Codeforces 1150E（树、线段树）

要点

• 括号序列平衡度即树深度的性质
• 相当于中序遍历，则两点间最浅的地方即是LCA的性质
• 线段树维护\(d(a) + d(c) - 2*d(lca(a,c))\)，一层层剥，思考维护这个量需要什么，结果维护一大堆。

#include <cstdio>
#include <iostream>
#include <algorithm>
using namespace std;

const int maxn = 2e5 + 5;

int n, q;
char s[maxn];

class SegmentTree {
public:
    #define ls (p << 1)
    #define rs (p << 1 | 1)

    struct node {
        int l, r;
        int depth;//its root is s[l]
        int ans;//d(a) + d(c) - 2d(lca(a, c))
        int lmax;//d(a) - 2d(lca(a, c))
        int rmax;//d(c) - 2d(lca(a, c))
        int mxd;//max depth
        int mnd;//min depth

        void init(int val) {
            mxd = mnd = depth = val;
            lmax = rmax = -val;
            ans = 0;
        }
    }t[maxn * 3];

    void Push_up(int p) {
        // b = lca(a, c), a <= b <= c
        t[p].depth = t[ls].depth + t[rs].depth;
        t[p].mxd = max(t[ls].mxd, t[rs].mxd + t[ls].depth);
        t[p].mnd = min(t[ls].mnd, t[rs].mnd + t[ls].depth);
        t[p].lmax = max(t[ls].lmax, t[rs].lmax - t[ls].depth);//a, b both in l or r
        t[p].lmax = max(t[p].lmax, t[ls].mxd - 2 * (t[rs].mnd + t[ls].depth));//a in l and b in r
        t[p].rmax = max(t[ls].rmax, t[rs].rmax - t[ls].depth);//b, c both in l or r
        t[p].rmax = max(t[p].rmax, t[rs].mxd + t[ls].depth - 2 * t[ls].mnd);//b int l and c in r
        t[p].ans = max(t[ls].ans, t[rs].ans);//a,b,c all in l or r
        t[p].ans = max(t[p].ans, max(t[ls].lmax + t[rs].mxd + t[ls].depth, t[ls].mxd + t[rs].rmax - t[ls].depth));//ab in l, c in r || a in l, bc in r
    }

    void Build(int l, int r, int p) {
        t[p].l = l, t[p].r = r;
        if (l == r) {
            t[p].init(s[l] == '(' ? 1 : -1);
            return;
        }
        int mid = (l + r) >> 1;
        Build(l, mid, ls);
        Build(mid + 1, r, rs);
        Push_up(p);
    }

    void Modify(int l, int r, int p) {
        if (t[p].l == t[p].r) {
            t[p].init(s[l] == '(' ? 1 : -1);
            return;
        }
        int mid = (t[p].l + t[p].r) >> 1;
        if (l <= mid)   Modify(l, r, ls);
        if (mid < r)    Modify(l, r, rs);
        Push_up(p);
    }
}tree;

int main() {
    scanf("%d %d", &n, &q);
    scanf("%s", s + 1);

    n = (n - 1) << 1;
    tree.Build(1, n, 1);
    printf("%d\n", tree.t[1].ans);

    for (int a, b; q; q--) {
        scanf("%d %d", &a, &b);
        if (s[a] != s[b]) {
            swap(s[a], s[b]);
            tree.Modify(a, a, 1);
            tree.Modify(b, b, 1);
        }
        printf("%d\n", tree.t[1].ans);
    }
}