 1 2 3 ( ) (2) / 44

38 A 1: def fnct(n) 2: print "a" 3: if n > 2 4: fnct(n-1) 5: fnct(n-2) 6: end 7: end fnct(n) a ( ) (2) / 44

39 n a n a n a n a ( ) (2) / 44

40 A 1: def fnct(n) 2: print "a" 3: if n > 2 4: fnct(n-1) 5: fnct(n-2) 6: end 7: end f n = fnct(n) a ( ) (2) / 44

41 A 1: def fnct(n) 2: print "a" 3: if n > 2 4: fnct(n-1) 5: fnct(n-2) 6: end 7: end 2 n 1 a 4 5 ( ) (2) / 44

42 A 1: def fnct(n) 2: print "a" 3: if n > 2 4: fnct(n-1) 5: fnct(n-2) 6: end 7: end f n = { 1 (n 2 ) 1 + f n 1 + f n 2 (n 3 ) ( ) (2) / 44

43 Euclid Euclid 1: def gcd(a, b) # precondition: a >= b 2: print "G" 3: if b == 0 4: return a 5: else 6: gcd(b, a % b) 7: end 8: end a % b = a b gcd(a, b) G ( ) ( ) (2) / 44

44 Euclid (1) a b G ( ) (2) / 44

45 Euclid (2) a b G ( ) (2) / 44

46 Euclid Euclid 1: def gcd(a, b) # precondition: a >= b 2: print "G" 3: if b == 0 4: return a 5: else 6: gcd(b, a % b) 7: end 8: end g n = max {gcd(a, b) G } a 1,b n g n = b n g n ( ) (2) / 44

47 Euclid a, b 1 a b a a % b 2 a = bq + r ( 0 r < b) a % b = r a b q = a r b r = 1 r b b b > 0 q q 1 a a b r < b 2 2 a a b > r = a bq a b < a 2 2 a r 2 a = 2 ( ) n n n 2 = n 2 ( ) (2) / 44

48 Euclid a, b 1 a b a a % b 2 a = bq + r ( 0 r < b) a % b = r a b q = a r b r = 1 r b b b > 0 q q 1 a a b r < b 2 2 a a b > r = a bq a b < a 2 2 a r 2 a = 2 ( ) n n n 2 = n 2 ( ) (2) / 44

49 Euclid a, b 1 a b a a % b 2 a = bq + r ( 0 r < b) a % b = r a b q = a r b r = 1 r b b b > 0 q q 1 a a b r < b 2 2 a a b > r = a bq a b < a 2 2 a r 2 a = 2 ( ) n n n 2 = n 2 ( ) (2) / 44

50 Euclid a, b 1 a b a a % b 2 a = bq + r ( 0 r < b) a % b = r a b q = a r b r = 1 r b b b > 0 q q 1 a a b r < b 2 2 a a b > r = a bq a b < a 2 2 a r 2 a = 2 ( ) n n n 2 = n 2 ( ) (2) / 44

51 Euclid a, b 1 a b a a % b 2 a = bq + r ( 0 r < b) a % b = r a b q = a r b r = 1 r b b b > 0 q q 1 a a b r < b 2 2 a a b > r = a bq a b < a 2 2 a r 2 a = 2 ( ) n n n 2 = n 2 ( ) (2) / 44

52 Euclid a, b 1 a b a a % b 2 a = bq + r ( 0 r < b) a % b = r a b q = a r b r = 1 r b b b > 0 q q 1 a a b r < b 2 2 a a b > r = a bq a b < a 2 2 a r 2 a = 2 ( ) n n n 2 = n 2 ( ) (2) / 44

53 Euclid Euclid 1: def gcd(a, b) # precondition: a >= b 2: print "G" 3: if b == 0 4: return a 5: else 6: gcd(b, a % b) 7: end 8: end g n = gcd(a, b) G a, b... ( b = n) ( ) (2) / 44

