「题解」kuangbin 最小生成树

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  • POJ-1251 Jungle Roads (水题，%c)
  • POJ-1287 Networking (水)
  • POJ-2031 Building a Space Station (%f浮点数尴尬精度，两球间距离)
  • POJ-2421 Constructing Roads (一些边已建好，简单处理一下)
  • ZOJ-1586 QS Network (处理一下边权)
  • HDU-1233 还是畅通工程 (水)
  • HDU-1875 畅通工程再续 (浮点数，条件连边)
  • HDU-1301 Jungle Roads (重复 -> POJ-1251)
  • POJ-2349 Arctic Network (第 K 大边)
  • POJ-1751 Highways (求最小生成树的边，加 0 权边)
  • UVA-10147 Highways (多组的 POJ-1751) +
  • POJ-1258 Agri-Net (水)
  • POJ-3026 Borg Maze (BFS + Kruskal，用最短路建图，垃圾输入)
  • POJ-1789 Truck History (Prim，Kruskal 超时)
  • POJ-1679 The Unique MST (次小生成树)

代码最后附简单题解

模板

Kruskal

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 * Kruskal 算法求 MST
 */


#include <algorithm>
using namespace std;

const int MAXN = 110;    //最大点数
const int MAXM = 10000;  //最大边数
int F[MAXN];             //并查集使用
struct  {
  int u, v, w;
} edge[MAXM];  //存储边的信息，包括起点/终点/权值
int tol;       //边数，加边前赋值为 0
void addedge(int u, int v, int w) {
  edge[tol].u = u;
  edge[tol].v = v;
  edge[tol++].w = w;
}  //排序函数，讲边按照权值从小到大排序
bool cmp(Edge a, Edge b) { return a.w < b.w; }
int find(int x) {
  if (F[x] == -1)
    return x;
  else
    return F[x] = find(F[x]);
}  //传入点数，返回最小生成树的权值，如果不连通返回 -1
int Kruskal(int n) {
  memset(F, -1, sizeof(F));
  sort(edge, edge + tol, cmp);
  int cnt = 0;  //计算加入的边数

  int ans = 0;
  for (int i = 0; i < tol; i++) {
    int u = edge[i].u;
    int v = edge[i].v;
    int w = edge[i].w;
    int t1 = find(u);
    int t2 = find(v);
    if (t1 != t2) {
      ans += w;
      F[t1] = t2;
      cnt++;
    }
    if (cnt == n - 1) break;
  }
  if (cnt < n - 1)
    return -1;  //不连通
  else
    return ans;
}

Prim

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 * Prim 求 MST
 * 耗费矩阵 cost[][]，标号从 0 开始，0～n-1
 * 返回最小生成树的权值，返回 -1 表示原图不连通
 */


const int INF = 0x3f3f3f3f;
const int MAXN = 110;
bool vis[MAXN];
int lowc[MAXN];
//点是 0 n-1
int Prim(int cost[][MAXN], int n) {
  int ans = 0;
  memset(vis, false, sizeof(vis));
  vis[0] = true;
  for (int i = 1; i < n; i++) lowc[i] = cost[0][i];
  for (int i = 1; i < n; i++) {
    int minc = INF;
    int p = -1;
    for (int j = 0; j < n; j++)
      if (!vis[j] && minc > lowc[j]) {
        minc = lowc[j];
        p = j;
      }
    if (minc == INF) return -1;  //原图不连通
    ans += minc;
    vis[p] = true;
    for (int j = 0; j < n; j++) {
      if (!vis[j] && lowc[j] > cost[p][j]) lowc[j] = cost[p][j];
    }
  }
  return ans;
}

SecondMST (Prim)

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 * 次小生成树
 * 求最小生成树时，用数组 Max[i][j] 来表示 MST 中 i 到 j 最大边权
 * 求完后，直接枚举所有不在 MST 中的边，替换掉最大边权的边，更新答案
 * 点的编号从 0 开始
 */

#include <algorithm>
#include <cmath>
using namespace std;

const int MAXN = 110;
const int INF = 0x3f3f3f3f;
bool vis[MAXN];
int lowc[MAXN];
int pre[MAXN];
int Max[MAXN][MAXN];
// Max[i][j] 表示在最小生成树中从 i 到 j 的路径中的最大边权
bool used[MAXN][MAXN];
int Prim(int cost[][MAXN], int n) {
  int ans = 0;
  memset(vis, false, sizeof(vis));
  memset(Max, 0, sizeof(Max));
  memset(used, false, sizeof(used));
  vis[0] = true;
  pre[0] = -1;
  for (int i = 1; i < n; i++) {
    lowc[i] = cost[0][i];
    pre[i] = 0;
  }
  lowc[0] = 0;
  for (int i = 1; i < n; i++) {
    int minc = INF;
    int p = -1;

    for (int j = 0; j < n; j++)
      if (!vis[j] && minc > lowc[j]) {
        minc = lowc[j];
        p = j;
      }
    if (minc == INF) return -1;
    ans += minc;
    vis[p] = true;
    used[p][pre[p]] = used[pre[p]][p] = true;
    for (int j = 0; j < n; j++) {
      if (vis[j] && j != p)
        Max[j][p] = Max[p][j] = max(Max[j][pre[p]], lowc[p]);
      if (!vis[j] && lowc[j] > cost[p][j]) {
        lowc[j] = cost[p][j];
        pre[j] = p;
      }
    }
  }
  return ans;
}

int smst(int cost[][MAXN], int n, int ans) {
  int Min = INF;
  for (int i = 0; i < n; i++)
    for (int j = i + 1; j < n; j++)
      if (cost[i][j] != INF && !used[i][j]) {
        Min = min(Min, ans + cost[i][j] - Max[i][j]);
      }
  if (Min == INF) return -1;  //不存在
  return Min;
}

