网络流24题の详解
\(1.\)洛谷P2756 飞行员配对方案问题
题意：派出最多架的飞机，并且每架飞机上都是一个英国飞行员和外籍飞行员
分析：经典的二分图匹配问题，将英国飞行员当做二分图的左部，外籍飞行员当做二分图的右部。可以用匈牙利算法求解，但这里使用网络流(最大流)。
考虑如何建图：
设立一个源点\(st\)，一个汇点\(ed\)。
\(I.\)让源点\(st\)向二分图的左部建一条流量为\(1\)的边，
\(II.\)让二分图的右部向汇点\(ed\)建一条流量为\(1\)的边，
\(III.\)如果英国飞行员\(u\)可以和外籍飞行员\(v\)配合，那么\(u\)和\(v\)建一条流量为\(1\)的边。
最后跑一遍最大流即可得到最大匹配数。
如何输出配对方案：
遍历所有的英国飞行员所连的边，如果此边\((u,v)\)流量不是\(1\)而是\(0\)说明\((u,v)\)配对。注意源点向英国飞行员所连的边流量会为0，但此时不应输出。
#include <bits/stdc++.h>
using namespace std;
typedef long long LL;
typedef unsigned long long ULL;
typedef pair<int, int> pii;
const int maxn = 300 + 5;
const int inf = 0x3f3f3f3f;
int n, m, u, v, st, ed, cnt, ans, head[maxn];
int flag, maxflow, vis[maxn], cur[maxn], deep[maxn];
struct Graph {
int v, w, next;
} edge[2 * maxn * maxn];
void addedge(int u, int v, int w) {
edge[cnt].v = v;
edge[cnt].w = w;
edge[cnt].next = head[u];
head[u] = cnt++;
}
int bfs(int st, int ed) {
for (int i = 0; i <= n + m + 5; i++) vis[i] = 0, cur[i] = head[i], deep[i] = inf;
queue<int> q;
q.push(st);
deep[st] = 0;
while (!q.empty()) {
int u = q.front();
q.pop();
for (int i = head[u]; ~i; i = edge[i].next) {
int v = edge[i].v;
if (deep[v] > deep[u] + 1 && edge[i].w) {
deep[v] = deep[u] + 1;
if (!vis[v]) {
vis[v] = 1;
q.push(v);
}
}
}
}
return deep[ed] != inf;
}
int dfs(int u, int ed, int flow) {
if (u == ed) {
flag = 1;
maxflow += flow;
return flow;
}
int now = 0, used = 0;
for (int i = cur[u]; ~i; i = edge[i].next) {
int v = edge[i].v;
cur[u] = i;
if (deep[v] == deep[u] + 1 && edge[i].w) {
if (now = dfs(v, ed, min(flow - used, edge[i].w))) {
used += now;
edge[i].w -= now;
edge[i ^ 1].w += now;
if (used == flow) break;
}
}
}
return used;
}
int dinic(int st, int ed) {
while (bfs(st, ed)) {
flag = 1;
while (flag) flag = 0, dfs(st, ed, inf);
}
return maxflow;
}
int main() {
memset(head, -1, sizeof head);
scanf("%d %d", &n, &m);
st = 0, ed = m + 1;
while (~scanf("%d %d", &u, &v)) {
if (u == -1 && v == -1) break;
addedge(u, v, 1), addedge(v, u, 0);
}
for (int i = 1; i <= n; i++) addedge(st, i, 1), addedge(i, st, 0);
for (int i = n + 1; i <= m; i++) addedge(i, ed, 1), addedge(ed, i, 0);
ans = dinic(st, ed);
if (!ans) return 0 * puts("No Solution!");
printf("%d\n", ans);
for (int i = 1; i <= n; i++) {
for (int j = head[i]; ~j; j = edge[j].next) {
if (edge[j].v != st && edge[j].w == 0 && edge[j ^ 1].w == 1)
printf("%d %d\n", i, edge[j].v);
}
}
return 0;
}
\(2.\)洛谷P2764 最小路径覆盖问题
做这道题首先我们要知道一个关于二分图的公式，
最小路径覆盖数 = 图的顶点数 - 图相应的二分图的最大匹配数
图转化为相应的二分图：对每个点拆点，拆成入点\(x_1\)(\(i\))和出点\(x_2\)(\(i+n\))，将这些入点作为二分图的左部，出点作为二分图的右部。
像\(1\)中一样建立源点与汇点，源点与左部相连，右部与汇点相连。
对于边(\(u,v\))，我们将\(u\)连向\(v+n\)即可。
跑一遍最大流，得到最大匹配数，则最小路径覆盖数就得到了。
如何输出路径：遍历所有的边(\(u,v\))(正向边即可)，如果这条边的流量为\(0\)，说明这条边一定在某条路径上。我们用非路径压缩的并查集来维护这些关系，如果此边流量为\(0\)，则将\(v\)并向\(u\)。那么最后一定有祖先为自己的点，\(dfs\)这些点即可。判断边的时候要注意源点的影响，
#include <bits/stdc++.h>
using namespace std;
typedef long long LL;
typedef unsigned long long ULL;
typedef pair<int, int> pii;
const int maxn = 300 + 5;
const int maxm = 12000 + 5;
const int inf = 0x3f3f3f3f;
int n, m, cnt, pre[maxn], head[maxn];
int u, v, st, ed, flag, maxflow, deep[maxn], vis[maxn], cur[maxn];
struct Graph {
int u, v, w, next;
} edge[maxm << 1];
void addedge(int u, int v, int w) {
edge[cnt].u = u;
edge[cnt].v = v;
edge[cnt].w = w;
edge[cnt].next = head[u];
head[u] = cnt++;
}
int bfs(int st, int ed) {
for (int i = 0; i <= n + n + 5; i++) deep[i] = inf, vis[i] = 0, cur[i] = head[i];
queue<int> q;
q.push(st);
deep[st] = 0;
while (!q.empty()) {
int u = q.front();
q.pop();
vis[u] = 0;
for (int i = head[u]; ~i; i = edge[i].next) {
int v = edge[i].v, w = edge[i].w;
if (w && deep[v] > deep[u] + 1) {
deep[v] = deep[u] + 1;
if (!vis[v]) {
vis[v] = 1;
q.push(v);
}
}
}
}
return deep[ed] != inf;
}
int dfs(int u, int ed, int flow) {
if (u == ed) {
flag = 1;
maxflow += flow;
return flow;
}
int now = 0, used = 0;
for (int i = cur[u]; ~i; i = edge[i].next) {
cur[u] = i;
int v = edge[i].v;
if (edge[i].w && deep[v] == deep[u] + 1) {
now = dfs(v, ed, min(flow - used, edge[i].w));
if (!now) continue;
edge[i].w -= now;
edge[i ^ 1].w += now;
used += now;
if (used == flow) break;
}
}
return used;
}
void dinic(int st, int ed) {
while (bfs(st, ed)) {
flag = 1;
while (flag) {
flag = 0;
dfs(st, ed, inf);
}
}
}
void dfs(int x) {
