Bridges

AtCoder

IDarc143_d

Time2000ms

Memory256MB

Difficulty—

We have two sequences $A_1,\ldots, A_M$ and $B_1,\ldots,B_M$ consisting of intgers between $1$ and $N$ (inclusive). For a string of length $M$ consisting of `0` and `1`, consider the following undirected graph with $2N$ vertices and $(M+N)$ edges corresponding to that string. * If the $i$\-th character of the string is `0`, the $i$\-th edge connects Vertex $A_i$ and Vertex $(B_i+N)$. * If the $i$\-th character of the string is `1`, the $i$\-th edge connects Vertex $B_i$ and Vertex $(A_i+N)$. * The $(j+M)$\-th edge connects Vertex $j$ and Vertex $(j+N)$. Here, $i$ and $j$ are integers such that $1 \leq i \leq M$ and $1 \leq j \leq N$, and the vertices are numbered $1$ to $2N$. Find one string of length $M$ consisting of `0` and `1` such that the corresponding undirected graph has the minimum number of bridges. Notes on bridges A bridge is an edge of a graph whose removal increases the number of connected components. ## Constraints * $1 \leq N \leq 2 \times 10^5$ * $1 \leq M \leq 2 \times 10^5$ * $1 \leq A_i, B_i \leq N$ ## Input Input is given from Standard Input in the following format: $N$ $M$ $A_1$ $A_2$ $\cdots$ $A_M$ $B_1$ $B_2$ $\cdots$ $B_M$ [samples]

Samples

Input #1

2 2
1 1
2 2

Output #1

01

The graph corresponding to `01` has no bridges.
In the graph corresponding to `00`, the edge connecting Vertex $1$ and Vertex $3$ and the edge connecting Vertex $2$ and Vertex $4$ are bridges, so `00` does not satisfy the requirement.

Input #2

6 7
1 1 2 3 4 4 5
2 3 3 4 5 6 6

Output #2

API Response (JSON)

{
  "problem": {
    "name": "Bridges",
    "description": {
      "content": "We have two sequences $A_1,\\ldots, A_M$ and $B_1,\\ldots,B_M$ consisting of intgers between $1$ and $N$ (inclusive). For a string of length $M$ consisting of `0` and `1`, consider the following undirec",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc143_d"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "We have two sequences $A_1,\\ldots, A_M$ and $B_1,\\ldots,B_M$ consisting of intgers between $1$ and $N$ (inclusive).\nFor a string of length $M$ consisting of `0` and `1`, consider the following undirec...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments