P4501:[ZJOI2018]胖 – 题解

主要思路

可以比较显然的看出来是个区间扩展问题,单调的向两边进行扩展。

正常暴力可以考虑 \(n^2\) 进行扩展,然而这题显然有单调性,所以我们可以试着进行二分优化掉一个 \(n\)。二分出一个范围,然后再用 ST 表或者是线段树之类的东西去查一下区域内的最远距离然后就可以进行判断了。

代码有点长而且还有点复杂。

代码

// P4501.cpp
#include <bits/stdc++.h>

using namespace std;

const int MAX_N = 2e5 + 200;

typedef long long ll;

ll n, m, K, ai[MAX_N], li[MAX_N], log2_[MAX_N], rk[MAX_N], tmp[MAX_N], dist[MAX_N];

inline char nc()
{
    static char buf[1 << 20], *p1 = buf, *p2 = buf;
    return p1 == p2 && (p2 = (p1 = buf) + fread(buf, 1, 1 << 20, stdin), p1 == p2) ? EOF : *p1++;
}

int read()
{
    int x = 0, f = 1;
    char ch = nc();
    while (!isdigit(ch))
    {
        if (ch == '-')
            f = -1;
        ch = nc();
    }
    while (isdigit(ch))
        x = (x << 3) + (x << 1) + ch - '0', ch = nc();
    return x * f;
}

struct STable
{
    ll mat[20][MAX_N];

    void init()
    {
        for (int i = 1; i < 20; i++)
            for (int j = 1; j + (1 << i) - 1 <= K; j++)
                mat[i][j] = min(mat[i - 1][j], mat[i - 1][j + (1 << (i - 1))]);
    }

    ll query(int l, int r)
    {
        int d = log2_[r - l + 1];
        return min(mat[d][l], mat[d][r - (1 << d) + 1]);
    }
} st[2];

int lower(int val)
{
    int l = 1, r = K, res = K + 1;
    while (l <= r)
    {
        int mid = (l + r) >> 1;
        if (tmp[mid] < val)
            l = mid + 1;
        else
            r = mid - 1, res = mid;
    }
    return res;
}

int upper(int val)
{
    int l = 1, r = K, res = K + 1;
    while (l <= r)
    {
        int mid = (l + r) >> 1;
        if (tmp[mid] <= val)
            l = mid + 1;
        else
            r = mid - 1, res = mid;
    }
    return res;
}

int main()
{
    n = read(), m = read();
    for (int i = 2; i <= n; i++)
        dist[i] = dist[i - 1] + read(), log2_[i] = log2_[i >> 1] + 1;
    while (m--)
    {
        ll ans = 0;
        K = read();
        for (int i = 1; i <= K; i++)
            ai[i] = read(), li[i] = read(), tmp[i] = ai[i];
        sort(tmp + 1, tmp + 1 + K);
        for (int i = 1; i <= K; i++)
            rk[tmp[i]] = i;
        for (int i = 1; i <= K; i++)
        {
            st[0].mat[0][rk[ai[i]]] = li[i] - dist[ai[i]];
            st[1].mat[0][rk[ai[i]]] = li[i] + dist[ai[i]];
        }
        st[0].init(), st[1].init();
        for (int i = 1; i <= K; i++)
        {
            int l = 1, r = ai[i] - 1, lft = ai[i], rig = ai[i];
            while (l <= r)
            {
                int mid = (l + r) >> 1;
                ll cdist = li[i] + dist[ai[i]] - dist[mid], diff = ai[i] - mid;
                ll ql = 0x3f3f3f3f3f3f3f3f, qr = ql;

                int qlft = lower(max(1LL, mid - diff));
                int qrig = upper(mid) - 1;
                if (qlft <= qrig)
                    ql = st[0].query(qlft, qrig);

                qlft = lower(mid);
                qrig = upper(ai[i] - 1) - 1;

                if (qlft <= qrig)
                    qr = st[1].query(qlft, qrig);
                bool flag = (ql + dist[mid] > cdist) && (qr - dist[mid] > cdist);
                if (flag)
                    r = mid - 1, lft = mid;
                else
                    l = mid + 1;
            }
            l = ai[i] + 1, r = n;
            while (l <= r)
            {
                int mid = (l + r) >> 1;
                ll cdist = li[i] + dist[mid] - dist[ai[i]], diff = mid - ai[i];
                ll ql = 0x3f3f3f3f3f3f3f3f, qr = ql;

                int qlft = lower(mid);
                int qrig = upper(min(mid + diff - 1, 1LL * n)) - 1;
                if (qlft <= qrig)
                    qr = st[1].query(qlft, qrig);

                qlft = lower(ai[i] + 1);
                qrig = upper(mid) - 1;

                if (qlft <= qrig)
                    ql = st[0].query(qlft, qrig);
                bool flag = (ql + dist[mid] > cdist) && (qr - dist[mid] > cdist);
                if (mid + diff <= n)
                {
                    qrig = upper(mid + diff) - 1;
                    if (tmp[qrig] == mid + diff && st[1].mat[0][qrig] - dist[mid] < cdist)
                        flag = false;
                }
                if (flag)
                    l = mid + 1, rig = mid;
                else
                    r = mid - 1;
            }
            ans += rig - lft + 1;
        }
        printf("%lld\n", ans);
    }
    return 0;
}

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