printf("%d ", x);
vis[x] = 1;
for (int i = head[x]; ~i; i = edge[i].next) {
if (!vis[edge[i].v] && edge[i].w == 0 && edge[i].v > n) {
dfs(edge[i].v - n);
}
}
}
int main() {
scanf("%d %d", &n, &m);
memset(head, -1, sizeof head);
for (int i = 1; i <= m; i++) {
scanf("%d %d", &u, &v);
addedge(u, v + n, 1), addedge(v + n, u, 0);
}
st = 0, ed = n + n + 1;
for (int i = 1; i <= n; i++) addedge(st, i, 1), addedge(i, st, 0);
for (int i = n + 1; i <= n + n; i++) addedge(i, ed, 1), addedge(ed, i, 0);
dinic(st, ed);
for (int i = 1; i <= n; i++) pre[i] = i;
for (int j = 0; j <= cnt; j++) {
if (edge[j].u >= 1 && edge[j].u <= n && edge[j].v > n && edge[j].v <= n + n && edge[j].w == 0) {
// if (pre[edge[j].v - n] != pre[edge[j].u])
pre[edge[j].v - n] = pre[edge[j].u];
}
}
memset(vis, 0, sizeof vis);
for (int i = 1; i <= n; i++) {
if (pre[i] == i) {
dfs(i);
puts("");
}
}
printf("%d\n", n - maxflow);
return 0;
}
\(3.\)洛谷P2765 魔术球问题
分析：因为柱子数是固定不变的，我们可以将\(n\)根柱子理解为\(n\)条路径。那么问题就可以转化为不断向路径中加入数，直到加入某个数使得路径数\(>\)\(n\)。从\(1\)开始枚举直到不满足条件。
如何建图：
每次新枚举一个数\(x\)，那么它有可能是一个柱子的开头，所以我们从源点向\(x\)建一条流量为\(1\)的边，它也有可能是某个柱子的最后一个数，所以\(x\)向汇点连一条流量为\(1\)的边。然后现在就是考虑数和数之间的边，题目限制\(x\)只有和相邻数相加为完全平方数时才能放入，所以枚举\(y\)\(\in\)\([1,x-1]\)，如果\(y\)能与\(x\)形成完全平方数就和\(x\)建边(\(x\)作为右部)。
然后在残余网络上跑最大流得到每次的最大匹配数，每次让总顶点数减去其再判断即可。
#include <bits/stdc++.h>
using namespace std;
typedef long long LL;
typedef unsigned long long ULL;
typedef pair<int, int> pii;
const int maxn = 7000 + 5;
const int maxm = 100 + 5;
const int inf = 0x3f3f3f3f;
int n, cnt, head[maxn];
int st, ed, ans, now, maxf, flag, deep[maxn], vis[maxn], cur[maxn], path[maxn];
struct Graph {
int v, w, next;
} edge[200000];
void addedge(int u, int v, int w) {
edge[cnt].v = v;
edge[cnt].w = w;
edge[cnt].next = head[u];
head[u] = cnt++;
}
int bfs(int st, int ed) {
for (int i = 0; i <= ed; i++) deep[i] = inf, vis[i] = 0, cur[i] = head[i];
queue<int> q;
q.push(st);
deep[st] = 0;
while (!q.empty()) {
int u = q.front();
q.pop();
vis[u] = 0;
for (int i = head[u]; ~i; i = edge[i].next) {
int v = edge[i].v;
if (edge[i].w && deep[v] > deep[u] + 1) {
deep[v] = deep[u] + 1;
if (!vis[v]) {
vis[v] = 1;
q.push(v);
}
}
}
}
return deep[ed] != inf;
}
int dfs(int u, int ed, int flow) {
if (u == ed) {
flag = 1;
maxf += flow;
return flow;
}
int now = 0, used = 0;
for (int i = cur[u]; ~i; i = edge[i].next) {
int v = edge[i].v;
cur[u] = i;
if (edge[i].w && deep[v] == deep[u] + 1) {
now = dfs(v, ed, min(flow - used, edge[i].w));
if (!now) continue;
edge[i].w -= now;
edge[i ^ 1].w += now;
used += now;
if (used == flow) break;
}
}
return used;
}
void dinic(int st, int ed) {
while (bfs(st, ed)) {
flag = 1;
while (flag) {
flag = 0;
dfs(st, ed, inf);
}
}
}
int main() {
scanf("%d", &n);
memset(head, -1, sizeof head);
st = 0, ed = 5001;
while (true) {
ans++, now++;
addedge(st, now, 1), addedge(now, st, 0);
for (int i = 1; i <= now - 1; i++) {
if (sqrt(i + now) == (int)(sqrt(i + now)))
addedge(i, now + 2000, 1), addedge(now + 2000, i, 0);
}
addedge(now + 2000, ed, 1), addedge(ed, now + 2000, 0);
maxf = 0, dinic(st, ed);
ans -= maxf;
if (ans > n) break;
}
printf("%d\n", now - 1);
for (int i = 1; i <= now - 1; i++) {
for (int j = head[i]; ~j; j = edge[j].next) {
if (!edge[j].w) {
path[i] = edge[j].v - 2000;
break;
}
}
}
memset(vis, 0, sizeof vis);
for (int i = 1; i <= now - 1; i++) {
if (vis[i]) continue;
int temp = i;
while (temp != -2000) {
vis[temp] = 1;
printf("%d ", temp);
temp = path[temp];
}
puts("");
}
return 0;
}
\(4.\)洛谷P2763 试题库问题
分析：
二分匹配，只不过左部可以匹配多个右部。
让题的类型\(x\in[1,n]\)作为二分图的左部，让题\(y\in[n+1,n+1+m]\)作为二分图的右部。
源点向题的类型建流量为题类型所需题数的边，题向汇点建流量为\(1\)的边(题只能用在某个类型一次)，题的类型与题建流量为\(1\)的边。跑一遍最大流即可。
输出方案：枚举题的类型所连的边，判断流量是否为\(0\)即是否匹配上即可。
#include <bits/stdc++.h>
using namespace std;
typedef long long LL;
typedef unsigned long long ULL;
typedef pair<int, int> pii;
const int maxn = 20000 + 5;
const int maxm = 100 + 5;
const int inf = 0x3f3f3f3f;
int k, n, m, x, y;
int st, ed, cnt, head[maxn];
int maxf, flag, deep[maxn], cur[maxn], vis[maxn];
struct Graph {
int v, w, next;
} edge[maxn << 1];
void addedge(int u, int v, int w) {
edge[cnt].v = v;
edge[cnt].w = w;
edge[cnt].next = head[u];
head[u] = cnt++;
}
int bfs(int st, int ed) {
for (int i = 0; i <= ed; i++) vis[i] = 0, deep[i] = inf, cur[i] = head[i];
queue<int> q;
q.push(st);
deep[st] = 0;
while (!q.empty()) {
int u = q.front();
q.pop();
vis[u] = 0;
for (int i = head[u]; ~i; i = edge[i].next) {
int v = edge[i].v;
if (edge[i].w && deep[v] > deep[u] + 1) {
deep[v] = deep[u] + 1;
if (!vis[v]) {
vis[v] = 1;
q.push(v);
}
}
}
}
return deep[ed] != inf;
}
int dfs(int u, int ed, int flow) {
if (u == ed) {
flag = 1;
maxf += flow;
return flow;
}
int now = 0, used = 0;
for (int i = cur[u]; ~i; i = edge[i].next) {
cur[u] = head[i];
int v = edge[i].v;
if (edge[i].w && deep[v] == deep[u] + 1) {
now = dfs(v, ed, min(flow - used, edge[i].w));
if (!now) continue;
edge[i].w -= now;
edge[i ^ 1].w += now;
used += now;
if (used == flow) break;
}
}
return used;
}
void dinic(int st, int ed) {
while (bfs(st, ed)) {
flag = 1;
while (flag) {
flag = 0;
dfs(st, ed, inf);
}
}
}
int main() {
memset(head, -1, sizeof head);
scanf("%d %d", &k, &n);
st = 0, ed = 20002;
for (int i = 1; i <= k; i++) {
scanf("%d", &x);
addedge(st, i, x), addedge(i, st, 0);
m += x;
}
for (int i = 1; i <= n; i++) {
scanf("%d", &x);
for (int j = 1; j <= x; j++) {
scanf("%d", &y);
addedge(y, i + k, 1), addedge(i + k, y, 0);
}
addedge(i + k, ed, 1), addedge(ed, i + k, 0);
}
dinic(st, ed);
for (int i = 1; i <= k; i++) {
printf("%d: ", i);
for (int j = head[i]; ~j; j = edge[j].next) {
if (edge[j].w == 0 && edge[j].v != st) printf("%d ", edge[j].v - k);
}
puts("");
}
return 0;
}
\(5.\)洛谷P3254 圆桌问题
与\(4\)是同样的问题，二分图匹配，左部可以匹配多个右部，此题不细讲。
输出方案同\(4\)。
#include <bits/stdc++.h>
using namespace std;
typedef long long LL;
typedef unsigned long long ULL;
typedef pair<int, int> pii;
const int maxn = 500 + 5;
const int maxm = 100 + 5;
const int inf = 0x3f3f3f3f;
int n, m, x, now, cnt, sum, head[500];
int st, ed, maxf, flag, cur[maxn], deep[maxn], vis[maxn];
struct Graph {
int v, w, next;
} edge[maxn * maxn];
void addedge(int u, int v, int w) {
edge[cnt].v = v;
edge[cnt].w = w;
edge[cnt].next = head[u];
head[u] = cnt++;
}
int bfs(int st, int ed) {
for (int i = 0; i <= ed; i++) vis[i] = 0, deep[i] = inf, cur[i] = head[i];
queue<int> q;
q.push(st);
deep[st] = 0;
while (!q.empty()) {
int u = q.front();
q.pop();
vis[u] = 0;
for (int i = head[u]; ~i; i = edge[i].next) {
int v = edge[i].v;
if (edge[i].w && deep[v] > deep[u] + 1) {
deep[v] = deep[u] + 1;
if (!vis[v]) {
q.push(v);
vis[v] = 1;
}
}
}
}
return deep[ed] != inf;
}
int dfs(int u, int ed, int flow) {
if (u == ed) {
flag = 1;
maxf += flow;
return flow;
}
int now = 0, used = 0;
for (int i = cur[u]; ~i; i = edge[i].next) {
cur[u] = i;
int v = edge[i].v;
if (edge[i].w && deep[v] == deep[u] + 1) {
now = dfs(v, ed, min(flow - used, edge[i].w));
if (!now) continue;
edge[i].w -= now;
edge[i ^ 1].w += now;
used += now;
if (used == flow) break;
}
}
return used;
}
void dinic(int st, int ed) {
while (bfs(st, ed)) {
flag = 1;
while (flag) {
flag = 0;
dfs(st, ed, inf);
}
}
}
int main() {
scanf("%d %d", &n, &m);
st = 0, ed = n + m + 2;
memset(head, -1, sizeof head);
for (int i = 1; i <= n; i++) {
scanf("%d", &x), sum += x;
addedge(st, i, x), addedge(i, st, 0);
for (int j = 1; j <= m; j++) {
addedge(i, j + n, 1), addedge(j + n, i, 0);
}
}
for (int i = 1; i <= m; i++) {
scanf("%d", &x);
addedge(i + n, ed, x), addedge(ed, i + n, 0);
}
dinic(st, ed);
if (maxf == sum) puts("1");
else return 0 * puts("0");
for (int i = 1; i <= n; i++) {
for (int j = head[i]; ~j; j = edge[j].next) {
if (edge[j].w == 0 && edge[j].v != st) {
printf("%d ", edge[j].v - n);
}
}
puts("");
}
return 0;
}
\(6.\)洛谷P2766 最长不下降子序列问题
分析：
给定正整数序列\(x_1,...,x_n\) 。
\(1)\)：计算其最长不下降子序列的长度\(s\)。
对于第一个询问，这是经典的\(LIS\)问题，我们使用\(dp\)即可求解，在这里应使用\(o(n^2)\)的算法来求解而不用\(o(nlogn)\)的算法，因为这样方便接下来两个询问的建图。
这里\(dp[i]\)表示以\(x[i]\)结尾的最长不下降子序列长度。
\(2)\)：计算从给定的序列中最多可取出多少个长度为s的不下降子序列。
显然，因为是取出即意味着每个数只能用一次。所以我们将每个点(数)拆为两个点，分别为入点和出点，然后入点与出点之间的流量为\(1\)，这样我们就满足了一个点只能取一次的条件。
考虑什么样的子序列能满足条件：这个子序列必然是以\(dp[i]==1\)的\(x[i]\)作为开头，且以\(dp[i]==s\)的\(x[i]\)作为结尾。(假设我取\(dp[i]==2\)的\(x[i]\)作为开头，那么这个子序列最长也就是\(s-1\)的长度。)
所以建立源点和汇点，源点向\(dp[i]==1\)的\(x[i]\)建流量为\(1\)的边，\(dp[i]==s\)的\(x[i]\)向汇点建流量为\(1\)的边。然后让所有\(dp[i]+1==dp[j]\)\((i<j)\)的\(x[i]\)和\(x[j]\)建流量为1的边。
最后跑一遍最大流即可解决\(2)\)
\(3)\)：如果允许在取出的序列中多次使用\(x_1\)和\(x_n\)，则从给定序列中最多可取出多少个长度为s的不下降子序列。
这句话翻译一下就是\(x_1\)和\(x_n\)可以无限取。
我们知道\(dp[1]\)是一定等于\(1\)的，也就是说\(x_1\)必定可以作为子序列的开头，所以我们修改\(x_1\)入点和出点之间的流量为\(inf\)。对于\(dp[n]\)来说，只有\(dp[n]==s\)时，我们才用修改它的流量，因为如果\(dp[n]=\not s\)，那么无论如何只要取了\(x_n\)就一定不会形成长度为\(s\)的子序列。
跑最大流即可
#include <bits/stdc++.h>
using namespace std;
typedef long long LL;
typedef unsigned long long ULL;
typedef pair<int, int> pii;
const int maxn = 500 + 5;
const int maxm = 100 + 5;
const int inf = 0x3f3f3f3f;
int n, ans, a[maxn], dp[maxn];
int st, ed, cnt, head[maxn << 3];
int maxf, flag, deep[maxn << 3], cur[maxn << 3], vis[maxn << 3];
struct Graph {
int v, w, next;
} edge[maxn << 3];
void addedge(int u, int v, int w) {
edge[cnt].v = v;
edge[cnt].w = w;
edge[cnt].next = head[u];
head[u] = cnt++;
}
int bfs(int st, int ed) {
for (int i = 0; i <= ed; i++) vis[i] = 0, deep[i] = inf, cur[i] = head[i];
queue<int> q;
q.push(st);
deep[st] = 0;
while (!q.empty()) {
int u = q.front();
q.pop();
vis[u] = 0;
for (int i = head[u]; ~i; i = edge[i].next) {
int v = edge[i].v;
if (edge[i].w && deep[v] > deep[u] + 1) {
deep[v] = deep[u] + 1;
if (!vis[v]) {
vis[v] = 1;
q.push(v);
}
}
}
}
return deep[ed] != inf;
}
int dfs(int u, int ed, int flow) {
if (u == ed) {
flag = 1;
maxf += flow;
return flow;
}
int now = 0, used = 0;
for (int i = head[u]; ~i; i = edge[i].next) {
cur[u] = head[i];
int v = edge[i].v;
if (edge[i].w && deep[v] == deep[u] + 1) {
now = dfs(v, ed, min(flow - used, edge[i].w));
if (!now) continue;
edge[i].w -= now;
edge[i ^ 1].w += now;
used += now;
if (used == flow) break;
}
}
return used;
}
void dinic(int st, int ed) {
while (bfs(st, ed)) {
flag = 1;
while (flag) {
flag = 0;
dfs(st, ed, inf);
}
}
}
int main() {
scanf("%d", &n);
for (int i = 1; i <= n; i++) scanf("%d", a + i);
ans = 0;
for (int i = 1; i <= n; i++) {
dp[i] = 1;
for (int j = 1; j < i; j++) {
if (a[i] >= a[j]) dp[i] = max(dp[i], dp[j] + 1);
}
ans = max(ans, dp[i]);
}
printf("%d\n", ans);
memset(head, -1, sizeof head);
st = 0, ed = n + n + 5;
for (int i = 1; i <= n; i++) {
addedge(i, i + n, 1), addedge(i + n, i, 0);
if (dp[i] == 1) addedge(st, i, 1), addedge(i, st, 0);
if (dp[i] == ans) addedge(i + n, ed, 1), addedge(ed, i + n, 0);
}
for (int i = 1; i <= n; i++) {
for (int j = 1; j < i; j++) {
if (a[j] <= a[i] && dp[j] + 1 == dp[i]) addedge(j + n, i, 1), addedge(i, j + n, 0);
}
}
dinic(st, ed);
printf("%d\n", maxf);
for (int i = 0; i <= cnt; i += 2) edge[i].w += edge[i ^ 1].w, edge[i ^ 1].w = 0;
addedge(1, n + 1, inf), addedge(n + 1, 1, 0);
addedge(st, 1, inf), addedge(1, st, 0);
if (dp[n] == ans) addedge(n, n * 2, inf), addedge(n * 2, n, 0), addedge(n * 2, ed, inf), addedge(ed, n * 2, 0);
maxf = 0;
dinic(st, ed);
printf("%d\n", maxf);
return 0;
}
\(7.\)洛谷P2774 方格取数问题
分析：
这道题逆着思考就比较容易了：删去权值和最小的点集，使得剩下的点不互斥。
考虑 最大流\(=\)最小割。
我们先将给定的图按题意进行黑白染色，那么样例染色之后就如下图：
so,我们让源点与黑点建流量为黑点点权的边，白点与汇点建流量为白点点权的边，黑点与其相邻的白点建流量为\(inf\)的边。那么我们会得到这样的图：
我们对这样的图跑最大流，就意味着每一条从源点到汇点的路径中，我们会删掉一条非\(inf\)边，删掉的这条边恰好是最大流量又恰好是最小割(删掉的点权值最小)。
所以答案就显然了，\(ans=sum-maxflow\)。
#include <bits/stdc++.h>
using namespace std;
typedef long long LL;
typedef unsigned long long ULL;
typedef pair<int, int> pii;
const int maxn = 100 + 5;
const int maxm = 100 + 5;
const int inf = 0x3f3f3f3f;
int n, m, it, sum, f[maxn][maxn], pic[maxn][maxn];
int cnt, head[maxn * maxn];
int st, ed, maxf, flag, cur[maxn * maxn], vis[maxn * maxn], deep[maxn * maxn];
struct Graph {
int v, w, next;
} edge[maxn * maxn];
void addedge(int u, int v, int w) {
edge[cnt].v = v;
edge[cnt].w = w;
edge[cnt].next = head[u];
head[u] = cnt++;
}
int get(int x, int y) {
return m * (x - 1) + y;
}
int bfs(int st, int ed) {
for (int i = 0; i <= ed; i++) cur[i] = head[i], vis[i] = 0, deep[i] = inf;
queue<int> q;
q.push(st);
deep[st] = 0;
while (!q.empty()) {
int u = q.front();
q.pop();
vis[u] = 0;
for (int i = head[u]; ~i; i = edge[i].next) {
int v = edge[i].v;
if (edge[i].w && deep[v] > deep[u] + 1) {
deep[v] = deep[u] + 1;
if (!vis[v]) {
q.push(v);
vis[v] = 1;
}
}
}
}
return deep[ed] != inf;
}
int dfs(int u, int ed, int flow) {
if (u == ed) {
maxf += flow;
flag = 1;
return flow;
}
int now = 0, used = 0;
for (int i = cur[u]; ~i; i = edge[i].next) {
cur[u] = i;
int v = edge[i].v;
if (edge[i].w && deep[v] == deep[u] + 1) {
now = dfs(v, ed, min(flow - used, edge[i].w));
if (!now) continue;
edge[i].w -= now;
edge[i ^ 1].w += now;
used += now;
if (used == flow) break;
