题目链接

问题分析

数据范围唬人的……毕竟 \(|Scc|\leqslant 100\) 。难写点罢了。

令 \(f_i\) 表示 \(i\) 到 \(T\) 的期望步数，那么有 \(f_i=1+\frac{1}{Deg_i}\sum\limits_{(i,v)\in E}f_v\) 。

那么只要一个 \(Scc\) 内高斯消元， \(Scc\) 之间按反着的拓扑序做即可。

不好好学tarjan又被教育了

参考程序

我也不知道为什么要写得这么翔

#include <cstdio>
#include <vector>
#include <cstring>

//#define __DEBUG__

#define Maxn 10010
#define Maxm 1000010
#define MaxScc 110
struct edge {
    int To, Next;
    edge() {}
    edge(int _To, int _Next) : To(_To), Next(_Next) {}
};
edge Edge[Maxm];
int Start[Maxn], UsedEdge;
int n, m, s, t, P[Maxn], Ref[Maxn];
int Dfn[Maxn], Low[Maxn], Time, Vis[Maxn];
int Stack[Maxn];

edge rEdge[Maxm];
int rStart[Maxn], rUsedEdge;

std::vector<int> Scc[Maxn];
int SccNum[Maxn];
int Index[Maxn], Degree[Maxn];
int L, R, Queue[Maxn];

long double Ans[Maxn];
long double A[MaxScc][MaxScc];

inline void AddEdge(int x, int y);
inline void rAddEdge(int x, int y);
void Dfs(int u);
inline void GetScc(int t);
inline void Go();
inline void Build(int u);
inline void Gauss(int n);
inline void Pan(int u);

int main() {
    scanf("%d%d%d%d", &n, &m, &s, &t);
    for (int i = 1, u, v; i <= m; ++i) {
        scanf("%d%d", &u, &v);
        if (u != t) {
            AddEdge(v, u);
            rAddEdge(u, v);
            ++P[u];
        }
    }
    GetScc(t);
    if (Dfn[s] == 0) {
        printf("INF\n");
        return 0;
    }
    Pan(s);
    for (int i = 1; i <= n; ++i)
        if (Vis[i] && !Dfn[i]) {
            printf("INF\n");
            return 0;
        }
    memset(Vis, 0, sizeof(Vis));
    Go();
    printf("%.3Lf\n", Ans[s]);
#ifdef __DEBUG__ 
    for (int i = 1; i <= n; ++i) printf("%.3Lf\n", Ans[i]);
#endif
    return 0;
}

inline void AddEdge(int x, int y) {
    Edge[++UsedEdge] = edge(y, Start[x]);
    Start[x] = UsedEdge;
    return;
}

inline void rAddEdge(int x, int y) {
    rEdge[++rUsedEdge] = edge(y, rStart[x]);
    rStart[x] = rUsedEdge;
    return;
}

inline void GetScc(int t) {
    Dfs(t);
#ifdef __DEBUG__
    printf("Num = %d\n", SccNum[0]);
    for (int i = 1; i <= SccNum[0]; ++i) {
        printf("  %d : ", i);
        for (int j = 0; j < SccNum[i]; ++j)
            printf(" %d", Scc[i][j]);
        printf("\n");
    }
#endif
    for (int i = 1; i <= SccNum[0]; ++i) 
        for (int j = 0; j < SccNum[i]; ++j)
            Index[Scc[i][j]] = i;
    for (int u = 1; u <= n; ++u) 
        for (int t = Start[u]; t; t = Edge[t].Next) {
            int v = Edge[t].To;
            if (Index[u] != Index[v]) 
                ++Degree[Index[v]];
        }
    L = R = 0;
    for (int i = 1; i <= SccNum[0]; ++i) 
        if (Degree[i] == 0) 
            Queue[++R] = i;
    return;
}

void Dfs(int u) {
    Dfn[u] = Low[u] = ++Time;
    Vis[u] = 1;
    Stack[++Stack[0]] = u;
    for (int t = Start[u]; t; t = Edge[t].Next) {
        int v = Edge[t].To;
        if (!Dfn[v]) Dfs(v), Low[u] = std::min(Low[v], Low[u]);
        else if (Vis[v]) Low[u] = std::min(Low[u], Low[v]);
    }
    if (Low[u] != Dfn[u]) return;
    ++SccNum[0];
    while (Stack[Stack[0]] != u) {
        Vis[Stack[Stack[0]]] = 0;
        Scc[SccNum[0]].push_back(Stack[Stack[0]--]);
    }
    Vis[Stack[Stack[0]]] = 0;
    Scc[SccNum[0]].push_back(Stack[Stack[0]--]);
    SccNum[SccNum[0]] = Scc[SccNum[0]].size();
    return;
}

inline void Go() {
    while (L < R) {
        int u = Queue[++L];
        Build(u);
        Gauss(SccNum[u]);
#ifdef __DEBUG__
        printf(">>>>>>>>>> Scc %d <<<<<<<<<<\n", u);
#endif
        for (int i = 0; i < SccNum[u]; ++i) {
            Ans[Scc[u][i]] = A[i + 1][SccNum[u] + 1];
            for (int t = Start[Scc[u][i]]; t; t = Edge[t].Next) {
                int v = Index[Edge[t].To];
                if (v != u) {
                    --Degree[v];
                    if (Degree[v] == 0) Queue[++R] = v;
                }
            }
        }
        Ans[t] = 0;
    }
    return;
}

inline void Build(int S) {
    memset(A, 0, sizeof(A));
    for (int i = 0; i < SccNum[S]; ++i) Ref[Scc[S][i]] = i + 1;
    for (int i = 0; i < SccNum[S]; ++i) {
        A[i + 1][i + 1] = A[i + 1][SccNum[S] + 1] = 1;
        int u = Scc[S][i];
        for (int t = rStart[u]; t; t = rEdge[t].Next) {
            int v = rEdge[t].To;
            if (Index[v] == S) A[i + 1][Ref[v]] += -1.0 / P[u];
            else A[i + 1][SccNum[S] + 1] += Ans[v] / P[u];
        }
    }
#ifdef __DEBUG__
    printf("========== Part ===========\n");
    for (int i = 0; i < SccNum[S]; ++i) printf("%d ", Scc[S][i]); printf("\n");
    for (int i = 1; i <= SccNum[S]; ++i) {
        for (int j = 1; j <= SccNum[S] + 1; ++j)
            printf("%6.3Lf ", A[i][j]);
        printf("\n");
    }
#endif
    return;
}

inline void Gauss(int n) {
    for (int i = 1; i <= n; ++i) {
        long double Tmp = A[i][i];
        for (int j = 1; j <= n + 1; ++j) A[i][j] /= Tmp;
        for (int j = 1; j <= n; ++j) {
            if (i == j) continue;
            Tmp = A[j][i];
            for (int k = 1; k <= n + 1; ++k) A[j][k] -= A[i][k] * Tmp;
        }
    }
    return;
}

inline void Pan(int u) {
    Vis[u] = 1;
    for (int t = rStart[u]; t; t = rEdge[t].Next) {
        int v = rEdge[t].To;
        if (Vis[v]) continue;
        Pan(v);
    }
    return;
}