代码

POJ-1251 Jungle Roads

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// POJ-1251 Jungle Roads
// https://vjudge.net/problem/POJ-1251


#include <algorithm>
#include <cstdio>
using namespace std;

const int MAXN = 110;    //最大点数
const int MAXM = 10000;  //最大边数
int F[MAXN];             //并查集使用
struct  {
  int u, v, w;
} edge[MAXM];  //存储边的信息，包括起点/终点/权值
int tol;       //边数，加边前赋值为 0
void addedge(int u, int v, int w) {
  edge[tol].u = u;
  edge[tol].v = v;
  edge[tol++].w = w;
}
//排序函数，讲边按照权值从小到大排序
bool cmp(Edge a, Edge b) { return a.w < b.w; }
int find(int x) {
  if (F[x] == -1)
    return x;
  else
    return F[x] = find(F[x]);
}
//传入点数，返回最小生成树的权值，如果不连通返回 -1
int Kruskal(int n) {
  memset(F, -1, sizeof(F));
  sort(edge, edge + tol, cmp);
  int cnt = 0;  //计算加入的边数
  int ans = 0;
  for (int i = 0; i < tol; i++) {
    int u = edge[i].u;
    int v = edge[i].v;
    int w = edge[i].w;
    int t1 = find(u);
    int t2 = find(v);
    if (t1 != t2) {
      ans += w;
      F[t1] = t2;
      cnt++;
    }
    if (cnt == n - 1) break;
  }
  if (cnt < n - 1)
    return -1;  //不连通
  else
    return ans;
}

int main() {
  int n;
  while (~scanf("%d", &n)) {
    if (n == 0) break;
    tol = 0;
    for (int i = 0; i < n - 1; i++) {
      char x;
      int m;
      scanf(" %c%d", &x, &m);
      for (int j = 0; j < m; j++) {
        char y;
        int w;
        scanf(" %c%d", &y, &w);
        addedge(x - 'A', y - 'A', w);
      }
    }
    int ans = Kruskal(n);
    printf("%dn", ans);
  }
  return 0;
}

// 裸题，scanf %c 读入尴尬
// scanf(" %c%d", &x, &m);

POJ-1287 Networking

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// POJ-1287 Networking
// https://vjudge.net/problem/POJ-1287


#include <algorithm>
#include <cstdio>
using namespace std;

const int MAXN = 110;    //最大点数
const int MAXM = 10000;  //最大边数
int F[MAXN];             //并查集使用
struct  {
  int u, v, w;
} edge[MAXM];  //存储边的信息，包括起点/终点/权值
int tol;       //边数，加边前赋值为 0
void addedge(int u, int v, int w) {
  edge[tol].u = u;
  edge[tol].v = v;
  edge[tol++].w = w;
}
//排序函数，讲边按照权值从小到大排序
bool cmp(Edge a, Edge b) { return a.w < b.w; }
int find(int x) {
  if (F[x] == -1)
    return x;
  else
    return F[x] = find(F[x]);
}
//传入点数，返回最小生成树的权值，如果不连通返回 -1
int Kruskal(int n) {
  memset(F, -1, sizeof(F));
  sort(edge, edge + tol, cmp);
  int cnt = 0;  //计算加入的边数
  int ans = 0;
  for (int i = 0; i < tol; i++) {
    int u = edge[i].u;
    int v = edge[i].v;
    int w = edge[i].w;
    int t1 = find(u);
    int t2 = find(v);
    if (t1 != t2) {
      ans += w;
      F[t1] = t2;
      cnt++;
    }
    if (cnt == n - 1) break;
  }
  if (cnt < n - 1)
    return -1;  //不连通
  else
    return ans;
}

int main() {
  int n, m;
  while (~scanf("%d", &n)) {
    if (n == 0) break;
    tol = 0;
    scanf("%d", &m);
    for (int i = 0; i < m; i++) {
      int a, b, c;
      scanf("%d%d%d", &a, &b, &c);
      addedge(a, b, c);
    }
    int ans = Kruskal(n);
    printf("%dn", ans);
  }
  return 0;
}

// 裸题

POJ-2031 Building a Space Station

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// POJ-2031 Building a Space Station
// https://vjudge.net/problem/POJ-2031

#include <string.h>
#include <algorithm>
#include <cmath>
#include <cstdio>
using namespace std;

const int MAXN = 110;    //最大点数
const int MAXM = 10000;  //最大边数
int F[MAXN];             //并查集使用
struct  {
  int u, v;
  double w;
} edge[MAXM];  //存储边的信息，包括起点/终点/权值
int tol;       //边数，加边前赋值为 0
void addedge(int u, int v, double w) {
  edge[tol].u = u;
  edge[tol].v = v;
  edge[tol++].w = w;
}
//排序函数，讲边按照权值从小到大排序
bool cmp(Edge a, Edge b) { return a.w < b.w; }
int find(int x) {
  if (F[x] == -1)
    return x;
  else
    return F[x] = find(F[x]);
}
//传入点数，返回最小生成树的权值，如果不连通返回 -1
double Kruskal(int n) {
  memset(F, -1, sizeof(F));
  sort(edge, edge + tol, cmp);
  int cnt = 0;  //计算加入的边数
  double ans = 0;
  for (int i = 0; i < tol; i++) {
    int u = edge[i].u;
    int v = edge[i].v;
    double w = edge[i].w;
    int t1 = find(u);
    int t2 = find(v);
    if (t1 != t2) {
      ans += w;
      F[t1] = t2;
      cnt++;
    }
    if (cnt == n - 1) break;
  }
  if (cnt < n - 1)
    return -1;  //不连通
  else
    return ans;
}

double len(double x, double y, double z, double xx, double yy, double zz) {
  return sqrt((xx - x) * (xx - x) + (yy - y) * (yy - y) + (zz - z) * (zz - z));
}
double cirlen(double x, double y, double z, double r, double xx, double yy,
              double zz, double rr) {
  double ll = len(x, y, z, xx, yy, zz);
  if (ll - r - rr < 1e-5) {
    return 0;
  } else {
    return ll - r - rr;
  }
}
double xp[MAXN];
double yp[MAXN];
double zp[MAXN];
double rp[MAXN];

int main() {
  int n;
  while (~scanf("%d", &n)) {
    if (n == 0) break;
    tol = 0;
    for (int i = 0; i < n; i++) {
      scanf("%lf%lf%lf%lf", &xp[i], &yp[i], &zp[i], &rp[i]);
    }
    for (int i = 0; i < n; i++) {
      for (int j = i + 1; j < n; j++) {
        addedge(i, j,
                cirlen(xp[i], yp[i], zp[i], rp[i], xp[j], yp[j], zp[j], rp[j]));
      }
    }
    double ans = Kruskal(n);
    printf("%.3fn", ans);
  }
  return 0;
}

// 浮点数的最小生成树，球之间的距离，距离小于零，就建一条权为 0 的边
// POJ尴尬精度 %f

POJ-2421 Constructing Roads

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// POJ-2421 Constructing Roads
// https://vjudge.net/problem/POJ-2421

#include <string.h>
#include <algorithm>
#include <cstdio>
using namespace std;

const int MAXN = 110;    //最大点数
const int MAXM = 10000;  //最大边数
int F[MAXN];             //并查集使用
struct  {
  int u, v, w;
} edge[MAXM];  //存储边的信息，包括起点/终点/权值
int tol;       //边数，加边前赋值为 0
void addedge(int u, int v, int w) {
  edge[tol].u = u;
  edge[tol].v = v;
  edge[tol++].w = w;
}  //排序函数，讲边按照权值从小到大排序
bool cmp(Edge a, Edge b) { return a.w < b.w; }
int find(int x) {
  if (F[x] == -1)
    return x;
  else
    return F[x] = find(F[x]);
}  //传入点数，返回最小生成树的权值，如果不连通返回 -1
int Kruskal(int n) {
  memset(F, -1, sizeof(F));
  sort(edge, edge + tol, cmp);
  int cnt = 0;  //计算加入的边数

  int ans = 0;
  for (int i = 0; i < tol; i++) {
    int u = edge[i].u;
    int v = edge[i].v;
    int w = edge[i].w;
    int t1 = find(u);
    int t2 = find(v);
    if (t1 != t2) {
      ans += w;
      F[t1] = t2;
      cnt++;
    }
    if (cnt == n - 1) break;
  }
  if (cnt < n - 1)
    return -1;  //不连通
  else
    return ans;
}

int mapx[MAXN][MAXN];

int main() {
  int N;
  scanf("%d", &N);
  for (int i = 1; i <= N; i++) {
    for (int j = 1; j <= N; j++) {
      scanf("%d", &mapx[i][j]);
    }
  }

  int Q;
  scanf("%d", &Q);
  for (int i = 0; i < Q; i++) {
    int a, b;
    scanf("%d%d", &a, &b);
    mapx[a][b] = 0;
    mapx[b][a] = 0;
  }
  for (int i = 1; i <= N; i++) {
    for (int j = i + 1; j <= N; j++) {
      if (i == j) continue;
      addedge(i, j, min(mapx[i][j], mapx[j][i]));
    }
  }
  int ans = Kruskal(N);
  printf("%dn", ans);
}

// 有一些路已建好，把边权改为 0 就好了

ZOJ-1586 QS Network

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// ZOJ-1586 QS Network
// https://vjudge.net/problem/ZOJ-1586

#include <string.h>
#include <algorithm>
#include <cstdio>
using namespace std;

const int MAXN = 1100;     //最大点数
const int MAXM = 2000000;  //最大边数
int F[MAXN];               //并查集使用

int codec[MAXN];  // *&*

struct Edge {
  int u, v, w;
} edge[MAXM];  //存储边的信息，包括起点/终点/权值
int tol;       //边数，加边前赋值为 0
void addedge(int u, int v, int w) {
  edge[tol].u = u;
  edge[tol].v = v;
  edge[tol++].w = w;
}
//排序函数，讲边按照权值从小到大排序
bool cmp(Edge a, Edge b) { return a.w < b.w; }
int find(int x) {
  if (F[x] == -1)
    return x;
  else
    return F[x] = find(F[x]);
}
//传入点数，返回最小生成树的权值，如果不连通返回 -1
int Kruskal(int n) {
  memset(F, -1, sizeof(F));
  sort(edge, edge + tol, cmp);
  int cnt = 0;  //计算加入的边数
  int ans = 0;
  for (int i = 0; i < tol; i++) {
    int u = edge[i].u;
    int v = edge[i].v;
    int w = edge[i].w;
    int t1 = find(u);
    int t2 = find(v);
    if (t1 != t2) {
      ans += w;