}
}
return used;
}
void dinic(int st, int ed) {
while (bfs(st, ed)) {
flag = 1;
while (flag) {
flag = 0;
dfs(st, ed, inf);
}
}
}
int main() {
scanf("%d %d", &n, &m);
for (int i = 1; i <= n; i++)
for (int j = 1; j <= m; j++) scanf("%d", &pic[i][j]), sum += pic[i][j];
st = 0, ed = n * m + 3;
memset(head, -1, sizeof head);
for (int i = 1; i <= n; i++) {
for (int j = 1; j <= m; j++) {
if (!f[i - 1][j] && !f[i][j - 1]) addedge(st, get(i, j), pic[i][j]), addedge(get(i, j), st, 0), f[i][j] = 1;
}
}
for (int i = 1; i <= n; i++)
for (int j = 1; j <= m; j++) if (!f[i][j])
addedge(get(i, j), ed, pic[i][j]), addedge(ed, get(i, j), 0), f[i][j] = 2;
for (int i = 1; i <= n; i++) {
for (int j = 1; j <= m; j++) {
if (f[i][j] == 2) continue;
it = get(i, j);
if (j >= 2) addedge(it, get(i, j - 1), inf), addedge(get(i, j - 1), it, 0);
if (j <= m - 1) addedge(it, get(i, j + 1), inf), addedge(get(i, j + 1), it, 0);
if (i >= 2) addedge(it, get(i - 1, j), inf), addedge(get(i - 1, j), it, 0);
if (i <= n - 1) addedge(it, get(i + 1, j), inf), addedge(get(i + 1, j), it, 0);
}
}
dinic(st, ed);
printf("%d\n", sum - maxf);
return 0;
}
\(8.\)洛谷P1251 餐巾计划问题
分析：
对于这题我们可以换一个角度来思考：将每天(点)分成早上(入点)和晚上(入点)，早上用餐巾。
建图：
\(1.\)每天早上都会用掉\(x\)条新餐巾，也就意味着晚上至少会有\(x\)条旧餐巾。
所以源点向每天晚上建流量为\(x\)费用为\(0\)的边。
\(2.\)因为每天一定要有\(x\)条新餐巾，所以早上向汇点建流量为\(x\)费用为\(0\)的边。
\(3.\)每天早上可以购买很多条新餐巾来满足当天条件，所以源点向早上建流量为\(inf\)费用为\(p\)的边。
\(4.\)对于晚上的旧餐巾有三种转移：
一：留着不洗放到下一天，即对下一天晚上建容量为\(inf\)费用为\(0\)的边。
二：慢洗，可以洗很多条旧餐巾，所以向洗好的那天早上建容量为\(inf\)费用为\(s\)的边。
三：快洗，同上，建容量为\(inf\)费用为\(f\)的边。
最后跑一遍最小费用最大流即可，最小费用即是答案
#include <bits/stdc++.h>
using namespace std;
typedef long long LL;
typedef unsigned long long ULL;
typedef pair<int, int> pii;
const int maxn = 4000 + 5;
const int maxm = 100 + 5;
const int inf = 0x3f3f3f3f;
int n, x, cnt, head[maxn];
LL a, b, c, d, e, minc, dis[maxn];
int st, ed, maxf, vis[maxn];
struct Graph {
int v, w, next;
LL c;
} edge[maxn << 3];
void addedge(int u, int v, int w, LL c) {
edge[cnt].v = v;
edge[cnt].w = w;
edge[cnt].c = c;
edge[cnt].next = head[u];
head[u] = cnt++;
}
int spfa(int st, int ed) {
for (int i = 0; i <= ed; i++) dis[i] = inf, vis[i] = 0;
queue<int> q;
q.push(st);
dis[st] = 0;
while (!q.empty()) {
int u = q.front();
q.pop();
vis[u] = 0;
for (int i = head[u]; ~i; i = edge[i].next) {
int v = edge[i].v;
if (edge[i].w && dis[v] > dis[u] + edge[i].c) {
dis[v] = dis[u] + edge[i].c;
if (!vis[v]) {
vis[v] = 1;
q.push(v);
}
}
}
}
return dis[ed] != inf;
}
int dfs(int u, int ed, int flow) {
if (u == ed) {
vis[ed] = 1;
// maxf += flow;
return flow;
}
vis[u] = 1;
int now = 0, used = 0;
for (int i = head[u]; ~i; i = edge[i].next) {
int v = edge[i].v;
if ((v == ed || !vis[v]) && edge[i].w && dis[v] == dis[u] + edge[i].c) {
now = dfs(v, ed, min(flow - used, edge[i].w));
if (!now) continue;
edge[i].w -= now;
edge[i ^ 1].w += now;
used += now;
minc += now * edge[i].c;
}
}
return used;
}
void mcmf(int st, int ed) {
while (spfa(st, ed)) {
vis[ed] = 1;
while (vis[ed]) {
memset(vis, 0, sizeof vis);
dfs(st, ed, inf);
}
}
}
int main() {
scanf("%d", &n);
st = 0, ed = n + n + 3;
memset(head, -1, sizeof head);
for (int i = 1; i <= n; i++) {
scanf("%d", &x);
addedge(i, ed, x, 0), addedge(ed, i, 0, 0);
addedge(st, i + n, x, 0), addedge(i + n, st, 0, 0);
}
scanf("%lld %lld %lld %lld %lld", &a, &b, &c, &d, &e);
for (int i = 1; i <= n; i++) {
addedge(st, i, inf, a), addedge(i, st, 0, -a);
if (i + 1 <= n) addedge(i + n, i + 1 + n, inf, 0), addedge(i + 1 + n, i + n, 0, 0);
if (i + d <= n) addedge(i + n, i + d, inf, e), addedge(i + d, i + n, 0, -e);
if (i + b <= n) addedge(i + n, i + b, inf, c), addedge(i + b, i + n, 0, -c);
}
mcmf(st, ed);
// printf("%d\n", maxf);
printf("%lld\n", minc);
return 0;
}
\(9.\)洛谷P4013 数字梯形问题
分析：
先对题目中的三种询问翻译成人话：
从第一行选\(m\)个点，第一行的每个点选一次，再选\(m\)条路径走到梯形底，求得到的最大价值。
\(I.\)梯形中的每个点只选一次组成的\(m\)条路径
\(II.\)梯形中的每个点可以选无数次，但不能有路径相交(边只能选一次)
\(III.\)无限制
对于\(I\)，点只能取一次的情况，将点拆成入点与出点，入点与出点间流量为\(1\)即可满足条件。源点到第一层的点和最后一层的点到汇点的流量均为\(1\)，费用分别为\(0\)与负的点权。
对于\(II\)，点可以无限取所以不必拆点了，由于边只能选一次，那么就将边的流量设为\(1\)就好了。源点到第一层的点流量为\(1\)，最后一层的点到汇点流量为\(inf\)，费用分别为\(0\)与负的点权。
对于\(III\)，边流量都设为\(inf\)即可。
点与点之间的费用设为负的跑最大费用最大流即可。
#include <bits/stdc++.h>
using namespace std;
typedef long long LL;
typedef unsigned long long ULL;
typedef pair<int, int> pii;
