D. Relatively Prime Graph

Codeforces

IDCF1009D

Time2000ms

Memory256MB

Difficulty—

brute forceconstructive algorithmsgraphsgreedymath

English · Original

Chinese · Translation

Formal · Original

Let's call an undirected graph $G = (V, E)$ _relatively prime_ if and only if for each edge $(v, u) \in E$ $GCD(v, u) = 1$ (the greatest common divisor of $v$ and $u$ is $1$). If there is no edge between some pair of vertices $v$ and $u$ then the value of $GCD(v, u)$ doesn't matter. The vertices are numbered from $1$ to $|V|$. Construct a _relatively prime_ graph with $n$ vertices and $m$ edges such that it is connected and it contains neither self-loops nor multiple edges. If there exists no valid graph with the given number of vertices and edges then output _"Impossible"_. If there are multiple answers then print any of them. ## Input The only line contains two integers $n$ and $m$ ($1 \le n, m \le 10^5$) — the number of vertices and the number of edges. ## Output If there exists no valid graph with the given number of vertices and edges then output _"Impossible"_. Otherwise print the answer in the following format: The first line should contain the word _"Possible"_. The $i$\-th of the next $m$ lines should contain the $i$\-th edge $(v_i, u_i)$ of the resulting graph ($1 \le v_i, u_i \le n, v_i \neq u_i$). For each pair $(v, u)$ there can be no more pairs $(v, u)$ or $(u, v)$. The vertices are numbered from $1$ to $n$. If there are multiple answers then print any of them. [samples] ## Note Here is the representation of the graph from the first example: ![image](https://espresso.codeforces.com/501a835f442401d928aaaee2ff94c95a4900a18e.png)

