uva 10808 - Rational Resistors(基尔霍夫定律+高斯消元)

题目链接：uva 10808 - Rational Resistors

题目大意：给出一个博阿含n个节点，m条导线的电阻网络，求节点a和b之间的等效电阻。

解题思路：基尔霍夫定律，任何一点的电流向量为0。就是说有多少电流流入该节点，就有多少电流流出。
对于每次询问的两点间等效电阻，先判断说两点是否联通，不连通的话绝逼是1/0（无穷大）。联通的话，将同一个联通分量上的节点都扣出来，假设电势作为变元，然后根据基尔霍夫定律列出方程，因为对于每个节点的电流向量为0，所以每个节点都有一个方程，所有与该节点直接连接的都会有电流流入，并且最后总和为0，（除了a，b两点，一个为1，一个为-1）。用高斯消元处理，但是这样列出的方程组不能准确求出节点的电势，只能求出各个节点之间电势的关系。所以我们将a点的电势置为0，那么用求出的b点电势减去0就是两点间的电压，又因为电流设为1，所以等效电阻就是电压除以电流。

#include <cstdio>
#include <cstring>
#include <algorithm>

using namespace std;

typedef long long type;

struct Fraction {
    type member; // 分子；
    type denominator; // 分母；

    Fraction (type member = 0, type denominator = 1);
    void operator = (type x) { this->set(x, 1); }
    Fraction operator * (const Fraction& u);
    Fraction operator / (const Fraction& u);
    Fraction operator + (const Fraction& u);
    Fraction operator - (const Fraction& u);

    Fraction operator *= (const Fraction& u) { return *this = *this * u; }
    Fraction operator /= (const Fraction& u) { return *this = *this / u; }
    Fraction operator += (const Fraction& u) { return *this = *this + u; }
    Fraction operator -= (const Fraction& u) { return *this = *this - u; }

    void set(type member, type denominator);
};

inline type gcd (type a, type b) {
    return b == 0 ? (a > 0 ? a : -a) : gcd(b, a % b);
}

inline type lcm (type a, type b) {
    return a / gcd(a, b) * b;
}

/*Code*/
/////////////////////////////////////////////////////
const int maxn = 105;
typedef long long ll;
typedef Fraction Mat[maxn][maxn];

int N, M, f[maxn];
Mat G, A;

bool cmp (Fraction& a, Fraction& b) {
    return a.member * b.denominator < b.member * a.denominator;
}

inline int getfar (int x) {
    return x == f[x] ? x : f[x] = getfar(f[x]);
}

inline void link (int u, int v) {
    int p = getfar(u);
    int q = getfar(v);
    f[p] = q;
}

void init () {
    scanf("%d%d", &N, &M);

    for (int i = 0; i < N; i++) {
        f[i] = i;
        for (int j = 0; j < N; j++)
            G[i][j] = 0;
    }

    int u, v;
    ll R;
    for (int i = 0; i < M; i++) {
        scanf("%d%d%lld", &u, &v, &R);

        if (u == v)
            continue;

        link(u, v);
        G[u][v] += Fraction(1, R);
        G[v][u] += Fraction(1, R);
    }
}

Fraction gauss_elimin (int u, int v, int n) {

    /*
    printf("\n");

    for (int i = 0; i < n; i++) {
        for (int j = 0; j <= n; j++)
            printf("%lld/%lld ", A[i][j].member,  A[i][j].denominator);
        printf("\n");
    }
    */

    for (int i = 0; i < n; i++) {
        int r;

        for (int j = i; j < n; j++)
            if (A[j][i].member) {
                r = j;
                break;
            }

        if (r != i) {
            for (int j = 0; j <= n; j++)
                swap(A[i][j], A[r][j]);
        }

        if (A[i][i].member == 0)
            continue;

        for (int j = i + 1; j < n; j++) {
            Fraction t = A[j][i] / A[i][i];
            for (int k = 0; k <= n; k++)
                A[j][k] -= A[i][k] * t;
        }
    }

    for (int i = n-1; i >= 0; i--) {
        for (int j = i+1; j < n; j++) {
            if (A[j][j].member)
                A[i][n] -= A[i][j] * A[j][n] / A[j][j];
        }
    }