const int maxn = 5000 + 5;
const int maxm = 50 + 5;
const int inf = 0x3f3f3f3f;
int n, m, now, a[maxm][maxm], b[maxm][maxm];
int cnt, head[maxn];
int st, ed, maxf, minc, dis[maxn], vis[maxn];
struct Graph {
int v, w, c, next;
} edge[maxn << 2];
void addedge(int u, int v, int w, int c) {
edge[cnt].v = v;
edge[cnt].w = w;
edge[cnt].c = c;
edge[cnt].next = head[u];
head[u] = cnt++;
}
void add(int u, int v, int w, int c) {
addedge(u, v, w, c);
addedge(v, u, 0, -c);
}
int spfa(int st, int ed) {
for (int i = 0; i <= ed; i++) dis[i] = inf, vis[i] = 0;
queue<int> q;
q.push(st);
dis[st] = 0;
while (!q.empty()) {
int u = q.front();
q.pop();
vis[u] = 0;
for (int i = head[u]; ~i; i = edge[i].next) {
int v = edge[i].v;
if (edge[i].w && dis[v] > dis[u] + edge[i].c) {
dis[v] = dis[u] + edge[i].c;
if (!vis[v]) {
vis[v] = 1;
q.push(v);
}
}
}
}
return dis[ed] != inf;
}
int dfs(int u, int ed, int flow) {
if (u == ed) {
vis[ed] = 1;
// maxf += flow;
return flow;
}
vis[u] = 1;
int now = 0, used = 0;
for (int i = head[u]; ~i; i = edge[i].next) {
int v = edge[i].v;
if ((v == ed || !vis[v]) && edge[i].w && dis[v] == dis[u] + edge[i].c) {
now = dfs(v, ed, min(flow - used, edge[i].w));
if (!now) continue;
edge[i].w -= now;
edge[i ^ 1].w += now;
used += now;
minc += now * edge[i].c;
}
}
return used;
}
void mcmf(int st, int ed) {
while (spfa(st, ed)) {
vis[ed] = 1;
while (vis[ed]) {
memset(vis, 0, sizeof vis);
dfs(st, ed, inf);
}
}
}
int main() {
scanf("%d %d", &n, &m);
for (int i = 1; i <= m; i++) {
for (int j = 1; j <= i + n - 1; j++) {
scanf("%d", &a[i][j]);
b[i][j] = ++now;
}
}
st = 0, ed = 5000;
memset(head, -1, sizeof head);
for (int i = 1; i <= n; i++) add(st, b[1][i], 1, 0);
for (int i = 1; i <= m; i++) {
for (int j = 1; j <= i + n - 1; j++) {
add(b[i][j], b[i][j] + 2500, 1, -a[i][j]);
if (i != m) add(b[i][j] + 2500, b[i + 1][j], 1, 0), add(b[i][j] + 2500, b[i + 1][j + 1], 1, 0);
}
}
for (int i = 1; i <= n + m - 1; i++) add(b[m][i] + 2500, ed, 1, 0);
mcmf(st, ed);
printf("%d\n", -minc);
minc = 0, cnt = 0, memset(head, -1, sizeof head);
for (int i = 1; i <= n; i++) add(st, b[1][i], 1, 0);
for (int i = 1; i <= m; i++) {
for (int j = 1; j <= i + n - 1; j++) {
if (i != m) add(b[i][j], b[i + 1][j], 1, -a[i][j]), add(b[i][j], b[i + 1][j + 1], 1, -a[i][j]);
}
}
for (int i = 1; i <= n + m - 1; i++) add(b[m][i], ed, inf, -a[m][i]);
mcmf(st, ed);
printf("%d\n", -minc);
minc = 0, cnt = 0, memset(head, -1, sizeof head);
for (int i = 1; i <= n; i++) add(st, b[1][i], 1, 0);
for (int i = 1; i <= m; i++) {
for (int j = 1; j <= i + n - 1; j++) {
if (i != m) add(b[i][j], b[i + 1][j], inf, -a[i][j]), add(b[i][j], b[i + 1][j + 1], inf, -a[i][j]);
}
}
for (int i = 1; i <= n + m - 1; i++) add(b[m][i], ed, inf, -a[m][i]);
mcmf(st, ed);
printf("%d\n", -minc);
return 0;
}
\(10.\)洛谷P4015 运输问题
分析：
类似二分匹配，只不过匹配带权了。
将\(a\)仓库作为左部，\(b\)仓库作为右部，源点\(\rightarrow a\)建流量为\(a\)仓库货物量，费用为\(0\)的边，\(b \rightarrow\)汇点建流量为\(b\)仓库所需货物量，费用为\(0\)。
\(a\)仓库与\(b\)仓库建流量为\(inf\)费用为相应费用的边。
跑一遍最小费用最大流即可得第一个询问的答案。
第二个询问是最大费用最大流，那么我们将原先的正向边与反向边的费用取反，并把流量回归再跑最小费用最大流，最后将答案取反就是最大费用。
#include <bits/stdc++.h>
using namespace std;
typedef long long LL;
typedef unsigned long long ULL;
typedef pair<int, int> pii;
const int maxn = 100 + 5;
const int maxm = 50000 + 5;
const int inf = 0x3f3f3f3f;
int n, m, w;
int st, ed, cnt, vis[maxn << 1], head[maxn << 1];
LL dis[maxn << 1], mincost;
struct Graph {
int v, w, c, next;
} edge[maxm * maxn << 1];
void addedge(int u, int v, int w, int c) {
edge[cnt].v = v;
edge[cnt].w = w;
edge[cnt].c = c;
edge[cnt].next = head[u];
head[u] = cnt++;
}
int spfa(int st, int ed) {
for (int i = 0; i <= n + m + 5; i++) dis[i] = 1e17;
memset(vis, 0, sizeof vis);
queue<int> q;
q.push(st);
dis[st] = 0;
while (!q.empty()) {
int u = q.front();
q.pop();
vis[u] = 0;
for (int i = head[u]; ~i; i = edge[i].next) {
int v = edge[i].v, c = edge[i].c;
if (edge[i].w && dis[v] > dis[u] + c) {
dis[v] = dis[u] + c;
if (!vis[v]) {
vis[v] = 1;
q.push(v);
}
}
}
}
return dis[ed] != 1e17;
}
int dfs(int u, int flow, int ed) {
if (u == ed) {
vis[ed] = 1;
return flow;
}
vis[u] = 1;
int now = 0, used = 0;
for (int i = head[u]; ~i; i = edge[i].next) {
int v = edge[i].v;
if ((!vis[v] || v == ed) && dis[v] == dis[u] + edge[i].c && edge[i].w) {