[{"iden":"statement","content":"我们称一个无向图 $G = (V, E)$ 为 _互质图_，当且仅当对于每条边 $(v, u) \\in E$，都有 $\\gcd(v, u) = 1$（即 $v$ 和 $u$ 的最大公约数为 1）。如果某对顶点 $v$ 和 $u$ 之间没有边，则 $\\gcd(v, u)$ 的值无关紧要。顶点编号从 $1$ 到 $|V|$。\n\n请构造一个包含 $n$ 个顶点和 $m$ 条边的 _互质图_，要求该图连通，且不包含自环或重边。\n\n如果不存在满足条件的图，则输出 _\"Impossible\"_。\n\n如果存在多个答案，输出任意一个即可。\n\n输入仅一行，包含两个整数 $n$ 和 $m$（$1 \\leq n, m \\leq 10^5$）—— 分别表示顶点数和边数。\n\n如果不存在满足条件的图，则输出 _\"Impossible\"_。\n\n否则，请按以下格式输出答案：\n\n第一行输出单词 _\"Possible\"_。\n\n接下来的 $m$ 行中，第 $i$ 行输出第 $i$ 条边 $(v_i, u_i)$（$1 \\leq v_i, u_i \\leq n$，且 $v_i \\neq u_i$）。对于每对 $(v, u)$，最多只能出现一次 $(v, u)$ 或 $(u, v)$。顶点编号从 $1$ 到 $n$。\n\n如果存在多个答案，输出任意一个即可。\n\n第一个示例的图表示如下：\n\n"},{"iden":"input","content":"输入仅一行，包含两个整数 $n$ 和 $m$（$1 \\leq n, m \\leq 10^5$）—— 分别表示顶点数和边数。"},{"iden":"output","content":"如果不存在满足条件的图，则输出 _\"Impossible\"_。否则请按以下格式输出答案：第一行输出单词 _\"Possible\"_。接下来的 $m$ 行中，第 $i$ 行输出第 $i$ 条边 $(v_i, u_i)$（$1 \\leq v_i, u_i \\leq n$，且 $v_i \\neq u_i$）。对于每对 $(v, u)$，最多只能出现一次 $(v, u)$ 或 $(u, v)$。顶点编号从 $1$ 到 $n$。如果存在多个答案，输出任意一个即可。"},{"iden":"examples","content":"输入\n5 6\n输出\nPossible\n2 5\n3 2\n5 1\n3 4\n4 1\n5 4\n\n输入\n6 12\n输出\nImpossible"},{"iden":"note","content":"第一个示例的图表示如下： "}] ```json [{"iden":"statement","content":"我们称一个无向图 $G = (V, E)$ 为 _互质图_，当且仅当对于每条边 $(v, u) \\in E$，都有 $\\gcd(v, u) = 1$（即 $v$ 和 $u$ 的最大公约数为 1）。如果某对顶点 $v$ 和 $u$ 之间没有边，则 $\\gcd(v, u)$ 的值无关紧要。顶点编号从 $1$ 到 $|V|$。\n\n请构造一个包含 $n$ 个顶点和 $m$ 条边的 _互质图_，要求该图连通，且不包含自环或重边。\n\n如果不存在满足条件的图，则输出 _\"Impossible\"_。\n\n如果存在多个答案，输出任意一个即可。\n\n输入仅一行，包含两个整数 $n$ 和 $m$（$1 \\leq n, m \\leq 10^5$）—— 分别表示顶点数和边数。\n\n如果不存在满足条件的图，则输出 _\"Impossible\"_。\n\n否则，请按以下格式输出答案：\n\n第一行输出单词 _\"Possible\"_。\n\n接下来的 $m$ 行中，第 $i$ 行输出第 $i$ 条边 $(v_i, u_i)$（$1 \\leq v_i, u_i \\leq n$，且 $v_i \\neq u_i$）。对于每对 $(v, u)$，最多只能出现一次 $(v, u)$ 或 $(u, v)$。顶点编号从 $1$ 到 $n$。\n\n如果存在多个答案，输出任意一个即可。\n\n第一个示例的图表示如下：\n\n"},{"iden":"input","content":"输入仅一行，包含两个整数 $n$ 和 $m$（$1 \\leq n, m \\leq 10^5$）—— 分别表示顶点数和边数。"},{"iden":"output","content":"如果不存在满足条件的图，则输出 _\"Impossible\"_。否则请按以下格式输出答案：第一行输出单词 _\"Possible\"_。接下来的 $m$ 行中，第 $i$ 行输出第 $i$ 条边 $(v_i, u_i)$（$1 \\leq v_i, u_i \\leq n$，且 $v_i \\neq u_i$）。对于每对 $(v, u)$，最多只能出现一次 $(v, u)$ 或 $(u, v)$。顶点编号从 $1$ 到 $n$。如果存在多个答案，输出任意一个即可。"},{"iden":"examples","content":"输入\n5 6\n输出\nPossible\n2 5\n3 2\n5 1\n3 4\n4 1\n5 4\n\n输入\n6 12\n输出\nImpossible"},{"iden":"note","content":"第一个示例的图表示如下： "}] ```

**Definitions** Let $ n, m \in \mathbb{Z}^+ $ with $ 1 \leq n, m \leq 10^5 $. Let $ V = \{1, 2, \dots, n\} $ be the set of vertices. Let $ E \subseteq \binom{V}{2} $ be a set of undirected edges (no self-loops, no multiple edges). A graph $ G = (V, E) $ is *relatively prime* if for every edge $ (u, v) \in E $, $ \gcd(u, v) = 1 $. The graph is *connected* if there exists a path between every pair of vertices in $ V $. **Constraints** 1. $ |E| = m $ 2. $ G = (V, E) $ is connected 3. For all $ (u, v) \in E $, $ \gcd(u, v) = 1 $ 4. $ u \ne v $ for all $ (u, v) \in E $ **Objective** Determine whether there exists a set $ E $ satisfying the above constraints. - If yes, output "Possible" followed by $ m $ distinct edges $ (u_i, v_i) \in E $. - If no, output "Impossible".

Samples

Input #1

5 6

Output #1

Possible
2 5
3 2
5 1
3 4
4 1
5 4

Input #2

6 12

Output #2

Impossible

API Response (JSON)

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