LG5066 [Ynoi2014] 人人本着正义之名

发布于 2021-07-20  2.26k 次阅读


题目链接

Statement

维护一个长为 $n$ 的 0/1 序列 $a$,有 $m$ 个操作:

  • 1 l r:把区间 $[l,~r]$ 的数变成 $0$。
  • 2 l r:把区间 $[l,~r]$ 的数变成 $1$。
  • 3 l r:$[l,~r-1]$ 内所有数 $a_i$,变为 $a_i$ 与 $a_{i+1}$ 按位或的值,这些数同时进行这个操作。
  • 4 l r:$[l+1,~r]$ 内所有数 $a_i$,变为 $a_i$ 与 $a_{i-1}$ 按位或的值,这些数同时进行这个操作。
  • 5 l r:$[l,~r-1]$ 内所有数 $a_i$,变为 $a_i$ 与 $a_{i+1}$ 按位与的值,这些数同时进行这个操作。
  • 6 l r:$[l+1,~r]$ 内所有数 $a_i$,变为 $a_i$ 与 $a_{i-1}$ 按位与的值,这些数同时进行这个操作。
  • 7 l r:查询区间 $[l,~r]$ 内的元素和。

强制在线。

$n,~m \le 3 \times 10^6$

Solution

我们显然无法使用平衡树直接维护每个点的单点信息,因为题目中给出的操作对于区间中的每一个元素均与它两边的元素进行了运算,难以直接维护每个单点的信息。

所以我们考虑将原序列拆分为一些区间,使之可以较方便地处理题目中给出的这些操作。

一个比较显然的想法是将原序列拆分为若干段同色的极长段。比如将 $0,~1,~1,~0,~0,~1$ 划分为 $[0]~[1,~1]~[0,~0]~[1]$。容易发现这样划分之后,如果一个操作覆盖了某一个区间,我们可以方便地计算出该操作对该区间的影响:

  • 1 操作: 所覆盖的 1 区间均变为 0 区间。
  • 2 操作: 所覆盖的 0 区间均变为 1 区间。
  • 3 操作: 所覆盖的 0 区间右端点 $-1$,所覆盖的 1 区间左端点 $-1$。
  • 4 操作: 所覆盖的 0 区间左端点 $+1$,所覆盖的 1 区间右端点 $+1$。
  • 5 操作: 所覆盖的 0 区间左端点 $-1$,所覆盖的 1 区间右端点 $-1$。
  • 6 操作: 所覆盖的 0 区间右端点 $+1$,所覆盖的 1 区间左端点 $+1$。

同时,由于我们需要在下放标记前计算出某个询问区间对应的元素和,所以我们需要保证每一个区间均为极长区间,也就是不能让空区间留在平衡树中。

由于在整个过程中于平衡树上出现的区间数量是 $O(n+m)$ 级别的,所以我们可以在每个平衡树结点上记录其子树内最短的 0/1 区间的长度,在该长度达到 0 时暴力地向下搜索找到空区间并删除,对于每个空区间我们需要花费 $O(\log n)$ 的时间复杂度,因此总共花费在删除空区间操作上的时间是 $O((n+m)\log n)$ 级别的,可以接受。

根据上面提到的方法维护平衡树即可。

Code

编写代码用时 3h,调试用时 2h,使用指针版 FhqTreap,由于卡常使用了内存池并省去了回收内存操作。

下面是一些注意事项和建议:

  • 写代码时务必多画图,仔细考虑每一个操作的边界情况。(即操作区间的左右端点所在区间是否需要被修改的情况)
  • 由于代码较长建议每写完几个操作就进行简单的测试以检验其正确性。
  • 如果 WA 并且找不到显然的错误,可以在每次操作结束后输出整棵平衡上的所有区间的信息,配合手动模拟或是暴力代码测试几组数据。

View on GitHub

/**
 * @author Macesuted (i@macesuted.moe)
 * @copyright Copyright (c) 2021
 * @brief
 *      My solution: https://www.macesuted.cn/article/lg5066/
 */