      F[t1] = t2;
      cnt++;
    }
    if (cnt == n - 1) break;
  }
  if (cnt < n - 1)
    return -1;  //不连通
  else
    return ans;
}
int main() {
  int T;
  scanf("%d", &T);
  while (T--) {
    tol = 0;
    int n;
    scanf("%d", &n);
    for (int i = 1; i <= n; i++) {
      scanf("%d", &codec[i]);
    }
    for (int i = 1; i <= n; i++) {
      for (int j = 1; j <= n; j++) {
        int w;
        scanf("%d", &w);
        if (i < j) addedge(i, j, w + codec[i] + codec[j]);
      }
    }
    int ans = Kruskal(n);
    printf("%dn", ans);
  }
  return 0;
}

// 加边时，加一下适配器的价格

HDU-1233 还是畅通工程

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// HDU-1233 还是畅通工程
// https://vjudge.net/problem/HDU-1233

#include <string.h>
#include <algorithm>
#include <cstdio>
using namespace std;
const int MAXN = 110;    //最大点数
const int MAXM = 10000;  //最大边数
int F[MAXN];             //并查集使用
struct Edge {
  int u, v, w;
} edge[MAXM];  //存储边的信息，包括起点/终点/权值
int tol;       //边数，加边前赋值为 0
void addedge(int u, int v, int w) {
  edge[tol].u = u;
  edge[tol].v = v;
  edge[tol++].w = w;
}  //排序函数，讲边按照权值从小到大排序
bool cmp(Edge a, Edge b) { return a.w < b.w; }
int find(int x) {
  if (F[x] == -1)
    return x;
  else
    return F[x] = find(F[x]);
}  //传入点数，返回最小生成树的权值，如果不连通返回 -1
int Kruskal(int n) {
  memset(F, -1, sizeof(F));
  sort(edge, edge + tol, cmp);
  int cnt = 0;  //计算加入的边数

  int ans = 0;
  for (int i = 0; i < tol; i++) {
    int u = edge[i].u;
    int v = edge[i].v;
    int w = edge[i].w;
    int t1 = find(u);
    int t2 = find(v);
    if (t1 != t2) {
      ans += w;
      F[t1] = t2;
      cnt++;
    }
    if (cnt == n - 1) break;
  }
  if (cnt < n - 1)
    return -1;  //不连通
  else
    return ans;
}
int main() {
  int n;
  while (~scanf("%d", &n)) {
    if (n == 0) break;
    tol = 0;
    int m = n * (n - 1) / 2;
    for (int i = 0; i < m; i++) {
      int a, b, w;
      scanf("%d%d%d", &a, &b, &w);
      addedge(a, b, w);
    }
    int ans = Kruskal(n);
    printf("%dn", ans);
  }
  return 0;
}

// 裸题

HDU-1875 畅通工程再续

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// HDU-1875 畅通工程再续
// https://vjudge.net/problem/HDU-1875

#include <string.h>
#include <algorithm>
#include <cmath>
#include <cstdio>
using namespace std;

const int MAXN = 310;    //最大点数
const int MAXM = 10000;  //最大边数
int F[MAXN];             //并查集使用
struct Edge {
  int u, v;
  double w;
} edge[MAXM];  //存储边的信息，包括起点/终点/权值
int tol;       //边数，加边前赋值为 0
void addedge(int u, int v, double w) {
  edge[tol].u = u;
  edge[tol].v = v;
  edge[tol++].w = w;
}
//排序函数，讲边按照权值从小到大排序
bool cmp(Edge a, Edge b) { return a.w < b.w; }
int find(int x) {
  if (F[x] == -1)
    return x;
  else
    return F[x] = find(F[x]);
}
//传入点数，返回最小生成树的权值，如果不连通返回 -1
double Kruskal(int n) {
  memset(F, -1, sizeof(F));
  sort(edge, edge + tol, cmp);
  int cnt = 0;  //计算加入的边数
  double ans = 0;
  for (int i = 0; i < tol; i++) {
    int u = edge[i].u;
    int v = edge[i].v;
    double w = edge[i].w;
    int t1 = find(u);
    int t2 = find(v);
    if (t1 != t2) {
      ans += w;
      F[t1] = t2;
      cnt++;
    }
    if (cnt == n - 1) break;
  }
  if (cnt < n - 1)
    return -1;  //不连通
  else
    return ans;
}

double len(double x, double y, double xx, double yy) {
  return sqrt((xx - x) * (xx - x) + (yy - y) * (yy - y));
}

double xp[MAXN];
double yp[MAXN];

int main() {
  int T;
  scanf("%d", &T);
  while (T--) {
    int n;
    scanf("%d", &n);
    if (n == 0) break;
    tol = 0;
    for (int i = 1; i <= n; i++) {
      scanf("%lf%lf", &xp[i], &yp[i]);
    }
    for (int i = 1; i <= n; i++) {
      for (int j = i + 1; j <= n; j++) {
        double xly = len(xp[i], yp[i], xp[j], yp[j]);
        if (xly >= 10 && xly <= 1000) addedge(i, j, xly * 100);
      }
    }
    double ans = Kruskal(n);
    if (ans > 0) {
      printf("%.1fn", ans);

    } else {
      printf("oh!n");
    }
  }
  return 0;
}

// 浮点数，条件连边

HDU-1301 Jungle Roads

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// HDU-1301 Jungle Roads
// https://vjudge.net/problem/HDU-1301

#include <string.h>
#include <algorithm>
#include <cstdio>
using namespace std;

const int MAXN = 110;    //最大点数
const int MAXM = 10000;  //最大边数
int F[MAXN];             //并查集使用
struct Edge {
  int u, v, w;
} edge[MAXM];  //存储边的信息，包括起点/终点/权值
int tol;       //边数，加边前赋值为 0
void addedge(int u, int v, int w) {
  edge[tol].u = u;
  edge[tol].v = v;
  edge[tol++].w = w;
}
//排序函数，讲边按照权值从小到大排序
bool cmp(Edge a, Edge b) { return a.w < b.w; }
int find(int x) {
  if (F[x] == -1)
    return x;
  else
    return F[x] = find(F[x]);
}
//传入点数，返回最小生成树的权值，如果不连通返回 -1
int Kruskal(int n) {
  memset(F, -1, sizeof(F));
  sort(edge, edge + tol, cmp);
  int cnt = 0;  //计算加入的边数
  int ans = 0;
  for (int i = 0; i < tol; i++) {
    int u = edge[i].u;
    int v = edge[i].v;
    int w = edge[i].w;
    int t1 = find(u);
    int t2 = find(v);
    if (t1 != t2) {
      ans += w;
      F[t1] = t2;
      cnt++;
    }
    if (cnt == n - 1) break;
  }
  if (cnt < n - 1)
    return -1;  //不连通
  else
    return ans;
}

int main() {
  int n;
  while (~scanf("%d", &n)) {
    if (n == 0) break;
    tol = 0;
    for (int i = 0; i < n - 1; i++) {
      char x;
      int m;
      scanf(" %c%d", &x, &m);
      for (int j = 0; j < m; j++) {
        char y;
        int w;
        scanf(" %c%d", &y, &w);
        addedge(x - 'A', y - 'A', w);
      }
    }
    int ans = Kruskal(n);
    printf("%dn", ans);
  }
  return 0;
}

// 竟然有重复的题，一模一样 -> POJ-1251
// 裸题，scanf %c 读入尴尬
// scanf(" %c%d", &x, &m);

POJ-2349 Arctic Network

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// POJ-2349 Arctic Network
// https://vjudge.net/problem/POJ-2349

#include <string.h>
#include <algorithm>
#include <cmath>
#include <cstdio>
#include <vector>
using namespace std;

const int MAXN = 510;     //最大点数
const int MAXM = 250000;  //最大边数
int F[MAXN];              //并查集使用
vector<double> vd;
struct Edge {
  int u, v;
  double w;
} edge[MAXM];  //存储边的信息，包括起点/终点/权值
int tol;       //边数，加边前赋值为 0
void addedge(int u, int v, double w) {
  edge[tol].u = u;
  edge[tol].v = v;
  edge[tol++].w = w;
}
//排序函数，讲边按照权值从小到大排序
bool cmp(Edge a, Edge b) { return a.w < b.w; }

int find(int x) {
  if (F[x] == -1)
    return x;
  else
    return F[x] = find(F[x]);
}
//传入点数，返回最小生成树的权值，如果不连通返回 -1
double Kruskal(int n) {
  memset(F, -1, sizeof(F));
  sort(edge, edge + tol, cmp);
  int cnt = 0;  //计算加入的边数
  double ans = 0;
  for (int i = 0; i < tol; i++) {
    int u = edge[i].u;
    int v = edge[i].v;
    double w = edge[i].w;
    int t1 = find(u);
    int t2 = find(v);
    if (t1 != t2) {
      ans = max(ans, w);
      vd.push_back(w);
      // ans += w;
      F[t1] = t2;
      cnt++;
    }
    if (cnt == n - 1) break;
  }
  if (cnt < n - 1)
    return -1;  //不连通
  else
    return ans;
}

double len(double x, double y, double xx, double yy) {
  return sqrt((xx - x) * (xx - x) + (yy - y) * (yy - y));
}

double xp[MAXN];
double yp[MAXN];

int main() {
  int T;
  scanf("%d", &T);
  while (T--) {
    vd.clear();
    int n, s;
    scanf("%d%d", &s, &n);
    if (n == 0) break;
    tol = 0;
    for (int i = 1; i <= n; i++) {
      scanf("%lf%lf", &xp[i], &yp[i]);
    }
    for (int i = 1; i <= n; i++) {
      for (int j = i + 1; j <= n; j++) {
        double xly = len(xp[i], yp[i], xp[j], yp[j]);
        addedge(i, j, xly);
      }
    }
    double ans = Kruskal(n);
    ans = vd[vd.size() - s];
    printf("%.2fn", ans);
  }
  return 0;
}

// 最小生成树的第 K 大边

POJ-1751 Highways

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// POJ-1751 Highways
// https://vjudge.net/problem/POJ-1751

#include <string.h>
#include <algorithm>
#include <cmath>
#include <cstdio>
#include <vector>
using namespace std;

const int MAXN = 1000;     //最大点数
const int MAXM = 1000000;  //最大边数
vector<int> vx;
vector<int> vy;

int F[MAXN];  //并查集使用
struct Edge {
  int u, v;
  double w;
} edge[MAXM];  //存储边的信息，包括起点/终点/权值
int tol;       //边数，加边前赋值为 0
void addedge(int u, int v, double w) {
  edge[tol].u = u;
  edge[tol].v = v;
  edge[tol++].w = w;
}
//排序函数，讲边按照权值从小到大排序
bool cmp(Edge a, Edge b) { return a.w < b.w; }
int find(int x) {
  if (F[x] == -1)
    return x;
  else
    return F[x] = find(F[x]);
}
//传入点数，返回最小生成树的权值，如果不连通返回 -1
double Kruskal(int n) {
  memset(F, -1, sizeof(F));
  sort(edge, edge + tol, cmp);
  int cnt = 0;  //计算加入的边数
  double ans = 0;
  for (int i = 0; i < tol; i++) {
    int u = edge[i].u;
    int v = edge[i].v;
    double w = edge[i].w;
    int t1 = find(u);
    int t2 = find(v);
    if (t1 != t2) {
      ans += w;
      if (w > 1e-5) {
        vx.push_back(u);
        vy.push_back(v);
      }

      F[t1] = t2;
      cnt++;
    }
    if (cnt == n - 1) break;
  }
  if (cnt < n - 1)
    return -1;  //不连通
  else
    return ans;
}

double len(double x, double y, double xx, double yy) {
  return sqrt((xx - x) * (xx - x) + (yy - y) * (yy - y));
}

double xp[MAXN];
double yp[MAXN];

int main() {
  int n;
  scanf("%d", &n);
  tol = 0;
  for (int i = 1; i <= n; i++) {
    scanf("%lf%lf", &xp[i], &yp[i]);
  }
  for (int i = 1; i <= n; i++) {
    for (int j = i + 1; j <= n; j++) {
      double xly = len(xp[i], yp[i], xp[j], yp[j]);
      addedge(i, j, xly * 100);
    }
  }
  int q;
  scanf("%d", &q);
  for (int i = 0; i < q; i++) {
    int a, b;
    scanf("%d%d", &a, &b);
    addedge(a, b, 0);
  }
  double ans = Kruskal(n);

  for (int i = 0; i < vx.size(); i++) {
    printf("%d %dn", vx[i], vy[i]);
  }

  return 0;
}

// 求最小生成树的边，加 0 权边

UVA-10147 Highways

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// UVA-10147 Highways
// https://vjudge.net/problem/UVA-10147

#include <string.h>
#include <algorithm>
#include <cmath>
#include <cstdio>
#include <vector>
using namespace std;

const int MAXN = 1000;     //最大点数
const int MAXM = 1000000;  //最大边数
vector<int> vx;
vector<int> vy;

int F[MAXN];  //并查集使用
struct Edge {
  int u, v;
  double w;
} edge[MAXM];  //存储边的信息，包括起点/终点/权值
int tol;       //边数，加边前赋值为 0
void addedge(int u, int v, double w) {
  edge[tol].u = u;
  edge[tol].v = v;
  edge[tol++].w = w;
}
//排序函数，讲边按照权值从小到大排序
bool cmp(Edge a, Edge b) { return a.w < b.w; }
int find(int x) {
  if (F[x] == -1)
    return x;
  else
    return F[x] = find(F[x]);
}
//传入点数，返回最小生成树的权值，如果不连通返回 -1
double Kruskal(int n) {
  memset(F, -1, sizeof(F));
  sort(edge, edge + tol, cmp);
  int cnt = 0;  //计算加入的边数
  double ans = 0;
  for (int i = 0; i < tol; i++) {
    int u = edge[i].u;
    int v = edge[i].v;
    double w = edge[i].w;
    int t1 = find(u);
    int t2 = find(v);
    if (t1 != t2) {
      ans += w;
      if (w > 0) {
        vx.push_back(u);
        vy.push_back(v);
      }

      F[t1] = t2;
      cnt++;
    }
    if (cnt == n - 1) break;
  }
  if (cnt < n - 1)
    return -1;  //不连通
  else
    return ans;
}

double len(double x, double y, double xx, double yy) {
  return sqrt((xx - x) * (xx - x) + (yy - y) * (yy - y));
}

double xp[MAXN];
double yp[MAXN];

int main() {
  int T;
  scanf("%d", &T);
  while (T--) {
    vx.clear();
    vy.clear();

    int n;
    scanf("%d", &n);
    tol = 0;
    for (int i = 1; i <= n; i++) {
      scanf("%lf%lf", &xp[i], &yp[i]);
    }
    for (int i = 1; i <= n; i++) {
      for (int j = i + 1; j <= n; j++) {
        double xly = len(xp[i], yp[i], xp[j], yp[j]);
        addedge(i, j, xly * 100);
      }
    }
    int q;
    scanf("%d", &q);
    for (int i = 0; i < q; i++) {
      int a, b;
      scanf("%d%d", &a, &b);
      addedge(a, b, 0);
    }
    double ans = Kruskal(n);

    for (int i = 0; i < vx.size(); i++) {
      printf("%d %dn", vx[i], vy[i]);
    }
    if (vx.size() == 0) printf("No new highways needn");
    if (T != 0) printf("n");
  }

  return 0;
}

// 求最小生成树的边，加 0 权边

POJ-1258 Agri-Net

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// POJ-1258 Agri-Net
// https://vjudge.net/problem/POJ-1258

