<p>On an <code>2 x 3</code> board, there are five tiles labeled from <code>1</code> to <code>5</code>, and an empty square represented by <code>0</code>. A <strong>move</strong> consists of choosing <code>0</code> and a 4-directionally adjacent number and swapping it.</p>

<p>The state of the board is solved if and only if the board is <code>[[1,2,3],[4,5,0]]</code>.</p>

<p>Given the puzzle board <code>board</code>, return <em>the least number of moves required so that the state of the board is solved</em>. If it is impossible for the state of the board to be solved, return <code>-1</code>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/06/29/slide1-grid.jpg" style="width: 244px; height: 165px;" />
<pre>
<strong>Input:</strong> board = [[1,2,3],[4,0,5]]
<strong>Output:</strong> 1
<strong>Explanation:</strong> Swap the 0 and the 5 in one move.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/06/29/slide2-grid.jpg" style="width: 244px; height: 165px;" />
<pre>
<strong>Input:</strong> board = [[1,2,3],[5,4,0]]
<strong>Output:</strong> -1
<strong>Explanation:</strong> No number of moves will make the board solved.
</pre>

<p><strong class="example">Example 3:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/06/29/slide3-grid.jpg" style="width: 244px; height: 165px;" />
<pre>
<strong>Input:</strong> board = [[4,1,2],[5,0,3]]
<strong>Output:</strong> 5
<strong>Explanation:</strong> 5 is the smallest number of moves that solves the board.
An example path:
After move 0: [[4,1,2],[5,0,3]]
After move 1: [[4,1,2],[0,5,3]]
After move 2: [[0,1,2],[4,5,3]]
After move 3: [[1,0,2],[4,5,3]]
After move 4: [[1,2,0],[4,5,3]]
After move 5: [[1,2,3],[4,5,0]]
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>board.length == 2</code></li>
	<li><code>board[i].length == 3</code></li>
	<li><code>0 &lt;= board[i][j] &lt;= 5</code></li>
	<li>Each value <code>board[i][j]</code> is <strong>unique</strong>.</li>
</ul>

template<typename T>
inline int serialize(const T& board) {
    return board[0][0] * 100000 + board[0][1] *  10000 + board[0][2] *   1000
         + board[1][0] *    100 + board[1][1] *     10 + board[1][2] *      1;
}
template<typename T>
inline pair<int, int> deserialize(int x, T& board) {
    int factor = 100000;
    int zr = -1, zc;
    for (int row = 0; row < 2; ++row) {
        for (int col = 0; col < 3; ++col) {
            board[row][col] = x / factor;
            x %= factor;
            if (board[row][col] == 0) {
                zr = row;
                zc = col;
            }
            factor /= 10;
        }
    }
    if (zr == -1) {
        stringstream ss;
        ss << "No 0 for " << x << ": ";
        for (const auto& row : board) {
            for (const auto& cell : row) {
                ss << cell << ", ";
            }
        }
        throw logic_error(ss.str());
    }
    return make_pair(zr, zc);
}

unordered_map<int, int> solution;

class Solution {
public:
    static void init() {
        solution.reserve(720); // == 6!, aka number of arrangements of 6 numbers
        solution[123450] = 0;
        vector<int> frontier;
        vector<int> nextFrontier;
        frontier.push_back(123450);
        int moves = 1;

        while (!frontier.empty()) {
            nextFrontier.clear();
            for (int x : frontier) {
                fillNeighbors(x, moves, nextFrontier);
            }
            swap(frontier, nextFrontier);
            ++moves;
        }
    }

    static void fillNeighbors(int x, int moves, vector<int>& out) {
        array<array<int, 3>, 2> board;
        auto [zr, zc] = deserialize(x, board);

        for (const auto& [r, c] : {make_pair(zr-1,zc), make_pair(zr+1,zc), make_pair(zr,zc-1), make_pair(zr,zc+1)}) {
            if (r < 0 || c < 0 || r >= 2 || c >= 3) { continue; }
            swap(board[zr][zc], board[r][c]);
            int num = serialize(board);
            if (solution.try_emplace(num, moves).second) {
                out.push_back(num);
            }
            swap(board[zr][zc], board[r][c]);
        }
    }

    int slidingPuzzle(vector<vector<int>>& board) {
        if (solution.empty()) {
            init();
        }
        auto it = solution.find(serialize(board));
        return it != solution.end() ? it->second : -1;
    }
};
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