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「Comet OJ 58D」菜菜种菜

题目链接

第 ii 个点，给定 xi≤i≤yix_i\le i\le y_i 和权值 aia_i

qq 个询问，回答

∑ai[L≤i≤R ∧ xi<L ∧ yi>R] \sum a_i[L\le i \le R\ \wedge\ x_i < L\ \wedge \ y_i>R]

n,q≤106n,q\le 10^6

首先这肯定不是一个图论问题

把所有土地排成一行，从左到右第 ii 个编号为 ii 。容易发现，一块土地 ii 能不能作为家，只跟他能直接到达的土地中，在它左边最靠右的一块（记作 ，xix_i 不存在则为 00 ）和在它右面最靠左的一块（记作 yiy_i ，不存在则为 n+1n+1 ）

询问 L,RL, R ，回答

f(L,R)=∑ai[L≤i≤R ∧ xi<L ∧ yi>R] f(L, R) = \sum a_i[L\le i \le R\ \wedge\ x_i < L\ \wedge \ y_i>R]

记

s(a,b,c,d,e,f)=∑ai[a≤i≤b ∧ c≤xi≤d ∧ e≤yi≤f] s(a,b,c,d,e,f)=\sum a_i[a\le i \le b\ \wedge\ c\le x_i \le d\ \wedge\ e\le y_i\le f]\\
f(L,R)=+s(L,R,0,L−1,R+1,n+1)=+s(L,R,0,n+1,0,n+1)−s(L,R,L,n+1,0,n+1)−s(L,R,0,n+1,0,R)+s(L,R,L,n+1,0,R) \begin{aligned} f(L,R) = &+s(L,R,0,L-1,R+1, n+1)\\ = &+s(L,R,0,n+1, 0, n+1)\\ &-s(L, R,L,n+1,0,n+1)\\ &-s(L,R,0,n+1,0,R)\\ &+s(L,R,L,n+1,0,R) \end{aligned}

一个容斥，最后因为 xi≥L,yi≤Rx_i\ge L, y_i \le R 被减了两次，需要加回来。

由 xi<i<yix_i<i<y_i

可得

L≤i≤R,xi≥L⇔i≤R,xi≥LL≤i≤R,yi≤R⇔i≥L,yi≤RL≤i≤R,xi≥L,yi≤R⇔xi≥L,yi≤R \begin{aligned} L\le i\le R, x_i\ge L&\Leftrightarrow i\le R,x_i\ge L\\ L\le i\le R, y_i\le R&\Leftrightarrow i\ge L,y_i\le R\\ L\le i\le R, x_i\ge L,y_i\le R&\Leftrightarrow x_i\ge L, y_i \le R \end{aligned}

所以问题变成了三个二维数点，用扫描线和树状数组即可解决

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#include <algorithm>
#include <bitset>
#include <cassert>
#include <cmath>
#include <cstdio>
#include <cstring>
#include <deque>
#include <iostream>
#include <map>
#include <queue>
#include <random>
#include <set>
#include <stack>
#include <string>
#include <utility>
#include <vector>
// #pragma GCC optimize("Ofast")
// #pragma GCC optimize("inline")
// #pragma GCC optimize("-fgcse")
// #pragma GCC optimize("-fgcse-lm")
// #pragma GCC optimize("-fipa-sra")
// #pragma GCC optimize("-ftree-pre")
// #pragma GCC optimize("-ftree-vrp")
// #pragma GCC optimize("-fpeephole2")
// #pragma GCC optimize("-ffast-math")
// #pragma GCC optimize("-fsched-spec")
// #pragma GCC optimize("unroll-loops")
// #pragma GCC optimize("-falign-jumps")
// #pragma GCC optimize("-falign-loops")
// #pragma GCC optimize("-falign-labels")
// #pragma GCC optimize("-fdevirtualize")
// #pragma GCC optimize("-fcaller-saves")
// #pragma GCC optimize("-fcrossjumping")
// #pragma GCC optimize("-fthread-jumps")
// #pragma GCC optimize("-funroll-loops")
// #pragma GCC optimize("-fwhole-program")
// #pragma GCC optimize("-freorder-blocks")
// #pragma GCC optimize("-fschedule-insns")
// #pragma GCC optimize("inline-functions")
// #pragma GCC optimize("-ftree-tail-merge")
// #pragma GCC optimize("-fschedule-insns2")
// #pragma GCC optimize("-fstrict-aliasing")
// #pragma GCC optimize("-fstrict-overflow")
// #pragma GCC optimize("-falign-functions")
// #pragma GCC optimize("-fcse-skip-blocks")
// #pragma GCC optimize("-fcse-follow-jumps")
// #pragma GCC optimize("-fsched-interblock")
// #pragma GCC optimize("-fpartial-inlining")
// #pragma GCC optimize("no-stack-protector")
// #pragma GCC optimize("-freorder-functions")
// #pragma GCC optimize("-findirect-inlining")
// #pragma GCC optimize("-fhoist-adjacent-loads")
// #pragma GCC optimize("-frerun-cse-after-loop")
// #pragma GCC optimize("inline-small-functions")
// #pragma GCC optimize("-finline-small-functions")
// #pragma GCC optimize("-ftree-switch-conversion")
// #pragma GCC optimize("-foptimize-sibling-calls")
// #pragma GCC optimize("-fexpensive-optimizations")
// #pragma GCC optimize("-funsafe-loop-optimizations")
// #pragma GCC optimize("inline-functions-called-once")
// #pragma GCC optimize("-fdelete-null-pointer-checks")
#define rep(i, l, r) for (int i = (l); i <= (r); ++i)
#define per(i, l, r) for (int i = (l); i >= (r); --i)
using std::cerr;
using std::endl;
using std::make_pair;
using std::pair;
typedef pair<int, int> pii;
typedef long long ll;
typedef unsigned int ui;
typedef unsigned long long ull;

// #define DEBUG 1  //调试开关
struct IO {
#define MAXSIZE (1 << 20)
#define isdigit(x) (x >= '0' && x <= '9')
    char buf[MAXSIZE], *p1, *p2;
    char pbuf[MAXSIZE], *pp;
#if DEBUG
#else
    IO() : p1(buf), p2(buf), pp(pbuf) {}
    ~IO() { fwrite(pbuf, 1, pp - pbuf, stdout); }
#endif
    inline char gc() {
#if DEBUG  //调试，可显示字符
        return getchar();
#endif
        if (p1 == p2) p2 = (p1 = buf) + fread(buf, 1, MAXSIZE, stdin);
        return p1 == p2 ? -1 : *p1++;
    }
    inline bool blank(char ch) { return ch == ' ' || ch == '\n' || ch == '\r' || ch == '\t'; }
    template <class T>
    inline void read(T &x) {
        register double tmp = 1;
        register bool sign = 0;
        x = 0;
        register char ch = gc();
        for (; !isdigit(ch); ch = gc())
            if (ch == '-') sign = 1;
        for (; isdigit(ch); ch = gc()) x = x * 10 + (ch - '0');
        if (ch == '.')
            for (ch = gc(); isdigit(ch); ch = gc()) tmp /= 10.0, x += tmp * (ch - '0');
        if (sign) x = -x;
    }
    inline void read(char *s) {
        register char ch = gc();
        for (; blank(ch); ch = gc())
            ;
        for (; !blank(ch); ch = gc()) *s++ = ch;
        *s = 0;
    }
    inline void read(char &c) {
        for (c = gc(); blank(c); c = gc())
            ;
    }
    inline void push(const char &c) {
#if DEBUG  //调试，可显示字符
        putchar(c);
#else
        if (pp - pbuf == MAXSIZE) fwrite(pbuf, 1, MAXSIZE, stdout), pp = pbuf;
        *pp++ = c;
#endif
    }
    template <class T>
    inline void write(T x) {
        if (x < 0) x = -x, push('-');  // 负数输出
        static T sta[35];
        T top = 0;
        do {
            sta[top++] = x % 10, x /= 10;
        } while (x);
        while (top) push(sta[--top] + '0');
    }
    inline void write(const char *s) {
        while (*s != '\0') push(*(s++));
    }
    template <class T>
    inline void write(T x, char lastChar) {
        write(x), push(lastChar);
    }
} io;

