Portal
Portal1: Codeforces
Portal2: Luogu
Description
In mathematical terms, the sequence (F_n) of Fibonacci numbers is defined by the recurrence relation
DZY loves Fibonacci numbers very much. Today DZY gives you an array consisting of (n) integers: (a1, a2, cdots , an). Moreover, there are (m) queries, each query has one of the two types:
-
Format of the query "
1 l r". In reply to the query, you need to add (F_i - l + 1) to each element ai, where (l le i le r). -
Format of the query "
2 l r". In reply to the query you should output the value of modulo (1000000009 (10^9 + 9)).
Help DZY reply to all the queries.
Input
The first line of the input contains two integers (n) and (m (1 le n, m le 300000)). The second line contains (n) integers (a_1, a_2, cdots , a_n (1 le ai le 10^9)) — initial array (a) .
Then, (m) lines follow. A single line describes a single query in the format given in the statement. It is guaranteed that for each query inequality (1 le l le r le n) holds.
Output
For each query of the second type, print the value of the sum on a single line.
Sample Input
4 4
1 2 3 4
1 1 4
2 1 4
1 2 4
2 1 3
Sample Output
17
12
Sample Explain
After the first query, (a = [2, 3, 5, 7]).
For the second query, (sum = 2 + 3 + 5 + 7 = 17).
After the third query, (a = [2, 4, 6, 9]).
For the fourth query, (sum = 2 + 4 + 6 = 12).
Description in Chinese
题目让我们求给你一个序列，支持区间加Fibonacci数列前r - l + 1项和查询区间和。
Solution
一些约定：把斐波那契数列的前两个数(F_1 = 1, F_2 = 1)换成另两个数，仍满足(F_n = F_{n - 1} + F_{n - 2}(n > 2))的数列称为广义斐波那契数列。
Fibonacci数列有一些性质：
性质(1). (F_n = (sum^{n - 2}_{i = 1}{F_i}) + F_2(n > 2))；
证明如下：
首先将前几项Fibonacci数列展开。
F(1) = 1
F(2) = 1
F(3) = F(1) + F(2)
F(4) = F(2) + F(3) = F(2) + F(1) + F(2)
F(5) = F(3) + F(4) = F(3) + F(2) + F(1) + F(2)
F(6) = F(4) + F(5) = F(4) + F(3) + F(2) + F(1) + F(2)
......
在(F_n = F_{n - 1} + F_{n - 2})中，我们可以把(F_{n - 1})按式子展开，可得(F_n = sum^{n - 3}_{i = 1} + F_2 + F_{n - 2})，即(F_n = (sum^{n - 2}_{i = 1}{F_i}) + F_2(n > 2))，跟原式一模一样，故原式正确性得证。
性质(2). 一个广义斐波那契数列数列(f_i)， 当(f_1 = x, f_2 = y)时，则有(f_n = x imes f_{n - 1} + y imes f_{n - 2})
证明如下：
这个性质与性质1类似，证明方法也与性质1类似，列举几个：
f(1) = x
f(2) = y
f(3) = f(1) + f(2) = x × F(1)
f(4) = f(2) + f(3) = x × F(1) + y × F(2)
f(5) = f(3) + f(4) = x × F(2) + y × F(3)
f(6) = f(4) + f(5) = x × F(3) + y × F(4)
......
把上述规律推广到代数式：
证毕。
性质(3): 任意两段不同的广义斐波那契数列段相加（逐项相加），所得的数列任然是广义斐波那契数列。
这个性质易证。
这题我们维护一棵线段树，线段树需要维护(L)至(R)区间的广义斐波那契数列的第一项，第二项与区间的和。
下传标记时，我们可以在左区间加广义斐波那契数列的前两项，在右区间可以求出总和再加上总和就行了，时间复杂( ext{O(n log n)})。
Code
#include<iostream>
#include<algorithm>
#include<cstdio>
#include<cstring>
#include<cmath>
using namespace std;
typedef long long LL;
const int MAXN = 300005, MAXM = 1200005, mod = 1e9 + 9;
struct node {
int c1, c2, sum;
} tree[MAXM];
int n, m, opt, x, y, a[MAXN], f[MAXN];
inline int add(int x, int y) {//两项相加并取模
int ret = x + y;
if (ret < 0) return ret += mod; else return ret % mod;
}
inline int calc1(int x, int y, int len) {//计算斐波那契
if (len == 1) return x; else
if (len == 2) return y; else return ((LL)x * f[len - 2] + (LL)y * f[len - 1]) % mod;
}
inline int calc2(int x, int y, int len) {//计算总和
if (len == 1) return x; else
if (len == 2) return add(x, y); else return add(calc1(x, y, len + 2), -y);
}
inline void pushup(int rt) {
tree[rt].sum = add(tree[rt << 1].sum, tree[rt << 1 | 1].sum);
}
inline void pushdown(int rt, int l, int r) {//下传标记
if (tree[rt].c1) {
int mid = l + r >> 1;
tree[rt << 1].c1 = add(tree[rt << 1].c1, tree[rt].c1);
tree[rt << 1].c2 = add(tree[rt << 1].c2, tree[rt].c2);
tree[rt << 1].sum = add(tree[rt << 1].sum, calc2(tree[rt].c1, tree[rt].c2, mid - l + 1));
int x = calc1(tree[rt].c1, tree[rt].c2, mid - l + 2), y = calc1(tree[rt].c1, tree[rt].c2, mid - l + 3);
tree[rt << 1 | 1].c1 = add(tree[rt << 1 | 1].c1, x);
tree[rt << 1 | 1].c2 = add(tree[rt << 1 | 1].c2, y);
tree[rt << 1 | 1].sum = add(tree[rt << 1 | 1].sum, calc2(x, y, r - mid));
tree[rt].c1 = 0; tree[rt].c2 = 0;
}
}
inline void update(int rt, int l, int r, int ansl, int ansr) {//线段树区间更新
if (ansl <= l && r <= ansr) {
tree[rt].c1 = add(tree[rt].c1, f[l - ansl + 1]);
tree[rt].c2 = add(tree[rt].c2, f[l - ansl + 2]);
tree[rt].sum = add(tree[rt].sum, calc2(f[l - ansl + 1], f[l - ansl + 2], r - l + 1));
return ;
}
pushdown(rt, l, r);
int mid = l + r >> 1;
if (ansl <= mid) update(rt << 1, l, mid, ansl, ansr);
if (ansr > mid) update(rt << 1 | 1, mid + 1, r, ansl, ansr);
pushup(rt);
}
inline int query(int rt, int l, int r, int ansl, int ansr) {//线段树区间查询
int ret = 0;
if (ansl <= l && r <= ansr) {
ret = tree[rt].sum;
return ret;
}
pushdown(rt, l, r);
int mid = l + r >> 1;
if (ansl <= mid) ret = add(ret, query(rt << 1, l, mid, ansl, ansr));
if (ansr > mid) ret = add(ret, query(rt << 1 | 1, mid + 1, r, ansl, ansr));
return ret;
}
int main() {
scanf("%d%d", &n, &m);
for (int i = 1; i <= n; i++) {
scanf("%d", &x);
a[i] = add(a[i - 1], x);
}
f[1] = 1; f[2] = 1;
for (int i = 3; i <= n + 2; i++)
f[i] = add(f[i - 1], f[i - 2]);
for (int i = 1; i <= m; i++) {
scanf("%d%d%d", &opt, &x, &y);
if (opt == 1) update(1, 1, n, x, y); else printf("%d
", add(query(1, 1, n, x, y), a[y] - a[x - 1]));
}
return 0;
}