P4899:[IOI2018] werewolf 狼人题解

解法

这道题是 Kruskal 重构树的裸题。我们先来考虑从\(s\)出发的人形状态,我们要找到一个点点权小于\(L\)的点,就相当于在 Kruskal 重构树上倍增找到最后一个小于\(L\)的点。我们把符合人形规律的 Krukskal 树记为 Tl,其中构造它的方式就是把原来图上所有边的边权赋值为端点的最大值,然后按最小生成树的方式去构造重构树;符合狼形规律的 Kruskal 重构树被记为 Tu,其中构造它的方式就是将端点编号最小值赋为边权,然后按最大生成树的方式构造。之后,我们再到主席树中求交集即可,如果有交集那么查询结果为一个不为零的数,输出\(1\)即可;没有交集那么查询结果即为\(0\),输出\(0\)。

代码

// P4899.cpp
#include <bits/stdc++.h>
using namespace std;

const int MAX_N = 401000, MAX_M = 401000, INF = 0x3f3f3f3f;

int n, m, q, mapping[MAX_N];

struct edge
{
    int from, to, nxt;
} source[MAX_M];

struct Ufs
{
    int buff[MAX_N];

    void init(int amount)
    {
        for (int i = 1; i <= amount; i++)
            buff[i] = i;
    }

    int find(int x) { return x == buff[x] ? x : buff[x] = find(buff[x]); }
} Uu, Ul;

struct graph
{
    int head[MAX_N], current, lft[MAX_N], rig[MAX_N], anti[MAX_N], dfn;
    int fa[20][MAX_N], val[MAX_N];
    edge edges[MAX_M];

    graph() { memset(head, -1, sizeof(head)); }

    void addpath(int u, int v)
    {
        edges[current].to = v, edges[current].from = u;
        edges[current].nxt = head[u], head[u] = current++;
    }

    void initialize(int u)
    {
        if (u <= n)
            anti[++dfn] = u, lft[u] = dfn, val[u] = u;
        else
            lft[u] = dfn + 1;
        for (int i = 1; i < 20; i++)
            fa[i][u] = fa[i - 1][fa[i - 1][u]];
        for (int i = head[u]; i != -1; i = edges[i].nxt)
            fa[0][edges[i].to] = u, initialize(edges[i].to);
        rig[u] = dfn;
    }

    int find_max(int u, int upper)
    {
        for (int i = 19; i >= 0; i--)
            if (val[fa[i][u]] <= upper)
                u = fa[i][u];
        return u;
    }

    int find_min(int u, int lower)
    {
        for (int i = 19; i >= 0; i--)
            if (val[fa[i][u]] >= lower)
                u = fa[i][u];
        return u;
    }
} Tu, Tl;

bool compare_u(const edge &a, const edge &b) { return max(a.from, a.to) < max(b.from, b.to); }

bool compare_l(const edge &a, const edge &b) { return min(a.from, a.to) > min(b.from, b.to); }

void kruskal_u()
{
    int tot = n;
    sort(source + 1, source + 1 + m, compare_u);
    Uu.init(n << 1);
    for (int i = 1; i <= m; i++)
    {
        int a = Uu.find(source[i].from), b = Uu.find(source[i].to);
        int v = max(source[i].from, source[i].to);
        if (a != b)
        {
            Uu.buff[a] = Uu.buff[b] = ++tot;
            Tu.val[tot] = v;
            Tu.addpath(tot, a), Tu.addpath(tot, b);
        }
    }
    Tu.initialize(tot);
}

void kruskal_l()
{
    int tot = n;
    sort(source + 1, source + 1 + m, compare_l);
    Ul.init(n << 1);
    for (int i = 1; i <= m; i++)
    {
        int a = Ul.find(source[i].from), b = Ul.find(source[i].to);
        int v = min(source[i].from, source[i].to);
        if (a != b)
        {
            Ul.buff[a] = Ul.buff[b] = ++tot;
            Tl.val[tot] = v;
            Tl.addpath(tot, a), Tl.addpath(tot, b);
        }
    }
    Tl.initialize(tot);
}

struct sustainable_segment_node
{
    int sum, lson, rson;
};

struct sustainable_segment_tree
{
    sustainable_segment_node nodes[MAX_N << 5];
    int roots[MAX_N], tot;

    int update(int qx, int l, int r, int pre, int val)
    {
        int p = ++tot;
        nodes[p] = nodes[pre], nodes[p].sum += val;
        if (l == r)
            return p;
        int mid = (l + r) >> 1;
        if (qx <= mid)
            nodes[p].lson = update(qx, l, mid, nodes[pre].lson, val);
        else
            nodes[p].rson = update(qx, mid + 1, r, nodes[pre].rson, val);
        return p;
    }

    int query(int ql, int qr, int l, int r, int p, int pre)
    {
        if (ql <= l && r <= qr)
            return nodes[p].sum - nodes[pre].sum;
        int mid = (l + r) >> 1, ans = 0;
        if (ql <= mid)
            ans += query(ql, qr, l, mid, nodes[p].lson, nodes[pre].lson);
        if (mid < qr)
            ans += query(ql, qr, mid + 1, r, nodes[p].rson, nodes[pre].rson);
        return ans;
    }
} seg;

int main()
{
    scanf("%d%d%d", &n, &m, &q);
    for (int i = 1; i <= m; i++)
        scanf("%d%d", &source[i].from, &source[i].to), source[i].from++, source[i].to++;
    kruskal_l(), kruskal_u();
    Tu.val[0] = INF;
    for (int i = 1; i <= n; i++)
        mapping[Tl.anti[i]] = i;
    seg.roots[0] = 0;
    for (int i = 1; i <= n; i++)
        seg.roots[i] = seg.update(mapping[Tu.anti[i]], 1, n, seg.roots[i - 1], 1);
    while (q--)
    {
        int s, e, l, r;
        scanf("%d%d%d%d", &s, &e, &l, &r), s++, e++, l++, r++;
        int x = Tl.find_min(s, l), y = Tu.find_max(e, r);
        int res = seg.query(Tl.lft[x], Tl.rig[x], 1, n, seg.roots[Tu.rig[y]], seg.roots[Tu.lft[y] - 1]);
        puts(res ? "1" : "0");
    }
    return 0;
}

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Kal0rona

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