cf-Sasha and Array

题目链接 http://codeforces.com/problemset/problem/719/E

 

解题思路

矩阵上的线段树。

因为矩阵有分配律（A+B）C = AC + BC，所以计算总和时直接把增量矩阵乘上去就行了。用矩阵快速幂。

fib的计算尽量拉到主函数计算。

 

代码

#include<stdio.h>
#include<string.h>#define MAX_SIZE 100010
const int MOD_NUM = 1e9 + 7;
typedef long long ll;
struct mat {
    mat() {}
    mat(int a, int b, int c, int d)
    { ans[0][0] = a; ans[0][1] = b; ans[1][0] = c; ans[1][1] = d; } 
    int ans[2][2];
    bool JudgeOne()
    {
        for(int i=0; i<2; i++)
        for(int j=0; j<2; j++)
            if(i == j && ans[i][j] != 1 || i != j && ans[i][j] != 0)
                return false;
        return true;
    }
    void SetOne()
    {
        for(int i=0; i<2; i++)
            for(int j=0; j<2; j++)
            ans[i][j] = (i == j) ? 1 : 0;
    }
    mat operator *(const mat &b)const
    {
        mat c(0,0,0,0);
        for(int k=0; k<2; k++)
        for(int i=0; i<2; i++)
            for(int j=0; j<2; j++)
            c.ans[i][j] = (c.ans[i][j] + (ll)ans[i][k] * b.ans[k][j]) % MOD_NUM;
        return c;
    }
    mat operator +(const mat &b)const
    {
        mat c;
        for(int i=0; i<2; i++)
        for(int j=0; j<2; j++)
            c.ans[i][j] = (ans[i][j] + b.ans[i][j]) % MOD_NUM; 
        return c;
    }
};
struct node {
    mat sum; 
    mat tempt;    
}tree[4*MAX_SIZE];
mat a[MAX_SIZE];
mat temp;
mat cal_fib(int x)
{
    mat c; mat t(0,1,1,1);
    c.SetOne();
    while(x) {
        if(x & 1) c = c * t;
        t = t * t;
        x  = x >> 1;
    }
    return c;
}
void build(int l, int r, int root)
{
    tree[root].tempt.SetOne();
    if(l == r) {
        tree[root].sum = a[l];
        return ;
    }
    int mid = (l + r) / 2;
    build(l, mid, root*2);
    build(mid+1, r, root*2+1);
    tree[root].sum = tree[root*2].sum + tree[root*2+1].sum;
}
void down(int root)
{
    tree[root*2].sum = tree[root*2].sum * tree[root].tempt;
    tree[root*2].tempt = tree[root*2].tempt * tree[root].tempt;
    tree[root*2+1].sum = tree[root*2+1].sum * tree[root].tempt;
    tree[root*2+1].tempt = tree[root*2+1].tempt * tree[root].tempt;
    tree[root].tempt.SetOne();
}
void query_1(int l, int r, int L, int R, int root, mat value)
{
    if(L == l && r == R) {
        tree[root].tempt = tree[root].tempt * value;
        tree[root].sum = tree[root].sum * value;
        return ;
    }
    if(!tree[root].tempt.JudgeOne()) {
        down(root);
    }
    int mid = (L + R) / 2;
    if(r <= mid) query_1(l, r, L, mid, root*2, value);
    else if(l > mid) query_1(l, r, mid+1, R, root*2+1, value);
    else {
        query_1(l, mid, L, mid, root*2, value);
        query_1(mid+1, r, mid+1, R, root*2+1, value);
    }
    tree[root].sum = tree[root*2].sum + tree[root*2+1].sum;
}
int query_2(int l, int r, int L, int R, int root)
{
    if(L == l && r == R) {
        return tree[root].sum.ans[0][1];
    }
    if(!tree[root].tempt.JudgeOne()) {
        down(root);
    }
    int ret = 0;
    int mid = (L + R) / 2;
    if(r <= mid) ret = query_2(l, r, L, mid, root*2);
    else if(l > mid) ret = query_2(l, r, mid+1, R, root*2+1);
    else {
        ret = query_2(l, mid, L, mid, root*2) + query_2(mid+1, r, mid+1, R, root*2+1) % MOD_NUM;
    }
    return ret % MOD_NUM;
}
int main()
{
    int n, m, d;
    scanf("%d%d", &n, &m);
    for(int i=1; i<=n; i++) {
        mat base(0, 1, 1, 1);
        scanf("%d", &d); a[i] = cal_fib(d);
    }
    build(1, n, 1);
    for(int i=0; i<m; i++) {
        int type, l, r, x;
        scanf("%d", &type);
        if(type == 1) {
            mat base(0, 1, 1, 1);
            scanf("%d%d%d", &l, &r, &x);
            temp = cal_fib(x);
            query_1(l, r, 1, n, 1, temp);
        }
        else {
            scanf("%d%d", &l, &r);
            printf("%d\n", query_2(l, r, 1, n, 1));
        }
    }
    return 0;
}