now = dfs(v, min(flow - used, edge[i].w), ed);
if (!now) continue;
used += now;
edge[i].w -= now;
edge[i ^ 1].w += now;
mincost += edge[i].c * now;
if (used == flow) break;
}
}
return used;
}
void mcmf(int st, int ed) {
while (spfa(st, ed)) {
vis[ed] = 1;
while (vis[ed]) {
memset(vis, 0, sizeof vis);
dfs(st, inf, ed);
}
}
}
int main() {
scanf("%d %d", &n, &m);
st = 0, ed = n + m + 1;
memset(head, -1, sizeof head);
for (int i = 1; i <= n; i++) {
scanf("%d", &w);
addedge(st, i, w, 0), addedge(i, st, 0, 0);
}
for (int i = 1; i <= m; i++) {
scanf("%d", &w);
addedge(i + n, ed, w, 0), addedge(ed, i + n, 0, 0);
}
for (int i = 1; i <= n; i++) {
for (int j = 1; j <= m; j++) {
scanf("%d", &w);
addedge(i, j + n, inf, w), addedge(j + n, i, 0, -w);
}
}
mcmf(st, ed);
printf("%lld\n", mincost);
for (int i = 0; i <= cnt; i += 2) {
edge[i].w += edge[i ^ 1].w;
edge[i ^ 1].w = 0;
edge[i].c = -edge[i].c;
edge[i ^ 1].c = -edge[i ^ 1].c;
}
mincost = 0;
mcmf(st, ed);
printf("%lld\n", -mincost);
return 0;
}
\(11.\)洛谷P4014 分配问题
与\(10\)相同是二分图带权匹配问题，输出最小以及最大费用。
#include <bits/stdc++.h>
using namespace std;
typedef long long LL;
typedef unsigned long long ULL;
typedef pair<int, int> pii;
const int maxn = 100 + 5;
const int maxm = 50000 + 5;
const int inf = 0x3f3f3f3f;
int n, w;
int st, ed, cnt, dis[maxn << 1], vis[maxn << 1], head[maxn << 1];
int maxflow, mincost;
struct Graph {
int v, w, c, next;
} edge[maxm * maxn << 1];
void addedge(int u, int v, int w, int c) {
edge[cnt].v = v;
edge[cnt].w = w;
edge[cnt].c = c;
edge[cnt].next = head[u];
head[u] = cnt++;
}
int spfa(int st, int ed) {
memset(dis, inf, sizeof dis);
memset(vis, 0, sizeof vis);
queue<int> q;
q.push(st);
dis[st] = 0;
while (!q.empty()) {
int u = q.front();
q.pop();
vis[u] = 0;
for (int i = head[u]; ~i; i = edge[i].next) {
int v = edge[i].v, c = edge[i].c;
if (dis[v] > dis[u] + c && edge[i].w) {
dis[v] = dis[u] + c;
if (!vis[v]) {
q.push(v);
vis[v] = 1;
}
}
}
}
if (dis[ed] != inf) return 1;
return 0;
}
int dfs(int x, int flow, int ed) {
if (x == ed) {
vis[ed] = 1;
maxflow += flow;
return flow;
}
vis[x] = 1;
int now = 0, used = 0;
for (int i = head[x]; ~i; i = edge[i].next) {
int v = edge[i].v;
if ((!vis[v] || v == ed) && edge[i].w && dis[v] == dis[x] + edge[i].c) {
now = dfs(v, min(flow - used, edge[i].w), ed);
if (now != 0) mincost += edge[i].c * now, edge[i].w -= now, edge[i ^ 1].w += now, used += now;
if (used == flow) break;
}
}
return used;
}
void mcmf(int st, int ed) {
while (spfa(st, ed)) {
vis[ed] = 1;
while (vis[ed]) {
memset(vis, 0, sizeof vis);
dfs(st, inf, ed);
}
}
}
int main() {
scanf("%d", &n);
st = 0, ed = n + n + 1;
memset(head, -1, sizeof head);
for (int i = 1; i <= n; i++) {
addedge(st, i, 1, 0), addedge(i, st, 0, 0);
addedge(i + n, ed, 1, 0), addedge(ed, i + n, 0, 0);
for (int j = 1; j <= n; j++) {
scanf("%d", &w);
addedge(i, j + n, 1, w), addedge(j + n, i, 0, -w);
}
}
mincost = 0;
mcmf(st, ed);
printf("%d\n", mincost);
for (int i = 0; i <= cnt; i += 2) edge[i].w += edge[i ^ 1].w, edge[i ^ 1].w = 0, edge[i].c = -edge[i].c, edge[i ^ 1].c = -edge[i ^ 1].c;
mincost = 0;
mcmf(st, ed);
printf("%d\n", -mincost);
return 0;
}
\(12.\)洛谷P4016 负载平衡问题
分析：最后要求所有仓库的货物量一样，那么每个仓库的最终货物量就是货物总和的平均数\(average\)。
设每个仓库货物量\(w[i]\)
如果一个仓库的货物量\(> ave\)，说明这个仓库要流出流量使得别的仓库到达平均值。
如果一个仓库的货物量\(< ave\)，说明这个仓库要流入到达平均值。
所以最大流就是流入\(< ave\)的仓库的流量。
从源点对货物量\(> ave\)的仓库建流量为\(w[i]-ave\)，费用为\(0\)的边。
从货物量\(< ave\)的仓库建流量为\(ave-w[i]\)，费用为\(0\)的边。
按照题意对每个货物的两端建流量为\(inf\)，费用为\(1\)的边。
跑一遍最小费用最大流即可。
#include <bits/stdc++.h>
using namespace std;
typedef long long LL;
typedef unsigned long long ULL;
typedef pair<int, int> pii;
const int maxn = 100 + 5;
const int maxm = 50000 + 5;
const int inf = 0x3f3f3f3f;
int n, ave, a[maxn];
int st, ed, cnt, dis[maxn], vis[maxn], head[maxn];
int maxflow, mincost;
struct node {
int u, e;
} pre[maxn << 1];
struct Graph {
int v, w, c, next;
} edge[maxn * maxn << 2];
void addedge(int u, int v, int w, int c) {
edge[cnt].v = v;
edge[cnt].w = w;
edge[cnt].c = c;
edge[cnt].next = head[u];
head[u] = cnt++;
}
int spfa(int st, int ed) {
memset(dis, inf, sizeof dis);
memset(vis, 0, sizeof vis);
queue<int> q;
q.push(st);
dis[st] = 0;
while (!q.empty()) {
int u = q.front();
q.pop();
vis[u] = 0;
for (int i = head[u]; ~i; i = edge[i].next) {
