莫队

莫队

• 解决区间统计问题 如这题

• 分块 + 排序 + 加减操作移动 + 统计

• 复杂度： \(O(N\sqrt{N})\)

• 思想：通过分块和排序后，减少相邻区间的移动操作次数，并在区间移动过程中进行区间中统计

• 本质：二维扫描线，在二维平面上求曼哈顿距离最小生成树

• 带修莫队即加入时间一维，变为三维扫描线

• 树上莫队可以用括号序转为序列上的操作

例题 ：

P3901

SP3267

[国家集训队]小Z的袜子 /【模板】莫队

[国家集训队]数颜色 / 维护队列 /【模板】带修莫队

【模板】回滚莫队&不删除莫队

[WC2013]糖果公园

各种莫队的模板

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

#define ll long long

ll read() {
  ll a = 0, f = 1; char c = getchar();
  while(c > '9' || c < '0') {if(c == '-') f = -1; c = getchar();}
  while(c >= '0' && c <= '9') {a = a * 10 + c - '0'; c = getchar();}
  return a * f;
}

const int MAXN = 5e5 + 14;

ll n, m, xl, xr, block;
ll ans;
ll a[MAXN];
ll cnt[MAXN];
ll s[MAXN], w[MAXN];
struct question {
  int l, r, bo, id;
  #define l(x) ask[x].l
  #define r(x) ask[x].r
  #define bo(x) ask[x].bo
  #define id(x) ask[x].id
} ask[MAXN];

bool cmp(question a, question b) {
  if(a.bo == b.bo) return a.r < b.r;
  else return a.bo < b.bo;
}
void add(int x) {
  if(cnt[a[x]] > 0) ans += cnt[a[x]];
  cnt[a[x]]++;
}
void del(int x) {
  cnt[a[x]]--;
  if(cnt[a[x]] > 0) ans -= cnt[a[x]];
}
ll gcd(ll x, ll y) {
  if(y == 0) return x;
  else return gcd(y, x % y);
}

int main() {
  
  n = read(), m = read(); block = sqrt(n);
  for (int i = 1; i <= n; i++) a[i] = read();
  for (int i = 1; i <= m; i++) l(i) = read(), r(i) = read(), bo(i) = l(i) / block, id(i) = i;

  sort(ask + 1, ask + m + 1, cmp);

  for (int i = 1; i <= m; i++) {
    while(xr > r(i)) del(xr--);
    while(xr < r(i)) add(++xr);
    while(xl < l(i)) del(xl++);
    while(xl > l(i)) add(--xl);
    if(xl == xr) {
      s[id(i)] = 0; w[id(i)] = 1;continue;
    } 
    s[id(i)] = ans; w[id(i)] = (xr - xl + 1) * (xr - xl) / 2;
    if(s[id(i)] == 0) w[id(i)] = 1;
    else {
      ll x = gcd(s[id(i)], w[id(i)]);
      s[id(i)] /= x; w[id(i)] /= x;
    }
  }
  
