Independent Set Game

AtCoder

IDarc210_d

Time2000ms

Memory256MB

Difficulty—

You are given a simple undirected graph $G$ with $N$ vertices and $M$ edges, where vertices are numbered $1$ to $N$. The $i$\-th edge connects vertices $u_i$ and $v_i$. Initially, no vertices of $G$ are colored. Alice and Bob will play a game using $G$. Alice and Bob perform the following actions in order for $t=1,\ldots,N$: * If $t$ is odd, Alice chooses one vertex that is not yet colored and colors that vertex black. * If $t$ is even, Bob chooses one vertex that is not yet colored and colors that vertex white. After all $N$ actions are completed, if the set of all vertices colored black is an independent set of $G$, Alice is the winner; otherwise, Bob is the winner. Output the name of the winner when both players play optimally. You are given $T$ test cases; solve each of them. What is an independent set A set $S$ consisting of vertices of a simple undirected graph $G$ is an independent set of $G$ if there does not exist a pair of vertices $u$ and $v$ connected by an edge such that $u\in S$ and $v\in S$. ## Constraints * $1\leq T\leq 2\times 10^5$ * $2\leq N\leq 2\times 10^5$ * $0\leq M\leq 2\times 10^5$ * $1\leq u_i < v_i \leq N$ * The given graph is a simple undirected graph. * The sum of $N$ over all test cases is at most $2\times 10^5$. * The sum of $M$ over all test cases is at most $2\times 10^5$. ## Input The input is given from Standard Input in the following format: $T$ $\mathrm{case}_1$ $\mathrm{case}_2$ $\vdots$ $\mathrm{case}_T$ Each case is given in the following format: $N$ $M$ $u_1$ $v_1$ $\vdots$ $u_M$ $v_M$ [samples]

Samples

Input #1

Output #1

Bob
Alice
Bob

The graph given in each test case is illustrated below:
![image](https://img.atcoder.jp/arc210/d904a1191b2a90eea4c58343a1def228.png)

API Response (JSON)

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  "problem": {
    "name": "Independent Set Game",
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      "content": "You are given a simple undirected graph $G$ with $N$ vertices and $M$ edges, where vertices are numbered $1$ to $N$. The $i$\\-th edge connects vertices $u_i$ and $v_i$. Initially, no vertices of $G$ a",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
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    "difficulty": "None",
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    "is_sync": true,
    "sync_url": null,
    "sign": "arc210_d"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given a simple undirected graph $G$ with $N$ vertices and $M$ edges, where vertices are numbered $1$ to $N$. The $i$\\-th edge connects vertices $u_i$ and $v_i$.\nInitially, no vertices of $G$ a...",
      "is_translate": false,
      "language": "English"
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}

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