int v = edge[i].v, c = edge[i].c;
if (dis[v] > dis[u] + c && edge[i].w) {
dis[v] = dis[u] + c;
pre[v].u = u;
pre[v].e = i;
if (!vis[v]) {
q.push(v);
vis[v] = 1;
}
}
}
}
if (dis[ed] != inf) return dis[ed];
return 0;
}
void mcmf(int st, int ed) {
int minw, res;
while (res = spfa(st, ed)) {
minw = inf;
for (int i = ed; i != st; i = pre[i].u) minw = min(minw, edge[pre[i].e].w);
for (int i = ed; i != st; i = pre[i].u) edge[pre[i].e].w -= minw, edge[pre[i].e ^ 1].w += minw;
maxflow += minw;
mincost += minw * res;
}
}
int main() {
scanf("%d", &n);
for (int i = 1; i <= n; i++) scanf("%d", a + i), ave += a[i];
ave /= n;
st = 0, ed = n + 1;
memset(head, -1, sizeof head);
for (int i = 1; i <= n; i++) {
if (a[i] > ave) addedge(st, i, a[i] - ave, 0), addedge(i, st, 0, 0);
else addedge(i, ed, ave - a[i], 0), addedge(ed, i, 0, 0);
if (i != 1 && i != n) {
addedge(i, i + 1, inf, 1), addedge(i + 1, i, 0, -1);
addedge(i, i - 1, inf, 1), addedge(i - 1, i, 0, -1);
}
}
addedge(1, n, inf, 1), addedge(n, 1, 0, -1);
addedge(1, 2, inf, 1), addedge(2, 1, 0, -1);
addedge(n, 1, inf, 1), addedge(1, n, 0, -1);
addedge(n, n - 1, inf, 1), addedge(n - 1, n, 0, -1);
mcmf(st, ed);
printf("%d\n", mincost);
return 0;
}
\(13.\)洛谷P2770 航空路线问题
分析：
实际上题目就是让我们求两条从\(1\)到\(n\)的路径，并且两条路径没有重点(除了\(1\)和\(n\))。
限制一个点的选取数量，那么我们队这个点进行拆点即可，入点与出点流量为\(1\)。
特别的\(1\)和\(n\)号点流量为\(2\)，因为可以用两次。
两点之间有边\(<u,v>\)，那么就让\(u\)的出点连向\(v\)的入点即可，流量为\(1\)。
建好图后，跑一遍最大流，有两种情况是有解的。
\(I.\)最大流为2，肯定有解
\(II.\)最大流为1，且有\(1\)直通\(n\)的路径。(特判)
输出路径两次\(dfs\)即可，每次\(dfs\)残量为\(0\)的边(记得\(break\))，\(dfs\)一次能得到一条路径，再将第二次倒序输出并且不输出\(n\)号点即可。
#include <bits/stdc++.h>
using namespace std;
#define pb push_back
typedef long long LL;
typedef unsigned long long ULL;
typedef pair<int, int> pii;
const int maxn = 200 + 5;
const int maxm = 50 + 5;
const int inf = 0x3f3f3f3f;
int n, m, u, v, flag;
int cnt, head[maxn];
int st, ed, maxf, minc, dis[maxn], vis[maxn];
string pic[111], str, ptr;
map<string, int> p;
vector<int> ve;
struct Graph {
int v, w, c, next;
} edge[maxn << 3];
void addedge(int u, int v, int w, LL c) {
edge[cnt].v = v;
edge[cnt].w = w;
edge[cnt].c = c;
edge[cnt].next = head[u];
head[u] = cnt++;
}
void add(int u, int v, int w, int c) {
addedge(u, v, w, c);
addedge(v, u, 0, -c);
}
int spfa(int st, int ed) {
for (int i = 0; i <= ed; i++) dis[i] = inf, vis[i] = 0;
queue<int> q;
q.push(st);
dis[st] = 0;
while (!q.empty()) {
int u = q.front();
q.pop();
vis[u] = 0;
for (int i = head[u]; ~i; i = edge[i].next) {
int v = edge[i].v;
if (edge[i].w && dis[v] > dis[u] + edge[i].c) {
dis[v] = dis[u] + edge[i].c;
if (!vis[v]) {
vis[v] = 1;
q.push(v);
}
}
}
}
return dis[ed] != inf;
}
int dfs(int u, int ed, int flow) {
if (u == ed) {
vis[ed] = 1;
maxf += flow;
return flow;
}
vis[u] = 1;
int now = 0, used = 0;
for (int i = head[u]; ~i; i = edge[i].next) {
int v = edge[i].v;
if ((v == ed || !vis[v]) && edge[i].w && dis[v] == dis[u] + edge[i].c) {
now = dfs(v, ed, min(flow - used, edge[i].w));
if (!now) continue;
edge[i].w -= now;
edge[i ^ 1].w += now;
used += now;
minc += now * edge[i].c;
}
}
return used;
}
void mcmf(int st, int ed) {
while (spfa(st, ed)) {
vis[ed] = 1;
while (vis[ed]) {
memset(vis, 0, sizeof vis);
dfs(st, ed, inf);
}
}
}
void dfs1(int u, int ed) {
ve.pb(u);
if (u == ed) {
return ;
}
for (int i = head[u]; ~i; i = edge[i].next) {
if (!edge[i].w) {
dfs1(edge[i].v + n, ed);
edge[i].w = 1;
break;
}
}
}
int main() {
cin >> n >> m;
memset(head, -1, sizeof head);
for (int i = 1; i <= n; i++) {
cin >> pic[i];
p[pic[i]] = i;
if (i == 1 || i == n) add(i, i + n, 2, -1);
else add(i, i + n, 1, -1);
}
for (int i = 1; i <= m; i++) {
cin >> str >> ptr;
u = p[str], v = p[ptr];
if (u > v) swap(u, v);
if (u == 1 && v == n) flag = 1;
add(u + n, v, 1, 0);
}
st = 1, ed = n << 1;
mcmf(st, ed);
if (maxf == 1 && flag) {
cout << 2 << '\n' << pic[1] << '\n' << pic[n] << '\n' << pic[1] << '\n';
} else if (maxf == 2){
cout << -minc - 2 << '\n';
ve.clear();
dfs1(1 + n, n + n);
for (auto it : ve) if (it > n) cout << pic[it - n] << '\n';
ve.clear();
dfs1(1 + n, n + n);
reverse(ve.begin(), ve.end());
for (auto it : ve) if (it > n && it != n * 2) cout << pic[it - n] << '\n';
} else {
cout << "No Solution!" << '\n';
}
return 0;
}
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