54 Euclid (1) g n = gcd(a, b) G = 1 + gcd(b, a % b) G a % b = 0 g n = 2 ( gcd(b, a % b) ) a % b 0 ( ) (2) / 44

55 Euclid (1) g n = gcd(a, b) G = 1 + gcd(b, a % b) G a % b = 0 g n = 2 ( gcd(b, a % b) ) a % b 0 ( ) (2) / 44

56 Euclid (1) g n = gcd(a, b) G = 1 + gcd(b, a % b) G a % b = 0 g n = 2 ( gcd(b, a % b) ) a % b 0 ( ) (2) / 44

57 Euclid (1) g n = gcd(a, b) G = 1 + gcd(b, a % b) G a % b = 0 g n = 2 ( gcd(b, a % b) ) a % b 0 ( ) (2) / 44

58 Euclid (1) g n = gcd(a, b) G = 1 + gcd(b, a % b) G a % b = 0 g n = 2 ( gcd(b, a % b) ) a % b 0 ( ) (2) / 44

59 Euclid (1) g n = gcd(a, b) G = 1 + gcd(b, a % b) G a % b = 0 g n = 2 ( gcd(b, a % b) ) a % b 0 ( ) (2) / 44

60 Euclid (2) g n = gcd(a, b) G = 1 + gcd(b, a % b) G = gcd(a % b, b % (a % b)) G max a 1,b b/2 {gcd(a, b ) G } = 2 + g b/2 = 2 + g n/2 n 1 g n 2 + g n/2 { = 1 n = 0 g n 2 + g n/2 n 1 ( ) (2) / 44

61 Euclid (2) g n = gcd(a, b) G = 1 + gcd(b, a % b) G = gcd(a % b, b % (a % b)) G max a 1,b b/2 {gcd(a, b ) G } = 2 + g b/2 = 2 + g n/2 n 1 g n 2 + g n/2 { = 1 n = 0 g n 2 + g n/2 n 1 ( ) (2) / 44

62 Euclid (2) g n = gcd(a, b) G = 1 + gcd(b, a % b) G = gcd(a % b, b % (a % b)) G max a 1,b b/2 {gcd(a, b ) G } = 2 + g b/2 = 2 + g n/2 n 1 g n 2 + g n/2 { = 1 n = 0 g n 2 + g n/2 n 1 ( ) (2) / 44

63 Euclid (2) g n = gcd(a, b) G = 1 + gcd(b, a % b) G = gcd(a % b, b % (a % b)) G max a 1,b b/2 {gcd(a, b ) G } = 2 + g b/2 = 2 + g n/2 n 1 g n 2 + g n/2 { = 1 n = 0 g n 2 + g n/2 n 1 ( ) (2) / 44

64 Euclid (2) g n = gcd(a, b) G = 1 + gcd(b, a % b) G = gcd(a % b, b % (a % b)) G max a 1,b b/2 {gcd(a, b ) G } = 2 + g b/2 = 2 + g n/2 n 1 g n 2 + g n/2 { = 1 n = 0 g n 2 + g n/2 n 1 ( ) (2) / 44

65 Euclid (2) g n = gcd(a, b) G = 1 + gcd(b, a % b) G = gcd(a % b, b % (a % b)) G max a 1,b b/2 {gcd(a, b ) G } = 2 + g b/2 = 2 + g n/2 n 1 g n 2 + g n/2 { = 1 n = 0 g n 2 + g n/2 n 1 ( ) (2) / 44

66 Euclid (2) g n = gcd(a, b) G = 1 + gcd(b, a % b) G = gcd(a % b, b % (a % b)) G max a 1,b b/2 {gcd(a, b ) G } = 2 + g b/2 = 2 + g n/2 n 1 g n 2 + g n/2 { = 1 n = 0 g n 2 + g n/2 n 1 ( ) (2) / 44

67 Euclid (2) g n = gcd(a, b) G = 1 + gcd(b, a % b) G = gcd(a % b, b % (a % b)) G max a 1,b b/2 {gcd(a, b ) G } = 2 + g b/2 = 2 + g n/2 n 1 g n 2 + g n/2 { = 1 n = 0 g n 2 + g n/2 n 1 ( ) (2) / 44

68 Collatz 1: def collatz(n) 2: print n 3: if n % 2 == 0 4: collatz(n/2) 5: else 6: collatz(3*n+1) 7: end 8: end Collatz ( ) n collatz(n) 1 n (Oliveira e Silva 10) ( ) (2) / 44

69 1 2 3 ( ) (2) / 44

70 ( ) (2) / 44

71 ( ) (2) / 44

72 ( ) TA OK OK ( ) (2) / 44

73 1 2 3 ( ) (2) / 44

離散数理工学 第 2回 数え上げの基礎:漸化式の立て方

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