[最小生成树] Codeforces 632F Educational Codeforces Round 9 F. Magic Matrix & SRM 687 div1 AllGraphCuts

Magic Matrix

把a矩阵当成邻接矩阵，设b为两点间路径最大值的最小值，那么ai,j≥bi,j，然后ai,j≤max(ai,k1,ak1,k2,⋯,akm,j)，所以ai,j≤bi,j
那么a=b，求b只要取最小生成树即可

#include<cstdio>#include<cstdlib>#include<algorithm>using namespace std;inline char nc(){  static char buf[100000],*p1=buf,*p2=buf;  return p1==p2&&(p2=(p1=buf)+fread(buf,1,100000,stdin),p1==p2)?EOF:*p1++;}inline void read(int &x){  char c=nc(),b=1;  for (;!(c>='0' && c<='9');c=nc()) if (c=='-') b=-1;  for (x=0;c>='0' && c<='9';x=x*10+c-'0',c=nc()); x*=b;}const int N=2505;struct edge{  int u,v,w,next;}G[N<<1];int head[N],inum;inline void add(int u,int v,int w){  int p=++inum; G[p].u=u; G[p].v=v; G[p].w=w; G[p].next=head[u]; head[u]=p;}#define V G[p].vint n;int w[N][N];int dis[N],pre[N],vst[N];inline void Prim(){  dis[1]=0; for (int i=2;i<=n;i++) dis[i]=1<<30;  for (int i=1;i<=n;i++){    int k=0;    for (int j=1;j<=n;j++)      if (!vst[j])    if (!k || dis[j]<dis[k])      k=j;    vst[k]=1; if (pre[k]) add(pre[k],k,w[pre[k]][k]),add(k,pre[k],w[k][pre[k]]);    for (int j=1;j<=n;j++)      if (!vst[j] && dis[j]>w[k][j])    dis[j]=w[k][j],pre[j]=k;  }}int b[N][N];inline void dfs(int u,int fa,int z){  for (int p=head[u];p;p=G[p].next)    if (V!=fa)      b[z][V]=max(b[z][u],G[p].w),dfs(V,u,z);}int main(){  freopen("t.in","r",stdin);  freopen("t.out","w",stdout);  read(n); for (int i=1;i<=n;i++) for (int j=1;j<=n;j++) read(w[i][j]);  for (int i=1;i<=n;i++) if (w[i][i]) return printf("NOT MAGIC\n"),0;  for (int i=1;i<=n;i++) for (int j=1;j<=n;j++) if (w[i][j]!=w[j][i]) return printf("NOT MAGIC\n"),0;  Prim();  for (int i=1;i<=n;i++) dfs(i,0,i);  for (int i=1;i<=n;i++) for (int j=1;j<=n;j++) if (w[i][j]!=b[i][j]) return printf("NOT MAGIC\n"),0;  printf("MAGIC\n");  return 0;}

PS.还可以压位过，O(n3W)

