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POJ 1966 Cable TV Network 无向图的点连通度

热度:13   发布时间:2024-01-13 18:15:57.0

求无向图的点连通度,一般的方法就是转化为网络流来求解

构建网络流模型:

G为无向图:

(1)G图中的每个顶点V变成N网中的两个顶点V'V'',顶点V'V''有一条弧容量为1

(2)原图G中的每条边e=(U,V),在N网中有两条弧e'=(U'',V'),e''=(V'',U')与之对应,e'e''容量均为无穷;

(3)某点为源点,枚举汇点,求最大流。

其中源点的确立有一个条件,就是这个点不能和其他的所有点都相邻,如果都相邻,显然是无法求出最小割的。 所以在确立源点的时候不能直接随意选点。

转载了一段关于为什么只要枚举汇点:
假设点连通度为k,那么如果你枚举源汇,我们构图的时候是拆点的,所有最小割中最小的一定是k,对应k个点,割边为这k个点拆点后对应的边,那么,这个最小割把所有点点分成了两部分,S和T集合,S和T集合中的点都是不连通的,而S集合中的点都是连通的,T集合中的点也都是连通的,对于任意一个点i属于S,任意一个点j属于T,要使得他们不连通,在图中删除的点都为k,不可能更小了,否则最小割值比k小,而如果他们同时属于S集合或者T集合,使得他们不连通,要删除的点大于k,因为他们现在还连通,于是,如果我们指定一个源点,枚举汇点,如果这个源点与汇点同时在刚才那个最小割的S或T集合中,要使他们不连通,删除的点必然大于k,如果他们一个属于S,一个属于T,那么使他们不连通,要删除的点就是k个,所以,只要枚举汇点就行了



#include<iostream>
#include<algorithm>
#include<iomanip>
#include<cstring>
#include<string>
#include<cstdio>
#include<cmath>
#include<queue>
#include<map>
#include<set>
#define MAXN 2222
#define MAXM 222222
#define INF 1000000000
using namespace std;
struct node
{int ver;    // vertexint cap;    // capacityint flow;   // current flow in this arcint next, rev;
}edge[MAXM];
int dist[MAXN], numbs[MAXN], src, des, n;
int head[MAXN], e;
void add(int x, int y, int c)
{       //e记录边的总数edge[e].ver = y;edge[e].cap = c;edge[e].flow = 0;edge[e].rev = e + 1;        //反向边在edge中的下标位置edge[e].next = head[x];   //记录以x为起点的上一条边在edge中的下标位置head[x] = e++;           //以x为起点的边的位置//反向边edge[e].ver = x;edge[e].cap = 0;  //反向边的初始网络流为0edge[e].flow = 0;edge[e].rev = e - 1;edge[e].next = head[y];head[y] = e++;
}
void rev_BFS()
{int Q[MAXN], qhead = 0, qtail = 0;for(int i = 1; i <= n; ++i){dist[i] = MAXN;numbs[i] = 0;}Q[qtail++] = des;dist[des] = 0;numbs[0] = 1;while(qhead != qtail){int v = Q[qhead++];for(int i = head[v]; i != -1; i = edge[i].next){if(edge[edge[i].rev].cap == 0 || dist[edge[i].ver] < MAXN)continue;dist[edge[i].ver] = dist[v] + 1;++numbs[dist[edge[i].ver]];Q[qtail++] = edge[i].ver;}}
}
void init()
{e = 0;memset(head, -1, sizeof(head));
}
int maxflow()
{int u, totalflow = 0;int Curhead[MAXN], revpath[MAXN];for(int i = 1; i <= n; ++i)Curhead[i] = head[i];u = src;while(dist[src] < n){if(u == des)     // find an augmenting path{int augflow = INF;for(int i = src; i != des; i = edge[Curhead[i]].ver)augflow = min(augflow, edge[Curhead[i]].cap);for(int i = src; i != des; i = edge[Curhead[i]].ver){edge[Curhead[i]].cap -= augflow;edge[edge[Curhead[i]].rev].cap += augflow;edge[Curhead[i]].flow += augflow;edge[edge[Curhead[i]].rev].flow -= augflow;}totalflow += augflow;u = src;}int i;for(i = Curhead[u]; i != -1; i = edge[i].next)if(edge[i].cap > 0 && dist[u] == dist[edge[i].ver] + 1)break;if(i != -1)     // find an admissible arc, then Advance{Curhead[u] = i;revpath[edge[i].ver] = edge[i].rev;u = edge[i].ver;}else        // no admissible arc, then relabel this vertex{if(0 == (--numbs[dist[u]]))break;    // GAP cut, Important!Curhead[u] = head[u];int mindist = n;for(int j = head[u]; j != -1; j = edge[j].next)if(edge[j].cap > 0)mindist = min(mindist, dist[edge[j].ver]);dist[u] = mindist + 1;++numbs[dist[u]];if(u != src)u = edge[revpath[u]].ver;    // Backtrack}}return totalflow;
}
int main()
{int m, u, v, w;int x[1555], y[1555];int mp[55][55];while(scanf("%d%d", &n, &m) != EOF){memset(mp, 0, sizeof(mp));src = n + 1;for(int i = 1; i <= m; i++){scanf(" (%d, %d)", &u, &v);u++; v++;x[i] = u;y[i] = v;mp[u][v] = mp[v][u] = 1;}for(int i = 1; i <= n; i++){bool flag = false;for(int j = i + 1; j <= n; j++){if(mp[i][j] == 0){flag = true;break;}}if(flag){src = n + i;break;}}int N = n;n = 2 * n;int ans = INF;for(int j = 1; j <= N; j++){if(src - N == j) continue;des = j;init();for(int i = 1; i <= N; i++){u = i;v = i + N;w = 1;add(u, v, w);}for(int i = 1; i <= m; i++){u = x[i];v = y[i];w = INF;add(u + N, v, w);add(v + N, u, w);}rev_BFS();ans = min(ans, maxflow());}if(ans >= INF) ans = N;printf("%d\n", ans);}return 0;
}


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