const int N = 1000010;

struct Node {
    ui i, x, y, v;
} p[N];
struct Q {
    ui id, l, r;
    ll ans;
} ask[N];

bool cmp_l(const Q &a, const Q &b) { return a.l < b.l; }

bool cmp_r(const Q &a, const Q &b) { return a.r < b.r; }

bool cmp_id(const Q &a, const Q &b) { return a.id < b.id; }

bool cmp_y(const Node &a, const Node &b) { return a.y < b.y; }

ll pre[N], c[N + 5];

// #define debug(x) cerr << x << endl
#define debug(x)

inline int lowbit(int x) { return x & (-x); }

void add(int i, ll x) {
    ++i;
    for (; i <= N; i += lowbit(i)) {
        debug(i);
        c[i] += x;
    }
}
ll sum(int i) {
    ++i;
    ll ret = 0;
    for (; i > 0; i -= lowbit(i)) ret += c[i];
    return ret;
}

ll query(int l, int r) { return sum(r) - sum(l - 1); }

int main() {
#ifdef LOCAL
    freopen("input", "r", stdin);
#endif
    int n, m, q;
    io.read(n), io.read(m), io.read(q);
    rep(i, 1, n) io.read(p[i].v), p[i].x = 0, p[i].y = n + 1, p[i].i = i, pre[i] = pre[i - 1] + p[i].v;
    while (m--) {
        ui u, v;
        io.read(u), io.read(v);
        if (v < u)
            p[u].x = std::max(p[u].x, v);
        else
            p[u].y = std::min(p[u].y, v);
    }
    rep(i, 1, q) io.read(ask[i].l), io.read(ask[i].r), ask[i].ans = pre[ask[i].r] - pre[ask[i].l - 1], ask[i].id = i;

    debug("read over");

    // i <= R, x_i >= L
    std::sort(ask + 1, ask + 1 + q, cmp_r);
    int ptr = 1;
    rep(i, 1, q) {
        while (ptr <= n && p[ptr].i <= ask[i].r) add(p[ptr].x, p[ptr].v), ++ptr;
        ask[i].ans -= query(ask[i].l, n + 1);
    }

    debug("i <= R, x_i >= L");

    // x >= L, y <= R
    std::memset(c, 0, sizeof(c));
    ptr = 1;
    std::sort(p + 1, p + 1 + n, cmp_y);
    rep(i, 1, q) {
        while (ptr <= n && p[ptr].y <= ask[i].r) add(p[ptr].x, p[ptr].v), ++ptr;
        ask[i].ans += query(ask[i].l, n + 1);
    }

    // y <= R, i >= L
    std::memset(c, 0, sizeof(c));
    ptr = 1;
    rep(i, 1, q) {
        while (ptr <= n && p[ptr].y <= ask[i].r) add(p[ptr].i, p[ptr].v), ++ptr;
        ask[i].ans -= query(ask[i].l, n + 1);
    }

    ll ans = 0;
    rep(i, 1, q) ans ^= ask[i].id * ask[i].ans;
    io.write(ans);
    return 0;
}

「Comet OJ 58D」菜菜种菜

https://gesrua.xyz/archives/题解/Comet-OJ/菜菜种菜

作者

Gesrua

发布于

2019-08-17

更新于

2020-11-21

许可协议

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