#include <string.h>
#include <algorithm>
#include <cstdio>
using namespace std;

const int MAXN = 110;    //最大点数
const int MAXM = 10000;  //最大边数
int F[MAXN];             //并查集使用
struct Edge {
  int u, v, w;
} edge[MAXM];  //存储边的信息，包括起点/终点/权值
int tol;       //边数，加边前赋值为 0
void addedge(int u, int v, int w) {
  edge[tol].u = u;
  edge[tol].v = v;
  edge[tol++].w = w;
}  //排序函数，讲边按照权值从小到大排序
bool cmp(Edge a, Edge b) { return a.w < b.w; }
int find(int x) {
  if (F[x] == -1)
    return x;
  else
    return F[x] = find(F[x]);
}  //传入点数，返回最小生成树的权值，如果不连通返回 -1
int Kruskal(int n) {
  memset(F, -1, sizeof(F));
  sort(edge, edge + tol, cmp);
  int cnt = 0;  //计算加入的边数

  int ans = 0;
  for (int i = 0; i < tol; i++) {
    int u = edge[i].u;
    int v = edge[i].v;
    int w = edge[i].w;
    int t1 = find(u);
    int t2 = find(v);
    if (t1 != t2) {
      ans += w;
      F[t1] = t2;
      cnt++;
    }
    if (cnt == n - 1) break;
  }
  if (cnt < n - 1)
    return -1;  //不连通
  else
    return ans;
}

int main() {
  int n;
  while (~scanf("%d", &n)) {
    tol = 0;
    for (int i = 1; i <= n; i++) {
      for (int j = 1; j <= n; j++) {
        int w;
        scanf("%d", &w);
        addedge(i, j, w);
      }
    }
    int ans = Kruskal(n);
    printf("%dn", ans);
  }
  return 0;
}

// 裸题

POJ-3026 Borg Maze

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// POJ-3026 Borg Maze
// https://vjudge.net/problem/POJ-3026

#include <string.h>
#include <algorithm>
#include <cstdio>
#include <queue>
using namespace std;

const int MAXN = 1100;     //最大点数
const int MAXM = 1000000;  //最大边数
int F[MAXN];               //并查集使用
struct Edge {
  int u, v, w;
} edge[MAXM];  //存储边的信息，包括起点/终点/权值
int tol;       //边数，加边前赋值为 0
void addedge(int u, int v, int w) {
  edge[tol].u = u;
  edge[tol].v = v;
  edge[tol++].w = w;
}  //排序函数，讲边按照权值从小到大排序
bool cmp(Edge a, Edge b) { return a.w < b.w; }
int find(int x) {
  if (F[x] == -1)
    return x;
  else
    return F[x] = find(F[x]);
}  //传入点数，返回最小生成树的权值，如果不连通返回 -1
int Kruskal(int n) {
  memset(F, -1, sizeof(F));
  sort(edge, edge + tol, cmp);
  int cnt = 0;  //计算加入的边数

  int ans = 0;
  for (int i = 0; i < tol; i++) {
    int u = edge[i].u;
    int v = edge[i].v;
    int w = edge[i].w;
    int t1 = find(u);
    int t2 = find(v);
    if (t1 != t2) {
      ans += w;
      F[t1] = t2;
      cnt++;
    }
    if (cnt == n - 1) break;
  }
  if (cnt < n - 1)
    return -1;  //不连通
  else
    return ans;
}

int mapx[MAXN][MAXN];

int dirx[10] = {0, 1, 0, -1};
int diry[10] = {1, 0, -1, 0};

int n, m;
void bfs(int x, int y) {
  bool vis[MAXN][MAXN];
  memset(vis, false, sizeof(vis));
  queue<pair<pair<int, int>, int> > Q;
  Q.push(make_pair(make_pair(x, y), 0));
  vis[x][y] = true;
  while (!Q.empty()) {
    int xt = Q.front().first.first;
    int yt = Q.front().first.second;
    int times = Q.front().second;
    Q.pop();
    if (mapx[x][y] != mapx[xt][yt] && mapx[xt][yt] != 0) {
      addedge(mapx[x][y], mapx[xt][yt], times);
    }
    for (int i = 0; i < 4; i++) {
      int tx = xt + dirx[i];
      int ty = yt + diry[i];

      if (tx >= 0 && ty >= 0 && tx < n && ty < m && !vis[tx][ty]) {
        if (mapx[tx][ty] != -1) {
          Q.push(make_pair(make_pair(tx, ty), times + 1));
          vis[tx][ty] = true;
        }
      }
    }
  }
}

int main() {
  int T;
  scanf("%d", &T);
  char str[100];
  while (T--) {
    tol = 0;
    memset(mapx, -1, sizeof(mapx));
    int cnt = 0;
    scanf("%d%d", &n, &m);
    gets(str);
    for (int i = 0; i < m; i++) {
      gets(str);
      for (int j = 0; j < n; j++) {
        if (str[j] == 'A' || str[j] == 'S') {
          mapx[i][j] = ++cnt;
        } else if (str[j] == ' ') {
          mapx[i][j] = 0;
        }
      }
    }
    for (int i = 0; i < m; i++) {
      for (int j = 0; j < n; j++) {
        if (mapx[i][j] > 0) {
          bfs(i, j);
        }
      }
    }
    int ans = Kruskal(cnt);
    printf("%dn", ans);
  }
  return 0;
}

// BFS + Kruskal
// 用无权图最短路权值加边
// 一开始 MAXN 和 MAXM 开小了，wa...wa...wa...