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

namespace io {
#define SIZE (1 << 20)
char ibuf[SIZE], *iS, *iT, obuf[SIZE], *oS = obuf, *oT = oS + SIZE - 1, c, qu[55];
int f, qr;
inline void flush(void) { return fwrite(obuf, 1, oS - obuf, stdout), oS = obuf, void(); }
inline char getch(void) { return (iS == iT ? (iT = (iS = ibuf) + fread(ibuf, 1, SIZE, stdin), (iS == iT ? EOF : *iS++)) : *iS++); }
inline void putch(char x) {
    *oS++ = x;
    if (oS == oT) flush();
    return;
}
string getstr(void) {
    string s = "";
    char c = getch();
    while (c == ' ' || c == '\n' || c == '\r' || c == '\t' || c == EOF) c = getch();
    while (!(c == ' ' || c == '\n' || c == '\r' || c == '\t' || c == EOF)) s.push_back(c), c = getch();
    return s;
}
void putstr(string str, int begin = 0, int end = -1) {
    if (end == -1) end = str.size();
    for (register int i = begin; i < end; i++) putch(str[i]);
    return;
}
template <typename T>
inline T read() {
    register T x = 0;
    for (f = 1, c = getch(); c < '0' || c > '9'; c = getch())
        if (c == '-') f = -1;
    for (x = 0; c <= '9' && c >= '0'; c = getch()) x = x * 10 + (c & 15);
    return x * f;
}
template <typename T>
inline void write(const T& t) {
    register T x = t;
    if (!x) putch('0');
    if (x < 0) putch('-'), x = -x;
    while (x) qu[++qr] = x % 10 + '0', x /= 10;
    while (qr) putch(qu[qr--]);
    return;
}
struct Flusher_ {
    ~Flusher_() { flush(); }
} io_flusher_;
}  // namespace io
using io::getch;
using io::getstr;
using io::putch;
using io::putstr;
using io::read;
using io::write;

#define maxn 3000005
#define INF 0x3f3f3f3f

class FhqTreap {
   private:
    struct Node {
        int rank;
        int col;             // 区间颜色
        int sum;             // 子树和
        int siz[2];          // 子树内 0/1 区间数量
        int tl, tr;          // 区间左右端点
        int deltaL, deltaR;  // 1 区间左右端点偏移量
        int minLen[2];       // 子树内最短 0/1 区间长度
        Node *l, *r;         // 平衡树左右孩子
        Node(void) {}
        Node(int _rank, int _col, int _tl, int _tr) {
            rank = _rank;
            col = _col;
            tl = _tl, tr = _tr;
            l = r = NULL;
            reset();
        }
        // 重新计算该点的单点信息
        void reset(void) {
            siz[col] = 1, siz[!col] = 0;
            sum = col * (tr - tl + 1);
            deltaL = deltaR = 0;
            minLen[col] = tr - tl + 1, minLen[!col] = INF;
            return;
        }
        Node operator+(const Node& b) const {
            Node a = *this;
            a.sum += b.sum;
            a.siz[0] += b.siz[0], a.siz[1] += b.siz[1];
            a.minLen[0] = min(a.minLen[0], b.minLen[0]), a.minLen[1] = min(a.minLen[1], b.minLen[1]);
            return a;
        }
    };

    Node pool[10000000];
    int tail = 0;

    Node* root;
    queue<Node*> needErase;  // 需要被删除的结点

    static const long long kMul = 0x9ddfea08eb382d69ULL;
    long long Seed = 1145141919810;
    inline long long fingerPrint(long long x) {
        return x *= kMul, x ^= x >> 47, x *= kMul, x ^= x >> 47, x *= kMul, x ^= x >> 47, x * kMul;
    }
    inline int frand(void) { return Seed += fingerPrint(Seed); }

    inline Node* newNode(int val, int l, int r) { return &(pool[tail++] = Node(frand(), val, l, r)); }

    inline void update(Node* p, int dl, int dr) {
        if (p == NULL) return;
        p->deltaL += dl, p->deltaR += dr;
        p->minLen[0] -= dl + dr, p->minLen[1] += dl + dr;
        p->sum += (dl + dr) * p->siz[1];
        if (p->col == 1)
            p->tl -= dl, p->tr += dr;
        else
            p->tl += dr, p->tr -= dl;
        return;
    }
    void findEmpty(Node* p) {
        if (p == NULL) return;
        if (p->minLen[0] && p->minLen[1]) return;
        pushDown(p);
        if (p->tl > p->tr) needErase.push(p);
        findEmpty(p->l), findEmpty(p->r);
        return;
    }
    void eraseEmpty(void) {
        while (!needErase.empty()) {
            Node* p = needErase.front();
            needErase.pop();
            Node *nodeL = NULL, *treeM = NULL, *nodeR = NULL, *treeR = NULL;
            split1(root, root, treeR, p->tl - 1), split1(treeR, treeM, treeR, p->tl);
            split2(root, root, nodeL, p->tl - 2);
            if (p == treeM)
                nodeR = (p->l == NULL) ? p->r : p->l;
            else
                nodeR = treeM, nodeR->l = nodeR->r = NULL;
            // delete p;
            if (nodeL != NULL && nodeR != NULL) nodeL->tr = nodeR->tr, nodeL->reset(), /* delete nodeR, */ nodeR = NULL;
            merge(root, root, nodeL), merge(root, root, nodeR), merge(root, root, treeR);
        }
        return;
    }
    // void destroy(Node* p) {
    //     if (p == NULL) return;
    //     if (p->l != NULL) destroy(p->l);
    //     if (p->r != NULL) destroy(p->r);
    //     delete p;
    //     return;
    // }
    void pushDown(Node* p) {
        if (!p->deltaL && !p->deltaR) return;
        if (p->l != NULL) update(p->l, p->deltaL, p->deltaR);
        if (p->r != NULL) update(p->r, p->deltaL, p->deltaR);
        p->deltaL = p->deltaR = 0;
        return;
    }
    inline void pushUp(Node* p) {
        p->reset();
        if (p->l != NULL) *p = *p + *(p->l);
        if (p->r != NULL) *p = *p + *(p->r);
        return;
    }

