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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,不论什么一点的电流向量为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。先推断说两点是否联通。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,如果电势作为变元,然后依据基尔霍夫定律列出方程,由于对于每一个节点的电流向量为0。所以每一个节点都有一个方程,全部与该节点直接连接的都会有电流流入。而且最后总和为0,(除了a,b两点,一个为1,一个为-1)。用高斯消元处理,可是这样列出的方程组不能准确求出节点的电势,仅仅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,又由于电流设为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; }

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