CodeForces 527D Clique Problem 树状数组

对于i∈S，有xi和wi。问S的最大子集其中每对i,j满足|xi-xj|>=wi+wj

首先可以干掉绝对值。然后变形得xi-wi>=wj+xj

所以对每个点建线段[xi-wi,xi+wi]。

即对左端点xi-wi，统计比xi-wi小的右端点组成的最大子集的大小，使用树状数组维护即可，查询到的数存入右端点。

问题转化为对于每条线段[xi-wi,xi+wi]，更新树状数组xi+wi，值为[1,xi-wi]最大数+1，则最后答案即为树状数组中的最大值。

#include <cstdio>
#include <map>
#include <algorithm>
using namespace std;
#define FOR(i,j,k) for(i=j;i<=k;i++)
#define N 200001
int c[N * 2], v[N * 2];
int n, sz = 0;
inline int lowbit(int x) { return x & -x; }
inline int sum(int x) {
    int s = 0;
    for (; x >= 1; x -= lowbit(x)) s = max(c[x], s);
    return s; 
}
inline void update(int i, int x) {
    for (; i <= sz; i += lowbit(i)) c[i] =  max(c[i], x);
}
struct Ask {
    int x, l, r;
    friend bool operator < (const Ask &a, const Ask &b) {
        return a.x < b.x;
    }
} q[N];
map<int, int> hash;
int main(){
    int w, i;
    scanf("%d",&n);
    FOR(i,1,n) {
        scanf("%d%d", &q[i].x, &w);
        q[i].l = q[i].x - w; q[i].r = q[i].x + w;
    }
    sort(q + 1, q + 1 + n);
    FOR(i,1,m) {
        if(!hash.count(q[i].l)) hash[q[i].l] = ++sz;
        if(!hash.count(q[i].r)) hash[q[i].r] = ++sz;
    }
    FOR(i,1,n) update(hash[q[i].r], sum(hash[q[i].l]) + 1);
    printf("%d\n", sum(sz)); 
    return 0;
}

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 G. It is required to find a subset of vertices C of the maximum size such that any two of them are connected by an edge in graph G. 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 n distinct points on a line. Let the i-th point have the coordinate xi and weight wi. Let's form graphG, whose vertices are these points and edges connect exactly the pairs of points (i, j), such that the distance between them is not less than the sum of their weights, or more formally: |xi - xj| ≥ wi + wj.

Find the size of the maximum clique in such graph.

Input

The first line contains the integer n (1 ≤ n ≤ 200 000) — the number of points.

Each of the next n lines contains two numbers xi, wi (0 ≤ xi ≤ 109, 1 ≤ wi ≤ 109) — the coordinate and the weight of a point. All xi are different.

Output

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

Sample test(s)

input

4
2 3
3 1
6 1
0 2

output

3

Note

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.