    // 将对应区间左端点 <= p 的部分划分到 tree1,左端点 > p 的部分划分到 tree2
    void split1(Node* root, Node*& tree1, Node*& tree2, int p) {
        if (root == NULL) return tree1 = tree2 = NULL, void();
        pushDown(root);
        if (root->tl <= p)
            tree1 = root, split1(root->r, tree1->r, tree2, p);
        else
            tree2 = root, split1(root->l, tree1, tree2->l, p);
        return pushUp(root);
    }
    // 将对应区间右端点 <= p 的部分划分到 tree1,右端点 > p 的部分划分到 tree2
    void split2(Node* root, Node*& tree1, Node*& tree2, int p) {
        if (root == NULL) return tree1 = tree2 = NULL, void();
        pushDown(root);
        if (root->tr <= p)
            tree1 = root, split2(root->r, tree1->r, tree2, p);
        else
            tree2 = root, split2(root->l, tree1, tree2->l, p);
        return pushUp(root);
    }
    void merge(Node*& root, Node* tree1, Node* tree2) {
        if (tree1 == NULL) return root = tree2, void();
        if (tree2 == NULL) return root = tree1, void();
        if (tree1->rank < tree2->rank)
            pushDown(root = tree1), merge(root->r, tree1->r, tree2);
        else
            pushDown(root = tree2), merge(root->l, tree1, tree2->l);
        return pushUp(root);
    }

    void print(Node* root) {
        if (root == NULL) return;
        pushDown(root);
        if (root->l != NULL) print(root->l);
        cerr << '[' << root->tl << '~' << root->tr << ':' << root->col << ']';
        if (root->r != NULL) print(root->r);
        return;
    }

