/
count-the-number-of-square-free-subsets.cpp
162 lines (154 loc) · 5.2 KB
/
count-the-number-of-square-free-subsets.cpp
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
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
// Time: O(n + m * 2^p)
// Space: O(m * 2^p)
// number theory, combinatorics, bitmasks, dp
class Solution {
public:
int squareFreeSubsets(vector<int>& nums) {
const int MAX_NUM = *max_element(cbegin(nums), cend(nums));
const auto& linear_sieve_of_eratosthenes = [](int n) { // Time: O(n), Space: O(n)
vector<int> spf(n + 1, -1);
vector<int> primes;
for (int i = 2; i <= n; ++i) {
if (spf[i] == -1) {
spf[i] = i;
primes.emplace_back(i);
}
for (const auto& p : primes) {
if (i * p > n || p > spf[i]) {
break;
}
spf[i * p] = p;
}
}
return primes;
};
const auto& PRIMES = linear_sieve_of_eratosthenes(MAX_NUM);
vector<int> MASKS(MAX_NUM + 1);
for (int x = 0; x <= MAX_NUM; ++x) {
for (int i = 0, y = x; i < size(PRIMES); ++i) {
if (y % PRIMES[i]) {
continue;
}
if (y % (PRIMES[i] * PRIMES[i]) == 0) {
MASKS[x] = 0;
break;
}
MASKS[x] |= (1 << i);
y /= PRIMES[i];
}
}
static const int MOD = 1e9 + 7;
const auto& powmod = [&](int a, int b) {
a %= MOD;
int64_t result = 1;
while (b) {
if (b & 1) {
result = result * a % MOD;
}
a = int64_t(a) * a % MOD;
b >>= 1;
}
return result;
};
unordered_map<int, int> cnt;
for (const auto& x : nums) {
++cnt[x];
}
vector<int> arr;
for (const auto& [k, _] : cnt) {
if (k != 1) {
arr.emplace_back(k);
}
}
vector<int> dp(1 << size(PRIMES), 1);
for (const auto& x : arr) {
if (!MASKS[x]) {
continue;
}
for (int mask = size(dp) - 1; mask >= 0; --mask) {
if ((MASKS[x] & mask) == 0) {
dp[mask | MASKS[x]] = (dp[mask | MASKS[x]] + static_cast<int64_t>(cnt[x]) * dp[mask]) % MOD;
}
}
}
return cnt.count(1) ? ((dp.back() * powmod(2, cnt[1]) - 1) + MOD) % MOD: ((dp.back() - 1) + MOD) % MOD;
}
};
// Time: O(n + m * 2^p)
// Space: O(m * 2^p)
// number theory, combinatorics, bitmasks, memoization
class Solution2 {
public:
int squareFreeSubsets(vector<int>& nums) {
const int MAX_NUM = *max_element(cbegin(nums), cend(nums));
const auto& linear_sieve_of_eratosthenes = [](int n) { // Time: O(n), Space: O(n)
vector<int> spf(n + 1, -1);
vector<int> primes;
for (int i = 2; i <= n; ++i) {
if (spf[i] == -1) {
spf[i] = i;
primes.emplace_back(i);
}
for (const auto& p : primes) {
if (i * p > n || p > spf[i]) {
break;
}
spf[i * p] = p;
}
}
return primes;
};
const auto& PRIMES = linear_sieve_of_eratosthenes(MAX_NUM);
vector<int> MASKS(MAX_NUM + 1);
for (int x = 0; x <= MAX_NUM; ++x) {
for (int i = 0, y = x; i < size(PRIMES); ++i) {
if (y % PRIMES[i]) {
continue;
}
if (y % (PRIMES[i] * PRIMES[i]) == 0) {
MASKS[x] = 0;
break;
}
MASKS[x] |= (1 << i);
y /= PRIMES[i];
}
}
static const int MOD = 1e9 + 7;
const auto& powmod = [&](int a, int b) {
a %= MOD;
int64_t result = 1;
while (b) {
if (b & 1) {
result = result * a % MOD;
}
a = int64_t(a) * a % MOD;
b >>= 1;
}
return result;
};
unordered_map<int, int> cnt;
for (const auto& x : nums) {
++cnt[x];
}
vector<int> arr;
for (const auto& [k, _] : cnt) {
if (k != 1) {
arr.emplace_back(k);
}
}
vector<vector<int>> dp(size(arr), vector<int>(1 << size(PRIMES), -1));
const function<int(int, int)> memoization = [&](int i, int mask) {
if (i == size(arr)) {
return 1;
}
if (dp[i][mask] == -1) {
dp[i][mask] = memoization(i + 1, mask);
if (MASKS[arr[i]] && (MASKS[arr[i]] & mask) == MASKS[arr[i]]) {
dp[i][mask] = (dp[i][mask] + static_cast<int64_t>(cnt[arr[i]]) * memoization(i + 1, mask ^ MASKS[arr[i]])) % MOD;
}
}
return dp[i][mask];
};
return cnt.count(1) ? ((memoization(0, (1 << size(PRIMES)) - 1) * powmod(2, cnt[1]) - 1) + MOD) % MOD: ((memoization(0, (1 << size(PRIMES)) - 1) - 1) + MOD) % MOD;
}
};