poj_2112 网络最大流+二分法

题目大意

    有K台挤奶机和C头奶牛，都被视为物体，这K+C个物体之间存在路径。给出一个 (K+C)x(K+C) 的矩阵A，A[i][j]表示物体i和物体j之间的距离，有些物体之间可能没有直接通路。 
    每台挤奶机可以容纳m头奶牛去挤奶，且每个奶牛仅可以去往一台挤奶机。现在安排这C头奶牛去挤奶，每头奶牛会去往某个挤奶机，求出这C头奶牛去其对应挤奶机的路径长度的最大值的最小值。

题目分析

    “每头奶牛仅可以去往一台挤奶机，每台挤奶机最多有M头奶牛”这似乎是一个路径流量的问题，考虑使用网络流算法来解决。 
    那么，如何确定最长路径的最小值呢？可以先考虑简化问题，“给定一个最大距离L，能否分配这C头奶牛的挤奶机，使得每头奶牛到达挤奶机的距离都小于L”。 
    解决简化问题：虚拟一个源点和汇点。从源点引出C条容量分别为1的路径到达C头奶牛，再将K台机器分别引出一条容量为M的路径到达汇点。则问题转化为，构造C头奶牛到K台机器的网络路径，且从每头奶牛去往挤奶机的路径的容量为1，距离不超过L，使得网络从源点到汇点的最大流量为C，且C头奶牛走的路径的最大值最小。 
    然后，再通过二分法枚举最大距离L，找到最大距离的最小值。 
    具体实现的时候，使用Floyd算法求出奶牛到达挤奶机的最短路径长度；使用ISAP算法找出最大流；使用二分法确定最长的路径最小值。

实现(c++)

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#include<stdio.h> #include<string.h> #include<queue> #include<vector> #include<algorithm> using namespace std; #define MAX_NODE 235 #define MAX_EDGE_NUM 50000 #define INFINITE 1 << 20 #define min(a, b) a < b?a:b struct Edge{     int from;     int to;     int w;     int next;     int rev;     bool operator==(const pair<int, int>& p){         return from == p.first && to == p.second;     } };   Edge gEdges[MAX_EDGE_NUM]; int gFlow[MAX_NODE][MAX_NODE]; int gHead[MAX_NODE]; int gDist[MAX_NODE]; int gGap[MAX_NODE]; int gPre[MAX_NODE]; int gPath[MAX_NODE];   int gEdgeCount; int gSource, gDestination; void InsertEdge(int u, int v, int w){     Edge* it = find(gEdges, gEdges + gEdgeCount, pair<int, int>(u, v));     if (it != gEdges + gEdgeCount){         it->w = w;     }     else{         int e1 = gEdgeCount++;         gEdges[e1].from = u;         gEdges[e1].to = v;         gEdges[e1].w = w;         gEdges[e1].next = gHead[u];         gHead[u] = e1;           int e2 = gEdgeCount++;         gEdges[e2].from = v;         gEdges[e2].to = u;         gEdges[e2].w = 0;         gEdges[e2].next = gHead[v];         gHead[v] = e2;           gEdges[e1].rev = e2;         gEdges[e2].rev = e1;     }     gFlow[u][v] = w; }   void Bfs(){     memset(gGap, 0, sizeof(gGap));     memset(gDist, -1, sizeof(gDist));     gDist[gDestination] = 0;     gGap[0] = 1;     queue<int>Q;     Q.push(gDestination);     while (!Q.empty()){         int u = Q.front();         Q.pop();         for (int e = gHead[u]; e != -1; e = gEdges[e].next){             int v = gEdges[e].to;             if (gDist[v] >= 0)                 continue;             gDist[v] = gDist[u] + 1;             gGap[gDist[v]] ++;             Q.push(v);         }     } }   int ISAP(int n){     int e, d, u = gSource;     int ans = 0;     Bfs();     while (gDist[gSource] <= n){         if (u == gDestination){             int min_flow = INFINITE;             for (e = gPath[u]; u != gSource; e = gPath[u = gPre[u]]){                 min_flow = min(min_flow, gEdges[e].w);             }             u = gDestination;             for (e = gPath[u]; u != gSource; e = gPath[u = gPre[u]]){                 gEdges[e].w -= min_flow;                 gEdges[gEdges[e].rev].w += min_flow;                   gFlow[gPre[u]][u] -= min_flow;                 gFlow[u][gPre[u]] += min_flow;             }             ans += min_flow;         }         for (e = gHead[u]; e != -1; e = gEdges[e].next){             if (gEdges[e].w > 0 && gDist[u] == gDist[gEdges[e].to] + 1)                 break;         }         if (e >= 0){             gPre[gEdges[e].to] = u;             gPath[gEdges[e].to] = e;             u = gEdges[e].to;         }         else{             if (--gGap[gDist[u]] == 0)                 break;             d = n;             for (int e = gHead[u]; e != -1; e = gEdges[e].next)                 if (gEdges[e].w > 0)                     d = min(d, gDist[gEdges[e].to]);             gDist[u] = d + 1;             ++gGap[gDist[u]];             if (u != gSource)                 u = gPre[u];         }     }     return ans; }   int gMinDist[MAX_NODE][MAX_NODE]; void Floyd(int n){     for (int k = 1; k <= n; k++){         for (int i = 1; i <= n; i++){             for (int j = 1; j <= n; j++){                 if (gMinDist[i][j] > gMinDist[i][k] + gMinDist[k][j]){                     gMinDist[i][j] = gMinDist[i][k] + gMinDist[k][j];                 }             }         }     } }   void BuildGraph(int k, int c, int m, int max_dist){     memset(gHead, -1, sizeof(gHead));     gEdgeCount = 0;     gSource = 0;     gDestination = k + c + 1;           for (int i = k + 1; i <= k + c; i++){         InsertEdge(gSource, i, 1);     }     for (int i = k + 1; i <= k + c; i++){         for (int j = 1; j <= k; j++){             if (gMinDist[i][j] <= max_dist)                 InsertEdge(i, j, 1);         }           }       for (int i = 1; i <= k; i++){         InsertEdge(i, gDestination, m);     } } void print_graph(int n){     for (int u = 0; u <= n; u++){         printf("node %d links to ", u);         for (int e = gHead[u]; e != -1; e = gEdges[e].next)             printf("%d(flow = %d) ", gEdges[e].to, gEdges[e].w);         printf("\n");     } } int main(){     int k, c, m, d;     while (scanf("%d %d %d", &k, &c, &m) != EOF){         for (int i = 1; i <= k + c; i++){             for (int j = 1; j <= k + c; j++){                 scanf("%d", &gMinDist[i][j]);                 if (i != j && gMinDist[i][j] == 0)                     gMinDist[i][j] = INFINITE;             }         }           Floyd(k + c);         int  beg = 1, end = 40010;         while (beg < end){             int max_dist = (beg + end) / 2;             BuildGraph(k, c, m, max_dist);             //  print_graph(k + c + 1);             int max_flow = ISAP(k + c + 2);             if (max_flow == c)                 end = max_dist;             else                 beg = max_dist + 1;         }         printf("%d\n", end);     }     return 0; }