The document discusses using bitwise operators and dynamic programming to solve problems efficiently. It provides examples of using bitwise OR, AND, XOR operators to add, remove and check for elements in a set represented as a bitmask. It then describes several problems involving bitmasks and dynamic programming including finding the number of possible paths in a graph and the maximum score in a traveling salesman problem. The examples demonstrate how bitwise operators can be used to optimize set operations and dynamic programming can reduce exponential time complexity problems to polynomial time.

1. Bitwise DP
(Acka)

2. Bitwise Operator
BitwiseDP 2

3. Bitwise Operator
• A | B => A or B
• A & B => A and B
• A ^ B => A xor B
• A = 1001, B = 0101
• A | B = 1101, A & B = 0001, A ^ B = 1100
• ~A => ones complement of A =>
• A << x => left shift A as x
• A >> x => right shift A as x
• A = 00110101
• ~A = 11001010, A << 2 = 11010100, A >> 2 = 00001101
BitwiseDP 3

4. • S M .
• add x: S x . x
• remove x: S x . x
• check x: S x 1, 0
• toggle x: S x x , x
• all: S {1, 2, ..., 20} .
• empty: S .
• 1 ≤ x ≤ 20
• 1 ≤ M ≤ 3,000,000
BitwiseDP
Problem: https://www.acmicpc.net/problem/11723
4

5. • bool exist [21]
• exist[i]: i true, false
BitwiseDP
Problem: https://www.acmicpc.net/problem/11723
5

6. • .
BitwiseDP
Problem: https://www.acmicpc.net/problem/11723
6

7. • .
• stat = 50 = 0….000000000000000110010(2)
• stat: {2, 5, 6}
• i-1 : i /
• .
BitwiseDP
Problem: https://www.acmicpc.net/problem/11723
7

8. • add x
BitwiseDP
Problem: https://www.acmicpc.net/problem/11723
8

9. • add 4
BitwiseDP
Problem: https://www.acmicpc.net/problem/11723
9
…01010001(2)
…01011001(2)

10. • add 4
BitwiseDP
Problem: https://www.acmicpc.net/problem/11723
10
…01010001(2)
+ …00001000(2)
…01011001(2)

11. • add 4
BitwiseDP
Problem: https://www.acmicpc.net/problem/11723
11
…01011001(2)
+ …00001000(2)

12. • add 4
BitwiseDP
Problem: https://www.acmicpc.net/problem/11723
12
…01011001(2)
+ …00001000(2)
…01100001(2)

13. • add 4
BitwiseDP
Problem: https://www.acmicpc.net/problem/11723
13
…0101?001(2)
or …00001000(2)

14. • add 4
BitwiseDP
Problem: https://www.acmicpc.net/problem/11723
14
…0101?001(2)
or …00001000(2)
…0101?001(2)

15. • add 4
BitwiseDP
Problem: https://www.acmicpc.net/problem/11723
15
…0101?001(2)
or …00001000(2)
…0101?001(2)

16. • add 4
BitwiseDP
Problem: https://www.acmicpc.net/problem/11723
16
…0101?001(2)
or …00001000(2)
…0101?001(2)
1|0 = 1
0|0 = 0
0 or

17. • add 4
BitwiseDP
Problem: https://www.acmicpc.net/problem/11723
17
…0101?001(2)
or …00001000(2)
…0101?001(2)

18. • add 4
BitwiseDP
Problem: https://www.acmicpc.net/problem/11723
18
…0101?001(2)
or …00001000(2)
…01011001(2)
1|1 = 1
0|1 = 1
1 or 1

19. • add x = stat | (1 << (x – 1))
BitwiseDP
Problem: https://www.acmicpc.net/problem/11723
19
…0101?001(2)
or …00001000(2)
…01011001(2)

20. • remove 4
BitwiseDP
Problem: https://www.acmicpc.net/problem/11723
20
…0101?001(2)

21. • remove 4
BitwiseDP
Problem: https://www.acmicpc.net/problem/11723
21
…0101?001(2)
& …11110111(2)
…0101?001(2)

22. • remove 4
BitwiseDP
Problem: https://www.acmicpc.net/problem/11723
22
…0101?001(2)
& …11110111(2)
…0101?001(2)

23. • remove 4
BitwiseDP
Problem: https://www.acmicpc.net/problem/11723
23
…0101?001(2)
& …11110111(2)
…0101?001(2)
1&1 = 1
0&1 = 0
1 and

24. • remove 4
BitwiseDP
Problem: https://www.acmicpc.net/problem/11723
24
…0101?001(2)
& …11110111(2)
…0101?001(2)

25. • remove 4
BitwiseDP
Problem: https://www.acmicpc.net/problem/11723
25
…0101?001(2)
& …11110111(2)
…01010001(2)
0&0 = 0
1&0 = 0
0 and 0

26. • remove x = stat & ~(1 << (x – 1))
BitwiseDP
Problem: https://www.acmicpc.net/problem/11723
26
…0101?001(2)
& …11110111(2)
…01010001(2)

27. • bit[x] = 2x-1
• add: stat |= bit[x]
• remove: stat &= ~bit[x]
• check: stat & bit[x]
• toggle: stat ^= bit[x]
• all: stat = bit[21] - 1
• empty: stat = 0
BitwiseDP
Problem: https://www.acmicpc.net/problem/11723
27

28. • C/C++:
https://gist.github.com/Acka1357/61f2990267345c21319bbf4fb79fe5a2
• Java:
https://gist.github.com/Acka1357/2c243480be1edca69968a7ad300279cf
BitwiseDP
Problem: https://www.acmicpc.net/problem/11723
28

29. •
• N 0 9
• (0 .)
• 1 ≤ N ≤ 100
BitwiseDP
Problem: https://www.acmicpc.net/problem/1562
29

30. •
• D[i][j]: i j
• D[i][j] = D[i – 1][j – 1] + D[i – 1][j + 1]
BitwiseDP
Problem: https://www.acmicpc.net/problem/1562
30

31. •
• D[i][j]: i j
• D[i][j] = D[i – 1][j – 1] + D[i – 1][j + 1]
• + 0 ~ 9
BitwiseDP
Problem: https://www.acmicpc.net/problem/1562
31

32. • D[i][j][use0][use1][use2]…[use9]:
i j
use0, use1, use2 .. use9
(use0, use1 … use9 = 0 or 1)
BitwiseDP
Problem: https://www.acmicpc.net/problem/1562
32

33. • D[i][j][use0][use1][use2]…[use9]:
i j
use0, use1, use2 .. use9
(use0, use1 … use9 = 0 or 1)
• [use0][use1][use2]…[use9]: 2 * 2 * 2 * … * 2
=> 210
=> 0000000000(2) ~ 1111111111(2)
BitwiseDP
Problem: https://www.acmicpc.net/problem/1562
33

34. • D[i][j][used]: i j
used
• used: 0000000000(2) ~ 1111111111(2) => 210
• used k(0~9) bit: k
BitwiseDP
Problem: https://www.acmicpc.net/problem/1562
34

35. • D[i][j][used]: i j
used
• used: 0000000000(2) ~ 1111111111(2) => 210
• used k(0~9) bit: k
BitwiseDP
Problem: https://www.acmicpc.net/problem/1562
used: 101 = 0001100101(2)
5 0
8
35

36. • D[i][j][used]: i j
used
• used .
• D[i][j - 1][used] -> D[i + 1][j][used | 2j]
• D[i][j + 1][used] -> D[i + 1][j][used | 2j]
BitwiseDP
Problem: https://www.acmicpc.net/problem/1562
36

37. • : ∑ 𝐷 𝑁 𝑖 [1023]-
./0
• N (used == 1111111111(2))
BitwiseDP
Problem: https://www.acmicpc.net/problem/1562
37

38. • : O(N * 10 * 210)
• * *
• 1 ≤ N ≤ 100
BitwiseDP
Problem: https://www.acmicpc.net/problem/1562
38

39. • C/C++:
https://gist.github.com/Acka1357/ba689997d73246d77a8ccc33f876073f
• Java:
https://gist.github.com/Acka1357/1d65d7af446ce436b48b80a36386e108
BitwiseDP
Problem: https://www.acmicpc.net/problem/1562
39

40. • 1 N .
• N
• . ( )
•
Problem: https://www.acmicpc.net/problem/2098
BitwiseDP 40

41. •
• *
* …
• N * (N – 1) * (N – 2) * … * 1 => O(N!)
Problem: https://www.acmicpc.net/problem/2098
BitwiseDP 41

42. •
• *
* …
• N * (N – 1) * (N – 2) * … * 1 => O(N!)
• 2 ≤ N ≤ 16
Problem: https://www.acmicpc.net/problem/2098
BitwiseDP 42

43. •
• *
* …
• N * (N – 1) * (N – 2) * … * 1 => O(N!)
• 2 ≤ N ≤ 16
• 16! => 20,922,789,888,000
Problem: https://www.acmicpc.net/problem/2098
BitwiseDP 43

44. • 1 -> 2 -> 3 -> 4 -> ?
• 1 -> 3 -> 2 -> 4 -> ?
• : 1, 2, 3, 4 4
• : 4
1
Problem: https://www.acmicpc.net/problem/2098
BitwiseDP 44

45. • 1 -> 2 -> 3 -> 4 -> ?
• 1 -> 3 -> 2 -> 4 -> ?
•
• .
Problem: https://www.acmicpc.net/problem/2098
BitwiseDP 45

46. •
•
• /
Problem: https://www.acmicpc.net/problem/2098
BitwiseDP 46

47. •
•
• /
• (1 ) * (2 ) * (3
) * … * (N )
• 2N
Problem: https://www.acmicpc.net/problem/2098
BitwiseDP 47

48. • d[cur][visited]: i
visited
Problem: https://www.acmicpc.net/problem/2098
BitwiseDP 48

49. • d[cur][visited]: i
visited
Problem: https://www.acmicpc.net/problem/2098
BitwiseDP
visited: 101 = 01100101(2)
49

50. • d[cur][visited]: i
visited
Problem: https://www.acmicpc.net/problem/2098
BitwiseDP
7 1
8
50
visited: 101 = 01100101(2)

51. • d[cur][visited]: i
visited
• d[cur][visited] + cost[cur][next] -> d[next][visited + 2next]
• 2next : next
• next
• visited & 2next == 0
Problem: https://www.acmicpc.net/problem/2098
BitwiseDP 51

52. • d[cur][visited]
visited cur
d[cur][visited] .
Problem: https://www.acmicpc.net/problem/2098
BitwiseDP
10011010
10011010
10011010
10011010
00011010
10001010
10010010
10011000
min
52
cur visited
2,4,5
2,4,8
2,5,8
4,5,8
cur visited
8
5
4
2

53. • d[cur][visited]
visited cur
d[cur][visited] .
• visited .
Problem: https://www.acmicpc.net/problem/2098
BitwiseDP
10011010
10011010
10011010
10011010
00011010
10001010
10010010
10011000
min
53
cur visited
2,4,5
2,4,8
2,5,8
4,5,8
cur visited
8
5
4
2

54. • d[cur][visited]: i
visited
• d[cur][visited] = minpast ∈ visited (d[past][visited – 2cur]
+ cost[past][cur])
• 2cur : cur
• past cur , visited
• visited & 2past == 2past
Problem: https://www.acmicpc.net/problem/2098
BitwiseDP 54

55. •
• ( ) .
Problem: https://www.acmicpc.net/problem/2098
BitwiseDP 55

56. •
• ( ) .
Problem: https://www.acmicpc.net/problem/2098
BitwiseDP 56
P1 P2 P3 P4 P5 P6

57. •
• ( ) .
Problem: https://www.acmicpc.net/problem/2098
BitwiseDP 57
P1
P2
P3
P4
P5
P6

58. • : O(N * N * 2N)
• : O(N * 2N)
• / : O(N)
Problem: https://www.acmicpc.net/problem/2098
BitwiseDP 58

59. • C/C++:
https://gist.github.com/Acka1357/be750aba1f1950f50c93dce7d038752c
https://gist.github.com/Acka1357/2815446354910d6b4ac84164a776b6ee
• Java:
https://gist.github.com/Acka1357/25a34b9ea7e49ad148fe4cfe2fe6143b
Problem: https://www.acmicpc.net/problem/2098
BitwiseDP 59

60. • N X M
• 2 X 1, 1 X 2
•
• 1 ≤ N, M ≤ 14
Problem: https://www.acmicpc.net/problem/1648
BitwiseDP 60

61. • , .
Problem: https://www.acmicpc.net/problem/1648
BitwiseDP 61
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
… …
… …

62. • , 0 .
• (0, 1) (0, 10)
Problem: https://www.acmicpc.net/problem/1648
BitwiseDP 62
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
… …
… …

63. • (0, 1)
Problem: https://www.acmicpc.net/problem/1648
BitwiseDP 63
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
… …
… …

64. • 0 1 .
• 0 , M-1
Problem: https://www.acmicpc.net/problem/1648
BitwiseDP 64
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
… …
… …

65. • status = 20 + 21
Problem: https://www.acmicpc.net/problem/1648
BitwiseDP 65
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
… …
… …
0으로부터 한 줄의 상태:
0000000011(2)

66. • (0, 10)
Problem: https://www.acmicpc.net/problem/1648
BitwiseDP 66
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
… …
… …

67. • status = 20 ( )
Problem: https://www.acmicpc.net/problem/1648
BitwiseDP 67
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
… …
… …
0으로부터 한 줄의 상태:
0000000001(2)

68. • 1 , 0 ,
• 0 .
Problem: https://www.acmicpc.net/problem/1648
BitwiseDP 68
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
… …
… …
0으로부터 한 줄의 상태:
0000000001(2)

69. • 0 10 ,
• 10 .
Problem: https://www.acmicpc.net/problem/1648
BitwiseDP 69
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
… …
… …
1로부터 ‘한 줄’의 상태:
1000000000(2)

70. • i-1 0 -> i
• i (i + 1) (i + M) .
Problem: https://www.acmicpc.net/problem/1648
BitwiseDP 70
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
… …

71. • i , (i – 1)
• (i + M) .
Problem: https://www.acmicpc.net/problem/1648
BitwiseDP 71
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
… …

72. • i i M
(i – 1) 1, (i + M) 0
.
Problem: https://www.acmicpc.net/problem/1648
BitwiseDP 72
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
… …

73. • 12
Problem: https://www.acmicpc.net/problem/1648
BitwiseDP 73
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
… …
12부터 M칸의 상태:
1111001001(2)

74. • 12
• 11 , 22 .
Problem: https://www.acmicpc.net/problem/1648
BitwiseDP 74
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
… …
12부터 M칸의 상태:
1111001001(2)

75. • 12
• 12 , 13 .
Problem: https://www.acmicpc.net/problem/1648
BitwiseDP 75
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
… …
12부터 M칸의 상태:
1111001001(2)

76. • 13 M
• 12 M .
Problem: https://www.acmicpc.net/problem/1648
BitwiseDP 76
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
… …
12부터 M칸의 상태:
1111001001(2)

77. • status >> 1 (or status /= 2)
Problem: https://www.acmicpc.net/problem/1648
BitwiseDP 77
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
… …
13부터 M칸의 상태:
0111100100(2)

78. • 13
• 13 , 23 .
Problem: https://www.acmicpc.net/problem/1648
BitwiseDP 78
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
… …
12부터 M칸의 상태:
1111001001(2)

79. • 13
• (13, 14) (13, 23) .
Problem: https://www.acmicpc.net/problem/1648
BitwiseDP 79
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
… …
13부터 M칸의 상태:
0111100100(2)

80. • (13, 14) (13, 14) .
• 0 1 0 .
Problem: https://www.acmicpc.net/problem/1648
BitwiseDP 80
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
… …
13부터 M칸의 상태:
0111100100(2)

81. • (13, 14) ,
• 14 M .
Problem: https://www.acmicpc.net/problem/1648
BitwiseDP 81
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
… …
13부터 M칸의 상태:
0111100100(2)

82. • (status >> 1) + 20
• 14 M 14 0
Problem: https://www.acmicpc.net/problem/1648
BitwiseDP 82
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
… …
14부터 M칸의 상태:
0011110011(2)

83. • (13, 23) (13, 23) .
• 23 .
Problem: https://www.acmicpc.net/problem/1648
BitwiseDP 83
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
… …
13부터 M칸의 상태:
0111100100(2)

84. • (13, 23) ,
• 14 M .
Problem: https://www.acmicpc.net/problem/1648
BitwiseDP 84
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
… …
13부터 M칸의 상태:
0111100100(2)

85. • (status >> 1) + 2M-1
• 14 M 23 M-1 .
Problem: https://www.acmicpc.net/problem/1648
BitwiseDP 85
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
… …
14부터 M칸의 상태:
1011110010(2)

86. • D[i][status]: i – 1 ,
i M status
• 0 ≤ status < 2M
Problem: https://www.acmicpc.net/problem/1648
BitwiseDP 86

87. • i (if (status & 1) == 1)
• d[i + 1][status >> 1] += d[i][status]
Problem: https://www.acmicpc.net/problem/1648
BitwiseDP 87

88. • i (if (status & 1) == 1)
• d[i + 1][status >> 1] += d[i][status]
• i
•
•
Problem: https://www.acmicpc.net/problem/1648
BitwiseDP 88

89. • (if (status & 3) == 0)
• d[i + 1][(status >> 1) + 1] += d[i][status]
• (i (status & 1) == 0)
• d[i + 1][(status >> 1) + 2M-1] += d[i][status]
Problem: https://www.acmicpc.net/problem/1648
BitwiseDP 89

90. • (if (status & 3) == 0)
• d[i + 1][(status >> 1) + 1] += d[i][status]
• (i (status & 1) == 0)
• d[i + 1][(status >> 1) + 2M-1] += d[i][status]
• , i .
Problem: https://www.acmicpc.net/problem/1648
BitwiseDP 90

91. • d[i + 1][(status >> 1)] += d[i][status]
• if (status & 1) == 1
• d[i + 1][(status >> 1) + 1] += d[i][status]
• if (status & 3) == 0, and (i % M) != M – 1
• d[i + 1][(status >> 1) + 2M-1] += d[i][status]
• if (status & 1) == 0
Problem: https://www.acmicpc.net/problem/1648
BitwiseDP 91

92. • : d[N * M][0]
• N * M (N * M – 1) ,
• N * M
Problem: https://www.acmicpc.net/problem/1648
BitwiseDP 92

93. • : O(N * M * 2M)
• : O(N * M)
• : O(2M)
Problem: https://www.acmicpc.net/problem/1648
BitwiseDP 93

94. • C/C++:
https://gist.github.com/Acka1357/473f67f678c484b19e5d5daabde7522c
• Java:
https://gist.github.com/Acka1357/3406495ff3597161c4245d9989dc3fa7
Problem: https://www.acmicpc.net/problem/1648
BitwiseDP 94

95. • N X M
• 2 X 1, 1 X 2
• .
• .
•
• 1 ≤ N, M ≤ 14
Problem: https://www.acmicpc.net/problem/1657
BitwiseDP 95

96. • ==
• , .
Problem: https://www.acmicpc.net/problem/1657
BitwiseDP 96

97. • ==
• , .
• , .
Problem: https://www.acmicpc.net/problem/1657
BitwiseDP 97

98. • D[i][status]: i – 1 ( ) ,
i M status
• 0 ≤ status < 2M
Problem: https://www.acmicpc.net/problem/1657
BitwiseDP 98

99. • i (if (status & 1) == 1)
• d[i + 1][status >> 1] <- d[i][status]
Problem: https://www.acmicpc.net/problem/1657
BitwiseDP 99

100. • i (if (status & 1) == 1)
• d[i + 1][status >> 1] <- d[i][status]
• i
• d[i + 1][status >> 1] <- d[i][status]
• d[i + 1][(status >> 1) + 1] += d[i][status] ( )
• d[i + 1][(status >> 1) + 2M-1] += d[i][status] ( )
Problem: https://www.acmicpc.net/problem/1657
BitwiseDP 100

101. • d[i + 1][(status >> 1)] <- d[i][status]
• always possible
• d[i + 1][(status >> 1) + 1] += d[i][status]
• if (status & 3) == 0, and (i % M) != M – 1
• d[i + 1][(status >> 1) + 2M-1] += d[i][status]
• if (status & 1) == 0
Problem: https://www.acmicpc.net/problem/1657
BitwiseDP 101

102. • : d[N * M][0]
• N * M (N * M – 1) ( ) ,
• N * M
Problem: https://www.acmicpc.net/problem/1657
BitwiseDP 102

103. • : O(N * M * 2M)
• : O(N * M)
• : O(2M)
Problem: https://www.acmicpc.net/problem/1657
BitwiseDP 103

104. • C/C++:
https://gist.github.com/Acka1357/99cd03e1d98611f02840ab9708052c99
• Java:
https://gist.github.com/Acka1357/6d5dbdcc1528957a2896a703f6fa796c
Problem: https://www.acmicpc.net/problem/1657
BitwiseDP 104