Bichromization

AtCoder

IDkeyence2020_e

Time2000ms

Memory256MB

Difficulty—

We have a connected undirected graph with $N$ vertices and $M$ edges. Edge $i$ in this graph ($1 \leq i \leq M$) connects Vertex $U_i$ and Vertex $V_i$ bidirectionally. We are additionally given $N$ integers $D_1, D_2, ..., D_N$. Determine whether the conditions below can be satisfied by assigning a color - white or black - to each vertex and an integer weight between $1$ and $10^9$ (inclusive) to each edge in this graph. If the answer is yes, find one such assignment of colors and integers, too. * There is at least one vertex assigned white and at least one vertex assigned black. * For each vertex $v$ ($1 \leq v \leq N$), the following holds. * The minimum cost to travel from Vertex $v$ to a vertex whose color assigned is different from that of Vertex $v$ by traversing the edges is equal to $D_v$. Here, the cost of traversing the edges is the sum of the weights of the edges traversed. ## Constraints * $2 \leq N \leq 100,000$ * $1 \leq M \leq 200,000$ * $1 \leq D_i \leq 10^9$ * $1 \leq U_i, V_i \leq N$ * The given graph is connected and has no self-loops or multiple edges. * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $M$ $D_1$ $D_2$ $...$ $D_N$ $U_1$ $V_1$ $U_2$ $V_2$ $\vdots$ $U_M$ $V_M$ [samples]

Samples

Input #1

Output #1

BWWBB
4
3
1
5
2

Assume that we assign the colors and integers as the sample output, and let us consider Vertex $5$, for example. To travel from Vertex $5$, which is assigned black, to a vertex that is assigned white with the minimum cost, we should make these moves: Vertex $5$ $\to$ Vertex $4$ $\to$ Vertex $2$. The total cost of these moves is $7$, which satisfies the condition. We can also verify that the condition is satisfied for other vertices.

Input #2

Output #2

\-1

Input #3

Output #3

BBBW
1
1
1
2
1
1

API Response (JSON)

{
  "problem": {
    "name": "Bichromization",
    "description": {
      "content": "We have a connected undirected graph with $N$ vertices and $M$ edges. Edge $i$ in this graph ($1 \\leq i \\leq M$) connects Vertex $U_i$ and Vertex $V_i$ bidirectionally. We are additionally given $N$ i",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "keyence2020_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "We have a connected undirected graph with $N$ vertices and $M$ edges. Edge $i$ in this graph ($1 \\leq i \\leq M$) connects Vertex $U_i$ and Vertex $V_i$ bidirectionally. We are additionally given $N$ i...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments