P13508 [OOI 2024] Burenka and Pether

EuphoricStar · 2025-11-13 17:38:30 · 题解

首先直接在序列上不好做，因为要时刻考虑跳到的点 \le a_v。

所以转而从值域考虑，令 (u, v) \to (a_u, a_v)，设 b_i 为 i 在 a 中的位置。然后对于一个数 x，我们能求出一个 c_x 表示在序列上，位置 b_x 能跳到所有位置在 [b_x + 1, c_x] 之间且值 > x 的数，可以并查集 + set 求出。

那么现在一组询问 (x, y)，我们的策略是：每次 x 跳到最大的一个 z 满足 z \le y 且 b_z \in [b_x + 1, c_x]。

发现有 z \le y 的限制，无法直接倍增优化。

但是我们可以分治。具体地，设当前分治区间为 [l, r]，把所有跨过中点（即 x \le mid, y > mid）的询问的询问的 x 一直往右跳直到 > mid 为止，然后递归进子问题。这样的好处是，除了最后跳到 > mid 的那一步，我们不用考虑 z \le y 的限制，只用无脑跳到最远即可。

大致的思路就差不多是这样。具体实现，考虑对每个 x \in [l, mid] 求出 to_x, f_x 分别表示 x 往右能跳到的 \le mid 最远点和 > mid 的最近点。对于一个跨过中点的询问 (x, y)，不断令 x \to to_x 直到 f_x \le y（这一步可以倍增优化），然后求出 x 能跳到的 [mid + 1, y] 之间最大的 \le y 的点（这一步可以扫描线 + 线段树二分）。

每个询问只会在 \log n 个分治区间被考虑。时间复杂度 O((n + q) \log^2 n)。

:::info[代码]

#include <bits/stdc++.h>
#define pb emplace_back
#define fst first
#define scd second
#define mkp make_pair
#define mems(a, x) memset((a), (x), sizeof(a))

using namespace std;
using ll = long long;
using ull = unsigned long long;
using db = double;
using ldb = long double;
using pii = pair<int, int>;
using pll = pair<ll, ll>;

namespace IO {
    const int maxn = 1 << 20;

    char ibuf[maxn], *iS, *iT, obuf[maxn], *oS = obuf;

    inline char gc() {
        return (iS == iT ? iT = (iS = ibuf) + fread(ibuf, 1, maxn, stdin), (iS == iT ? EOF : *iS++) : *iS++);
    }

    template<typename T = int>
    inline T read() {
        char c = gc();
        T x = 0;
        bool f = 0;
        while (c < '0' || c > '9') {
            f |= (c == '-');
            c = gc();
        }
        while (c >= '0' && c <= '9') {
            x = (x << 1) + (x << 3) + (c ^ 48);
            c = gc();
        }
        return f ? ~(x - 1) : x;
    }

    inline int reads(char *s) {
        char c = gc();
        int len = 0;
        while (isspace(c)) {
            c = gc();
        }
        while (!isspace(c) && c != -1) {
            s[len++] = c;
            c = gc();
        }
        s[len] = '\0';
        return len;
    }

    inline string reads() {
        char c = gc();
        string s;
        while (isspace(c)) {
            c = gc();
        }
        while (!isspace(c) && c != -1) {
            s += c;
            c = gc();
        }
        return s;
    }

    inline void flush() {
        fwrite(obuf, 1, oS - obuf, stdout);
        oS = obuf;
    }

    struct Flusher {
        ~Flusher() {
            flush();
        }
    } AutoFlush;

    inline void pc(char ch) {
        if (oS == obuf + maxn) {
            flush();
        }
        *oS++ = ch;
    }

    template<typename T>
    inline void write(T x) {
        static char stk[64], *tp = stk;
        if (x < 0) {
            x = ~(x - 1);
            pc('-');
        }
        do {
            *tp++ = x % 10;
            x /= 10;
        } while (x);
        while (tp != stk) {
            pc((*--tp) | 48);
        }
    }

    inline void write(const char *s) {
        for (int i = 0; s[i]; ++i) {
            pc(s[i]);
        }
    }

    template<typename T>
    inline void writesp(T x) {
        write(x);
        pc(' ');
    }

    template<typename T>
    inline void writeln(T x) {
        write(x);
        pc('\n');
    }
}

using IO::read;
using IO::reads;
using IO::write;
using IO::pc;
using IO::writesp;
using IO::writeln;

const int maxn = 300100;
const int logn = 20;
const int inf = 0x3f3f3f3f;

int n, m, q, a[maxn], fa[maxn], b[maxn], c[maxn], ans[maxn], f[maxn];
vector<int> vc[maxn];

struct que {
    int o, x, y;
} qq[maxn];

int find(int x) {
    return fa[x] == x ? x : fa[x] = find(fa[x]);
}

int to[logn][maxn], st[logn][maxn], g[logn][maxn];
pii ls[maxn];

inline int qmax(int l, int r) {
    if (l > r) {
        return 0;
    }
    int k = __lg(r - l + 1);
    return max(st[k][l], st[k][r - (1 << k) + 1]);
}

inline int qmin(int l, int r) {
    if (l > r) {
        return inf;
    }
    int k = __lg(r - l + 1);
    return min(st[k][l], st[k][r - (1 << k) + 1]);
}

namespace SGT {
    int mn[maxn << 2];

    void build(int rt, int l, int r) {
        mn[rt] = inf;
        if (l == r) {
            return;
        }
        int mid = (l + r) >> 1;
        build(rt << 1, l, mid);
        build(rt << 1 | 1, mid + 1, r);
    }

    void update(int rt, int l, int r, int x, int y) {
        mn[rt] = min(mn[rt], y);
        if (l == r) {
            return;
        }
        int mid = (l + r) >> 1;
        (x <= mid) ? update(rt << 1, l, mid, x, y) : update(rt << 1 | 1, mid + 1, r, x, y);
    }

    int find(int rt, int l, int r, int ql, int qr, int x) {
        if (ql > qr || mn[rt] > x) {
            return -1;
        }
        if (l == r) {
            return l;
        }
        int mid = (l + r) >> 1;
        if (qr > mid) {
            int t = find(rt << 1 | 1, mid + 1, r, ql, qr, x);
            if (t != -1) {
                return t;
            }
        }
        if (ql <= mid) {
            int t = find(rt << 1, l, mid, ql, qr, x);
            if (t != -1) {
                return t;
            }
        }
        return -1;
    }
}