    /*
       Fraction U = A[u][n] / A[u][u];
       printf("%lld/%lld!\n", A[u][n].member, A[u][n].denominator);
       printf("%lld/%lld!\n", A[u][u].member, A[u][u].denominator);
       printf("%lld/%lld\n", U.member, U.denominator);

       Fraction V = A[v][n] / A[v][v];
       printf("%lld/%lld\n", V.member, V.denominator);
       */

    return A[u][n] / A[u][u] - A[v][n] / A[v][v];
}

Fraction solve (int u, int v) {
    int n = 0, hash[maxn];
    int hu, hv;

    for (int i = 0; i < N; i++) {
        if (i == u)
            hu = u;

        if (i == v)
            hv = v;

        if (getfar(i) == getfar(u))
            hash[n++] = i;
    }

    n++;
    for (int i = 0; i <= n; i++) {
        for (int j = 0; j <= n; j++)
            A[i][j] = 0;
    }

    for (int i = 0; i < n - 1; i++) {
        for (int j = 0; j < n - 1; j++) {
            if (i == j)
                continue;

            int p = hash[i];
            int q = hash[j];

            A[i][i] += G[p][q];
            A[i][j] -= G[p][q];
        }
    }

    A[hu][n] = 1;
    A[hv][n] = -1;
    A[n-1][0] = 1;
    return gauss_elimin (hu, hv, n);
}

int main () {
    int cas;
    scanf("%d", &cas);

    for (int kcas = 1; kcas <= cas; kcas++) {
        init();

        int Q, u, v;
        scanf("%d", &Q);
        printf("Case #%d:\n", kcas);
        for (int i = 0; i < Q; i++) {
            scanf("%d%d", &u, &v);

            printf("Resistance between %d and %d is ", u, v);
            if (getfar(u) == getfar(v)) {
                Fraction ans = solve(u, v);
                printf("%lld/%lld\n", ans.member, ans.denominator);
            } else
                printf("1/0\n");
        }
        printf("\n");
    }
    return 0;
}

/////////////////////////////////////////////////////

Fraction::Fraction (type member, type denominator) {
    this->set(member, denominator);
}

Fraction Fraction::operator * (const Fraction& u) {
    type tmp_p = gcd(member, u.denominator);
    type tmp_q = gcd(u.member, denominator);
    return Fraction( (member / tmp_p) * (u.member / tmp_q), (denominator / tmp_q) * (u.denominator / tmp_p) );
}

Fraction Fraction::operator / (const Fraction& u) {
    type tmp_p = gcd(member, u.member);
    type tmp_q = gcd(denominator, u.denominator);
    return Fraction( (member / tmp_p) * (u.denominator / tmp_q), (denominator / tmp_q) * (u.member / tmp_p));
}

Fraction Fraction::operator + (const Fraction& u) {
    type tmp_l = lcm (denominator, u.denominator);
    return Fraction(tmp_l / denominator * member + tmp_l / u.denominator * u.member, tmp_l);
}

Fraction Fraction::operator - (const Fraction& u) {
    type tmp_l = lcm (denominator, u.denominator);
    return Fraction(tmp_l / denominator * member - tmp_l / u.denominator * u.member, tmp_l);
}

void Fraction::set (type member, type denominator) {

    if (denominator == 0) {
        denominator = 1;
        member = 0;
    }

    if (denominator < 0) {
        denominator = -denominator;
        member = -member;
    }

    type tmp_d = gcd(member, denominator);
    this->member = member / tmp_d;
    this->denominator = denominator / tmp_d;
}