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
use crate::integer_square_root_with_binary_search_u64::isqrt;
/// Compute \pi(n)
/// O(N^{3/4}/log{N})
/// reference
/// - https://judge.yosupo.jp/submission/61553
pub fn prime_pi_fast_optimized(n: u64) -> u64 {
if n < 2 {
return 0;
}
if n == 2 {
return 1;
}
let half = |i: usize| (i - 1) >> 1;
let sqrt = isqrt(n) as usize;
let n = n as usize;
let mut size = (sqrt + 1) >> 1;
// for memory saving. do not have space for even numbers.
let mut small: Vec<usize> = (0..size).collect();
// j=1, 3, 5, 7, ..., k=0, 1, 2, 3
// -> unsieved count less than or equal to j is (j - 1) >> 1 = k.
let mut large: Vec<usize> =
(0..size).map(|i| half(n / (i << 1 | 1))).collect();
// (j - 1) >> 1 = k <-> (k << 1 | 1) = j
let mut unsieved_nums: Vec<usize> = (0..size).map(|i| i << 1 | 1).collect();
// 1initially, 1, 3, 5, ... (odd at most sqrt(n))
// unsieved_nums[..size] are odd integers which are still unsieved.
// (size will be updated in each iteration)
// unsieved_nums[size..] are no longer used.
let mut checked_or_sieved = vec![false; size];
// 1, 2 -> 0, 3, 4 -> 1, ... (because even numbers are skipped.)
let mut pi = 0;
for i in (3..=sqrt).step_by(2) {
if checked_or_sieved[half(i)] {
// sieved
continue;
}
let i2 = i * i;
if i2 * i2 > n {
break;
}
checked_or_sieved[half(i)] = true; // checked
for j in (i2..=sqrt).step_by(i << 1) {
checked_or_sieved[half(j)] = true;
}
// update large and unsieved_nums
let mut ptr = 0;
for k in 0..size {
let j = unsieved_nums[k];
if checked_or_sieved[half(j)] {
continue;
}
let border = j * i;
large[ptr] = large[k]
- if border <= sqrt {
large[small[border >> 1] - pi]
} else {
small[half(n / border)]
}
+ pi;
unsieved_nums[ptr] = j;
ptr += 1;
}
size = ptr;
let mut j = half(sqrt);
let mut k = sqrt / i - 1 | 1;
while k >= i {
let c = small[k >> 1] - pi;
let e = k * i >> 1;
while j >= e {
small[j] -= c;
j -= 1;
}
k -= 2;
}
pi += 1;
}
// be careful of overflow.
large[0] += if pi > 0 {
size + ((pi - 1) << 1)
} else {
// -1 << 1 == -2
size.saturating_sub(2)
// if size == 1,
// (size + ((pi - 1) << 1)) * (size - 1) >> 1 == 0
// regardless of `size + ((pi - 1) << 1)`
} * (size - 1)
>> 1;
for k in 1..size {
large[0] -= large[k];
}
for k in 1..size {
let q = unsieved_nums[k];
let n_q = n / q;
let e = small[half(n_q / q)] - pi;
if e < k + 1 {
break;
}
let mut t = 0;
for l in k + 1..=e {
t += small[half(n_q / unsieved_nums[l])];
}
large[0] += t - (e - k) * (pi + k - 1);
}
large[0] as u64 + 1
}
#[cfg(test)]
mod tests {
#[test]
fn test() {
use super::*;
use crate::test_fast_prime_counting::test_fast_prime_counting;
test_fast_prime_counting(&prime_pi_fast_optimized);
}
}