void dfs(int l, int r, vector<int> Q) {
    if (Q.empty()) {
        return;
    }
    int mid = (l + r) >> 1;
    vector<int> L, R, U;
    for (int i : Q) {
        if (qq[i].y <= mid) {
            L.pb(i);
        } else if (qq[i].x > mid) {
            R.pb(i);
        } else {
            U.pb(i);
        }
    }
    for (int i = l; i <= mid; ++i) {
        to[0][i] = 0;
    }
    int tot = 0;
    for (int i = l; i <= mid; ++i) {
        ls[++tot] = pii(b[i], i);
    }
    sort(ls + 1, ls + tot + 1);
    for (int i = 1; i <= tot; ++i) {
        st[0][i] = ls[i].scd;
    }
    for (int j = 1; (1 << j) <= tot; ++j) {
        for (int i = 1; i + (1 << j) - 1 <= tot; ++i) {
            st[j][i] = max(st[j - 1][i], st[j - 1][i + (1 << (j - 1))]);
        }
    }
    for (int i = l; i <= mid; ++i) {
        int x = upper_bound(ls + 1, ls + tot + 1, pii(b[i], i)) - ls;
        int y = upper_bound(ls + 1, ls + tot + 1, pii(c[i], n + 1)) - ls - 1;
        to[0][i] = qmax(x, y);
        if (to[0][i] <= i) {
            to[0][i] = 0;
        }
    }
    tot = 0;
    for (int i = mid + 1; i <= r; ++i) {
        ls[++tot] = pii(b[i], i);
    }
    sort(ls + 1, ls + tot + 1);
    for (int i = 1; i <= tot; ++i) {
        st[0][i] = ls[i].scd;
    }
    for (int j = 1; (1 << j) <= tot; ++j) {
        for (int i = 1; i + (1 << j) - 1 <= tot; ++i) {
            st[j][i] = min(st[j - 1][i], st[j - 1][i + (1 << (j - 1))]);
        }
    }
    for (int i = l; i <= mid; ++i) {
        int x = upper_bound(ls + 1, ls + tot + 1, pii(b[i], i)) - ls;
        int y = upper_bound(ls + 1, ls + tot + 1, pii(c[i], n + 1)) - ls - 1;
        f[i] = g[0][i] = qmin(x, y);
    }
    for (int j = 1; j < logn; ++j) {
        for (int i = l; i <= mid; ++i) {
            to[j][i] = to[j - 1][to[j - 1][i]];
            g[j][i] = min(g[j - 1][i], g[j - 1][to[j - 1][i]]);
        }
    }
    for (int i = l; i <= mid; ++i) {
        vector<int>().swap(vc[i]);
    }
    for (int i : U) {
        int x = qq[i].x, y = qq[i].y, z = 0;
        if (f[x] <= y) {
            vc[qq[i].x].pb(i);
            continue;
        }
        for (int j = logn - 1; ~j; --j) {
            if (to[j][x] && g[j][x] > y) {
                z += (1 << j);
                x = to[j][x];
            }
        }
        if (f[x] > y) {
            ans[i] = 0;
            continue;
        }
        ans[i] += z;
        qq[i].x = x;
        vc[qq[i].x].pb(i);
    }
    tot = 0;
    for (int i = l; i <= r; ++i) {
        ls[++tot] = pii(b[i], i);
    }
    sort(ls + 1, ls + tot + 1, greater<pii>());
    SGT::build(1, mid + 1, r);
    for (int i = 1; i <= tot; ++i) {
        int j = ls[i].scd;
        if (j > mid) {
            SGT::update(1, mid + 1, r, j, b[j]);
        } else {
            for (int k : vc[j]) {
                int x = qq[k].x, y = qq[k].y;
                int z = SGT::find(1, mid + 1, r, mid + 1, y, c[x]);
                assert(z != -1);
                ++ans[k];
                qq[k].x = z;
                if (z < y) {
                    R.pb(k);
                }
            }
        }
    }
    dfs(l, mid, L);
    dfs(mid + 1, r, R);
}

void solve() {
    n = read();
    m = read();
    read();
    for (int i = 1; i <= n; ++i) {
        a[i] = read();
        b[a[i]] = i;
    }
    for (int i = 1; i <= n; ++i) {
        fa[i] = i;
    }
    set<int> S;
    for (int i = 1; i <= n; ++i) {
        auto it = S.insert(b[i]).fst;
        if (next(it) != S.end() && *next(it) - b[i] <= m) {
            fa[b[i]] = *next(it);
        }
        if (it != S.begin() && b[i] - *prev(it) <= m) {
            fa[*prev(it)] = b[i];
        }
        c[i] = min(n, find(b[i]) + m);
    }
    q = read();
    vector<int> Q;
    for (int i = 1; i <= q; ++i) {
        qq[i].o = read();
        qq[i].x = a[read()];
        qq[i].y = a[read()];
        if (qq[i].x < qq[i].y) {
            Q.pb(i);
        }
    }
    dfs(1, n, Q);
    for (int i = 1; i <= q; ++i) {
        if (qq[i].o == 1) {
            ans[i] = min(ans[i], 1);
        }
        writeln(ans[i]);
    }
}

int main() {
    int T = 1;
    // scanf("%d", &T);
    while (T--) {
        solve();
    }
    return 0;
}

:::info