XOR Tree

AtCoder

IDapc001_f

Time2000ms

Memory256MB

Difficulty—

You are given a tree with $N$ vertices. The vertices are numbered $0$ through $N-1$, and the edges are numbered $1$ through $N-1$. Edge $i$ connects Vertex $x_i$ and $y_i$, and has a value $a_i$. You can perform the following operation any number of times: * Choose a simple path and a non-negative integer $x$, then for each edge $e$ that belongs to the path, change $a_e$ by executing $a_e ← a_e ⊕ x$ (⊕ denotes XOR). Your objective is to have $a_e = 0$ for all edges $e$. Find the minimum number of operations required to achieve it. ## Constraints * $2 ≤ N ≤ 10^5$ * $0 ≤ x_i,y_i ≤ N-1$ * $0 ≤ a_i ≤ 15$ * The given graph is a tree. * All input values are integers. ## Input Input is given from Standard Input in the following format: $N$ $x_1$ $y_1$ $a_1$ $x_2$ $y_2$ $a_2$ $:$ $x_{N-1}$ $y_{N-1}$ $a_{N-1}$ [samples]

Samples

Input #1

Output #1

3

The objective can be achieved in three operations, as follows:

*   First, choose the path connecting Vertex $1, 2$, and $x = 1$.
*   Then, choose the path connecting Vertex $2, 3$, and $x = 2$.
*   Lastly, choose the path connecting Vertex $0, 4$, and $x = 4$.

Input #2

2
1 0 0

Output #2

API Response (JSON)

{
  "problem": {
    "name": "XOR Tree",
    "description": {
      "content": "You are given a tree with $N$ vertices. The vertices are numbered $0$ through $N-1$, and the edges are numbered $1$ through $N-1$. Edge $i$ connects Vertex $x_i$ and $y_i$, and has a value $a_i$. You ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "apc001_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given a tree with $N$ vertices. The vertices are numbered $0$ through $N-1$, and the edges are numbered $1$ through $N-1$. Edge $i$ connects Vertex $x_i$ and $y_i$, and has a value $a_i$. You ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments