Sliding Edge on Torus

AtCoder

IDarc141_e

Time3000ms

Memory256MB

Difficulty—

There is an undirected graph with $N^2$ vertices. Initially, it has no edges. For each pair of integers $(i,\ j)$ such that $0 \leq i,\ j < N$, the graph has a corresponding vertex called $(i,\ j)$. You will get $Q$ queries, which should be processed in order. The $i$\-th query, which gives you four integers $a_i,\ b_i,\ c_i,\ d_i$, is as follows. * For each $k$ $(0 \leq k < N)$, add an edge between two vertices $((a_i+k) \bmod N,\ (b_i+k) \bmod N)$ and $((c_i+k) \bmod N,\ (d_i+k) \bmod N)$. Then, print the current number of connected components in the graph. ## Constraints * $2 \leq N \leq 2 \times 10^5$ * $1 \leq Q \leq 2 \times 10^5$ * $0 \leq a_i,\ b_i,\ c_i,\ d_i < N$ * $(a_i,\ b_i) \neq (c_i,\ d_i)$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $Q$ $a_1$ $b_1$ $c_1$ $d_1$ $a_2$ $b_2$ $c_2$ $d_2$ $\vdots$ $a_Q$ $b_Q$ $c_Q$ $d_Q$ [samples]

Samples

Input #1

Output #1

6
4
4

The first query adds an edge between $(0,\ 0),\ (1,\ 2)$, between $(1,\ 1),\ (2,\ 0)$, and between $(2,\ 2),\ (0,\ 1)$, changing the number of connected components from $9$ to $6$.

Input #2

Output #2

14
11
11

The graph after queries may not be simple.

Input #3

Output #3

API Response (JSON)

{
  "problem": {
    "name": "Sliding Edge on Torus",
    "description": {
      "content": "There is an undirected graph with $N^2$ vertices. Initially, it has no edges. For each pair of integers $(i,\\ j)$ such that $0 \\leq i,\\ j < N$, the graph has a corresponding vertex called $(i,\\ j)$. Y",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 3000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc141_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There is an undirected graph with $N^2$ vertices. Initially, it has no edges. For each pair of integers $(i,\\ j)$ such that $0 \\leq i,\\ j < N$, the graph has a corresponding vertex called $(i,\\ j)$.\nY...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments