Description

Transmission Gate

你需要維護一個長度為\(n \leq 300000\) 的數列，茲詞兩個操作：

1.給一個區間加上一個fibonacci數列，規定\(f[0] = 0, f[1] = 1, f[2] = 1\)

2.查詢一段區間的和。對1e9+9取模

操作個數m不超過300000.

Solution

? 這一題其實要考慮Fibonacci數列的兩個性質:

? (i)\(\sum_{i = 1}^{n} fib(i) = f(n + 2) - 1\)

? (ii)令\(S_i = S_{i - 1} + S_{i - 2}\), 其中\(S_1 = a, S_2 = b\)

? 那麽\(S_i = aFib(i - 1) + bFib(i - 2)\)

? 那麽就很好做了， 在線段樹中， 我們要只要記數列的前兩項就可以方便的對數列進行求和，pushdown等操作.

? 這題細節比較多，相關部分見代碼。

Code

#include<bits/stdc++.h>
using namespace std;
#define rep(i, a, b) for(int i = (a), i##_end_ = (b); i <= i##_end_; ++i)
#define drep(i, a, b) for(int i = (a), i##_end_ = (b); i >= i##_end_; --i)
#define clar(a, b) memset((a), (b), sizeof(a))
#define debug(...) fprintf(stderr, __VA_ARGS__)
#define Debug(s) debug("The massage in line %d, Function %s: %s\n", __LINE__, __FUNCTION__, s)
typedef long long LL;
typedef long double LD;
const int BUF_SIZE = (int)1e6 + 10;
struct fastIO {
    char buf[BUF_SIZE], buf1[BUF_SIZE];
    int cur, cur1;
    FILE *in, *out;
    fastIO() {
        cur = BUF_SIZE, in = stdin, out = stdout;
        cur1 = 0;
    }
    inline char getchar() {
        if(cur == BUF_SIZE) fread(buf, BUF_SIZE, 1, in), cur = 0;
        return *(buf + (cur++));
    }
    inline void putchar(char ch) {
        *(buf1 + (cur1++)) = ch;
        if (cur1 == BUF_SIZE) fwrite(buf1, BUF_SIZE, 1, out), cur1 = 0;
    }
    inline int flush() {
        if (cur1 > 0) fwrite(buf1, cur1, 1, out);
        return cur1 = 0;
    }
}IO;
#define getchar IO.getchar
#define putchar IO.putchar
int read() {
    char ch = getchar();
    int x = 0, flag = 1;
    for(;!isdigit(ch); ch = getchar()) if(ch == ‘-‘) flag *= -1;
    for(;isdigit(ch); ch = getchar()) x = x * 10 + ch - 48;
    return x * flag;
}
void write(int x) {
    if(x < 0) putchar(‘-‘), x = -x;
    if(x >= 10) write(x / 10);
    putchar(x % 10 + 48);
}
void putString(char s[], char EndChar = ‘\n‘) {
    rep(i, 0, strlen(s) - 1) putchar(*(s + i));
    if(~EndChar) putchar(EndChar);
}

#define Maxn 300009
const LL Mod = 1000000009;
int n, a[Maxn], m; LL f[Maxn];
struct node { LL x, y; };
LL FibDelta(LL F1, LL F2, LL len) {
    LL res = 0;
    if(len >= 1) (res += F1) %= Mod;
    if(len >= 2) (res += F2) %= Mod;
    if(len >= 3) (res += (F2 * (f[len + 1] - 2ll) % Mod + F1 * (f[len] - 1ll) % Mod) % Mod) %= Mod;
    return res;
}// It needs to analysis in different conditions
node FibForward(node s, int len) {
    LL c = s.x * f[len] % Mod + s.y * f[len + 1] % Mod, d = s.x * f[len + 1] % Mod + s.y * f[len + 2] % Mod;
    return (node){c % Mod, d % Mod};
}
namespace SGMT_tree {
    LL tree[Maxn << 2], beg[Maxn << 2][2], val[Maxn << 2];
#define lc(x) (x) << 1
#define rc(x) (x) << 1 | 1
#define ls rt << 1, l, mid
#define rs rt << 1 | 1, mid + 1, r
    void pushup(int rt) { tree[rt] = (1ll * tree[lc(rt)] + 1ll * tree[rc(rt)]) % Mod; }
    void pushdown(int rt, int l, int r) {
        int mid = (l + r) >> 1;
        if(val[rt]) {
            LL &a = beg[rt][0], &b = beg[rt][1];
            (beg[lc(rt)][0] += a) %= Mod, (beg[lc(rt)][1] += b) %= Mod;
            (tree[lc(rt)] += FibDelta(a, b, mid - l + 1)) %= Mod;
            val[lc(rt)] = 1;

