Codeforces 316E3 线段树 + 斐波那切数列 (看题解)

最关键的一点就是

f[ 0 ] * a[ 0 ] + f[ 1 ] * a[ 1 ] + ... + f[ n - 1] * a[ n  - 1]

f[ 1 ] * a[ 0 ] + f[ 2 ] * a[ 1 ] + ... + f[ n ] * a[ n  - 1]

f[ 2 ] * a[ 0 ] + f[ 3 ] * a[ 1 ] + ... + f[ n + 1] * a[ n  - 1]

......

这也是满足斐波那切的性质

也就是说，系数的斐波那切的多项式也能向斐波那切一样递推。

然后我们在线段树上保存系数为f[ 0 ]开始的多项式的值和系数为f[ 1 ]开始的多项式的值。

#include<bits/stdc++.h>
#define LL long long
#define fi first
#define se second
#define mk make_pair
#define PLL pair<LL, LL>
#define PLI pair<LL, int>
#define PII pair<int, int>
#define SZ(x) ((int)x.size())
#define ull unsigned long long

using namespace std;

const int N = 2e5 + ;
const int inf = 0x3f3f3f3f;
const LL INF = 0x3f3f3f3f3f3f3f3f;
const int mod = 1e9 + ;
const double eps = 1e-;
const double PI = acos(-);

void add(int& a, int b) {
    a += b; if(a >= mod) a -= mod;
}

int mat[N][][], f[N], g[N];

void print(int o) {
    puts("");
    for(int i = ; i < ; i++) {
        for(int j = ; j < ; j++) {
            printf("%d ", mat[o][i][j]);
        }
        puts("");
    }
}

#define lson l, mid, rt << 1
#define rson mid + 1, r, rt << 1 | 1
int f0[N << ], f1[N << ];
int lazy[N << ];

inline int getVal(int f0, int f1, int k) {
    if(k == ) return f0;
    else if(k == ) return f1;
    else return (1LL * mat[k - ][][] * f1 % mod + 1LL * mat[k - ][][] * f0 % mod) % mod;
}
void pull(int rt, int l, int r) {
    int mid = l + r >> ;
    f0[rt] = f0[rt << ];
    f1[rt] = f1[rt << ];
    add(f0[rt], getVal(f0[rt <<  | ], f1[rt <<  | ], mid - l + ));
    add(f1[rt], getVal(f0[rt <<  | ], f1[rt <<  | ], mid - l + ));
}
void push(int rt, int l, int r) {
    int mid = l + r >> ;
    if(lazy[rt]) {
        add(lazy[rt << ], lazy[rt]);
        add(lazy[rt <<  | ], lazy[rt]);
        add(f0[rt << ], 1LL * lazy[rt] * f[mid - l] % mod);
        add(f0[rt <<  | ], 1LL * lazy[rt] * f[r - mid - ] % mod);
        add(f1[rt << ], 1LL * lazy[rt] * g[mid - l] % mod);
        add(f1[rt <<  | ], 1LL * lazy[rt] * g[r - mid - ] % mod);
        lazy[rt] = ;
    }
}
void build(int l, int r, int rt) {
    if(l == r) {
        scanf("%d", &f0[rt]);
        f1[rt] = f0[rt];
        return;
    }
    int mid = l + r >> ;
    build(lson); build(rson);
    pull(rt, l, r);
}
void update(int L, int R, int d, int l, int r, int rt) {
    if(r < L || R < l) return;
    if(L <= l && r <= R) {
        add(f0[rt], 1LL * d * f[r - l] % mod);
        add(f1[rt], 1LL * d * g[r - l] % mod);
        add(lazy[rt], d);
        return ;
    }
    push(rt, l, r);
    int mid = l + r >> ;
    update(L, R, d, lson);
    update(L, R, d, rson);
    pull(rt, l, r);
}
int query(int L, int R, int l, int r, int rt) {
    if(r < L || R < l) return ;
    if(L <= l && r <= R) return getVal(f0[rt], f1[rt], l - L);
    push(rt, l, r);
    int mid = l + r >> ;
    return (query(L, R, lson) + query(L, R, rson)) % mod;
}

int n, m;
int main() {
    mat[][][] = ; mat[][][] = ;
    mat[][][] = ; mat[][][] = ;
    mat[][][] = mat[][][] = ;
    mat[][][] = ; mat[][][] = ;
    for(int o = ; o < N; o++) {
        for(int i = ; i < ; i++)
            for(int j = ; j < ; j++)
                for(int k = ; k < ; k++)
                    mat[o][i][j] = (mat[o][i][j] + 1LL * mat[][i][k] * mat[o - ][k][j] % mod) % mod;
    }
    f[] = f[] = ;
    for(int i = ; i < N; i++) f[i] = (f[i - ] + f[i - ]) % mod;
    for(int i = ; i < N; i++) add(f[i], f[i - ]);
    g[] = , g[] = ;
    for(int i = ; i < N; i++) g[i] = (g[i - ] + g[i - ]) % mod;
    for(int i = ; i < N; i++) add(g[i], g[i - ]);

    scanf("%d%d", &n, &m);
    build(, n, );
    while(m--) {
        int op;
        scanf("%d", &op);
        if(op == ) {
            int x, v;
            scanf("%d%d", &x, &v);
            int ret = query(x, x, , n, );
            update(x, x, -ret, , n, );
            update(x, x, v, , n, );
        } else if(op == ) {
            int L, R;
            scanf("%d%d", &L, &R);
            printf("%d\n", query(L, R, , n, ));
        } else {
            int L, R, d;
            scanf("%d%d%d", &L, &R, &d);
            update(L, R, d, , n, );
        }
    }
    return ;
}

/*
*/