POJ-1789 Truck History

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// POJ-1789 Truck History
// https://vjudge.net/problem/POJ-1789

#include <string.h>
#include <algorithm>
#include <cstdio>
using namespace std;

const int MAXN = 2010;  //最大点数
const int INF = 0x3f3f3f3f;
bool vis[MAXN];
int lowc[MAXN];
int cost[MAXN][MAXN];
//点是 0 n-1
int Prim(int cost[][MAXN], int n) {
  int ans = 0;
  memset(vis, false, sizeof(vis));
  vis[0] = true;
  for (int i = 1; i < n; i++) lowc[i] = cost[0][i];
  for (int i = 1; i < n; i++) {
    int minc = INF;
    int p = -1;
    for (int j = 0; j < n; j++)
      if (!vis[j] && minc > lowc[j]) {
        minc = lowc[j];
        p = j;
      }
    if (minc == INF) return -1;  //原图不连通
    ans += minc;
    vis[p] = true;
    for (int j = 0; j < n; j++) {
      if (!vis[j] && lowc[j] > cost[p][j]) lowc[j] = cost[p][j];
    }
  }
  return ans;
}
char str[MAXN][10];
int len(int a, int b) {
  int ans = 0;
  for (int i = 0; i < 7; i++) {
    if (str[a][i] != str[b][i]) {
      ans++;
    }
  }
  return ans;
}

int main() {
  int n;
  while (~scanf("%d", &n)) {
    if (n == 0) break;
    for (int i = 0; i < n; i++) {
      scanf("%s", str[i]);
    }
    for (int i = 0; i < n; i++) {
      for (int j = 0; j < n; j++) {
        if (i == j) continue;
        cost[i][j] = len(i, j);
      }
    }
    int ans = Prim(cost, n);
    printf("The highest possible quality is 1/%d.n", ans);
  }
  return 0;
}

// Kruskal 超时，用 Prim !

POJ-1679 The Unique MST

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// POJ-1679 The Unique MST
// https://vjudge.net/problem/POJ-1679

#include <string.h>
#include <algorithm>
#include <cstdio>
#include <cmath>
using namespace std;

const int MAXN = 110;
const int INF = 0x3f3f3f3f;
bool vis[MAXN];
int lowc[MAXN];
int pre[MAXN];
int Max[MAXN][MAXN];
// Max[i][j] 表示在最小生成树中从 i 到 j 的路径中的最大边权
bool used[MAXN][MAXN];
int Prim(int cost[][MAXN], int n) {
  int ans = 0;
  memset(vis, false, sizeof(vis));
  memset(Max, 0, sizeof(Max));
  memset(used, false, sizeof(used));
  vis[0] = true;
  pre[0] = -1;
  for (int i = 1; i < n; i++) {
    lowc[i] = cost[0][i];
    pre[i] = 0;
  }
  lowc[0] = 0;
  for (int i = 1; i < n; i++) {
    int minc = INF;
    int p = -1;

    for (int j = 0; j < n; j++)
      if (!vis[j] && minc > lowc[j]) {
        minc = lowc[j];
        p = j;
      }
    if (minc == INF) return -1;
    ans += minc;
    vis[p] = true;
    used[p][pre[p]] = used[pre[p]][p] = true;
    for (int j = 0; j < n; j++) {
      if (vis[j] && j != p)
        Max[j][p] = Max[p][j] = max(Max[j][pre[p]], lowc[p]);
      if (!vis[j] && lowc[j] > cost[p][j]) {
        lowc[j] = cost[p][j];
        pre[j] = p;
      }
    }
  }
  return ans;
}

int smst(int cost[][MAXN],int n,int ans)
{
    int Min=INF;
    for(int i=0;i<n;i++)
        for(int j=i+1;j<n;j++)
            if(cost[i][j]!=INF && !used[i][j])
            {
                Min=min(Min,ans+cost[i][j]-Max[i][j]);
            }
    if(Min==INF)return -1;//不存在
    return Min;
}

int costx[MAXN][MAXN];

int main() {
  int T;
  scanf("%d", &T);
  while (T--) {
    memset(costx, 0x3f, sizeof(costx));
    int n, m;
    scanf("%d%d", &n, &m);
    int sum = 0;
    for (int i = 0; i < n; i++) costx[i][i] = 0;
    for (int i = 0; i < m; i++) {
      int a, b, c;
      scanf("%d%d%d", &a, &b, &c);
      a--;
      b--;
      costx[a][b] = c;
      costx[b][a] = c;
    }
    int ans = Prim(costx, n);
    int ansMST = smst(costx, n,ans);

    if (ans == ansMST) {
      printf("Not Unique!n");
    } else {
      printf("%dn", ans);
    }
  }
  return 0;
}

// 次小生成树，不知道为什么最小生成树计数不行
// kuangbin 题解：https://www.cnblogs.com/kuangbin/p/3147329.html

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