            LL c = a * f[mid - l] + b * f[mid - l + 1], d = a * f[mid - l + 1] + b * f[mid - l + 2];
            c %= Mod, d %= Mod;
            (beg[rc(rt)][0] += c) %= Mod, (beg[rc(rt)][1] += d) %= Mod;
            (tree[rc(rt)] += FibDelta(c, d, r - mid)) %= Mod;
            val[rc(rt)] = 1;
            beg[rt][0] = beg[rt][1] = 0;
            val[rt] = 0;
        }
    }
    void build(int rt, int l, int r) {
        if(l == r) { 
            tree[rt] = a[l]; 
            return ; 
        }
        int mid = (l + r) >> 1;
        build(ls), build(rs);
        pushup(rt);
    }
    void modify(int rt, int l, int r, int x, int y, int p1, int p2) {
        if(x <= l && r <= y) {
            (beg[rt][0] += p1) %= Mod, (beg[rt][1] += p2) %= Mod;
            (tree[rt] += FibDelta(p1, p2, r - l + 1)) %= Mod;
            val[rt] = 1;
            return ;
        }

        int mid = (l + r) >> 1; 
        pushdown(rt, l, r);

        if(y <= mid)  modify(ls, x, y, p1, p2); 
        else if(mid + 1 <= x)  modify(rs, x, y, p1, p2);
        else {
            modify(ls, x, mid, p1, p2); /**/
            node z = FibForward((node){p1, p2}, mid - x);/**/
            //F[i] -> F[i + mid - x]: p1 -> z.x, p2 -> z.y
            modify(rs, mid + 1, y, z.x, z.y);/*The InterVal needs to move*/
        }
        pushup(rt);
    }
    int query(int rt, int l, int r, int x, int y) {
        if(x <= l && r <= y) return tree[rt];

        int mid = (l + r) >> 1; pushdown(rt, l, r);

        if(y <= mid) return query(ls, x, y);
        else if(mid + 1 <= x) return query(rs, x, y);
        else return (1ll * query(ls, x, y) + 1ll * query(rs, x, y)) % Mod;
    }
#undef lc
#undef rc
#undef ls
#undef rs
}
namespace INIT {
    void Main() {
        n = read(), m = read();

        f[1] = 1; f[2] = 1;
        rep(i, 3, n + 4) f[i] = (f[i - 1] * 1ll + f[i - 2]) % Mod;

        rep(i, 1, n) a[i] = read();

        SGMT_tree :: build(1, 1, n);
    }
}
namespace SOLVE {
    void Main() {
        rep(i, 1, m) {
            int opt = read();

            if(opt == 1) {
                int x = read(), y = read();
                SGMT_tree :: modify(1, 1, n, x, y, 1, 1);
            }
            if(opt == 2) {
                int x = read(), y = read();
                write(SGMT_tree :: query(1, 1, n, x, y)), putchar(‘\n‘);    
            }
        }
    }
}
int main() {
    freopen("CF446C.in", "r", stdin);
    freopen("CF446C.out", "w", stdout);

    INIT :: Main();
    SOLVE :: Main();
#ifdef Qrsikno
    debug("\nRunning time: %.3lf(s)\n", clock() * 1.0 / CLOCKS_PER_SEC);
#endif
    return IO.flush();
}
 

CF446C [DZY loves Fibonacci]