题目描述 In a far away kingdom lives a very greedy king. To defend his land, he built nn guard towers. Apart from the towers the kingdom has two armies, each headed by a tyrannical and narcissistic general. The generals can't stand each other, specifically, they will never let soldiers of two armies be present in one tower. During defence operations to manage a guard tower a general has to send part of his army to that tower. Each general asks some fee from the king for managing towers. As they live in a really far away kingdom, each general evaluates his fee in the following weird manner: he finds two remotest (the most distant) towers, where the soldiers of his army are situated and asks for the fee equal to the distance. Each tower is represented by a point on the plane with coordinates (x,y)(x,y) , and the distance between two points with coordinates (x_{1},y_{1})(x 1 ,y 1 ) and (x_{2},y_{2})(x 2 ,y 2 ) is determined in this kingdom as |x_{1}-x_{2}|+|y_{1}-y_{2}|∣x 1 −x 2 ∣+∣y 1 −y 2 ∣ . The greedy king was not exactly satisfied with such a requirement from the generals, that's why he only agreed to pay one fee for two generals, equal to the maximum of two demanded fees. However, the king is still green with greed, and among all the ways to arrange towers between armies, he wants to find the cheapest one. Each tower should be occupied by soldiers of exactly one army. He hired you for that. You should find the minimum amount of money that will be enough to pay the fees. And as the king is also very scrupulous, you should also count the number of arrangements that will cost the same amount of money. As their number can be quite large, it is enough for the king to know it as a remainder from dividing by 10^{9}+710 9 +7 . Two arrangements are distinct if the sets of towers occupied by soldiers of the first general are distinct. 输入格式 The first line contains an integer nn ( 2<=n<=50002<=n<=5000 ), nn is the number of guard towers. Then follow nn lines, each of which contains two integers x,yx,y — the coordinates of the ii -th tower (0<=x,y<=5000)(0<=x,y<=5000) . No two towers are present at one point. Pretest 6 is one of the maximal tests for this problem. 输出格式 Print on the first line the smallest possible amount of money that will be enough to pay fees to the generals. Print on the second line the number of arrangements that can be carried out using the smallest possible fee. This number should be calculated modulo 10000000071000000007 ( 10^{9}+710 9 +7 ). 题意翻译 题目大意 已知n座塔的坐标，n<=5000 把他们分成两组，使得同组内的两座塔的曼哈顿距离的最大值最小 在此前提下求出有多少种分组方案 mod10^9+7 输入格式 第一行一个n 后面n行为每个点的坐标 输出格式 第一行为最小花费 第二行为方案数 感谢@moye到碗里来 提供的翻译 输入输出样例 输入 #1复制 2 0 0 1 1 输出 #1复制 0 2 输入 #2复制 4 0 0 0 1 1 0 1 1 输出 #2复制 1 4 输入 #3复制 3 0 0 1000 1000 5000 5000 输出 #3复制 2000 2 说明/提示 In the first example there are only two towers, the distance between which is equal to 2. If we give both towers to one general, then we well have to pay 2 units of money. If each general receives a tower to manage, to fee will be equal to 0. That is the smallest possible fee. As you can easily see, we can obtain it in two ways.
法1：
比雪夫距离： (x,y)=(x+y,x-y)(x,y)=(x+y,x−y)，
曼哈顿距离：|x_1-x_2|+|y_1-y_2|∣x1−x2∣+∣y1−y2∣
切比雪夫距离：\max\{|x_1-x_2|,|y_1-y_2|\}max{∣x1−x2∣,∣y1−y2∣}。
- 用大矩形先把所有点包裹起来
- 在用2个正方形（分2种情况即2个对角线）把这些点给包裹起来。（扫描点，看那个点离角落比较近，就去那个正方体。）
- 重叠的有那几个点就有2^N次方个解决方案。（有些情况需要特判）
法2：
利用二分图+二分答案：
- 在超过了距离就连接条边
- 染色判断是否为二分图
- 有多少个连通块就有2^N次方个解决方案。