LU / RD Marking

AtCoder

IDarc166_c

Time2000ms

Memory256MB

Difficulty—

There is a grid with $H$ rows and $W$ columns. This grid has $H(W+1)$ vertical edges and $W(H+1)$ horizontal edges, for a total of $H(W+1) + W(H+1)$ (see also the figures at Sample Input/Output). Consider marking these edges by the following two kinds of operations. * **Operation (1)**: Choose a square whose left and upper edges are unmarked when performing this operation. Mark the left and upper edges of that square. * **Operation (2)**: Choose a square whose right and lower edges are unmarked when performing this operation. Mark the right and lower edges of that square. Find the number, modulo $998244353$, of possible sets of edges that are eventually marked when Operations (1) and (2) can be performed any number of times, possibly zero. You have $T$ test cases to solve. ## Constraints * $1\leq T\leq 2\times 10^5$ * $1\leq H, W\leq 10^6$ ## Input The input is given from Standard Input in the following format: $T$ $\text{case}_1$ $\vdots$ $\text{case}_T$ Each test case is given in the following format: $H$ $W$ [samples]

Samples

Input #1

2
1 1
2 3

Output #1

4
800

For the case $(H, W)=(1,1)$, there are four possible sets of edges that are eventually marked, as shown below. The marked edges are drawn in bold.
![image](https://img.atcoder.jp/arc166/bd84a15ada55c0df6d4e66863053bd59.png)
For the case $(H, W)=(2,3)$, the following sets of marked edges are **possible**:
![image](https://img.atcoder.jp/arc166/66c77b9132c38d82c36732966ff3ae4f.png)
On the other hand, the following sets of marked edges are **impossible**:
![image](https://img.atcoder.jp/arc166/0029e3cc83a0232231ddf6e280c7f9a8.png)

Input #2

3
123 456
654 321
1000000 1000000

Output #2

60549740
298307903
656009181

API Response (JSON)

{
  "problem": {
    "name": "LU / RD Marking",
    "description": {
      "content": "There is a grid with $H$ rows and $W$ columns. This grid has $H(W+1)$ vertical edges and $W(H+1)$ horizontal edges, for a total of $H(W+1) + W(H+1)$ (see also the figures at Sample Input/Output). Cons",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc166_c"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There is a grid with $H$ rows and $W$ columns.\nThis grid has $H(W+1)$ vertical edges and $W(H+1)$ horizontal edges, for a total of $H(W+1) + W(H+1)$ (see also the figures at Sample Input/Output). Cons...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments