Codeforces Round #306 (Div. 2) D

D. Regular Bridge

time limit per test

2 seconds

memory limit per test

256 megabytes

input

standard input

output

standard output

An undirected graph is called k-regular, if the degrees of all its vertices are equal k. An edge of a connected graph is called a bridge, if after removing it the graph is being split into two connected components.

Build a connected undirected k-regular graph containing at least one bridge, or else state that such graph doesn't exist.

Input

The single line of the input contains integer k (1 ≤ k ≤ 100) — the required degree of the vertices of the regular graph.

Output

Print "NO" (without quotes), if such graph doesn't exist.

Otherwise, print "YES" in the first line and the description of any suitable graph in the next lines.

The description of the made graph must start with numbers n and m — the number of vertices and edges respectively.

Each of the next m lines must contain two integers, a and b (1 ≤ a, b ≤ n, a ≠ b), that mean that there is an edge connecting the vertices a and b. A graph shouldn't contain multiple edges and edges that lead from a vertex to itself. A graph must be connected, the degrees of all vertices of the graph must be equal k. At least one edge of the graph must be a bridge. You can print the edges of the graph in any order. You can print the ends of each edge in any order.

The constructed graph must contain at most 106 vertices and 106 edges (it is guaranteed that if at least one graph that meets the requirements exists, then there also exists the graph with at most 106 vertices and at most 106 edges).

Sample test(s)

input

1

output

YES
2 1
1 2

Note

 

 比赛时竟然在C题卡了一会，操，C题就一个裸的DP，我竟然2B地往字符串去想。然后这D题很容易就想到只构造一条桥，然后点数就是（K+2）*2了，然而我连边时竟然连错了，SHIT，四题就这样溜了。

 

看过这图之后都会知道怎么做的，两边对称，中间一点是空的。偶数度是无解。

#include <iostream>
#include <cstdio>
#include <cstring>
#include <vector>
#include <utility>
using namespace std;

struct edge{
	int u,v;
	edge(){}
	edge(int uu,int vv){u=uu,v=vv;}
};

vector<edge>ans;


int main(){
	int k;
	while(scanf("%d",&k)!=EOF){
		if(k%2==0){
			puts("NO");
		}
		else{
			if(k==1){
				puts("YES");
				printf("%d %d
",2,1);
				printf("%d %d
",1,2);
				continue;
			}
			ans.clear();
			k+=2;
			int g=k/2;
			for(int i=0;i<k-1;i++){
				for(int j=1;j<g;j++){
					ans.push_back(edge(i,(i+j)%(k-1)));
					ans.push_back(edge(i+k,(i+j)%(k-1)+k));
				}
			}
			for(int i=1;i<k-1;i++){
				ans.push_back(edge(i,k-1));
				ans.push_back(edge(i+k,(k-1)+k));
			}
			ans.push_back(edge(0,k));
			puts("YES");
			int sz=ans.size();
			printf("%d %d
",k*2,sz);
			for(int i=0;i<sz;i++){
				printf("%d %d
",ans[i].u+1,ans[i].v+1);
			}
			
		}
	}
	return 0;
}