Loj 2052 矿区

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平面图转对偶图.

先把平面图转成对偶图.

方法是把每条无向边拆成两条有向边,规定每个面都在围成它的有向边的左侧,这样每条有向边就只会被用一次.

对于每个边,按照极角序找它转动后下一条边,这样会形成一个置换.

分解后得到的每个环就对应了每个面,即对偶图中的每个点,再在有公共边的面之间连边即可.

找出面的时候用叉积顺便算一下面积,面积为负数的就是外面无穷大的面.

然后以最外面无穷大的面为根,做一棵生成树出来,询问时按照最外围边对子树和容斥一下即可.

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//%std
#include<bits/stdc++.h>
using namespace std;
typedef long long ll;
inline ll read()
{
	ll out = 0, fh = 1;
	char jp = getchar();
	while ((jp > '9' || jp < '0') && jp != '-')
		jp = getchar();
	if (jp == '-')
		fh = -1, jp = getchar();
	while (jp >= '0' && jp <= '9')
		out = out * 10 + jp - '0', jp = getchar();
	return out * fh;
}
ll Abs(ll x)
{
	return x > 0 ? x : -x;
}
const double eps = 1e-10;
const int N = 4e6 + 10;
int n, m, q, rt, cnt, tid[N];
ll sum[N][2];
struct v2
{
	int x, y;
	v2(int x = 0, int y = 0) : x(x), y(y) {}
	v2 operator - (const v2 &rhs) const
	{
		return v2(x - rhs.x, y - rhs.y);
	}
	ll operator * (const v2 &rhs) const
	{
		return 1LL * x * rhs.y - 1LL * y * rhs.x;
	}
	double angle()
	{
		return atan2(y, x);
	}
} p[N];
double calc(int u, int v)
{
	return (p[v] - p[u]).angle();
}
struct Edge
{
	int u, v, id;
	double slop;
	Edge(int u = 0, int v = 0, int id = 0, double slop = 0) :
		u(u), v(v), id(id), slop(slop) {}
	friend bool operator < (Edge A, Edge B)
	{
		if (fabs(A.slop - B.slop) > eps)
			return A.slop < B.slop;
		return A.v < B.v;
	}
} E[N];
typedef vector<Edge>::iterator vit;
vector<Edge> vec[N];
namespace DualGraph
{
	int ecnt = 1, head[N], to[N], nx[N];
	void addedge(int u, int v)
	{
		++ecnt;
		to[ecnt] = v;
		nx[ecnt] = head[u];
		head[u] = ecnt;
	}
	int fa[N], vis[N], ontree[N];
	void dfs(int u, int F)
	{
		fa[u] = F, vis[u] = 1;
		for (int i = head[u]; i; i = nx[i])
		{
			int v = to[i];
			if (vis[v])
				continue;
			ontree[i] = ontree[i ^ 1] = 1;
			dfs(v, u);
			sum[u][0] += sum[v][0], sum[u][1] += sum[v][1];
		}
	}
	int np[N];
	void solve()
	{
		dfs(rt, 0);
		ll num = 0, deno = 0;
		for (int i = 1; i <= q; ++i)
		{
			ll c = (read() + num) % n + 1;
			for (int j = 1; j <= c; ++j)
			{
				ll x = (read() + num) % n + 1;
				np[j] = x;
			}
			np[c + 1] = np[1];
			num = deno = 0;
			for (int j = 1; j <= c; ++j)
			{
				int x = np[j], y = np[j + 1];
				Edge tmp = Edge(x, y, 0, calc(x, y));
				vit it = lower_bound(vec[x].begin(), vec[x].end(), tmp);
				int a = (*it).id, b = a ^ 1;
				if (!ontree[a])
					continue;
				a = tid[a], b = tid[b];
				if (fa[a] == b)
				{
					deno += sum[a][0];
					num += sum[a][1];
				}
				else if (fa[b] == a)
				{
					deno -= sum[b][0];
					num -= sum[b][1];
				}
			}
			num = Abs(num), deno = Abs(deno);
			ll g = __gcd(num, deno);
			num /= g, deno /= g;
			printf("%lld %lld\n", num, deno);
		}
	}
}
namespace PlanGraph
{
	int ecnt = 1, nx[N];
	void addedge(int u, int v)
	{
		++ecnt;
		E[ecnt] = Edge(u, v, ecnt, calc(u, v));
		vec[u].push_back(E[ecnt]);
	}
	void Init()
	{
		n = read(), m = read(), q = read();
		for (int i = 1; i <= n; ++i)
			p[i].x = read(), p[i].y = read();
		for (int i = 1; i <= m; ++i)
		{
			int u = read(), v = read();
			addedge(u, v);
			addedge(v, u);
		}
		for (int i = 1; i <= n; ++i)
			sort(vec[i].begin(), vec[i].end());
		for (int i = 2; i <= ecnt; ++i)
		{
			int v = E[i].v;
			vit it = lower_bound(vec[v].begin(), vec[v].end(), E[i ^ 1]);
			if (it == vec[v].begin())
				it = vec[v].end();
			--it;
			nx[i] = (*it).id;				
		}
		for (int i = 2; i <= ecnt; ++i)
			if (!tid[i])
			{
				tid[i] = ++cnt;
				ll Area = 0;
				v2 st = p[E[i].u];
				for (int x = nx[i]; x != i; x = nx[x])
				{
					tid[x] = cnt;
					v2 a = p[E[x].u], b = p[E[x].v];
					Area += (a - st) * (b - st);
				}
				if (Area < 0)
					rt = cnt;
				else
				{
					sum[cnt][0] = Area * 2;
					sum[cnt][1] = Area * Area;
				}
			}
		for (int i = 2; i <= ecnt; ++i)
			DualGraph::addedge(tid[i], tid[i ^ 1]);
	}
}
int main()
{
	PlanGraph::Init();
	DualGraph::solve();
	return 0;
}