题意
个点, 条边的图,边有费用。
次询问:若每条边的容量为 ,则从 到 流 个单位的流量的最小费用是多少,要求输出最简分数。
题解
首先设 表示每条边容量为 ,要从 向 流 的流量的费用。则有 。
根据费用流的算法流程: 表示每条边容量为 时,第 条增广路的费用(流量为y)。若只需要 的流量,则乘上比例系数 即可。
对于 ,设 。考虑对前 条增广路每条路流 的流量,费用为 ;对第 条增广路流 的流量,费用为 。
设
,则有
综上,设所有边容量为 ,做一次最小费用最大流求出所有的 。对于一次询问 ,则答案为 ,化成最简分数即可。
单组数据时间复杂度
#include <bits/stdc++.h>
using namespace std;
const int N = 50 + 5, M = 2 * 1000 + 5;
const long long Inf = 1e17;
using ll = long long;
using arr = int[N];
/*-----------------------------------------------------------------*/
int n, m, q, Ecnt, Fi[N];
ll dis[N], f[N];
struct Edge {
int nx, v, c, w;
} E[M];
inline void Add(int u, int v, int c, int w) {
E[++Ecnt] = (Edge){Fi[u], v, c, w}, Fi[u] = Ecnt;
}
class MCMF {
private:
static const int N = 2e5 + 5;
ll h[N];
int Flow[N], Pre[N], From[N];
struct Seg {
int Cnt, tr[N << 2];
inline void Cls() {
fill(tr, tr + 2 * Cnt + 1, 0);
Cnt = 0;
}
inline int Cmp(int a, int b) { return dis[a] < dis[b] ? a : b; }
inline void Set(int n) {
for (Cnt = 1; Cnt < n + 2; Cnt <<= 1)
;
tr[0] = n + 1;
}
inline void Mdy(int u, ll w) {
for (int i = u + Cnt; dis[tr[i]] > w; tr[i] = u, i >>= 1)
;
dis[u] = w;
}
inline void Del(int u) {
for (tr[u += Cnt] = 0, u >>= 1; u;
tr[u] = Cmp(tr[u << 1], tr[u << 1 | 1]), u >>= 1)
;
}
} zkw;
inline bool Dij(int n, int S, int T) {
for (int i = 0; i <= n; ++i)
dis[i] = Inf, Flow[i] = From[i] = Pre[i] = 0;
Flow[S] = 1e9;
dis[n + 1] = 0;
zkw.Cls(), zkw.Set(n);
zkw.Mdy(S, 0);
for (int _ = 2; _ <= n; ++_) {
int u = zkw.tr[1];
zkw.Del(u);
for (int i = Fi[u], v = E[i].v; i; v = E[i = E[i].nx].v)
if (E[i].c > 0 && dis[v] > dis[u] + E[i].w + h[u] - h[v]) {
zkw.Mdy(v, dis[u] + E[i].w + h[u] - h[v]);
From[v] = u, Pre[v] = i;
Flow[v] = min(Flow[u], E[i].c);
}
}
return dis[T] != Inf;
}
public:
int MF;
inline void Calc(int n, int S, int T) {
int Now;
for (int i = 1; i <= n; ++i)
h[i] = 0;
while (Dij(n, S, T)) {
MF += (Now = Flow[T]);
f[MF] = f[MF - Now] + Now * (dis[T] - h[S] + h[T]);
for (int v = T; v != S; v = From[v])
E[Pre[v]].c -= Now, E[Pre[v] ^ 1].c += Now;
for (int i = 1; i <= n; ++i)
h[i] += dis[i];
}
}
} D;
inline void CLS() {
Ecnt = 1;
D.MF = 0;
for (int i = 1; i <= n; ++i)
Fi[i] = f[i] = 0;
}
inline void Solve() {
CLS();
for (int u, v, w; m--;) {
scanf("%d%d%d", &u, &v, &w);
Add(u, v, 1, w), Add(v, u, 0, -w);
}
D.Calc(n, 1, n);
scanf("%d", &q);
for (ll u, v, k; q--;) {
scanf("%lld%lld", &u, &v);
if (u == 0 || (k = v / u) > D.MF || (v - u * k > 0 && !f[k + 1]))
puts("NaN");
else {
ll a = (v - k * u) * (f[k + 1] - f[k]) + u * f[k], b = v,
g = __gcd(a, b);
printf("%lld/%lld\n", a / g, b / g);
}
}
}
int main() {
while (~scanf("%d%d", &n, &m))
Solve();
return 0;
}