POJ 3241 Object Clustering（Manhattan MST）

题目链接：http://poj.org/problem?id=3241

Description

We have N (N ≤ 10000) objects, and wish to classify them into several groups by judgement of their resemblance. To simply the model, each object has 2 indexes a and b (a, b ≤ 500). The resemblance of object i and object j is defined by d_ij = |a_i - a_j| + |b_i - b_j|, and then we say i is d_ij resemble to j. Now we want to find the minimum value of X, so that we can classify the N objects into K (K < N) groups, and in each group, one object is at most X resemble to another object in the same group, i.e, for every object i, if i is not the only member of the group, then there exists one object j (i ≠ j) in the same group that satisfies d_ij ≤ X

Input

The first line contains two integers N and K. The following N lines each contain two integers a and b, which describe a object.

Output

A single line contains the minimum X.

题目大意：给n个点，两个点之间的距离为曼哈顿距离。要求把n个点分成k份，每份构成一个连通的子图（树），要求最大边最小。求最大边的最小值。

思路：实际上就是求平面上曼哈顿距离的最小生成树的第k大边（即减掉最大的k-1条边构成k份）。

资料：曼哈顿MST。复杂度O(nlogn)。

http://wenku.baidu.com/view/1e4878196bd97f192279e941.html

http://blog.csdn.net/huzecong/article/details/8576908

代码（79MS）：

 #include <cstdio>
 #include <iostream>
 #include <cstring>
 #include <algorithm>
 using namespace std;
 typedef long long LL;
 #define FOR(i, n) for(int i = 0; i < n; ++i)

 namespace Bilibili {
     const int MAXV = ;
     const int MAXE = MAXV * ;

     struct Edge {
         int u, v, cost;
         Edge(int u = , int v = , int cost = ):
             u(u), v(v), cost(cost) {}
         bool operator < (const Edge &rhs) const {
             return cost < rhs.cost;
         }
     };

     struct Point {
         int x, y, id;
         void read(int i) {
             id = i;
             scanf("%d%d", &x, &y);
         }
         bool operator < (const Point &rhs) const {
             if(x != rhs.x) return x < rhs.x;
             return y < rhs.y;
         }
     };

     Point p[MAXV];
     Edge edge[MAXE];
     int x_plus_y[MAXV], y_sub_x[MAXV];
     int n, k, ecnt;

     int hash[MAXV], hcnt;

     void get_y_sub_x() {
         for(int i = ; i < n; ++i) hash[i] = y_sub_x[i] = p[i].y - p[i].x;
         sort(hash, hash + n);
         hcnt = unique(hash, hash + n) - hash;
         for(int i = ; i < n; ++i) y_sub_x[i] = lower_bound(hash, hash + hcnt, y_sub_x[i]) - hash + ;
     }

     void get_x_plus_y() {
         for(int i = ; i < n; ++i) x_plus_y[i] = p[i].x + p[i].y;
     }

     int tree[MAXV];
     int lowbit(int x) {
         return x & -x;
     }

     void update_min(int &a, int b) {
         if(b == -) return ;
         if(a == - || x_plus_y[a] > x_plus_y[b])
             a = b;
     }

     void initBit() {
         memset(tree + , -, hcnt * sizeof(int));
     }

     void modify(int x, int val) {
         while(x) {
             update_min(tree[x], val);
             x -= lowbit(x);
         }
     }

     int query(int x) {
         int res = -;
         while(x <= hcnt) {
             update_min(res, tree[x]);
             x += lowbit(x);
         }
         return res;
     }

     void build_edge() {
         sort(p, p + n);
         get_x_plus_y();
         get_y_sub_x();
         initBit();
         for(int i = n - ; i >= ; --i) {
             int tmp = query(y_sub_x[i]);
             if(tmp != -) edge[ecnt++] = Edge(p[i].id, p[tmp].id, x_plus_y[tmp] - x_plus_y[i]);
             modify(y_sub_x[i], i);
         }
     }

     int fa[MAXV], ans[MAXV];

     int find_set(int x) {
         return fa[x] == x ? x : fa[x] = find_set(fa[x]);
     }

     int kruskal() {
         for(int i = ; i < n; ++i) fa[i] = i;
         sort(edge, edge + ecnt);
         int acnt = ;
         for(int i = ; i < ecnt; ++i) {
             int fu = find_set(edge[i].u), fv = find_set(edge[i].v);
             if(fu != fv) {
                 ans[acnt++] = edge[i].cost;
                 fa[fu] = fv;
             }
         }
         reverse(ans, ans + acnt);
         return ans[k - ];
     }

     void mymain() {
         scanf("%d%d", &n, &k);
         for(int i = ; i < n; ++i) p[i].read(i);

         build_edge();
         for(int i = ; i < n; ++i) swap(p[i].x, p[i].y);
         build_edge();
         for(int i = ; i < n; ++i) p[i].x = -p[i].x;
         build_edge();
         for(int i = ; i < n; ++i) swap(p[i].x, p[i].y);
         build_edge();

         printf("%d\n", kruskal());
     }
 }

 int main() {
     Bilibili::mymain();
 }