AllGraphCuts

全局最小割，联想到最小割树，ai,j就是最小割树上的最小边，同理可以转化成一颗最大生成树

// BEGIN CUT HERE  #include<conio.h>#include<sstream>// END CUT HERE  #include<cstdio>#include<cstdlib>#include<cstring>#include<cmath>#include<iostream>#include<algorithm>#include<vector>#include<map>#include<string>#include<set>#define pb push_back#define cl(x) memset(x,0,sizeof(x))using namespace std;typedef long long ll;const int N=2505;struct edge{  int u,v,w,next;}G[N<<1];int head[N],inum;inline void add(int u,int v,int w){  int p=++inum; G[p].u=u; G[p].v=v; G[p].w=w; G[p].next=head[u]; head[u]=p;}#define V G[p].vint n;int w[N][N];int dis[N],pre[N],vst[N];inline void Prim(){  dis[1]=0; for (int i=2;i<=n;i++) dis[i]=-1<<30;  for (int i=1;i<=n;i++){    int k=0;    for (int j=1;j<=n;j++)      if (!vst[j])    if (!k || dis[j]>dis[k])      k=j;    vst[k]=1; if (pre[k]) add(pre[k],k,w[pre[k]][k]),add(k,pre[k],w[k][pre[k]]);    for (int j=1;j<=n;j++)      if (!vst[j] && dis[j]<w[k][j])    dis[j]=w[k][j],pre[j]=k;  }}int b[N][N];inline void dfs(int u,int fa,int z){  for (int p=head[u];p;p=G[p].next)    if (V!=fa)      b[z][V]=min(b[z][u],G[p].w),dfs(V,u,z);}class AllGraphCuts{public:  vector <int> findGraph(vector <int> x){    n=sqrt(x.size());    for (int i=1;i<=n;i++) for (int j=1;j<=n;j++) w[i][j]=x[(i-1)*n+(j-1)];    for (int i=1;i<=n;i++) if (w[i][i]) return {-1};    for (int i=1;i<=n;i++) for (int j=1;j<=n;j++) if (w[i][j]!=w[j][i]) return {-1};    Prim();    for (int i=1;i<=n;i++) b[i][i]=1<<30,dfs(i,0,i),b[i][i]=0;    for (int i=1;i<=n;i++) for (int j=1;j<=n;j++) if (w[i][j]!=b[i][j]) return {-1};    vector<int> ans;    for (int p=1;p<=inum;p+=2)      ans.pb(G[p].w*n*n+(G[p].u-1)*n+(G[p].v-1));    return ans;  }  // BEGIN CUT HEREpublic:  void run_test(int Case) { if ((Case == -1) || (Case == 0)) test_case_0(); if ((Case == -1) || (Case == 1)) test_case_1(); if ((Case == -1) || (Case == 2)) test_case_2(); if ((Case == -1) || (Case == 3)) test_case_3(); if ((Case == -1) || (Case == 4)) test_case_4(); if ((Case == -1) || (Case == 5)) test_case_5(); }private:  template <typename T> string print_array(const vector<T> &_V) { ostringstream os; os << "{ "; for (typename vector<T>::const_iterator iter = _V.begin(); iter != _V.end(); ++iter) os << '\"' << *iter << "\","; os << " }"; return os.str(); }  void verify_case(int Case, const vector <int> &Expected, const vector <int> &Received) { cerr << "Test Case #" << Case << "..."; if (Expected == Received) cerr << "PASSED" << endl; else { cerr << "FAILED" << endl; cerr << "\tExpected: " << print_array(Expected) << endl; cerr << "\tReceived: " << print_array(Received) << endl; } }  void test_case_0() { int Arr0[] = {0,1,                     1,0}; vector <int> Arg0(Arr0, Arr0 + (sizeof(Arr0) / sizeof(Arr0[0]))); int Arr1[] = {6 }; vector <int> Arg1(Arr1, Arr1 + (sizeof(Arr1) / sizeof(Arr1[0]))); verify_case(0, Arg1, findGraph(Arg0)); }  void test_case_1() { int Arr0[] = {0,1,                     1,1}; vector <int> Arg0(Arr0, Arr0 + (sizeof(Arr0) / sizeof(Arr0[0]))); int Arr1[] = {-1 }; vector <int> Arg1(Arr1, Arr1 + (sizeof(Arr1) / sizeof(Arr1[0]))); verify_case(1, Arg1, findGraph(Arg0)); }  void test_case_2() { int Arr0[] = {0,2,2,                     2,0,2,                     2,2,0}; vector <int> Arg0(Arr0, Arr0 + (sizeof(Arr0) / sizeof(Arr0[0]))); int Arr1[] = {10, 11, 14 }; vector <int> Arg1(Arr1, Arr1 + (sizeof(Arr1) / sizeof(Arr1[0]))); verify_case(2, Arg1, findGraph(Arg0)); }  void test_case_3() { int Arr0[] = {0,1,2,3,4,                     1,0,2,3,4,                     1,2,0,3,4,                     1,2,3,0,4,                     1,2,3,4,0}; vector <int> Arg0(Arr0, Arr0 + (sizeof(Arr0) / sizeof(Arr0[0]))); int Arr1[] = {-1 }; vector <int> Arg1(Arr1, Arr1 + (sizeof(Arr1) / sizeof(Arr1[0]))); verify_case(3, Arg1, findGraph(Arg0)); }  void test_case_4() { int Arr0[] = {0,0,0,0}; vector <int> Arg0(Arr0, Arr0 + (sizeof(Arr0) / sizeof(Arr0[0]))); int Arr1[] = {2 }; vector <int> Arg1(Arr1, Arr1 + (sizeof(Arr1) / sizeof(Arr1[0]))); verify_case(4, Arg1, findGraph(Arg0)); }  void test_case_5() { int Arr0[] = {      0, 2545, 2348, 2993, 2606, 2623, 2013, 3001, 2626, 2440, 2495, 2262, 2342, 2617, 2773, 2662, 2736, 2827, 3044, 3082,      2545, 0, 2348, 2545, 2545, 2545, 2013, 2545, 2545, 2440, 2495, 2262, 2342, 2545, 2545, 2545, 2545, 2545, 2545, 2545,      2348, 2348, 0, 2348, 2348, 2348, 2013, 2348, 2348, 2348, 2348, 2262, 2342, 2348, 2348, 2348, 2348, 2348, 2348, 2348,      2993, 2545, 2348, 0, 2606, 2623, 2013, 2993, 2626, 2440, 2495, 2262, 2342, 2617, 2773, 2662, 2736, 2827, 2993, 2993,      2606, 2545, 2348, 2606, 0, 2606, 2013, 2606, 2606, 2440, 2495, 2262, 2342, 2606, 2606, 2606, 2606, 2606, 2606, 2606,      2623, 2545, 2348, 2623, 2606, 0, 2013, 2623, 2623, 2440, 2495, 2262, 2342, 2617, 2623, 2623, 2623, 2623, 2623, 2623,      2013, 2013, 2013, 2013, 2013, 2013, 0, 2013, 2013, 2013, 2013, 2013, 2013, 2013, 2013, 2013, 2013, 2013, 2013, 2013,      3001, 2545, 2348, 2993, 2606, 2623, 2013, 0, 2626, 2440, 2495, 2262, 2342, 2617, 2773, 2662, 2736, 2827, 3001, 3001,      2626, 2545, 2348, 2626, 2606, 2623, 2013, 2626, 0, 2440, 2495, 2262, 2342, 2617, 2626, 2626, 2626, 2626, 2626, 2626,      2440, 2440, 2348, 2440, 2440, 2440, 2013, 2440, 2440, 0, 2440, 2262, 2342, 2440, 2440, 2440, 2440, 2440, 2440, 2440,      2495, 2495, 2348, 2495, 2495, 2495, 2013, 2495, 2495, 2440, 0, 2262, 2342, 2495, 2495, 2495, 2495, 2495, 2495, 2495,      2262, 2262, 2262, 2262, 2262, 2262, 2013, 2262, 2262, 2262, 2262, 0, 2262, 2262, 2262, 2262, 2262, 2262, 2262, 2262,      2342, 2342, 2342, 2342, 2342, 2342, 2013, 2342, 2342, 2342, 2342, 2262, 0, 2342, 2342, 2342, 2342, 2342, 2342, 2342,      2617, 2545, 2348, 2617, 2606, 2617, 2013, 2617, 2617, 2440, 2495, 2262, 2342, 0, 2617, 2617, 2617, 2617, 2617, 2617,      2773, 2545, 2348, 2773, 2606, 2623, 2013, 2773, 2626, 2440, 2495, 2262, 2342, 2617, 0, 2662, 2736, 2773, 2773, 2773,      2662, 2545, 2348, 2662, 2606, 2623, 2013, 2662, 2626, 2440, 2495, 2262, 2342, 2617, 2662, 0, 2662, 2662, 2662, 2662,      2736, 2545, 2348, 2736, 2606, 2623, 2013, 2736, 2626, 2440, 2495, 2262, 2342, 2617, 2736, 2662, 0, 2736, 2736, 2736,      2827, 2545, 2348, 2827, 2606, 2623, 2013, 2827, 2626, 2440, 2495, 2262, 2342, 2617, 2773, 2662, 2736, 0, 2827, 2827,      3044, 2545, 2348, 2993, 2606, 2623, 2013, 3001, 2626, 2440, 2495, 2262, 2342, 2617, 2773, 2662, 2736, 2827, 0, 3044,      3082, 2545, 2348, 2993, 2606, 2623, 2013, 3001, 2626, 2440, 2495, 2262, 2342, 2617, 2773, 2662, 2736, 2827, 3044, 0    }    ; vector <int> Arg0(Arr0, Arr0 + (sizeof(Arr0) / sizeof(Arr0[0]))); int Arr1[] = {76801, 87602, 76403, 15604, 14005, 14406, 95607, 14408, 45609, 54010, 113211, 102812, 53613, 72414, 10015, 90416, 110417, 108818, 76819, 20022, 5223, 16824, 14025, 20426, 37227, 115628, 118829, 105230, 39631, 114032, 70833, 2434, 9235, 71636, 3637, 108438, 68439, 16443, 24444, 37245, 54046, 118447, 46048, 13249, 38450, 27651, 23652, 105253, 76054, 22455, 112056, 5657, 82458, 28859, 31264, 117665, 73666, 91667, 11268, 67269, 110870, 18471, 114872, 119673, 22074, 86075, 44876, 22477, 74478, 93679, 96085, 8486, 59687, 110088, 1689, 107290, 56091, 56092, 74893, 61294, 109295, 109296, 18097, 66498, 20899, 12506, 98907, 82908, 77309, 51710, 110111, 108112, 22513, 66514, 43315, 36516, 19717, 15318, 26519, 34127, 70128, 8929, 20930, 131, 81732, 80133, 29334, 68935, 53336, 81737, 36538, 57739, 35348, 96149, 25750, 19351, 15752, 4153, 80554, 107355, 96556, 70557, 66958, 48559, 33769, 81370, 55771, 15372, 61373, 72574, 11375, 2976, 66577, 108178, 57779, 590, 78991, 14992, 10593, 69794, 47395, 33796, 113397, 77398, 68999, 77811, 28212, 12613, 64614, 80215, 9816, 27017, 30618, 73819, 46632, 66633, 7834, 100635, 636, 7437, 4238, 76639, 87453, 16254, 10255, 31856, 50257, 18258, 3459, 73474, 69475, 63876, 41077, 3878, 28679, 13495, 56696, 109497, 101098, 116699, 60316, 107517, 12318, 98719, 51137, 91138, 81139, 108358, 119959, 106379 }; vector <int> Arg1(Arr1, Arr1 + (sizeof(Arr1) / sizeof(Arr1[0]))); verify_case(5, Arg1, findGraph(Arg0)); }  // END CUT HERE};// BEGIN CUT HEREint main(){  AllGraphCuts ___test;  ___test.run_test(5);  getch() ;  return 0;}// END CUT HERE