Unique XOR Path

AtCoder

IDagc060_b

Time2000ms

Memory256MB

Difficulty—

We have a grid with $N$ rows and $M$ columns. You want to write an integer between $0$ and $2^K-1$ in each square in the grid to satisfy the following condition. * Consider a path that starts at the top-left square, repeatedly moves right or down to an adjacent square, and ends at the bottom-right square. Such a path is said to be **good** if and only if the total $\mathrm{XOR}$ of the integers written on the squares visited (including the starting and ending points) is $0$. * There is exactly one good path, which is the path represented by a string $S$. The path represented by the string $S$ is a path that, for each $i$ ($1 \leq i \leq N+M-2$), the $i$\-th move is right if the $i$\-th character of $S$ is `R` and down if that character is `D`. Determine whether there is a way to write integers that satisfies the condition. For each input file, solve $T$ test cases. What is bitwise $\mathrm{XOR}$?The bitwise $\mathrm{XOR}$ of non-negative integers $A$ and $B$, $A \oplus B$, is defined as follows. * When $A \oplus B$ is written in binary, the $k$\-th lowest bit ($k \geq 0$) is $1$ if exactly one of the $k$\-th lowest bits of $A$ and $B$ is $1$, and $0$ otherwise. For instance, $3 \oplus 5 = 6$ (in binary: $011 \oplus 101 = 110$). Generally, the bitwise $\mathrm{XOR}$ of $k$ non-negative integers $p_1, p_2, p_3, \dots, p_k$ is defined as $(\dots ((p_1 \oplus p_2) \oplus p_3) \oplus \dots \oplus p_k)$, which can be proved to be independent of the order of $p_1, p_2, p_3, \dots, p_k$. ## Constraints * $1 \leq T \leq 100$ * $2 \leq N,M \leq 30$ * $1 \leq K \leq 30$ * $S$ is a string containing exactly $N-1$ `D`s and $M-1$ `R`s. * All numbers in the input are integers. ## Input The input is given from Standard Input in the following format: $T$ $case_1$ $case_2$ $\vdots$ $case_T$ Each case is in the following format: $N$ $M$ $K$ $S$ [samples]

Samples

Input #1

4
2 2 1
RD
4 3 1
RDDDR
15 20 18
DDRRRRRRRDDDDRRDDRDRRRRDDRDRDDRRR
20 15 7
DRRDDDDDRDDDRRDDRRRDRRRDDDDDRRRDD

Output #1

Yes
No
Yes
No

As an example, for the first case, you can make the grid as follows.

11
00

API Response (JSON)

{
  "problem": {
    "name": "Unique XOR Path",
    "description": {
      "content": "We have a grid with $N$ rows and $M$ columns. You want to write an integer between $0$ and $2^K-1$ in each square in the grid to satisfy the following condition. *   Consider a path that starts at th",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "agc060_b"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "We have a grid with $N$ rows and $M$ columns. You want to write an integer between $0$ and $2^K-1$ in each square in the grid to satisfy the following condition.\n\n*   Consider a path that starts at th...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments