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알고리즘 연합캠프 세미나 1-B (Bitwise DP)

H
HYUNJEONG KIM

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.

Engineering
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Bitwise DP (Acka)
Bitwise Operator BitwiseDP 2
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
• 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
• bool exist [21] • exist[i]: i true, false BitwiseDP Problem: https://www.acmicpc.net/problem/11723 5
• . BitwiseDP Problem: https://www.acmicpc.net/problem/11723 6
• . • stat = 50 = 0….000000000000000110010(2) • stat: {2, 5, 6} • i-1 : i / • . BitwiseDP Problem: https://www.acmicpc.net/problem/11723 7
• add x BitwiseDP Problem: https://www.acmicpc.net/problem/11723 8
• add 4 BitwiseDP Problem: https://www.acmicpc.net/problem/11723 9 …01010001(2) …01011001(2)
• add 4 BitwiseDP Problem: https://www.acmicpc.net/problem/11723 10 …01010001(2) + …00001000(2) …01011001(2)
• add 4 BitwiseDP Problem: https://www.acmicpc.net/problem/11723 11 …01011001(2) + …00001000(2)
• add 4 BitwiseDP Problem: https://www.acmicpc.net/problem/11723 12 …01011001(2) + …00001000(2) …01100001(2)
• add 4 BitwiseDP Problem: https://www.acmicpc.net/problem/11723 13 …0101?001(2) or …00001000(2)
• add 4 BitwiseDP Problem: https://www.acmicpc.net/problem/11723 14 …0101?001(2) or …00001000(2) …0101?001(2)
• add 4 BitwiseDP Problem: https://www.acmicpc.net/problem/11723 15 …0101?001(2) or …00001000(2) …0101?001(2)
• 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
• add 4 BitwiseDP Problem: https://www.acmicpc.net/problem/11723 17 …0101?001(2) or …00001000(2) …0101?001(2)
• 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
• add x = stat | (1 << (x – 1)) BitwiseDP Problem: https://www.acmicpc.net/problem/11723 19 …0101?001(2) or …00001000(2) …01011001(2)
• remove 4 BitwiseDP Problem: https://www.acmicpc.net/problem/11723 20 …0101?001(2)
• remove 4 BitwiseDP Problem: https://www.acmicpc.net/problem/11723 21 …0101?001(2) & …11110111(2) …0101?001(2)
• remove 4 BitwiseDP Problem: https://www.acmicpc.net/problem/11723 22 …0101?001(2) & …11110111(2) …0101?001(2)
• 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
• remove 4 BitwiseDP Problem: https://www.acmicpc.net/problem/11723 24 …0101?001(2) & …11110111(2) …0101?001(2)
• 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
• remove x = stat & ~(1 << (x – 1)) BitwiseDP Problem: https://www.acmicpc.net/problem/11723 26 …0101?001(2) & …11110111(2) …01010001(2)
• 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
• C/C++: https://gist.github.com/Acka1357/61f2990267345c21319bbf4fb79fe5a2 • Java: https://gist.github.com/Acka1357/2c243480be1edca69968a7ad300279cf BitwiseDP Problem: https://www.acmicpc.net/problem/11723 28
• • N 0 9 • (0 .) • 1 ≤ N ≤ 100 BitwiseDP Problem: https://www.acmicpc.net/problem/1562 29
• • 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
• • 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
• 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
• 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
• 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
• 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
• 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
• : ∑ 𝐷 𝑁 𝑖 [1023]- ./0 • N (used == 1111111111(2)) BitwiseDP Problem: https://www.acmicpc.net/problem/1562 37
• : O(N * 10 * 210) • * * • 1 ≤ N ≤ 100 BitwiseDP Problem: https://www.acmicpc.net/problem/1562 38
• C/C++: https://gist.github.com/Acka1357/ba689997d73246d77a8ccc33f876073f • Java: https://gist.github.com/Acka1357/1d65d7af446ce436b48b80a36386e108 BitwiseDP Problem: https://www.acmicpc.net/problem/1562 39
• 1 N . • N • . ( ) • Problem: https://www.acmicpc.net/problem/2098 BitwiseDP 40
• • * * … • N * (N – 1) * (N – 2) * … * 1 => O(N!) Problem: https://www.acmicpc.net/problem/2098 BitwiseDP 41
• • * * … • N * (N – 1) * (N – 2) * … * 1 => O(N!) • 2 ≤ N ≤ 16 Problem: https://www.acmicpc.net/problem/2098 BitwiseDP 42
• • * * … • 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
• 1 -> 2 -> 3 -> 4 -> ? • 1 -> 3 -> 2 -> 4 -> ? • : 1, 2, 3, 4 4 • : 4 1 Problem: https://www.acmicpc.net/problem/2098 BitwiseDP 44
• 1 -> 2 -> 3 -> 4 -> ? • 1 -> 3 -> 2 -> 4 -> ? • • . Problem: https://www.acmicpc.net/problem/2098 BitwiseDP 45
• • • / Problem: https://www.acmicpc.net/problem/2098 BitwiseDP 46
• • • / • (1 ) * (2 ) * (3 ) * … * (N ) • 2N Problem: https://www.acmicpc.net/problem/2098 BitwiseDP 47
• d[cur][visited]: i visited Problem: https://www.acmicpc.net/problem/2098 BitwiseDP 48
• d[cur][visited]: i visited Problem: https://www.acmicpc.net/problem/2098 BitwiseDP visited: 101 = 01100101(2) 49
• d[cur][visited]: i visited Problem: https://www.acmicpc.net/problem/2098 BitwiseDP 7 1 8 50 visited: 101 = 01100101(2)
• 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
• 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
• 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
• 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
• • ( ) . Problem: https://www.acmicpc.net/problem/2098 BitwiseDP 55
• • ( ) . Problem: https://www.acmicpc.net/problem/2098 BitwiseDP 56 P1 P2 P3 P4 P5 P6
• • ( ) . Problem: https://www.acmicpc.net/problem/2098 BitwiseDP 57 P1 P2 P3 P4 P5 P6
• : O(N * N * 2N) • : O(N * 2N) • / : O(N) Problem: https://www.acmicpc.net/problem/2098 BitwiseDP 58
• 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
• N X M • 2 X 1, 1 X 2 • • 1 ≤ N, M ≤ 14 Problem: https://www.acmicpc.net/problem/1648 BitwiseDP 60
• , . 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 … … … …
• , 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 … … … …
• (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 … … … …
• 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 … … … …
• 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)
• (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 … … … …
• 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)
• 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)
• 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)
• 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 … …
• 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 … …
• 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 … …
• 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)
• 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)
• 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)
• 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)
• 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)
• 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)
• 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)
• (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)
• (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)
• (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)
• (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)
• (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)
• (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)
• D[i][status]: i – 1 , i M status • 0 ≤ status < 2M Problem: https://www.acmicpc.net/problem/1648 BitwiseDP 86
• i (if (status & 1) == 1) • d[i + 1][status >> 1] += d[i][status] Problem: https://www.acmicpc.net/problem/1648 BitwiseDP 87
• i (if (status & 1) == 1) • d[i + 1][status >> 1] += d[i][status] • i • • Problem: https://www.acmicpc.net/problem/1648 BitwiseDP 88
• (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
• (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
• 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
• : d[N * M][0] • N * M (N * M – 1) , • N * M Problem: https://www.acmicpc.net/problem/1648 BitwiseDP 92
• : O(N * M * 2M) • : O(N * M) • : O(2M) Problem: https://www.acmicpc.net/problem/1648 BitwiseDP 93
• C/C++: https://gist.github.com/Acka1357/473f67f678c484b19e5d5daabde7522c • Java: https://gist.github.com/Acka1357/3406495ff3597161c4245d9989dc3fa7 Problem: https://www.acmicpc.net/problem/1648 BitwiseDP 94
• N X M • 2 X 1, 1 X 2 • . • . • • 1 ≤ N, M ≤ 14 Problem: https://www.acmicpc.net/problem/1657 BitwiseDP 95
• == • , . Problem: https://www.acmicpc.net/problem/1657 BitwiseDP 96
• == • , . • , . Problem: https://www.acmicpc.net/problem/1657 BitwiseDP 97
• D[i][status]: i – 1 ( ) , i M status • 0 ≤ status < 2M Problem: https://www.acmicpc.net/problem/1657 BitwiseDP 98
• i (if (status & 1) == 1) • d[i + 1][status >> 1] <- d[i][status] Problem: https://www.acmicpc.net/problem/1657 BitwiseDP 99
• 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
• 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
• : d[N * M][0] • N * M (N * M – 1) ( ) , • N * M Problem: https://www.acmicpc.net/problem/1657 BitwiseDP 102
• : O(N * M * 2M) • : O(N * M) • : O(2M) Problem: https://www.acmicpc.net/problem/1657 BitwiseDP 103
• C/C++: https://gist.github.com/Acka1357/99cd03e1d98611f02840ab9708052c99 • Java: https://gist.github.com/Acka1357/6d5dbdcc1528957a2896a703f6fa796c Problem: https://www.acmicpc.net/problem/1657 BitwiseDP 104

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알고리즘 연합캠프 세미나 1-B (Bitwise DP)

  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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