Codeforces Round #457 (Div. 2) C. Jamie and Interesting Graph(构造)

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描述

Jamie has recently found undirected weighted graphs with the following
properties very interesting:

The graph is connected and contains exactly n vertices and m edges.
All edge weights are integers and are in range [1, 109] inclusive. The
length of shortest path from 1 to n is a prime number. The sum of
edges’ weights in the minimum spanning tree (MST) of the graph is a
prime number. The graph contains no loops or multi-edges. If you are
not familiar with some terms from the statement you can find
definitions of them in notes section.

Help Jamie construct any graph with given number of vertices and edges
that is interesting!

Input

First line of input contains 2 integers n, m — the required number of
vertices and edges.

Output

In the first line output 2 integers sp, mstw (1 ≤ sp, mstw ≤ 1014) —
the length of the shortest path and the sum of edges’ weights in the
minimum spanning tree.

In the next m lines output the edges of the graph. In each line output
3 integers u, v, w (1 ≤ u, v ≤ n, 1 ≤ w ≤ 109) describing the edge
connecting u and v and having weight w.

input

4 4

output

input

5 4

output

Note

The graph of sample 1: Shortest path sequence: {1, 2, 3, 4}. MST edges
are marked with an asterisk (*).

Definition of terms used in the problem statement:

A shortest path in an undirected graph is a sequence of vertices
(v1, v2, … , vk) such that vi is adjacent to vi + 1 1 ≤ i < k and
the sum of weight is minimized where w(i, j) is the edge weight
between i and j. (https://en.wikipedia.org/wiki/Shortest_path_problem)

A prime number is a natural number greater than 1 that has no positive
divisors other than 1 and itself.
(https://en.wikipedia.org/wiki/Prime_number)

A minimum spanning tree (MST) is a subset of the edges of a connected,
edge-weighted undirected graph that connects all the vertices
together, without any cycles and with the minimum possible total edge
weight. (https://en.wikipedia.org/wiki/Minimum_spanning_tree)

https://en.wikipedia.org/wiki/Multiple_edges

思路

要求构造一个图，它的最短路的权值和最小生成树的权值和一样，不能有重边

我们只需要把他处理成一条链，连上n-2条边权为1的边，然后剩下的边用一个权值非常大的值连起来。

代码

#include <cstdio>
#include <cstring>
#include <cstdlib>
#include <cmath>
#include <string>
#include <iostream>
#include <stack>
#include <queue>
#include <vector>
#include <algorithm>
#define mem(a,b) memset(a,b,sizeof(a))
#define inf 0x3f3f3f3f
using namespace std;
const int N=3e5+9;
int prime[N];
bool is_prime[N];
int tot=0;
void get_prime()
{mem(is_prime,true);is_prime[0]=is_prime[1]=false;for(int i=2; i<N; i++){if(is_prime[i]){prime[tot++]=i;for(int j=2*i; j<=N; j+=i)is_prime[j]=false;}}
}
int main()
{get_prime();int n,m;scanf("%d%d",&n,&m);int num=prime[lower_bound(prime,prime+tot,n-1)-prime];int cha=num-n+1;int t=m-n+1;printf("%d %d\n",num,num);for(int i=1; i<=n-1; i++){if(i==1)printf("%d %d %d\n",i,i+1,cha+1);elseprintf("%d %d %d\n",i,i+1,1);}for(int i=1; i<=n-1; i++){for(int j=i+2; j<=n; j++){if(t==0) return 0;t--;printf("%d %d %d\n",i,j,10000000);}}return 0;
}