【Clique Problem】

【Clique Problem】

题目描述

The clique problem is one of the most well-known NP-complete problems. Under some simplification it can be formulated as follows. Consider an undirected graph GG . It is required to find a subset of vertices CC of the maximum size such that any two of them are connected by an edge in graph GG . Sounds simple, doesn't it? Nobody yet knows an algorithm that finds a solution to this problem in polynomial time of the size of the graph. However, as with many other NP-complete problems, the clique problem is easier if you consider a specific type of a graph.

Consider nn distinct points on a line. Let the ii -th point have the coordinate x_{i}xi​ and weight w_{i}wi​ . Let's form graph GG , whose vertices are these points and edges connect exactly the pairs of points (i,j)(i,j) , such that the distance between them is not less than the sum of their weights, or more formally: |x_{i}-x_{j}|>=w_{i}+w_{j}∣xi​−xj​∣>=wi​+wj​ .

Find the size of the maximum clique in such graph.

输入输出格式

输入格式：

 

The first line contains the integer nn ( 1<=n<=2000001<=n<=200000 ) — the number of points.

Each of the next nn lines contains two numbers x_{i}xi​ , w_{i}wi​ ( 0<=x_{i}<=10^{9},1<=w_{i}<=10^{9}0<=xi​<=109,1<=wi​<=109 ) — the coordinate and the weight of a point. All x_{i}xi​ are different.

 

输出格式：

 

Print a single number — the number of vertexes in the maximum clique of the given graph.

 

输入输出样例

输入样例#1： 

4
2 3
3 1
6 1
0 2

输出样例#1：

3

说明

If you happen to know how to solve this problem without using the specific properties of the graph formulated in the problem statement, then you are able to get a prize of one million dollars!

The picture for the sample test.

 

简单翻译一下（语文较差，转载@Asougi_Kazuma的翻译）：

题目描述

      数轴上有n 个点，第i 个点的坐标为xi，权值为wi。两个点i,j之间存在一条边当且仅当 abs(xi-xj)>=wi+wj。 你需要求出这张图的最大团的点数。（团就是两两之间有边的顶点集合） 

输入数据

     第一行一个整数n，接下来n行每行两个整数 xi,wi。 

输出数据

     一行一个整数表示答案。 

样例输入

4 
2 3 
3 1 
6 1 
0 2 

样例输出

3

数据范围

       对于20%的数据，n<=10。 
       对于60%的数据，n<=1000。 
       对于100%的数据n<=200000，0<=|xi|,wi<=10^9。

那么这样粗粗一看，很简单啊，控制好左边的端点以及右边的端点，既然输入为x[i]与y[i]；
那么左端点就为x-y；右端点就为x+y；
快排sort从头到尾迅速排一遍，然后枚举暴力搜索勉强通过：

var
  n,i,x,y,R,ans:longint;
  pt:array[1..200000,1..2]of longint;//因为是二维数组，别开的过分大，也可以试试record
procedure sort(l,r:longint);//快排，不多说
var
  i,j,x,y: longint;
      begin
         i:=l;
         j:=r;
         x:=pt[(l+r) div 2,1];//注意，我们要以pt[i,1]作为标准来排序
         repeat
           while pt[i,1]<x do
            inc(i);
           while x<pt[j,1] do
            dec(j);
           if i<j then
             begin
                y:=pt[i,1];//一块换
                pt[i,1]:=pt[j,1];
                pt[j,1]:=y;
                y:=pt[i,2];
                pt[i,2]:=pt[j,2];
                pt[j,2]:=y;
                inc(i);
                j:=j-1;
             end;
         until i>j;
         if l<j then
           sort(l,j);
         if i<r then
           sort(i,r);
      end;
 begin
   readln(n);
   for i:=1 to n do
   begin
     readln(x,y);//重点
     pt[i,1]:=x-y;//左端点为x-w
     pt[i,2]:=x+y;//右端点为x+w
   end;
   sort(1,n);
   ans:=0;
   R:=-maxlongint;
   for i:=1 to n do
   begin
     if pt[i,1]>=R then//枚举+处理
     begin
       ans:=ans+1;
       R:=pt[i,2];
     end;
   end;
   writeln(ans);//输出
 end.

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