Jamie and Interesting Graph

Problem

　　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.

Examples

　　input

　　　　4 4

　　output

　　　　7 7
　　　　1 2 3
　　　　2 3 2
　　　　3 4 2
　　　　2 4 4

　　input

　　　　5 4

　　output

　　　　7 13
　　　　1 2 2
　　　　1 3 4
　　　　1 4 3
　　　　4 5 4

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, ... , v_k) such that vi is adjacent to v_i + 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

 

 

 

题目大意

　　构造一个n点m边的带权简单连通图G，要求点1到点n的最短路径与G的最小生成树权值都为质数。其中n范围为[2, 10⁵]，边权范围为[1, 10⁹]。

 

算法

　　特判n == 2的情况，否则使最短路径和最小生成树权值为2和100003。

　　点1与点n连权值为2的边。

　　点2与点n连权值为100004 – n的边。

　　点3到点n – 1与点n连权值为1的边。

　　在不重边不自环的前提下，将剩下的边数用完，权值相对设得大些即可，

　　时间复杂度：

　　　　O(m)

　　空间复杂度：

　　　　O(1)

 

代码

 1 def doit(n, m):
 2     if (n == 2):
 3         print(2, 2)
 4         print(1, 2, 2)
 5         return
 6     sp = 2
 7     mstw = 100003
 8     print(sp, mstw)
 9     print(1, n, sp)
10     print(2, n, mstw - n + 3 - sp)
11     for i in range(3, n):
12         print(i, n, 1)
13     for i in range(2, n):
14         for j in range(1, i):
15             if (m == n - 1):
16                 return
17             print(i, j, mstw)
18             m -= 1
19 
20 n, m = input().split()
21 doit(int(n), int(m))