Balancing Network

AtCoder

IDagc041_e

Time2000ms

Memory256MB

Difficulty—

A _balancing network_ is an abstract device built up of $N$ wires, thought of as running from left to right, and $M$ _balancers_ that connect pairs of wires. The wires are numbered from $1$ to $N$ from top to bottom, and the balancers are numbered from $1$ to $M$ from left to right. Balancer $i$ connects wires $x_i$ and $y_i$ ($x_i < y_i$). ![image](https://img.atcoder.jp/agc041/pic1-small-2acea94b.png) Each balancer must be in one of two states: _up_ or _down_. Consider a token that starts moving to the right along some wire at a point to the left of all balancers. During the process, the token passes through each balancer exactly once. Whenever the token encounters balancer $i$, the following happens: * If the token is moving along wire $x_i$ and balancer $i$ is in the down state, the token moves down to wire $y_i$ and continues moving to the right. * If the token is moving along wire $y_i$ and balancer $i$ is in the up state, the token moves up to wire $x_i$ and continues moving to the right. * Otherwise, the token doesn't change the wire it's moving along. Let a state of the balancing network be a string of length $M$, describing the states of all balancers. The $i$\-th character is `^` if balancer $i$ is in the up state, and `v` if balancer $i$ is in the down state. A state of the balancing network is called _uniforming_ if a wire $w$ exists such that, regardless of the starting wire, the token will always end up at wire $w$ and run to infinity along it. Any other state is called _non-uniforming_. You are given an integer $T$ ($1 \le T \le 2$). Answer the following question: * If $T = 1$, find any uniforming state of the network or determine that it doesn't exist. * If $T = 2$, find any non-uniforming state of the network or determine that it doesn't exist. Note that if you answer just one kind of questions correctly, you can get partial score. ## Constraints * $2 \leq N \leq 50000$ * $1 \leq M \leq 100000$ * $1 \leq T \leq 2$ * $1 \leq x_i < y_i \leq N$ * All input values are integers. ## Input Input is given from Standard Input in the following format: $N$ $M$ $T$ $x_1$ $y_1$ $x_2$ $y_2$ $:$ $x_M$ $y_M$ ## Partial Score * $800$ points will be awarded for passing the testset satisfying $T = 1$. * $800$ points will be awarded for passing the testset satisfying $T = 2$. [samples]

Samples

Input #1

Output #1

^^^^^

This state is uniforming: regardless of the starting wire, the token ends up at wire $1$.

![image](https://img.atcoder.jp/agc041/pic2-small-2acea94b.png)

Input #2

Output #2

v^^^^

This state is non-uniforming: depending on the starting wire, the token might end up at wire $1$ or wire $2$.

![image](https://img.atcoder.jp/agc041/pic3final-small-2acea94b.png)

Input #3

3 1 1
1 2

Output #3

\-1

A uniforming state doesn't exist.

Input #4

2 1 2
1 2

Output #4

\-1

A non-uniforming state doesn't exist.

API Response (JSON)

{
  "problem": {
    "name": "Balancing Network",
    "description": {
      "content": "A _balancing network_ is an abstract device built up of $N$ wires, thought of as running from left to right, and $M$ _balancers_ that connect pairs of wires. The wires are numbered from $1$ to $N$ fro",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "agc041_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "A _balancing network_ is an abstract device built up of $N$ wires, thought of as running from left to right, and $M$ _balancers_ that connect pairs of wires. The wires are numbered from $1$ to $N$ fro...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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