  for (int i = 1; i <= m; i++) {  
    printf("%lld/%lld\n", s[i], w[i]);
  }

  return 0;
}

//带修
//块大小为n^(2/3),复杂度O(n^(3/4))
const int MAXN = 2000010;
int n, m, qu, xt;
int block, B[MAXN];
int nl = 1, nr = 0, now = 0, a[MAXN], cnt[MAXN], tmp;
int c[MAXN][2];//0表示位置，1表示值
int ans[MAXN];
struct question
{
	int l, r, ti, id;
}ask[MAXN];
inline bool cmp(const question &a, const question &b)
{
	return B[a.l] ^ B[b.l] ? B[a.l] < B[b.l] : B[a.r] ^ B[b.r] ? B[a.r] < B[b.r] : a.ti < b.ti;
}
inline void add(int x)
{
	if(!cnt[a[x]]) tmp++;
	cnt[a[x]]++;
}
inline void del(int x)
{
	cnt[a[x]]--;
	if(!cnt[a[x]]) tmp--;
}
int main()
{
	scanf("%d %d", &n, &m);
	block = pow(n, 2.0 / 3.0);
	for (int i = 1; i <= n; i++) scanf("%d", &a[i]);
	for (int i = 1; i <= n; i++) B[i] = (i - 1) / block + 1;
	for (int i = 1; i <= m; i++)
	{
		char opt[3];
		scanf("%s", opt + 1);
		if(opt[1] == 'Q')
		{
			++qu;
			scanf("%d %d", &ask[qu].l, &ask[qu].r);
			ask[qu].id = qu; ask[qu].ti = xt;
		}
		else 
		{
			++xt;
			scanf("%d %d", &c[xt][0], &c[xt][1]);
		}
	}
	sort(ask + 1, ask + qu + 1, cmp);
	for (int i = 1; i <= qu; i++)
	{
		while(nl > ask[i].l) add(--nl);
		while(nl < ask[i].l) del(nl++);
		while(nr > ask[i].r) del(nr--);
		while(nr < ask[i].r) add(++nr);
		while(ask[i].ti > now)//往前更新
		{
			++now;
			if(c[now][0] >= nl && c[now][0] <= nr)
			{
				cnt[ a[c[now][0]] ]--;
				if(!cnt[ a[c[now][0]] ]) --tmp;
			}
			swap(a[ c[now][0] ], c[now][1]);
			if(c[now][0] >= nl && c[now][0] <= nr)
			{
				if(!cnt[ a[c[now][0]] ]) ++tmp;
				cnt[ a[c[now][0]] ]++;
			}

		}
		while(ask[i].ti < now)//往回撤销
		{
			if(c[now][0] >= nl && c[now][0] <= nr)
			{
				cnt[ a[c[now][0]] ]--;
				if(!cnt[ a[c[now][0]] ]) --tmp;
			}
			swap(a[ c[now][0] ], c[now][1]);
			if(c[now][0] >= nl && c[now][0] <= nr)
			{
				if(!cnt[ a[c[now][0]] ]) ++tmp;
				cnt[ a[c[now][0]] ]++;
			}
			--now;
		}
		ans[ ask[i].id ] = tmp;
	}
	for (int i = 1; i <= qu; i++)
	{
		printf("%d\n", ans[i]);
	}
	return 0;
}

//树上
#include <bits/stdc++.h>
using namespace std;
const int MAXN = 1000010;
const int LOG = 20;
int n, m, q, nl = 1, nr = 0, nt = 0;
long long vm[MAXN], wn[MAXN], a[MAXN], c[MAXN][2];
int block, B[MAXN];
int head[MAXN], nxt[MAXN << 1], v[MAXN << 1], cnt;
int qu, tx;
int dfn[MAXN], ind, fi[MAXN], se[MAXN];
int anc[MAXN][LOG], depth[MAXN];
int vis[MAXN];
int ct[MAXN];
long long res[MAXN], ans;
struct question
{
	int l, r, ti, id;
} ask[MAXN];
inline bool cmp(const question &a, const question &b)
{
	if(B[fi[a.l]] ^ B[fi[b.l]]) return B[fi[a.l]] < B[fi[b.l]];
	if(B[fi[a.r]] ^ B[fi[b.r]]) return B[fi[a.r]] < B[fi[b.r]];
	return a.ti < b.ti;
}
inline void add(int x, int y)
{
	nxt[++cnt] = head[x]; head[x] = cnt; v[cnt] = y;
	nxt[++cnt] = head[y]; head[y] = cnt; v[cnt] = x;
}
inline void dfsl(int u, int p, int d) 
{
    anc[u][0] = p; depth[u] = d;
    for (int i = head[u]; i; i = nxt[i]) 
    {
        if(v[i] == p) continue;
        dfsl(v[i], u, d + 1);
    }
}
inline void init() {
	dfsl(1, 0, 1);
	for (int j = 1; j < LOG; j++) 
		for (int i = 1; i <= n; i++) 
			anc[i][j] = anc[ anc[i][j - 1] ][j - 1];
}
inline void swim(int &x, int h) {
	for (int i = 0; h > 0; i++) 
	{
		if(h & 1) x = anc[x][i];
		h >>= 1;
	}
}
inline int LCA(int x, int y) 
{
	if(depth[x] < depth[y]) swap(x, y);
	if(y == 1) return 1;
	swim(x, depth[x] - depth[y]);
	if(x == y) return x;
	for (int i = LOG - 1; i >= 0; i--) 
	{
		if(anc[x][i] != anc[y][i]) 
		{
	   		x = anc[x][i];
	   		y = anc[y][i];
		}
	}
	return anc[x][0];
}
void dfs(int now, int f)
{
	dfn[++ind] = now; fi[now] = ind;
	for (int i = head[now]; i; i = nxt[i])
	{
		if(v[i] == f) continue;
		dfs(v[i], now);
	}
	dfn[++ind] = now; se[now] = ind;
}
inline void sol(int x)//原编号
{
	if(!vis[x]) ans += vm[a[x]] * wn[++ct[a[x]]];
	else ans -= vm[a[x]] * wn[ct[a[x]]--];
	vis[x] ^= 1;//翻转标记
}
void change()
{
	if(vis[c[nt][0]])
	{
		sol(c[nt][0]);
		swap(a[c[nt][0]], c[nt][1]);
		sol(c[nt][0]);
	}
	else swap(a[c[nt][0]], c[nt][1]);
}
int main()
{
	scanf("%d %d %d", &n, &m, &q);
	for (int i = 1; i <= m; i++) scanf("%lld", &vm[i]);
	for (int i = 1; i <= n; i++) scanf("%lld", &wn[i]);
	for (int i = 1; i < n; i++)
	{
		int x, y;
		scanf("%d %d", &x, &y);
		add(x, y);
	}
	for (int i = 1; i <= n; i++) scanf("%lld", &a[i]);
	init();
	dfs(1, 0);
	block = pow(ind, 2.0 / 3.0) + 1;
	for (int i = 1; i <= ind; i++) B[i] = (i - 1) / block + 1;
	tx = 0, qu = 0;
	for (int i = 1; i <= q; i++)
	{
		int opt; scanf("%d", &opt);
		if(opt == 0)
		{
			++tx;
			scanf("%lld %lld", &c[tx][0], &c[tx][1]);//0表示在原编号的位置，1表示修改的值
		}
		else 
		{
			++qu;
			scanf("%d %d", &ask[qu].l, &ask[qu].r);
			if(fi[ ask[qu].l ] > fi[ ask[qu].r ]) swap(ask[qu].l, ask[qu].r);
			ask[qu].id = qu; ask[qu].ti = tx;
		}
	}
	sort(ask + 1, ask + qu + 1, cmp);
	for (int i = 1; i <= qu; i++)
	{
		while(nl < fi[ask[i].l]) sol(dfn[nl++]);//dfn记录了原编号
		while(nl > fi[ask[i].l]) sol(dfn[--nl]);
		while(nr < fi[ask[i].r]) sol(dfn[++nr]);
		while(nr > fi[ask[i].r]) sol(dfn[nr--]);
		while(nt < ask[i].ti)
		{
			++nt;
   			//vis 记录一个 节点计算了几次
			change();
		}
		while(nt > ask[i].ti)
		{
			change();
			--nt;
		}
		int p = LCA(ask[i].l, ask[i].r);
		if(ask[i].l ^ p) sol(ask[i].l);
		if(ask[i].l ^ p && ask[i].r ^ p) sol(p);
		res[ask[i].id] = ans;
		if(ask[i].l ^ p) sol(ask[i].l);
		if(ask[i].l ^ p && ask[i].r ^ p) sol(p);
	}
	for (int i = 1; i <= qu; i++)
	{
		printf("%lld\n", res[i]);
	}
	return 0;
}

//回滚莫队
#include <bits/stdc++.h>
using namespace std;
const int MAXN = 200010;
int n, m, block, bn, B[MAXN];
int a[MAXN], nl, nr, ans, b[MAXN];
int Ans[MAXN];
int last[MAXN], nxt[MAXN], fir[MAXN];
int s[MAXN], st;//记录操作
inline int read()
{
    int a=0,f=1; char c=getchar();
    while(c<'0'||c>'9'){if(c=='-')f=-1;c=getchar();}
    while(c>='0'&&c<='9'){a=a*10+c-'0';c=getchar();}
    return a*f;
}
inline void print(int x)
{
    if(x<0)putchar('-'),x=-x;
    if(x<10)putchar(x+48);
    else print(x/10),putchar(x%10+48);
}
struct question
{
	int l, r, id;
} ask[MAXN];
inline bool cmp(const question &a, const question &b)
{
	return B[a.l] ^ B[b.l] ? B[a.l] < B[b.l] : a.r < b.r;
}
inline int max(const int &x, const int &y)
{
	return x > y ? x : y;
}
inline int min(const int &x, const int &y)
{
	return x < y ? x : y;
}
inline int calc(int l, int r)
{
	int res = 0;
	for (int i = l; i <= r; i++) fir[a[i]] = 0;
	for (int i = l; i <= r; i++)
	{
		if(!fir[a[i]]) fir[a[i]] = i;
		else res = max(res, i - fir[a[i]]);
	}
	return res;
}
int main()
{
	n = read(); block = pow(n, 1.0 / 2.0);
	for (int i = 1; i <= n; i++) a[i] = read(), b[i] = a[i], B[i] = (i - 1) / block + 1;
	sort(b + 1, b + n + 1);
	int un = unique(b + 1, b + n + 1) - b - 1;
	for (int i = 1; i <= n; i++) a[i] = lower_bound(b + 1, b + un + 1, a[i]) - b;
	m = read();
	for (int i = 1; i <= m; i++) ask[i].l = read(), ask[i].r = read(), ask[i].id = i;
	bn = B[n];
	sort(ask + 1, ask + m + 1, cmp);
	for (int i = 1, j = 1; i <= bn; i++)
	{
		int br = min(n, i * block); 
		nl = br + 1, nr = br, ans = 0;
		st = 0;
		for (;B[ask[j].l] == i; j++)
		{
			//last记录第一次出现，nxt记录最后一次出现
			if(B[ask[j].r] == i)//在同一个块内暴力计算
			{
				Ans[ask[j].id] = calc(ask[j].l, ask[j].r);
				continue;
			}
			while(nr < ask[j].r)//同一个块内r递增
			{
				++nr;
				nxt[a[nr]] = nr;
				if(!last[a[nr]]) last[a[nr]] = nr, s[++st] = a[nr];
				ans = max(ans, nr - last[a[nr]]);
			}
			int tmp = ans; //先保存一下，因为右区间的贡献不会被刷新，但左区间的会
			while(nl > ask[j].l)
			{
				--nl;
				if(nxt[a[nl]]) tmp = max(tmp, nxt[a[nl]] - nl);
				else nxt[a[nl]] = nl;
			}
			Ans[ask[j].id] = tmp;
			while(nl <= br)
			{
				if(nxt[a[nl]] == nl) nxt[a[nl]] = 0;//去掉左区间的贡献
				++nl;
			}
		}
		for (int i = 1; i <= st; i++) last[s[i]] = nxt[s[i]] = 0;//将上一个块处理的贡献清空
	}
	for (int i = 1; i <= m; i++) print(Ans[i]), putchar('\n');
	return 0;
}