[HNOI2011]括号修复——Splay

题面

　　 Bzoj2329

解析

　　 要支持区间翻转，就可以想到Splay了

　　但是要维护什么信息才能得到答案呢，将 “)” 看作1，”(“ 看作-1，记前缀最大和为$lx$， 后缀最小和为$rn$，那么 $ans = left lceil frac{lx}{2} ight ceil + left lceil left | frac{rn}{2} ight | ight ceil$, 因此维护前缀最大和$lx$， 后缀最小和$rn$，但由于有区间翻转与取反操作，还需要维护前缀最小和$ln$， 后缀最大和$rx$，更新的方式与Splay求最大子段和时一样，如图：

 

　　可以求出答案了，但还有一件事需要考虑，就是标记与标记之间的影响，也就是标记下放时的先后顺序。显然覆盖标记需要先下放，它需要把儿子的翻转与取反标记清0；接下来是先翻转还是先取反呢？实际上两个标记之间没有影响，无论哪个第二个下放都行，我按照题目顺序，把翻转标记第二个下放，取反标记最后下放

　代码：

#include<cstdio>
#include<iostream>
#include<algorithm>
#include<climits>
using namespace std;
const int maxn = 100006, inf = INT_MAX;

template<class T> void read(T &re)
{
    re=0;
    T sign=1;
    char tmp;
    while((tmp=getchar())&&(tmp<'0'||tmp>'9')) if(tmp=='-') sign=-1;
    re=tmp-'0';
    while((tmp=getchar())&&(tmp>='0'&&tmp<='9')) re=(re<<3)+(re<<1)+(tmp-'0');
    re*=sign;
}

int n, m, root;
int a[maxn];
char ch[maxn];

struct splay_tree{
    int s[2], fa, val, lx, rx, ln, rn, cov, rev, opp, sum, siz;
}tr[maxn];

void Cover(int now, int x)
{
    tr[now].val = x;
    tr[now].sum = tr[now].siz * tr[now].val;
    tr[now].lx = max(0, tr[now].sum);
    tr[now].rx = max(0, tr[now].sum);
    tr[now].ln = min(0, tr[now].sum);
    tr[now].rn = min(0, tr[now].sum);
    tr[now].cov = tr[now].val;
    tr[now].rev = tr[now].opp = 0;
}

void Reverse(int now)
{
    swap(tr[now].s[0], tr[now].s[1]);
    swap(tr[now].lx, tr[now].rx);
    swap(tr[now].ln, tr[now].rn);
    tr[now].rev ^= 1;
}

void Oppsit(int now)
{
    swap(tr[now].lx, tr[now].ln);
    swap(tr[now].rx, tr[now].rn);
    tr[now].lx *= -1;
    tr[now].rx *= -1;
    tr[now].ln *= -1;
    tr[now].rn *= -1;
    tr[now].val *= -1;
    tr[now].sum *= -1;
    tr[now].opp ^= 1;
}

void spread(int x)
{
    int ls = tr[x].s[0], rs = tr[x].s[1];
    if(tr[x].cov)
    {
        if(ls)
            Cover(ls, tr[x].cov);
        if(rs)
            Cover(rs, tr[x].cov);
        tr[x].cov = 0;
    }
    if(tr[x].rev)
    {
        if(ls)
            Reverse(ls);
        if(rs)
            Reverse(rs);
        tr[x].rev = 0;
    }
    if(tr[x].opp)
    {
        if(ls)
            Oppsit(ls);
        if(rs)
            Oppsit(rs);
        tr[x].opp = 0;
    }
}

void update(int x)
{
    int ls = tr[x].s[0], rs = tr[x].s[1];
    tr[x].sum = tr[ls].sum + tr[rs].sum + tr[x].val;
    tr[x].siz = tr[ls].siz + tr[rs].siz + 1;
    tr[x].lx = max(tr[ls].lx, tr[ls].sum + tr[x].val + tr[rs].lx);
    tr[x].rx = max(tr[rs].rx, tr[rs].sum + tr[x].val + tr[ls].rx);
    tr[x].ln = min(tr[ls].ln, tr[ls].sum + tr[x].val + tr[rs].ln);
    tr[x].rn = min(tr[rs].rn, tr[rs].sum + tr[x].val + tr[ls].rn);
}

void Rotate(int x)
{
    int y = tr[x].fa, z = tr[y].fa, k = (tr[y].s[1] == x), w = (tr[z].s[1] == y), son = tr[x].s[k^1];
    spread(y);spread(x);
    tr[y].s[k] = son;tr[son].fa = y;
    tr[x].s[k^1] = y;tr[y].fa = x;
    tr[z].s[w] = x;tr[x].fa = z;
    update(y);update(x);
}

void Splay(int x, int to)
{
    int y, z;
    while(tr[x].fa != to)
    {
        y = tr[x].fa;
        z = tr[y].fa;
        if(z != to)
            Rotate((tr[y].s[0] == x) ^ (tr[z].s[0] == y)? x: y);
        Rotate(x);
    }
    if(!to)
        root = x;
}

void Build(int l, int r, int ff)
{
    int mid = (l + r)>>1;
    if(l < mid)
        Build(l, mid - 1, mid);
    if(mid < r)
        Build(mid + 1, r, mid);
    if(ff)
        tr[ff].s[ff < mid] = mid;
    tr[mid].fa = ff;
    tr[mid].val = a[mid];
    update(mid);
}

int Find(int x)
{
    int now = root;
    while(1)
    {
        spread(now);
        int ls = tr[now].s[0], rs = tr[now].s[1];
        if(tr[ls].siz + 1 == x)
            return now;
        if(x <= tr[ls].siz)
            now = ls;
        else
            x -= tr[ls].siz + 1, now = rs;
    }
}

int main()
{
    read(n);read(m);
    scanf("%s", ch);
    for(int i = 1; i <= n; ++i)
        a[i+1] = (ch[i-1] == ')'? 1: -1);
    n += 2;
    Build(1, n, 0);
    tr[0].s[1] = root = (1 + n)>>1;
    for(int i = 1; i <= m; ++i)
    {
        //cout<<i<<endl;
        char opt[10];
        scanf("%s", opt);
        if(opt[0] == 'R')
        {
            int x, y;
            char c[2];
            read(x);read(y);
            scanf("%s", c);
            y += 2;
            x = Find(x);
            y = Find(y);
            Splay(x, 0);
            Splay(y, x);
            int now = tr[y].s[0];
            Cover(now, (c[0] == ')'? 1: -1));
            update(y);update(x);
        }
        else if(opt[0] == 'S')
        {
            int x, y;
            read(x);read(y);
            y += 2;
            x = Find(x);
            y = Find(y);
            Splay(x, 0);
            Splay(y, x);
            int now = tr[y].s[0];
            Reverse(now);
        }
        else if(opt[0] == 'I')
        {
            int x, y;
            read(x);read(y);
            y += 2;
            x = Find(x);
            y = Find(y);
            Splay(x, 0);
            Splay(y, x);
            int now = tr[y].s[0];
            Oppsit(now);
            update(y);update(x);
        }
        else
        {
            int x, y;
            read(x);read(y);
            y += 2;
            x = Find(x);
            y = Find(y);
            Splay(x, 0);
            Splay(y, x);
            int now = tr[y].s[0];
            printf("%d
", (tr[now].lx + 1) / 2 + (abs(tr[now].rn) + 1) / 2);
        }
    }
    return 0;
}