Label [高斯消元]

$Label$

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$\color{red}{正解部分}$

考虑一个未知点连向两个已知点, 边权为 $w_1, w_2$, 点权为 $a_x, a_1, a_2$,
则现在要做的就是使得 $w_1(a_x-a_1)^2 + w_2(a_x-a_2)^2$ 最小,
根据二次函数相关知识可得, 当 $a_x = \frac{w_1a_1+w_2a_2}{w_1+w_2}$ 时, 上式取得最小值,

推广到多个点, 当 $a_x = \frac{\sum w_ia_i}{\sum w_i}$ 时, 答案最小 .

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$\color{red}{实现部分}$

将所有 未知数 提出, 列出方程, 高斯消元 即可 .

#include<bits/stdc++.h>
#define reg register

int read(){
        char c;
        int s = 0, flag = 1;
        while((c=getchar()) && !isdigit(c))
                if(c == '-'){ flag = -1, c = getchar(); break ; }
        while(isdigit(c)) s = s*10 + c-'0', c = getchar();
        return s * flag;
}

const int maxn = 30005;
const double eps = 1e-12;

int N;
int M;
int cnt;
int num0;
int A[maxn];
int B[maxn];
int Mp[maxn];
int head[maxn];

double C[maxn];
double K[305][305];

struct Edge{ int nxt, to, w; } edge[maxn << 1];

struct EDGE{ int u, v, w; } E[maxn];

void Add(int from, int to, int w){ edge[++ num0] = (Edge){ head[from], to, w }; head[from] = num0; }

void Gays(){
        for(reg int i = 1; i <= cnt; i ++){
                int max_id = i;
                for(reg int j = i+1; j <= cnt; j ++)
                        if(fabs(K[j][i]) > fabs(K[max_id][i])) max_id = j;
                std::swap(K[max_id], K[i]);
                for(reg int j = cnt+1; j >= i; j --) K[i][j] /= K[i][i]; // !
                for(reg int j = i+1; j <= cnt; j ++){
                        if(fabs(K[j][i]) < eps) continue ;
                        for(reg int k = cnt+1; k >= i; k --) K[j][k] -= K[j][i] * K[i][k];
                }
        }
        for(reg int i = cnt; i >= 1; i --){
                for(reg int j = cnt; j > i; j --) K[i][cnt+1] -= K[i][j] * K[j][cnt+1];
                C[B[i]] = K[i][cnt+1];
        }
}

int main(){
        N = read(), M = read();
        for(reg int i = 1; i <= M; i ++){
                int u = read(), v = read(), w = read();
                E[i].u = u, E[i].v = v, E[i].w = w;
                Add(u, v, w), Add(v, u, w);
        }
        for(reg int i = 1; i <= N; i ++){
                C[i] = A[i] = read();
                if(A[i] == -1) B[++ cnt] = i, Mp[i] = cnt;
        }
        for(reg int i = 1; i <= N; i ++){
                if(~A[i]) continue ;
                double sum = 0;
                for(reg int j = head[i]; j; j = edge[j].nxt){
                        int to = edge[j].to;
                        if(to == i) continue ;
                        sum += edge[j].w;
                        if(A[to] == -1) K[Mp[i]][Mp[to]] -= edge[j].w;
                        else K[Mp[i]][cnt+1] += 1ll * edge[j].w * A[to];
                }
                for(reg int j = 1; j <= cnt+1; j ++) K[Mp[i]][j] /= sum;
                K[Mp[i]][Mp[i]] = 1;
        }
        Gays();
        double Ans = 0;
        for(reg int i = 1; i <= M; i ++) Ans += 1ll*E[i].w * (C[E[i].u] - C[E[i].v]) * (C[E[i].u] - C[E[i].v]);
        printf("%.12lf\n", Ans);
        return 0;
}