codeforces 718c 矩阵快速幂套线段树
题意 给出一个序列ai,定义斐波那契数列,求sigma f(ai)
还有可能令一段ai加上一个数字。
斐波那契数列可以有矩阵快速幂加速求出,复杂度logn,我们这里定义mat为加速矩阵,那么对一段求和便可以写成sigma mati,对于一段区间加上一个数字,实际上等同于给每一个mat在乘上一个矩阵。
实际上变成维护区间加和的题目,只不过每一个节点储存一个矩阵。
#include<iostream>
#include<cstring>
#include<algorithm>
#include<cmath>
#include<cstdlib>
#include<climits>
#include<stack>
#include<vector>
#include<queue>
#include<set>
#include<map>
//#include<regex>
#include<cstdio>
#define up(i,a,b) for(int i=a;i<b;i++)
#define dw(i,a,b) for(int i=a;i>b;i--)
#define upd(i,a,b) for(int i=a;i<=b;i++)
#define dwd(i,a,b) for(int i=a;i>=b;i--)
//#define local
typedef long long ll;
const double esp = 1e-6;
const double pi = acos(-1.0);
const int INF = 0x3f3f3f3f;
const int inf = 1e9;
using namespace std;
ll read()
{
char ch = getchar(); ll x = 0, f = 1;
while (ch<'0' || ch>'9') { if (ch == '-')f = -1; ch = getchar(); }
while (ch >= '0' && ch <= '9') { x = x * 10 + ch - '0'; ch = getchar(); }
return x * f;
}
typedef pair<int, int> pir;
#define lson l,mid,root<<1
#define rson mid+1,r,root<<1|1
#define lrt root<<1
#define rrt root<<1|1
const int N = 1e5 + 10;
const int mod = 1e9 + 7;
struct matrix {
ll a[2][2];
void init() { memset(a, 0, sizeof(a)); }
void initfib() { a[0][0] = a[0][1] = a[1][0] = 1; a[1][1] = 0; }
void one() { a[0][0] = a[1][1] = 1; a[0][1] = a[1][0] = 0; }
matrix operator +(const matrix &m)const {
matrix b; b.init();
up(i, 0, 2)up(j, 0, 2)b.a[i][j] = (a[i][j] + m.a[i][j]) % mod; return b;
}
matrix operator *(const matrix &m)const {
matrix b; b.init();
up(i, 0, 2)up(j, 0, 2)up(k, 0, 2)b.a[i][j] = (b.a[i][j] + a[i][k]* m.a[k][j] % mod) % mod;return b;
}
matrix operator ^(const ll &k)const
{
ll num = k;
matrix res, now; res.one(); now = *this;
while (num)
{
if (num & 1)res = res * now;
now = now * now;
num >>= 1;
}
return res;
}
}tree[N<<2],lazy[N<<2];
matrix isone;
bool same(matrix a, matrix b) { up(i, 0, 2)up(j, 0, 2)if (a.a[i][j] != b.a[i][j])return false; return true; }
void pushup(int root)
{
tree[root] = tree[lrt] + tree[rrt];
}
void pushdown(int root)
{
if (!same(isone, lazy[root]))
{
lazy[lrt] = lazy[lrt] * lazy[root];
lazy[rrt] = lazy[rrt] * lazy[root];
tree[lrt] = tree[lrt] * lazy[root];
tree[rrt] = tree[rrt] * lazy[root];
lazy[root] = isone;
}
}
void build(int l, int r, int root)
{
lazy[root].one();
if (l == r)
{
tree[root].one(); return;
}
int mid = (l + r) >> 1;
build(lson);
build(rson);
pushup(root);
}
void update(int l,int r,int root,int lf,int rt,matrix x)
{
if (lf <= l && r <= rt)
{
tree[root] = tree[root] * x;
lazy[root] = lazy[root] * x;
return;
}
pushdown(root);
int mid = (l + r) >> 1;
if (lf <= mid)update(lson, lf, rt, x);
if (rt > mid)update(rson, lf, rt, x);
pushup(root);
}
matrix querry(int l, int r, int root,int lf,int rt)
{
if (lf <= l && r <= rt)
{
return tree[root];
}
pushdown(root);
int mid=(l + r) >> 1;
matrix ans; ans.init();
if (lf <= mid)ans = ans + querry(lson, lf, rt);
if (rt > mid)ans = ans + querry(rson, lf, rt);
return ans;
}
int n, m;
ll a[N];
int main()
{
n = read(), m = read(); isone.one();
ll tp, l, r,x;
build(1, n, 1);
up(i, 0, n)a[i] = read();
matrix tmp; tmp.initfib();
up(i, 0, n)tmp = tmp ^ a[i], update(1,n, 1,i + 1, i + 1, tmp), tmp.initfib();
while (m--)
{
tp = read();
if (tp == 1) {
l = read(), r = read(), x = read(); matrix tmp; tmp.initfib(); tmp = tmp ^ x; update(1, n, 1, l, r, tmp);
}
else
{
l = read(), r = read(); printf("%lld\n", querry(1, n, 1, l,r).a[1][0]);
}
}
return 0;
}
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