   public:
    FhqTreap(void) { root = NULL; }
    void pushBack(int l, int r, int val) {
        Node* p = newNode(val, l, r);
        merge(root, root, p);
        return;
    }
    void opt1_2(int l, int r, int val) {
        Node *nodeL = NULL, *p = NULL, *nodeR = NULL, *treeR = NULL;
        split2(root, root, p, l - 1), split1(p, p, treeR, r);
        split1(p, nodeL, p, l - 1), split2(p, p, nodeR, r);
        if (nodeL != NULL) {
            if (nodeL->tr > r) nodeR = newNode(nodeL->col, r, nodeL->tr);
            nodeL->tr = l - 1, nodeL->reset();
        } else
            split2(root, root, nodeL, l - 2);
        if (nodeR != NULL)
            nodeR->tl = r + 1, nodeR->reset();
        else
            split1(treeR, nodeR, treeR, r + 1);
        // destroy(p);
        p = newNode(val, l, r);
        if (nodeL != NULL && p->col == nodeL->col) {
            p->tl = nodeL->tl, p->reset();
            // delete nodeL;
            nodeL = NULL;
        }
        if (nodeR != NULL && p->col == nodeR->col) {
            p->tr = nodeR->tr, p->reset();
            // delete nodeR;
            nodeR = NULL;
        }
        merge(root, root, nodeL), merge(root, root, p), merge(root, root, nodeR), merge(root, root, treeR);
        return;
    }
    void opt3(int l, int r) {
        r--;
        Node *treeM = NULL, *treeR = NULL, *node = NULL;
        split2(root, root, treeM, l - 1), split1(treeM, treeM, treeR, r);
        split1(treeM, node, treeM, l);
        if (node != NULL && node->col == 1)
            merge(root, root, node);
        else
            merge(treeM, node, treeM);
        node = NULL;
        split2(treeM, treeM, node, r - 1);
        if (node != NULL && node->col == 0) {
            if (node->tr > r)
                merge(treeR, node, treeR);
            else {
                merge(treeM, treeM, node);
                Node* nodeR = NULL;
                split1(treeR, nodeR, treeR, r + 1);
                merge(treeM, treeM, nodeR);
            }
        } else
            merge(treeM, treeM, node);
        if (treeM != NULL) update(treeM, 1, 0);
        merge(root, root, treeM), merge(root, root, treeR);
        findEmpty(root), eraseEmpty();
        return;
    }
    void opt4(int l, int r) {
        l++;
        Node *treeM = NULL, *treeR = NULL, *node = NULL;
        split2(root, root, treeM, l - 1), split1(treeM, treeM, treeR, r);
        split1(treeM, node, treeM, l);
        if (node != NULL && node->col == 0) {
            if (node->tl < l)
                merge(root, root, node);
            else {
                merge(treeM, node, treeM);
                Node* nodeL = NULL;
                split2(root, root, nodeL, l - 2);
                merge(treeM, nodeL, treeM);
            }
        } else
            merge(treeM, node, treeM);
        node = NULL;
        split2(treeM, treeM, node, r - 1);
        if (node != NULL && node->col == 1)
            merge(treeR, node, treeR);
        else
            merge(treeM, treeM, node);
        if (treeM != NULL) update(treeM, 0, 1);
        merge(root, root, treeM), merge(root, root, treeR);
        findEmpty(root), eraseEmpty();
        return;
    }
    void opt5(int l, int r) {
        r--;
        Node *treeM = NULL, *treeR = NULL, *node = NULL;
        split2(root, root, treeM, l - 1), split1(treeM, treeM, treeR, r);
        split1(treeM, node, treeM, l);
        if (node != NULL && node->col == 0)
            merge(root, root, node);
        else
            merge(treeM, node, treeM);
        node = NULL;
        split2(treeM, treeM, node, r - 1);
        if (node != NULL && node->col == 1) {
            if (node->tr > r)
                merge(treeR, node, treeR);
            else {
                merge(treeM, treeM, node);
                Node* nodeR = NULL;
                split1(treeR, nodeR, treeR, r + 1);
                merge(treeM, treeM, nodeR);
            }
        } else
            merge(treeM, treeM, node);
        if (treeM != NULL) update(treeM, 0, -1);
        merge(root, root, treeM), merge(root, root, treeR);
        findEmpty(root), eraseEmpty();
        return;
    }
    void opt6(int l, int r) {
        l++;
        Node *treeM = NULL, *treeR = NULL, *node = NULL;
        split2(root, root, treeM, l - 1), split1(treeM, treeM, treeR, r);
        split1(treeM, node, treeM, l);
        if (node != NULL && node->col == 1) {
            if (node->tl < l)
                merge(root, root, node);
            else {
                merge(treeM, node, treeM);
                Node* nodeL = NULL;
                split2(root, root, nodeL, l - 2);
                merge(treeM, nodeL, treeM);
            }
        } else
            merge(treeM, node, treeM);
        node = NULL;
        split2(treeM, treeM, node, r - 1);
        if (node != NULL && node->col == 0)
            merge(treeR, node, treeR);
        else
            merge(treeM, treeM, node);
        if (treeM != NULL) update(treeM, -1, 0);
        merge(root, root, treeM), merge(root, root, treeR);
        findEmpty(root), eraseEmpty();
        return;
    }
    int opt7(int l, int r) {
        Node *node = NULL, *p = NULL, *treeR = NULL;
        split2(root, root, p, l - 1), split1(p, p, treeR, r);
        int answer = 0;
        if (p != NULL) answer += p->sum;
        split1(p, node, p, l - 1);
        if (node != NULL) answer -= node->col * (l - node->tl);
        merge(p, node, p);
        node = NULL;
        split2(p, p, node, r);
        if (node != NULL) answer -= node->col * (node->tr - r);
        merge(p, p, node);
        merge(root, root, p), merge(root, root, treeR);
        return answer;
    }
    inline void print(void) { return print(root), cerr << endl, void(); }
};

FhqTreap treap;

int a[maxn];

int main() {
    int n = read<int>(), m = read<int>();
    for (register int i = 1; i <= n; i++) a[i] = read<int>();
    int last = 1;
    for (register int i = 2; i <= n; i++)
        if (a[i] != a[i - 1]) treap.pushBack(last, i - 1, a[i - 1]), last = i;
    treap.pushBack(last, n, a[n]);
    last = 0;
    while (m--) {
        int opt = read<int>(), l = last ^ read<int>(), r = last ^ read<int>();
        if (opt == 1)
            treap.opt1_2(l, r, 0);
        else if (opt == 2)
            treap.opt1_2(l, r, 1);
        else if (opt == 3)
            treap.opt3(l, r);
        else if (opt == 4)
            treap.opt4(l, r);
        else if (opt == 5)
            treap.opt5(l, r);
        else if (opt == 6)
            treap.opt6(l, r);
        else
            write(last = treap.opt7(l, r)), putch('\n');
    }
    return 0;
}

我缓慢吐出一串啊吧啊吧并不再想说话