<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>

class Solution {
public:
    int slidingPuzzle(vector<vector<int>>& board) {
        string target = "123450";
        string start;
        for (int i = 0; i < 2; ++i)
            for (int j = 0; j < 3; ++j)
                start += to_string(board[i][j]);
        vector<vector<int>> neighbors = {
            {1, 3}, {0, 2, 4}, {1, 5}, {0, 4}, {1, 3, 5}, {2, 4}
        };        
        auto heuristic = [](const string& s) {
            int distance = 0;
            for (int i = 0; i < 6; ++i) {
                if (s[i] == '0') continue;
                int val = s[i] - '1';
                distance += abs(i / 3 - val / 3) + abs(i % 3 - val % 3);
            }
            return distance;
        };
        using State = pair<int, pair<int, string>>;
        priority_queue<State, vector<State>, greater<State>> pq;
        pq.push({heuristic(start), {0, start}});
        unordered_set<string> visited;
        visited.insert(start);
        while (!pq.empty()) {
            auto [estCost, cur] = pq.top();
            pq.pop();
            int moves = cur.first;
            string state = cur.second;
            if (state == target) return moves;
            int zeroPos = state.find('0');
            for (int neighbor : neighbors[zeroPos]) {
                string newState = state;
                swap(newState[zeroPos], newState[neighbor]);
                if (visited.find(newState) == visited.end()) {
                    visited.insert(newState);
                    pq.push({moves + 1 + heuristic(newState), {moves + 1, newState}});
                }
            }
        }
        return -1;
    }
};

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