【BZOJ2095】[Poi2010]Bridges
题面
题解
首先可以想到二分答案，那么我们就是要求我们新图中给所有边定向是否存在欧拉回路。
而有向图存在欧拉回路的充要条件为所有顶点的入度等于出度且该图是连通图，我们考虑在这一点上做文章。
令一个点的入度减出度表示一个点的度数差\(\phi\)，首先我们随机定向，假设有两个点\(u\),\(v\)，此时我们从\(u\)连一条边向\(v\)。
那么我们每改变一次连边的方向，会使\(\phi_u\)减去\(2\)，\(\phi_v\)加上\(2\)。
如果此时存在一点\(x\)，\(\phi_x\)为奇数，那么显然无解。
考虑网络流。首先连边\(v\rightarrow u\)流量为\(1\)表示一次转换方向，接着我们对于所有\(\phi>0\)的点\(x\)连边\(S\rightarrow x\)，流量为\(\frac {\phi}2\)；否则对于所有\(\phi<0\)的点\(x\)连边\(x\rightarrow T\)，流量为\(-\frac {\phi}2\)。
直接判断满流即可。
代码
#include <iostream>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <cmath>
#include <algorithm>
#include <queue>
using namespace std;
inline int gi() {
register int data = 0, w = 1;
register char ch = 0;
while (!isdigit(ch) && ch != '-') ch = getchar();
if (ch == '-') w = -1, ch = getchar();
while (isdigit(ch)) data = 10 * data + ch - '0', ch = getchar();
return w * data;
}
const int MAX_N = 1005;
struct Graph { int to, cap, next; } e[MAX_N << 5];
int fir[MAX_N], e_cnt;
void clearGraph() { memset(fir, -1, sizeof(fir)); e_cnt = 0; }
void Add_Edge(int u, int v, int c) {
e[e_cnt] = (Graph){v, c, fir[u]}, fir[u] = e_cnt++;
e[e_cnt] = (Graph){u, 0, fir[v]}, fir[v] = e_cnt++;
}
struct Edge { int u, v, a, b; } c[MAX_N << 1];
int N, M, deg[MAX_N];
int level[MAX_N];
bool bfs(int s, int t) {
queue<int> que;
for (int i = 0; i <= N + 1; i++) level[i] = -1;
level[s] = 0, que.push(s);
while (!que.empty()) {
int x = que.front(); que.pop();
for (int i = fir[x]; ~i; i = e[i].next) {
int v = e[i].to; if (!e[i].cap || ~level[v]) continue;
level[v] = level[x] + 1, que.push(v);
}
}
return level[t] != -1;
}
int iter[MAX_N];
int dfs(int x, int t, int f) {
if (x == t) return f;
for (int &i = iter[x]; ~i; i = e[i].next) {
int v = e[i].to;
if (e[i].cap && level[v] > level[x]) {
int d = dfs(v, t, min(f, e[i].cap));
if (d) {
e[i].cap -= d;
e[i ^ 1].cap += d;
return d;
}
}
}
level[x] = -1;
return 0;
}
int dinic(int s, int t) {
int res = 0;
while (bfs(s, t)) {
for (int i = 0; i <= N + 1; i++) iter[i] = fir[i];
int f;
while ((f = dfs(s, t, 1e9))) res += f;
}
return res;
}
bool check(int mid) {
clearGraph();
for (int i = 1; i <= N; i++) deg[i] = 0;
for (int i = 1; i <= M; i++) {
if (c[i].a <= mid) --deg[c[i].u], ++deg[c[i].v];
if (c[i].b <= mid) Add_Edge(c[i].v, c[i].u, 1);
}
for (int i = 1; i <= N; i++) if (deg[i] & 1) return 0;
int cnt = 0, s = 0, t = N + 1;
for (int i = 1; i <= N; i++) {
if (deg[i] > 0) Add_Edge(s, i, deg[i] >> 1), cnt += deg[i] >> 1;
if (deg[i] < 0) Add_Edge(i, t, -deg[i] >> 1);
}
return dinic(s, t) == cnt;
}
int main () {
#ifndef ONLINE_JUDGE
freopen("cpp.in", "r", stdin);
#endif
N = gi(), M = gi();
int l = 1e9, r = 1;
for (int i = 1; i <= M; i++) {
c[i] = (Edge){gi(), gi(), gi(), gi()};
if (c[i].a > c[i].b) swap(c[i].u, c[i].v), swap(c[i].a, c[i].b);
l = min(l, c[i].a), r = max(r, c[i].b);
}
if (!check(r)) return puts("NIE") & 0;
int ans = r;
while (l <= r) {
int mid = (l + r) >> 1;
if (check(mid)) ans = mid, r = mid - 1;
else l = mid + 1;
}
printf("%d\n", ans);
return 0;
}