Skip to content

「二項係数の和（オンライン）」の空間計算量オーダーを改善 #107

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Merged
NotLeonian merged 1 commit into from
Jul 8, 2026
+58 −29
Merged

「二項係数の和（オンライン）」の空間計算量オーダーを改善 #107

Show file tree
Hide file tree
Changes from all commits
Commits
Show all changes
1 commit
Select commit
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
4 changes: 1 addition & 3 deletions docs/math/combinatorics/online-binomial-sum.md
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -8,8 +8,6 @@ documentation_of: math/combinatorics/online-binomial-sum.hpp
- 整数 $n,m$ と重み $r$ に対する二項係数の prefix sum を $\displaystyle F(n,m)=\sum_{i=0}^{n-1}r^i\binom{m}{i}$ とおく。
- 半開区間の左端、右端をそれぞれ $l,u$ とする。 $\displaystyle \sum_{i=l}^{u-1}r^i\binom{m}{i}$ をオンラインで求める。
- $\binom{m}{i}=0\;(i>m)$ として扱う。
- 前計算を行う長さを $k$ とする。 $m$ をバケットに分け、境界を $m_0$ とおいて $F(k,m_0)$ を前計算する。
- $d=m-m_0$ とおく。 $m=m_0+d$ では $\displaystyle F(n,m)=\sum_{j=0}^{d}r^j\binom{d}{j}F(n-j,m_0)$ を使う。
- $r=0$ や $r=-1$ でも $r+1$ による除算は行わない。

## 使い方
Expand All @@ -33,6 +31,6 @@ documentation_of: math/combinatorics/online-binomial-sum.hpp

`max_m` を $M$ とおく。

- コンストラクタ: 時間、空間 $O(M\sqrt M)$
- コンストラクタ: 時間 $O(M\sqrt M)$ 、空間 $O(M)$
- `binom_prefix_sum(n, m)`: 時間 $O(\sqrt M)$
- `binom_sum(l, u, m)`: 時間 $O(\sqrt M)$
83 changes: 57 additions & 26 deletions math/combinatorics/online-binomial-sum.hpp
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -3,10 +3,12 @@

// Σ_{i=l}^{u-1} r^i binom(m,i) をオンラインで求める。
// 0 <= l <= u と 0 <= m <= max_m を仮定する。
// m のバケット境界の累積和と、バケット内の二項係数を前計算する。
// m のバケット境界の累積和を n のバケット境界でサンプルし、
// バケット内の二項係数を前計算する。
// T は四則演算を持つ型で、std::numeric_limits<T>::is_integer が
// false の場合は T(1) から T(max_m) で除算できる。
// 計算量は前計算 O(max_m √max_m)、クエリ O(√max_m)。
// 時間計算量は前計算 O(max_m √max_m)、クエリ O(√max_m)。
// 空間計算量は O(max_m)。

#include <cassert>
#include <limits>
Expand All @@ -18,8 +20,10 @@ template <class T> struct OnlineBinomialSum {
T r;
std::vector<int> prefix_sum_offset;
std::vector<T> prefix_sum_table;
std::vector<T> prefix_term_table;
std::vector<int> weighted_binomial_offset;
std::vector<T> weighted_binomial_table;
std::vector<T> integer_inverse;

explicit OnlineBinomialSum(int m, T r = T(1)) : max_m(m), r(r) {
assert(max_m >= 0);
Expand Down Expand Up @@ -50,37 +54,35 @@ template <class T> struct OnlineBinomialSum {
d - 1];
}

const int bucket_count = max_m / bucket_size + 1;
prefix_sum_offset.assign(bucket_count + 1, 0);
for (int b = 0; b < bucket_count; ++b) {
const int base = b * bucket_size;
int last = base + bucket_size;
if (last > max_m + 1) {
last = max_m + 1;
}
prefix_sum_offset[b + 1] = prefix_sum_offset[b] + last + 1;
}
prefix_sum_table.assign(prefix_sum_offset[bucket_count], T());

std::vector<T> integer_inverse;
if constexpr (!std::numeric_limits<T>::is_integer) {
integer_inverse.assign(max_m + 1, T());
for (int i = 1; i <= max_m; ++i) {
integer_inverse[i] = T(1) / T(i);
}
}

const int bucket_count = max_m / bucket_size + 1;
prefix_sum_offset.assign(bucket_count + 1, 0);
for (int b = 0; b < bucket_count; ++b) {
prefix_sum_offset[b + 1] = prefix_sum_offset[b] + b + 1;
}

prefix_sum_table.assign(prefix_sum_offset[bucket_count], T());
prefix_term_table.assign(prefix_sum_offset[bucket_count], T());

for (int b = 0; b < bucket_count; ++b) {
const int base = b * bucket_size;
const int offset = prefix_sum_offset[b];
const int row_size = prefix_sum_offset[b + 1] - offset;
T sum = T();
T term = T(1);
prefix_sum_table[offset] = T();
for (int i = 0; i <= base; ++i) {
sum += term;
prefix_sum_table[offset + i + 1] = sum;
if (i % bucket_size == 0) {
const int q = i / bucket_size;
prefix_sum_table[offset + q] = sum;
prefix_term_table[offset + q] = term;
}
if (i < base) {
sum += term;
term *= r;
term *= T(base - i);
if constexpr (std::numeric_limits<T>::is_integer) {
Expand All @@ -90,9 +92,6 @@ template <class T> struct OnlineBinomialSum {
}
}
}
for (int i = base + 2; i < row_size; ++i) {
prefix_sum_table[offset + i] = sum;
}
}
}

Expand All @@ -106,15 +105,47 @@ template <class T> struct OnlineBinomialSum {
if (n > m) {
n = m + 1;
}

const int bucket = m / bucket_size;
const int base = bucket * bucket_size;
const int d = m - base;
const int prefix_offset = prefix_sum_offset[bucket];
const int weight_offset = weighted_binomial_offset[d];

int last_j = d;
if (last_j >= n) {
last_j = n - 1;
}
const int first_n = n - last_j;

int sample_index = first_n / bucket_size;
if (sample_index > bucket) {
sample_index = bucket;
}
const int sample_n = sample_index * bucket_size;
const int sample_offset = prefix_sum_offset[bucket] + sample_index;

T sum = prefix_sum_table[sample_offset];
T term = prefix_term_table[sample_offset];
T ans = T();
for (int j = 0; j <= d && j < n; ++j) {
ans += weighted_binomial_table[weight_offset + j] *
prefix_sum_table[prefix_offset + n - j];
for (int current_n = sample_n; current_n <= n; ++current_n) {
if (current_n >= first_n) {
const int j = n - current_n;
ans += weighted_binomial_table[weight_offset + j] * sum;
}
if (current_n < n) {
sum += term;
if (current_n < base) {
term *= r;
term *= T(base - current_n);
if constexpr (std::numeric_limits<T>::is_integer) {
term /= T(current_n + 1);
} else {
term *= integer_inverse[current_n + 1];
}
} else {
term = T();
}
}
}
return ans;
}
Expand Down
Loading