25087번 - Hamiltonian Tour 서브태스크스페셜 저지다국어

Hamilton is a Canadian city near Toronto, and a nice place to take a walking tour.

In this problem, Hamilton is represented by a grid of unit cells with $2R$ rows and $2C$ columns, where each cell is either empty (represented by *) or contains a building (represented by #). The cell on the $i$-th row and $j$-th column is represented by $A_{i,j}$ where $1≤i≤2R$ and $1≤j≤2C$. It is not possible to enter cells containing buildings and you can only move to an adjacent cell that shares a side with the current cell (not just a corner). The grid is such that if it is divided evenly into $2×2$ blocks of unit cells, then in each of those blocks, either all four cells are empty, or all four cells are occupied by a building. Let us represent the block formed by $A_{2i-1,2j-1}$, $A_{2i-1,2j}$, $A_{2i,2j-1}$, and $A_{2i,2j}$ cells as $B_{i,j}$ where $1≤i≤R$ and $1≤j≤C$.

Grace is a tourist in Hamilton and wants to visit all the empty cells in Hamilton. Grace is currently in cell $A_{1,1}$. Visiting the same cell twice could be boring for her. Hence, Grace wants to visit each of the empty cells exactly once and finally end in cell $A_{1,1}$. Can you help Grace by providing a string (consisting of directional moves {N, E, S, W} representing the unit moves to the north, east, south, or west respectively) which Grace can follow to visit every empty cell once and end again in $A_{1,1}$.

The first line of the input gives the number of test cases, $T$. $T$ test cases follow. The first line of each test case contains two integers $R$ and $C$. The next $R$ lines of each test case contains $C$ characters each.

The $j$-th character on the $i$-th of these lines represents the block $B_{i,j}$ formed by the following four cells: $A_{2i-1,2j-1}$, $A_{2i-1,2j}$, $A_{2i,2j-1}$, and $A_{2i,2j}$. If $B_{i,j}=$ #, all four of the cells in $B_{i,j}$ are occupied by a building. Otherwise, if $B_{i,j}=$ *, all four of the cells in $B_{i,j}$ are empty.

For each test case, output one line containing Case #x: y, where $x$ is the test case number (starting from 1) and $y$ is the answer to the problem as follows.

If there is no solution to the problem, $y$ should be IMPOSSIBLE. Otherwise, $y$ should be a sequence of characters from the set {N, E, S, W}, representing the unit moves (to the north, east, south, or west respectively) in a valid route, starting from $A_{1,1}$, as described in the statement above.

Note that your last move should take you to $A_{1,1}$; this move does not count as visiting the same cell twice.

If there are multiple valid solutions, you may output any one of them.

• $1≤T≤100$.
• $1≤R≤200$.
• $1≤C≤200$.
• All characters in the grid are from the set {#,*}.
• The first character of the first line of the input grid for each test case is a * character, i.e. $B_{1,1}=$*.

A block contains buildings if and only if the row number and column number of it are divisible by 2. i.e. $B_{i,j}=$ # $⟺((i \text{ mod } 2=0)$ and $(j \text{ mod }2=0))$.

No extra constraints.