1. 程式人生 > 其它 >「題解」洛谷 P6054 [RC-02] 開門大吉

「題解」洛谷 P6054 [RC-02] 開門大吉

一套題裡面的各個題是假的,可以處理出 \(w_{i,j}\) 為第 \(i\) 個人選第 \(j\) 套題的期望得分。

對於每個人來講,有 \(m\) 種選擇套題,只能選擇一個套題,要求價值最小,於是想到一個最小割的經典建圖:對於每個人,從 \(S\)\(T\) 連一條長度為 \(m\) 的鏈,邊的流量依次為這個人選擇第 \(i\) 套題的期望得分。

考慮怎樣描述限制,若割了 \(j\) 的第 \(l\) 條邊,那麼就不能割 \(i\) 的前 \((j+k-1)\) 條邊。

那麼就從 \(j\) 的第 \(l\) 條邊的起點,向 \(i\) 的前 \((j+k-1)\) 條邊的終點連一條 \(+\infty\)

但是要注意到,限制和限制之間是有傳遞性的,假若有 \(1\)\(2\)\(1\)\(2\)\(3\)\(1\),那麼還需要滿足 \(1\)\(3\)\(2\)

將一開始 \(S\)\(T\) 的若干條長度為 \(m\) 的鏈,連一條反向的流量為 \(+\infty\) 的邊,這樣限制和限制組成的限制也被滿足了。

最小割 建圖模型:每個位置 \(n\)\(1\),求最小代價,\(S\)\(T\) 連若干條長度為 \(n\),流量依次為每種選擇的代價,求最小割。

#include<iostream>
#include<cstdio>
#include<algorithm>
#include<vector>
#include<queue>
#define pb emplace_back
#define mp std::make_pair
#define fi first
#define se second
typedef long long ll;
typedef long double ld;
typedef unsigned long long ull;
typedef std::pair<int, int> pii;
typedef std::vector<int> vi;
const ll mod = 998244353;
ll Add(ll x, ll y) { return (x+y>=mod) ? (x+y-mod) : (x+y); }
ll Mul(ll x, ll y) { return x * y % mod; }
ll Mod(ll x) { return x < 0 ? (x + mod) : (x >= mod ? (x-mod) : x); }
ll cadd(ll &x, ll y) { return x = (x+y>=mod) ? (x+y-mod) : (x+y); }
ll cmul(ll &x, ll y) { return x = x * y % mod; }
template <typename T> T Max(T x, T y) { return x > y ? x : y; }
template<typename T, typename... T2> T Max(T x, T2 ...y) { return Max(x, y...); }
template <typename T> T Min(T x, T y) { return x < y ? x : y; }
template<typename T, typename... T2> T Min(T x, T2 ...y) { return Min(x, y...); }
template <typename T> T cmax(T &x, T y) { return x = x > y ? x : y; }
template <typename T> T cmin(T &x, T y) { return x = x < y ? x : y; }
template <typename T>
T &read(T &r) {
	r = 0; bool w = 0; char ch = getchar();
	while(ch < '0' || ch > '9') w = ch == '-' ? 1 : 0, ch = getchar();
	while(ch >= '0' && ch <= '9') r = r * 10 + (ch ^ 48), ch = getchar();
	return r = w ? -r : r;
}
template<typename T1, typename... T2>
void read(T1 &x, T2& ...y) { read(x); read(y...); }
const int N = 100010;
const int M = 500010;
const ll INF = 0x7fffffffffffffff;
const ld eps = 1e-10;
int n, m, p, y, c[N], pos[81][81];
ld f[81][81][81], w[81][81];
int tot, S, T, ent = 1, head[N], cur[N], dis[N];
struct Edge {
	int to, nxt;
	ld fl;
}e[M];
inline void add(int x, int y, ld z) {
	e[++ent].to = y; e[ent].fl = z; e[ent].nxt = head[x]; head[x] = ent;
	e[++ent].to = x; e[ent].fl = 0; e[ent].nxt = head[y]; head[y] = ent;
}
bool bfs() {
	for(int i = 1; i <= tot; ++i) dis[i] = -1, cur[i] = head[i];
	std::queue<int>q;
	q.push(S); dis[S] = 0;
	while(!q.empty()) {
		int x = q.front(); q.pop();
		for(int i = head[x]; i; i = e[i].nxt) {
			int v = e[i].to;
			if(dis[v] == -1 && e[i].fl > eps) {
				dis[v] = dis[x] + 1;
				q.push(v);
			}
		}
	}
	return dis[T] != -1;
}
bool qwqbfs() {
	for(int i = 1; i <= tot; ++i) dis[i] = -1;
	std::queue<int>q;
	q.push(S); dis[S] = 0;
	while(!q.empty()) {
		int x = q.front(); q.pop();
		for(int i = head[x]; i; i = e[i].nxt) {
			int v = e[i].to;
			if(dis[v] == -1 && e[i].fl == INF) {
				dis[v] = dis[x] + 1;
				q.push(v);
			}
		}
	}
	return dis[T] != -1;
}
ld dfs(int x, ld lim) {
	if(x == T) return lim;
	ld flow = 0;
	for(int i = cur[x]; i && flow < lim; i = e[i].nxt) {
		int v = e[i].to; cur[x] = i;
		if(dis[v] == dis[x] + 1 && e[i].fl > eps) {
			ld f = dfs(v, Min(e[i].fl, lim - flow));
			flow += f; e[i].fl -= f; e[i^1].fl += f;
		}
	}
	return flow;
}
ld dinic() {
	ld mxfl = 0;
	while(bfs())
		mxfl += dfs(S, INF);
	return mxfl;
}
void solve() { for(int i = 1; i <= tot; ++i) head[i] = 0; tot = 0; ent = 1;
	read(n, m, p, y);
	for(int i = 1; i <= p; ++i) read(c[i]);
	for(int j = 1; j <= m; ++j)
		for(int i = 1; i <= n; ++i)
			for(int k = 1; k <= p; ++k) {
				double qwq; scanf("%lf", &qwq);
				f[i][j][k] = qwq;
			}
	for(int i = 1; i <= n; ++i)
		for(int j = 1; j <= m; ++j) {
			ld t = 1; w[i][j] = 0;
			for(int k = 1; k <= p; ++k) {
				t *= f[i][j][k];
				w[i][j] += t * c[k];
			}
		}
	S = ++tot; T = ++tot;
	for(int i = 1; i <= n; ++i) {
		pos[i][0] = S;
		for(int j = 1; j <= m; ++j) {
			if(j < m) pos[i][j] = ++tot;
			else pos[i][j] = T;
			add(pos[i][j-1], pos[i][j], w[i][j]);
			add(pos[i][j], pos[i][j-1], INF);
		}
	}
	for(int o = 1; o <= y; ++o) {
		int i, j, k; read(i, j, k);
		for(int a = 1; a <= m; ++a)
			if(a-1+k >= 1 && a-1+k <= m)
				add(pos[j][a-1], pos[i][a-1+k], INF);
	}
	if(qwqbfs()) {
		puts("-1");
		return ;
	}
	double qwq = dinic();
	printf("%lf\n", qwq);
}
signed main() {
	int T; read(T);
	while(T--)
		solve();
	return 0;
}